Research ArticleRobust Linear Output Regulation Using Extended State Observer

Mehran Hosseini-Pishrobat 1 Jafar Keighobadi1

Atta Oveisi 2 and Tamara NestoroviT 2

1Faculty of Mechanical Engineering University of Tabriz 29 Bahman Tabriz Iran2Institute of Computational Engineering Ruhr-Universitat Bochum Universitatsstr 150 D-44801 Bochum Germany

Correspondence should be addressed to Atta Oveisi attaoveisirubde

Received 16 February 2018 Revised 31 May 2018 Accepted 5 June 2018 Published 12 August 2018

Academic Editor Francesco Riganti-Fulginei

Copyright copy 2018 Mehran Hosseini-Pishrobat et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper presents a disturbance rejection-based solution to the problem of robust output regulation The mismatch between theunderlying plant and its nominal mathematical model is formulated by two disturbance classes The first class is assumed to begenerated by an autonomous linear system while for the second class no specific dynamical structure is considered Accordinglythe robustness of the closed-loop system against the first disturbance class is achieved by following the internal model principleOn the other hand in the framework of disturbance rejection control an extended state observer (ESO) is designed to approximateand compensate for the second class ie unstructured disturbances As a result the proposed output regulation method can dealwith a vast range of uncertainties Finally the stability of the closed-loop system based on the proposed compound controller iscarried out via Lyapunov and centermanifold analyses and some results on the robust output regulation are drawnA representativesimulation example is also presented to show the effectiveness of the control method

1 Introduction

The theory of output regulation in its essence deals withthe problem of trackingdisturbance rejection of a class ofsignals generated by autonomous dynamic systems whileguaranteeing the closed-loop stability For linear systems thistheory is well-established and the solution of the associatedcontrol problem is obtained by the internal model principle[1 2] The principle is also instrumental in addressingthe problem of robust output regulation On this basisrobust design methods have been proposed in the literaturefor linear systems with uncertain parameters [3ndash6] Thesemethods consider the effect of external disturbances by anexogenous signal which belongs to the solution space of aparticular differential equation According to such a differ-ential equation the steady-state behavior of the underlyingplant should be considered to examine the solvability of theoutput regulation problem After determining the steady-state forms of the plant trajectories that satisfy the track-ingdisturbance rejection requirements the internal model-based design can be pursued [7] By this method a stabilizing

controller that incorporates a suitable internal model of theclass of the desired signals offers guaranteed asymptoticrejectiontracking of any signal from that class

In this paper we depart from the conventional formu-lation of the output regulation problem by considering theeffect of disturbances that cannot be modeled by a processsignal of a differential equation As a matter of fact in manypractical applications such modeling assumption on thedisturbances is restrictive In many systems the disturbancesare not only caused by exogenous factors but also fromendogenous factors such as structural variations parametricuncertainty and unmodeled dynamics The latter factorsusually appear as state-dependent disturbances in the systemrsquosmathematical model For example in many structural vibra-tion control problems the main source of the disturbance isthe unmodeled higher order modes Even though the domi-nant higher order modes can be obtained via suitable systemidentification methods a comprehensive dynamic model forthe disturbances is not feasible [8] As another example inmanymechanical and electromechanical systems parametric

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4095473 12 pageshttpsdoiorg10115520184095473

2 Mathematical Problems in Engineering

uncertainty friction and nonlinear effects bring about state-dependent disturbances that are not guaranteed to belong tothe solution space of any particular dynamic system [9] Onthis basis our goal in this paper is to extend the applicabilityof the output regulation method to such examples To thisend we recast the output regulation problem into the activedisturbance rejection framework [10 11] Active disturbancerejection control (ADRC) is a robust control method forsystems with large uncertainty and disturbances [10 12ndash14]The key idea of the ADRC is to treat a robust control problemas an estimation-rejection problem More specifically inthe first step all sorts of discrepancies between the physi-cal system and its nominal mathematical model includingparameter variations unknown nonlinearities and externaldisturbancesnoises are lumped into a total disturbance term[14] Next an estimator referred to as extended state observer(ESO) is applied to estimate the total disturbance and thenreject it in a closed-loop framework The active disturbancerejection control offers several promising advantages over theconventional control methods including the following (1)ADRC is an active robustification method that in comparisonto classical robust techniques (based on worst-case analysis)minimizes the conservatism in the design (2) ADRC canhandle large uncertaintydisturbances from both internaland external sources (3) ADRC does not require involvedaccurate systemmodeling this feature enables a disturbance-oriented design instead of a model-oriented one [12ndash14]More details about the ADRC and its various engineeringapplications can be found in [10 14] and the recent book [13]

In light of the above discussion active disturbancerejection provides a suitable framework to generalize theapplicability of the output regulation to the systems whosedisturbances do not necessarily satisfy a differential equa-tion The main contribution of this paper is reported interms of combining the merits of output regulation anddisturbance rejection control techniques to achieve a bettertracking performance in the presence of a general class ofdisturbanceuncertainty The class of considered systems isfairly general which includes linear systems subjected tomismatched disturbances as well as state-dependent uncer-tainties Following the methodology of ADRC we lump alldisturbancesuncertainties of the underlying system into atotal disturbance The total disturbance term is decomposedinto two parts The first part can be classically modeled asthe solution of a differential equation while the second parthas no specific dynamical structureThemodeled componentof the total disturbance is handled via standard method ofthe output regulation theory which entails embedding aninternal model into the closed-loop system This approachwill also guarantee output tracking of the reference signalsthat can be generated by the internal model The unmod-eled component of the total disturbance is handled by themethod of active disturbance rejection To this end thesystem dynamics is extended by appending an integratorchannel that transmits the uncertainty of the unmodeledtotal disturbance to its time derivative Meanwhile an ESO isdesigned that continuously monitors input-output signals ofthe extended system to estimate the unmodeled disturbancecomponent By combining this estimated value with the

nominal output regulation control themismatch disturbancerejection is achieved We note that since the ADRC usesan integral action to model the dynamical behavior of thedisturbance the method can be construed as an approximateoutput regulation technique [16] This point of view is alsoexplained in [17] where active disturbance rejection andoutput regulation are combined together for an improvedrobust vibration control of a MEMS gyroscope As a noveltyin the disturbance rejection aspect a newESOdesignmethodis proposed based on the convex programming To this endthe ESO error dynamics is presented as a Lurrsquoe system andits stability condition is phrased in terms of linear matrixinequalities (LMIs) Additionally the proposed ESO appliesnonlinear gain approach which enhances its immunity to themeasurement noises

We note that various methods have been reported inthe literature for the general problem of disturbance rejec-tionattenuation of control systems for example LMIs-basedL2 disturbance attenuation for nonlinear systems with inputdelays [18] operator-based disturbance attenuationrejection[19] disturbance observer-based disturbance attenuationfor stochastic systems [20] disturbance rejection based onthe equivalent-input-disturbance approach [21] ESO-baseddisturbance rejection controller for fully actuated Euler-Lagrange systems [22] and sliding mode observercontrollerhybrid control structure [15] With respect to these meth-ods the main advantages of our method lie in combiningthe strong points of the output regulation and the distur-bance rejection techniquesTheoutput regulation componentenables a guaranteed asymptotic rejectiontracking of the dis-turbancesreference signalswith knowndynamicsMoreoverfor such disturbancesreference signals no assumptions onthe energy boundedness nor control matching are requiredFor disturbances without known dynamic behavior the ESO-based disturbance rejection engages to fortify the outputregulation In other words our method attempts to get themost out of the available information about the disturbanceswith the primary aim of disturbance rejection rather thandisturbance attenuation

The remainder of this paper is organized as followsIn Section 2 the control problem is formulated in termsof the output regulation theory In Section 3 a nominalregulator is designed to handle the tracking problem aswell as the rejection of the disturbance components withknown dynamic characteristics In Section 4 to compensatefor the disturbances without known characteristics an ESOis developed A convex optimization-based design methodfor the ESO is proposed in Section 5 The stability of theclosed-loop system is investigated in Section 6 In Section 7 asimulation example based on the plotter system is elaboratedFinally the paper is concluded by Section 8

2 Formulation of the Control Problem

Consider the following square state space model (119905) = 119860119909 (119905) + 119861119906 (119905) + 120574 (119909 (119905) 119905) 119910 (119905) = 119862119909 (119905) (1)

Mathematical Problems in Engineering 3

where 119909(119905) isin R119899 is the state 119906(119905) isin R119898 is the controlinput 119910(119905) isin R119898 denotes the measured performanceoutput and 120574(119909(119905) 119905) isin R119899 times R+ 997888rarr R119899 is a sufficientlysmooth (differentiable) nonlinear function representing thetotal disturbance119860 119861 and 119862 are real matrices of appropriatedimensions satisfying the following assumption

Assumption 1 The pair (119860 119861) is controllable and (119862 119860) isobservable

The control goal is to asymptotically track a givenreference signal ie 119910119898(119905) isin R119898 while boundedness ofthe all other signals is guaranteed in the presence of thetotal disturbance 120574(119909(119905) 119905) The reference signal 119910119898(119905) isconsidered to be generated by the solution of a fixed linearautonomous dynamic system of the form (119905) = 119878119908 (119905) 119910119898 (119905) = 119862119898119908 (119905) (2)

where 119908(119905) isin R119903 and 119878 and 119862119898 are real-valued matriceswith proper dimensionsThe following assumption is taken toguarantee the persistence and boundedness of the referencesignal 119910119898(119905) [23]Assumption 2 All eigenvalues of the matrix 119878 have zero realparts with multiplicity one in the minimal polynomial

By setting 120590(119905) fl 120574(119909(119905) 119905) the system in (1) can berewritten as (119905) = 119860119909 (119905) + 119861119906 (119905) + 120590 (119905) 119910 (119905) = 119862119909 (119905) (3)

The systems in (1) and (2) are trajectory equivalent inthe sense that under the same initial values their statetrajectories coincide with each other [24 25] Therefore wecarry out the controller design and analysis on the basis of(2)

The conventional output regulation theory is based on theassumption that the perturbing disturbances belong to thesolution space of the dynamic system given in (2) Here wegeneralize this assumption by including disturbances whichdo not necessarily comply with this particular dynamics

Assumption 3 The total disturbance signal satisfies120590 (119905) = 119862120590119908 (119905) + 119861120585 (119905) (4)

where 119862120590 isin R119903times119903 is a given matrix with the pair (119862120590 119878) beingobservable The signal 120585(119905) isin R119898 is bounded and sufficientlydifferentiable such that 120585(119905) is bounded as well

Remark 4 The rationale behind Assumption 3 can beexplained as follows In many engineering systems only thedominant frequencies of the disturbances can be obtainedvia time andor frequency domain identification methods(see for example [8]) In such systems one can embed theknown dominant frequencies into the system (2) and model

the corresponding disturbance approximation error into theterm 120585() As another example consider the tracking controlproblems where the dominant dynamics of the underlyingsystems are linear Assuming that the norm of the state-dependent disturbances is smaller than the convergence rateof the linear part it is conceivable that for a successfultracking the frequency spectrum of the disturbances willcontain the modes of the reference signals Accordingly (4)forms a suitable basis for modeling of the disturbances Anexample of such modeling can be found in [9]

Let us define the tracking performance of the controlsystem as 119890 fl 119910 minus 119910119898 (5)

Note that here and after the dependence on time variable (119905)is dropped (unless necessary) in the formulation for the sakeof readability The following composite system is proposedregarding the underlying output regulation control problem119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 = 119878119908119890 = 119862119909 minus 119862119898119908 (6)

In order to examine the solvability of the output regulationproblem first we consider a steady-state condition in whichthe output tracking is achieved In such a steady-state condi-tion the trajectories of the underlying system should be well-defined to ensure that the solution of the output regulationproblem exists Denoting 119909 119906 120585 and 120590 fl 120574(119909 119905) as thesteady-state vectors of the state control unstructured andthe structured disturbances respectively it is easy to obtain = 119860119909 + 119861119906 + 120590120590 = 119862120590119908 + 119861120585119862119909 minus 119862119898119908 = 0 (7)

Setting the solutions as 119909 = 119883119908 and 119906 = 119880119908 minus 120585 for some119883 isin R119899times119903 and 119880 isin R119898times119903 we obtain the following Sylvester-type matrix equations from (7)119883119878 = 119860119883 + 119861119880 + 119862120590119862119883 minus 119862119898 = 0 (8)

Theorem 5 For any given matrices 119862119898 and 119862120590 there existunique matrices 119883 and 119880 satisfying (8) if and only if thefollowing assumption holds

Assumption 6 For all 119904 isin spec(119878) (spec() represents thevector of eigenvalues of the matrix ())

rank(119860 minus 119904119868 119861119862 0) = 119899 + 119898 (9)

Proof See [26] for the proof

4 Mathematical Problems in Engineering

From a geometrical point of view solutions of thematrix equations (8) mdashknown as the regulator equa-tionsmdashcorresponds to a zero-error controlled invariant man-ifold defined by M fl (119909 119908) isin R119899 times R119903 | 119909 minus 119883119908 = 0for the composite system (6) In other terms for an initialcondition of the form 119909(0) = 119883119908(0) under the control input119906 the trajectory 119909 will evolve on M for all 119905 isin R+ andthe tracking error (5) will be identically equal to zero Fromthis standpoint solving the formulated output regulationproblem amounts to rendering the manifold M globallyattractive by a suitable control function [7] Since exactdynamic characterization of the total disturbance signal is notavailable we pursue practical stabilization of the manifoldMfor the composite system (6)More specifically trajectories ofthe closed-loop system all should converge to a small compactset aroundM by approximately recovering the solutions of theregulator equations To this end we propose a disturbancerejection-based two-degrees-of-freedom control structureThe control system is composed of a nominal output regulatorequipped with an ESO for disturbance rejection Accordinglythe main design stages of this control system are as follows

Stage 1 The nominal controller solves the output regulationproblem for the given dynamical system in (5) with thedifference that the state equation is simplified to = 119860119909 + 119861119906119899 + 119862120590119908 (10)

Note that the corresponding control input 119906119899 isin R119898 isdesigned to drive the nominal closed-loop trajectories towardM by recovering the nominal part of the steady-state controlsignal defined by 119906119899 = 119880119908 This corresponds to thefeedforward control input required for tracking 119910119898 as well ascompensating for 119862120590119908Stage 2 On the other hand the disturbance rejection looputilizes an ESO that continuously monitors the input-outputsignals of the plant to provide an estimate of the disturbancedenoted by 120585 The obtained estimated value is then integratedwith the nominal control input to cancel out the disturbancesignal Accordingly the overall control input is in the form119906 = 119906119899 minus 120585 (11)

The configuration of the proposed control system is schemat-ically illustrated by the block diagram in Figure 1

3 Nominal Output Regulation

To solve the nominal output regulation problem we charac-terize the distance of the system trajectories from the targetmanifold M by a suitable coordinate Then by controllingthis coordinate to zero the output regulation will be accom-plished Considering the steady-state system trajectories onthe manifold an intuitive candidate for such a coordinate is119909minus119909 However owing to the direct dependence of the regula-tor equations on the solutions of 120585 this method is sensitive toperturbation (see [7]) Consequently the intuitive candidatemay render the control system vulnerable to the disturbance

Figure 1 Block diagram of the proposed control system

estimation errors Hence we follow a more robust designapproach based on the internal model principle For thispurpose setting 119901 = 119889119889119905 the following linear differentialoperator is considered

Γ (119901) = 119901119897 + 119897minus1sum119894=0

120572119897minus119894minus1119901119897minus119894minus1 (12)

where 120572119894|119897minus1119894=0 with 120572119897 fl 1 are the coefficients of the minimalpolynomial of the matrix 119878 in the ascending order Next weintroduce the following transformed variable119909119905 = Γ (119901) 119909 (13)

Proposition 7 119909119905 equiv 0 if and only if (119909 119908) isin M

Proof The motion of the system trajectories onM under thecontrol input 119906119899 is equivalent to the immersion of system (10)into (2) By properties of the minimal polynomial for anytrajectory of 119908 in (2) we have Γ(119901)119908 = 0 Thereby the setof state trajectories evolving on the manifold M correspondsto the kernel of the operator Γ(119901) ie Ker(Γ()) = 119909 R+ 997888rarr R119899 | Γ()119909 = 0 Thus the variable 119909119905 acts as anindicator function for the set Ker(Γ()) in the sense that a statetrajectory 119909(119905) evolves onM if and only if its transformation119909119905(119905) is equivalent to zero

In view of Proposition 7 the variable 119909119905 is a distancecoordinate characterizing attractivity of the manifold M fortrajectories of system (10) By embedding the internal modelof the reference signal this distance coordinate does notdepend on the solutions of the regulator equation andtherefore enables a robust design Interestingly based on thisargument the output regulation problem entails the problemof regulating 119909119905 to zero Hence by applying the operator (12)to the nominal state equation (10) we obtain the followingdynamics 119905 = 119860119909119905 + 119861119906119905 (14)

in which the following control transformation is introduced119906119905 fl Γ (119901) 119906119899 (15)

Since the variable 119909119905 is not available for feedback system (14)cannot be stabilized by the standard methods To overcomethis issue we obtain the following differential equation

Mathematical Problems in Engineering 5

describing the dynamic relation between the distance coor-dinate and the reference tracking through applying Γ(119901) tothe tracking error (5)

119890(119897) + 119897minus1sum119894=0

120572119897minus119894minus1119890(119897minus119894minus1) = 119862119909119905 (16)

This differential equation is converted to the following statespace representation simply by considering 119909119905 as an input119909119890 = Λ119909119890 + 119864119862119909119905119890 = 119866119909119890 (17)

in which 119909119890 = col(119890 119890(119897minus1)) andΛ = [[[[[[[

0 119868 00 0 sdot sdot sdot 0 d 119868minus1205720119868 minus1205721119868 minus120572119897minus1119868]]]]]]]

119864 = [[[[[0119868]]]]]

119866 = [119868 0 0]

(18)

Setting 119909119886119906119892 = col(119909119905 119909119890) while selecting 119890 as the fictitiousoutput both (14) and (17) are integrated into the followingaugmented system119886119906119892 = 119860119886119906119892119909119886119906119892 + 119861119886119906119892119906119905119890 = 119862119886119906119892119909119886119906119892 (19)

where

119860119886119906119892 = [ 119860 0119864119862 Λ] 119861119886119906119892 = [1198610] 119862119886119906119892 = [0 119866]

(20)

Proposition 8 Under Assumptions 1 and 6 the augmentedsystem (19) is both controllable and observable

Proof According to the Popov-Belevitch-Hautus (PBH) testthe following rank conditions should be verified for all 119904 isinspec(119860119886119906119892) ie

rank ([119904119868 minus 119860119886119906119892 119861119886119906119892]) = 119899 + 119898119897 (21)

rank([119904119868 minus 119860119886119906119892119862119886119906119892 ]) = 119899 + 119898119897 (22)

Controllability To assure the controllability of the augmentedsystem (19) the rank condition (21) should hold By expand-ing the corresponding matrix and after an appropriate rear-rangement of the block rows and columns the rank conditionis inherited over

rank((((

[[[[[[[[[[

119861 119904119868 minus 119860 0 0 00 minus119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 0 119904119868 minus119868 0 d0 0 0 119904119868 minus119868

]]]]]]]]]]))))

(23)

By the controllability condition of Assumption 1 the firstblock row has the rank of 119899 for all 119904 isin C Consequentlyowing to the identity matrices with different column indicesthe last 119897 minus 1 block rows have the rank 119898(119897 minus 1) Note that thesecond block row is independent of the other rows unlessfor 119904 isin spec(119878) Thereby to guarantee the controllabilitythe following rank condition needs to be satisfied for all 119904 isinspec(119878)

rank([119861 119904119868 minus 1198600 minus119862 ]) = 119899 + 119898 (24)

Clearly in view of Assumption 6 this condition is verified

Observability To show observability of the augmented system(17) the rank condition (22) needs to be verified Again weexpand the pertinent matrix rank condition as

rank((((

[[[[[[[[[[

119904119868 minus 119860 0 0 00 119904119868 minus119868 0 d minus119868119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 119868 0 0]]]]]]]]]]))))

(25)

Due to the observability condition of Assumption 1 the firstblock column has the rank 119899 for all 119904 isin C The last 119897 blockcolumns are independent due to the identity matrices withdifferent row indices Accordingly the overall rank of thematrix is 119899 + 119898119897 This completes the proof

In order to stabilize the augmented system (19) wepropose a dynamic output feedback controller of the form120581 = 1198651119909120581 + 1198652119890119906119905 = 1198653119909120581 (26)

where 119909120581 isin R119899120581 is the internal state of the controller and119865119894 119894 = 1 2 3 are real matrices of appropriate dimensions Thecontroller (26) in conjunction with the augmented system(19) yields a closed-loop system governed by the followingequation 119904 = 119860119904119909119904 (27)

6 Mathematical Problems in Engineering

Here 119909119904 = col(119909119886119906119892 119909120581) is the aggregated state vectorMoreover the system matrix is given as

119860 119904 = [ 119860119886119906119892 11986111988611990611989211986531198652119862119886119906119892 1198651 ] (28)

Owing to Proposition 8 there always exist matrices 11986511198652 and 1198653 such that the resultant closed-loop system isexponentially stable In view of this observation without lossof generality the following assumption is made

Assumption 9 The controller (26) is synthesized in such away that the matrix 119860119904 is Hurwitz stable

The proposed output regulation method uses the trans-formed control variable (15) in its feedback loop ie 119906119905 Con-sequently the primary control input is readily available byapplying the inverse of the operator Γ(119901) to the transformedvariable To this end the following state space realization isemployed 119906 = Λ119909119906 + 119864119906119905119906 = 119866119909119906 (29)

where 119909119906 = col(119906 119906(119897minus1)) and the system matrices are thesame as those of (17) The controller (26) together with thesystem (29) constitute the nominal output regulation loop ofthe overall control system

Remark 10 System (29) replicate a model of the exogenoussignal 119908 defined by (2) Therefore the proposed nominaloutput regulation has the internal model property which inturn guarantees asymptotic rejectiontracking of any signalbelonging to the solution space of (2) [7]

4 Extended State Observer

The output regulation technique presented in Section 3can handle the disturbances whose frequency spectraare embedded in the modes of system (2) For com-pensation of the disturbances without known frequencycharacteristicsmdashrepresented by the signal 120585(119905)mdashan ESO isdesigned Note that trivially based on the matched condi-tion the unstructured disturbance is assumed to belong tothe range space of 119861 otherwise the transfer function fromdisturbance to the output of interest cannot be set to null (seethe geometrical approach proposed in [27]) Considering thecomposite system (6) the following ESO is proposed119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 + 1198671120601 (119910 minus 119910) 119908 = 119878119908 + 1198672120601 (119910 minus 119910) 120585 = 1198673120601 (119910 minus 119910) (30)

where 119909 119908 and 120585 are the estimates of the states structuredand unstructured disturbances respectively Additionally theestimated system output is 119910 = 119862119909 In ESO dynamics 119867119894 119894 =1 2 3 are design matrices of appropriate dimensions Based

on the nonlinear observer proposed by Prasov and Khalil[28] for positive scalars 0 lt 120576 lt 1 and 119889 gt 0 the nonlineargain function of the observer ie 120601() is defined as120601 (119911) = [1206011 (1199111) 120601119898 (119911119898)]119879 120601119895 (119911119895) fl

120576119911119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 119889119911119895 + 119889 (120576 minus 1) sign (119911119895) 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 gt 119889119895 = 1 119898(31)

Remark 11 Thegain function (31) is a dead-zone nonlinearitydecreasing the observer gain when the estimation error fallsinside the zone [minus119889 119889] This allows adjusting a trade-offbetween fast state estimation and robustness to measurementnoises

In order to investigate convergence of the ESO (30)defining the error variables as 1205781 = 119909 minus 119909 1205782 = 119908 minus 119908 and1205783 fl 120585 minus 120585 we obtain the following differential equation forthe estimation error dynamics120578 = 1198600120578 + 119867120601 (120589) + 119876 120585120589 = 1198620120578 (32)

where 120578 = col(1205781 1205782 1205783) and the system matrices are givenby

1198600 = [[[119860 119862120590 1198610 119878 00 0 0]]]

119867 = [[[minus1198671minus1198672minus1198673]]]

119876 = [[[00119868]]]

1198620 = [119862 0 0]

(33)

Assumption 12 The unperturbed error dynamics given in120578 = 1198600120578 + 119867120601 (120589) 120589 = 1198620120578 (34)

has a globally exponentially stable equilibrium at the originThe feasibility of this assumption is addressed in Section 5

Theorem 13 Under Assumptions 3 and 12 the estimationerror of the ESO (30) is globally bounded and globally ulti-mately bounded in the sense that after a finite transient time1198790 gt 0 the following bound is valid10038171003817100381710038171205781003817100381710038171003817 le 1205880 + 1205731205850 (35)

where 1205880 gt 0 is arbitrarily small while 120573 gt 0 and 1205850 flsup119905ge0120585(119905)

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Mathematical Problems in Engineering 3

where 119909(119905) isin R119899 is the state 119906(119905) isin R119898 is the controlinput 119910(119905) isin R119898 denotes the measured performanceoutput and 120574(119909(119905) 119905) isin R119899 times R+ 997888rarr R119899 is a sufficientlysmooth (differentiable) nonlinear function representing thetotal disturbance119860 119861 and 119862 are real matrices of appropriatedimensions satisfying the following assumption

Assumption 1 The pair (119860 119861) is controllable and (119862 119860) isobservable

The control goal is to asymptotically track a givenreference signal ie 119910119898(119905) isin R119898 while boundedness ofthe all other signals is guaranteed in the presence of thetotal disturbance 120574(119909(119905) 119905) The reference signal 119910119898(119905) isconsidered to be generated by the solution of a fixed linearautonomous dynamic system of the form (119905) = 119878119908 (119905) 119910119898 (119905) = 119862119898119908 (119905) (2)

where 119908(119905) isin R119903 and 119878 and 119862119898 are real-valued matriceswith proper dimensionsThe following assumption is taken toguarantee the persistence and boundedness of the referencesignal 119910119898(119905) [23]Assumption 2 All eigenvalues of the matrix 119878 have zero realparts with multiplicity one in the minimal polynomial

By setting 120590(119905) fl 120574(119909(119905) 119905) the system in (1) can berewritten as (119905) = 119860119909 (119905) + 119861119906 (119905) + 120590 (119905) 119910 (119905) = 119862119909 (119905) (3)

The systems in (1) and (2) are trajectory equivalent inthe sense that under the same initial values their statetrajectories coincide with each other [24 25] Therefore wecarry out the controller design and analysis on the basis of(2)

The conventional output regulation theory is based on theassumption that the perturbing disturbances belong to thesolution space of the dynamic system given in (2) Here wegeneralize this assumption by including disturbances whichdo not necessarily comply with this particular dynamics

Assumption 3 The total disturbance signal satisfies120590 (119905) = 119862120590119908 (119905) + 119861120585 (119905) (4)

where 119862120590 isin R119903times119903 is a given matrix with the pair (119862120590 119878) beingobservable The signal 120585(119905) isin R119898 is bounded and sufficientlydifferentiable such that 120585(119905) is bounded as well

Remark 4 The rationale behind Assumption 3 can beexplained as follows In many engineering systems only thedominant frequencies of the disturbances can be obtainedvia time andor frequency domain identification methods(see for example [8]) In such systems one can embed theknown dominant frequencies into the system (2) and model

the corresponding disturbance approximation error into theterm 120585() As another example consider the tracking controlproblems where the dominant dynamics of the underlyingsystems are linear Assuming that the norm of the state-dependent disturbances is smaller than the convergence rateof the linear part it is conceivable that for a successfultracking the frequency spectrum of the disturbances willcontain the modes of the reference signals Accordingly (4)forms a suitable basis for modeling of the disturbances Anexample of such modeling can be found in [9]

Let us define the tracking performance of the controlsystem as 119890 fl 119910 minus 119910119898 (5)

Note that here and after the dependence on time variable (119905)is dropped (unless necessary) in the formulation for the sakeof readability The following composite system is proposedregarding the underlying output regulation control problem119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 = 119878119908119890 = 119862119909 minus 119862119898119908 (6)

In order to examine the solvability of the output regulationproblem first we consider a steady-state condition in whichthe output tracking is achieved In such a steady-state condi-tion the trajectories of the underlying system should be well-defined to ensure that the solution of the output regulationproblem exists Denoting 119909 119906 120585 and 120590 fl 120574(119909 119905) as thesteady-state vectors of the state control unstructured andthe structured disturbances respectively it is easy to obtain = 119860119909 + 119861119906 + 120590120590 = 119862120590119908 + 119861120585119862119909 minus 119862119898119908 = 0 (7)

Setting the solutions as 119909 = 119883119908 and 119906 = 119880119908 minus 120585 for some119883 isin R119899times119903 and 119880 isin R119898times119903 we obtain the following Sylvester-type matrix equations from (7)119883119878 = 119860119883 + 119861119880 + 119862120590119862119883 minus 119862119898 = 0 (8)

Theorem 5 For any given matrices 119862119898 and 119862120590 there existunique matrices 119883 and 119880 satisfying (8) if and only if thefollowing assumption holds

Assumption 6 For all 119904 isin spec(119878) (spec() represents thevector of eigenvalues of the matrix ())

rank(119860 minus 119904119868 119861119862 0) = 119899 + 119898 (9)

Proof See [26] for the proof

4 Mathematical Problems in Engineering

From a geometrical point of view solutions of thematrix equations (8) mdashknown as the regulator equa-tionsmdashcorresponds to a zero-error controlled invariant man-ifold defined by M fl (119909 119908) isin R119899 times R119903 | 119909 minus 119883119908 = 0for the composite system (6) In other terms for an initialcondition of the form 119909(0) = 119883119908(0) under the control input119906 the trajectory 119909 will evolve on M for all 119905 isin R+ andthe tracking error (5) will be identically equal to zero Fromthis standpoint solving the formulated output regulationproblem amounts to rendering the manifold M globallyattractive by a suitable control function [7] Since exactdynamic characterization of the total disturbance signal is notavailable we pursue practical stabilization of the manifoldMfor the composite system (6)More specifically trajectories ofthe closed-loop system all should converge to a small compactset aroundM by approximately recovering the solutions of theregulator equations To this end we propose a disturbancerejection-based two-degrees-of-freedom control structureThe control system is composed of a nominal output regulatorequipped with an ESO for disturbance rejection Accordinglythe main design stages of this control system are as follows

Stage 1 The nominal controller solves the output regulationproblem for the given dynamical system in (5) with thedifference that the state equation is simplified to = 119860119909 + 119861119906119899 + 119862120590119908 (10)

Note that the corresponding control input 119906119899 isin R119898 isdesigned to drive the nominal closed-loop trajectories towardM by recovering the nominal part of the steady-state controlsignal defined by 119906119899 = 119880119908 This corresponds to thefeedforward control input required for tracking 119910119898 as well ascompensating for 119862120590119908Stage 2 On the other hand the disturbance rejection looputilizes an ESO that continuously monitors the input-outputsignals of the plant to provide an estimate of the disturbancedenoted by 120585 The obtained estimated value is then integratedwith the nominal control input to cancel out the disturbancesignal Accordingly the overall control input is in the form119906 = 119906119899 minus 120585 (11)

The configuration of the proposed control system is schemat-ically illustrated by the block diagram in Figure 1

3 Nominal Output Regulation

To solve the nominal output regulation problem we charac-terize the distance of the system trajectories from the targetmanifold M by a suitable coordinate Then by controllingthis coordinate to zero the output regulation will be accom-plished Considering the steady-state system trajectories onthe manifold an intuitive candidate for such a coordinate is119909minus119909 However owing to the direct dependence of the regula-tor equations on the solutions of 120585 this method is sensitive toperturbation (see [7]) Consequently the intuitive candidatemay render the control system vulnerable to the disturbance

Figure 1 Block diagram of the proposed control system

estimation errors Hence we follow a more robust designapproach based on the internal model principle For thispurpose setting 119901 = 119889119889119905 the following linear differentialoperator is considered

Γ (119901) = 119901119897 + 119897minus1sum119894=0

120572119897minus119894minus1119901119897minus119894minus1 (12)

where 120572119894|119897minus1119894=0 with 120572119897 fl 1 are the coefficients of the minimalpolynomial of the matrix 119878 in the ascending order Next weintroduce the following transformed variable119909119905 = Γ (119901) 119909 (13)

Proposition 7 119909119905 equiv 0 if and only if (119909 119908) isin M

Proof The motion of the system trajectories onM under thecontrol input 119906119899 is equivalent to the immersion of system (10)into (2) By properties of the minimal polynomial for anytrajectory of 119908 in (2) we have Γ(119901)119908 = 0 Thereby the setof state trajectories evolving on the manifold M correspondsto the kernel of the operator Γ(119901) ie Ker(Γ()) = 119909 R+ 997888rarr R119899 | Γ()119909 = 0 Thus the variable 119909119905 acts as anindicator function for the set Ker(Γ()) in the sense that a statetrajectory 119909(119905) evolves onM if and only if its transformation119909119905(119905) is equivalent to zero

In view of Proposition 7 the variable 119909119905 is a distancecoordinate characterizing attractivity of the manifold M fortrajectories of system (10) By embedding the internal modelof the reference signal this distance coordinate does notdepend on the solutions of the regulator equation andtherefore enables a robust design Interestingly based on thisargument the output regulation problem entails the problemof regulating 119909119905 to zero Hence by applying the operator (12)to the nominal state equation (10) we obtain the followingdynamics 119905 = 119860119909119905 + 119861119906119905 (14)

in which the following control transformation is introduced119906119905 fl Γ (119901) 119906119899 (15)

Since the variable 119909119905 is not available for feedback system (14)cannot be stabilized by the standard methods To overcomethis issue we obtain the following differential equation

Mathematical Problems in Engineering 5

describing the dynamic relation between the distance coor-dinate and the reference tracking through applying Γ(119901) tothe tracking error (5)

119890(119897) + 119897minus1sum119894=0

120572119897minus119894minus1119890(119897minus119894minus1) = 119862119909119905 (16)

This differential equation is converted to the following statespace representation simply by considering 119909119905 as an input119909119890 = Λ119909119890 + 119864119862119909119905119890 = 119866119909119890 (17)

in which 119909119890 = col(119890 119890(119897minus1)) andΛ = [[[[[[[

0 119868 00 0 sdot sdot sdot 0 d 119868minus1205720119868 minus1205721119868 minus120572119897minus1119868]]]]]]]

119864 = [[[[[0119868]]]]]

119866 = [119868 0 0]

(18)

Setting 119909119886119906119892 = col(119909119905 119909119890) while selecting 119890 as the fictitiousoutput both (14) and (17) are integrated into the followingaugmented system119886119906119892 = 119860119886119906119892119909119886119906119892 + 119861119886119906119892119906119905119890 = 119862119886119906119892119909119886119906119892 (19)

where

119860119886119906119892 = [ 119860 0119864119862 Λ] 119861119886119906119892 = [1198610] 119862119886119906119892 = [0 119866]

(20)

Proposition 8 Under Assumptions 1 and 6 the augmentedsystem (19) is both controllable and observable

Proof According to the Popov-Belevitch-Hautus (PBH) testthe following rank conditions should be verified for all 119904 isinspec(119860119886119906119892) ie

rank ([119904119868 minus 119860119886119906119892 119861119886119906119892]) = 119899 + 119898119897 (21)

rank([119904119868 minus 119860119886119906119892119862119886119906119892 ]) = 119899 + 119898119897 (22)

Controllability To assure the controllability of the augmentedsystem (19) the rank condition (21) should hold By expand-ing the corresponding matrix and after an appropriate rear-rangement of the block rows and columns the rank conditionis inherited over

rank((((

[[[[[[[[[[

119861 119904119868 minus 119860 0 0 00 minus119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 0 119904119868 minus119868 0 d0 0 0 119904119868 minus119868

]]]]]]]]]]))))

(23)

By the controllability condition of Assumption 1 the firstblock row has the rank of 119899 for all 119904 isin C Consequentlyowing to the identity matrices with different column indicesthe last 119897 minus 1 block rows have the rank 119898(119897 minus 1) Note that thesecond block row is independent of the other rows unlessfor 119904 isin spec(119878) Thereby to guarantee the controllabilitythe following rank condition needs to be satisfied for all 119904 isinspec(119878)

rank([119861 119904119868 minus 1198600 minus119862 ]) = 119899 + 119898 (24)

Clearly in view of Assumption 6 this condition is verified

Observability To show observability of the augmented system(17) the rank condition (22) needs to be verified Again weexpand the pertinent matrix rank condition as

rank((((

[[[[[[[[[[

119904119868 minus 119860 0 0 00 119904119868 minus119868 0 d minus119868119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 119868 0 0]]]]]]]]]]))))

(25)

Due to the observability condition of Assumption 1 the firstblock column has the rank 119899 for all 119904 isin C The last 119897 blockcolumns are independent due to the identity matrices withdifferent row indices Accordingly the overall rank of thematrix is 119899 + 119898119897 This completes the proof

In order to stabilize the augmented system (19) wepropose a dynamic output feedback controller of the form120581 = 1198651119909120581 + 1198652119890119906119905 = 1198653119909120581 (26)

where 119909120581 isin R119899120581 is the internal state of the controller and119865119894 119894 = 1 2 3 are real matrices of appropriate dimensions Thecontroller (26) in conjunction with the augmented system(19) yields a closed-loop system governed by the followingequation 119904 = 119860119904119909119904 (27)

6 Mathematical Problems in Engineering

Here 119909119904 = col(119909119886119906119892 119909120581) is the aggregated state vectorMoreover the system matrix is given as

119860 119904 = [ 119860119886119906119892 11986111988611990611989211986531198652119862119886119906119892 1198651 ] (28)

Owing to Proposition 8 there always exist matrices 11986511198652 and 1198653 such that the resultant closed-loop system isexponentially stable In view of this observation without lossof generality the following assumption is made

Assumption 9 The controller (26) is synthesized in such away that the matrix 119860119904 is Hurwitz stable

The proposed output regulation method uses the trans-formed control variable (15) in its feedback loop ie 119906119905 Con-sequently the primary control input is readily available byapplying the inverse of the operator Γ(119901) to the transformedvariable To this end the following state space realization isemployed 119906 = Λ119909119906 + 119864119906119905119906 = 119866119909119906 (29)

where 119909119906 = col(119906 119906(119897minus1)) and the system matrices are thesame as those of (17) The controller (26) together with thesystem (29) constitute the nominal output regulation loop ofthe overall control system

Remark 10 System (29) replicate a model of the exogenoussignal 119908 defined by (2) Therefore the proposed nominaloutput regulation has the internal model property which inturn guarantees asymptotic rejectiontracking of any signalbelonging to the solution space of (2) [7]

4 Extended State Observer

The output regulation technique presented in Section 3can handle the disturbances whose frequency spectraare embedded in the modes of system (2) For com-pensation of the disturbances without known frequencycharacteristicsmdashrepresented by the signal 120585(119905)mdashan ESO isdesigned Note that trivially based on the matched condi-tion the unstructured disturbance is assumed to belong tothe range space of 119861 otherwise the transfer function fromdisturbance to the output of interest cannot be set to null (seethe geometrical approach proposed in [27]) Considering thecomposite system (6) the following ESO is proposed119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 + 1198671120601 (119910 minus 119910) 119908 = 119878119908 + 1198672120601 (119910 minus 119910) 120585 = 1198673120601 (119910 minus 119910) (30)

where 119909 119908 and 120585 are the estimates of the states structuredand unstructured disturbances respectively Additionally theestimated system output is 119910 = 119862119909 In ESO dynamics 119867119894 119894 =1 2 3 are design matrices of appropriate dimensions Based

on the nonlinear observer proposed by Prasov and Khalil[28] for positive scalars 0 lt 120576 lt 1 and 119889 gt 0 the nonlineargain function of the observer ie 120601() is defined as120601 (119911) = [1206011 (1199111) 120601119898 (119911119898)]119879 120601119895 (119911119895) fl

120576119911119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 119889119911119895 + 119889 (120576 minus 1) sign (119911119895) 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 gt 119889119895 = 1 119898(31)

Remark 11 Thegain function (31) is a dead-zone nonlinearitydecreasing the observer gain when the estimation error fallsinside the zone [minus119889 119889] This allows adjusting a trade-offbetween fast state estimation and robustness to measurementnoises

In order to investigate convergence of the ESO (30)defining the error variables as 1205781 = 119909 minus 119909 1205782 = 119908 minus 119908 and1205783 fl 120585 minus 120585 we obtain the following differential equation forthe estimation error dynamics120578 = 1198600120578 + 119867120601 (120589) + 119876 120585120589 = 1198620120578 (32)

where 120578 = col(1205781 1205782 1205783) and the system matrices are givenby

1198600 = [[[119860 119862120590 1198610 119878 00 0 0]]]

119867 = [[[minus1198671minus1198672minus1198673]]]

119876 = [[[00119868]]]

1198620 = [119862 0 0]

(33)

Assumption 12 The unperturbed error dynamics given in120578 = 1198600120578 + 119867120601 (120589) 120589 = 1198620120578 (34)

has a globally exponentially stable equilibrium at the originThe feasibility of this assumption is addressed in Section 5

Theorem 13 Under Assumptions 3 and 12 the estimationerror of the ESO (30) is globally bounded and globally ulti-mately bounded in the sense that after a finite transient time1198790 gt 0 the following bound is valid10038171003817100381710038171205781003817100381710038171003817 le 1205880 + 1205731205850 (35)

where 1205880 gt 0 is arbitrarily small while 120573 gt 0 and 1205850 flsup119905ge0120585(119905)

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

4 Mathematical Problems in Engineering

From a geometrical point of view solutions of thematrix equations (8) mdashknown as the regulator equa-tionsmdashcorresponds to a zero-error controlled invariant man-ifold defined by M fl (119909 119908) isin R119899 times R119903 | 119909 minus 119883119908 = 0for the composite system (6) In other terms for an initialcondition of the form 119909(0) = 119883119908(0) under the control input119906 the trajectory 119909 will evolve on M for all 119905 isin R+ andthe tracking error (5) will be identically equal to zero Fromthis standpoint solving the formulated output regulationproblem amounts to rendering the manifold M globallyattractive by a suitable control function [7] Since exactdynamic characterization of the total disturbance signal is notavailable we pursue practical stabilization of the manifoldMfor the composite system (6)More specifically trajectories ofthe closed-loop system all should converge to a small compactset aroundM by approximately recovering the solutions of theregulator equations To this end we propose a disturbancerejection-based two-degrees-of-freedom control structureThe control system is composed of a nominal output regulatorequipped with an ESO for disturbance rejection Accordinglythe main design stages of this control system are as follows

Stage 1 The nominal controller solves the output regulationproblem for the given dynamical system in (5) with thedifference that the state equation is simplified to = 119860119909 + 119861119906119899 + 119862120590119908 (10)

Note that the corresponding control input 119906119899 isin R119898 isdesigned to drive the nominal closed-loop trajectories towardM by recovering the nominal part of the steady-state controlsignal defined by 119906119899 = 119880119908 This corresponds to thefeedforward control input required for tracking 119910119898 as well ascompensating for 119862120590119908Stage 2 On the other hand the disturbance rejection looputilizes an ESO that continuously monitors the input-outputsignals of the plant to provide an estimate of the disturbancedenoted by 120585 The obtained estimated value is then integratedwith the nominal control input to cancel out the disturbancesignal Accordingly the overall control input is in the form119906 = 119906119899 minus 120585 (11)

The configuration of the proposed control system is schemat-ically illustrated by the block diagram in Figure 1

3 Nominal Output Regulation

To solve the nominal output regulation problem we charac-terize the distance of the system trajectories from the targetmanifold M by a suitable coordinate Then by controllingthis coordinate to zero the output regulation will be accom-plished Considering the steady-state system trajectories onthe manifold an intuitive candidate for such a coordinate is119909minus119909 However owing to the direct dependence of the regula-tor equations on the solutions of 120585 this method is sensitive toperturbation (see [7]) Consequently the intuitive candidatemay render the control system vulnerable to the disturbance

Figure 1 Block diagram of the proposed control system

estimation errors Hence we follow a more robust designapproach based on the internal model principle For thispurpose setting 119901 = 119889119889119905 the following linear differentialoperator is considered

Γ (119901) = 119901119897 + 119897minus1sum119894=0

120572119897minus119894minus1119901119897minus119894minus1 (12)

where 120572119894|119897minus1119894=0 with 120572119897 fl 1 are the coefficients of the minimalpolynomial of the matrix 119878 in the ascending order Next weintroduce the following transformed variable119909119905 = Γ (119901) 119909 (13)

Proposition 7 119909119905 equiv 0 if and only if (119909 119908) isin M

Proof The motion of the system trajectories onM under thecontrol input 119906119899 is equivalent to the immersion of system (10)into (2) By properties of the minimal polynomial for anytrajectory of 119908 in (2) we have Γ(119901)119908 = 0 Thereby the setof state trajectories evolving on the manifold M correspondsto the kernel of the operator Γ(119901) ie Ker(Γ()) = 119909 R+ 997888rarr R119899 | Γ()119909 = 0 Thus the variable 119909119905 acts as anindicator function for the set Ker(Γ()) in the sense that a statetrajectory 119909(119905) evolves onM if and only if its transformation119909119905(119905) is equivalent to zero

In view of Proposition 7 the variable 119909119905 is a distancecoordinate characterizing attractivity of the manifold M fortrajectories of system (10) By embedding the internal modelof the reference signal this distance coordinate does notdepend on the solutions of the regulator equation andtherefore enables a robust design Interestingly based on thisargument the output regulation problem entails the problemof regulating 119909119905 to zero Hence by applying the operator (12)to the nominal state equation (10) we obtain the followingdynamics 119905 = 119860119909119905 + 119861119906119905 (14)

in which the following control transformation is introduced119906119905 fl Γ (119901) 119906119899 (15)

Since the variable 119909119905 is not available for feedback system (14)cannot be stabilized by the standard methods To overcomethis issue we obtain the following differential equation

Mathematical Problems in Engineering 5

describing the dynamic relation between the distance coor-dinate and the reference tracking through applying Γ(119901) tothe tracking error (5)

119890(119897) + 119897minus1sum119894=0

120572119897minus119894minus1119890(119897minus119894minus1) = 119862119909119905 (16)

This differential equation is converted to the following statespace representation simply by considering 119909119905 as an input119909119890 = Λ119909119890 + 119864119862119909119905119890 = 119866119909119890 (17)

in which 119909119890 = col(119890 119890(119897minus1)) andΛ = [[[[[[[

0 119868 00 0 sdot sdot sdot 0 d 119868minus1205720119868 minus1205721119868 minus120572119897minus1119868]]]]]]]

119864 = [[[[[0119868]]]]]

119866 = [119868 0 0]

(18)

Setting 119909119886119906119892 = col(119909119905 119909119890) while selecting 119890 as the fictitiousoutput both (14) and (17) are integrated into the followingaugmented system119886119906119892 = 119860119886119906119892119909119886119906119892 + 119861119886119906119892119906119905119890 = 119862119886119906119892119909119886119906119892 (19)

where

119860119886119906119892 = [ 119860 0119864119862 Λ] 119861119886119906119892 = [1198610] 119862119886119906119892 = [0 119866]

(20)

Proposition 8 Under Assumptions 1 and 6 the augmentedsystem (19) is both controllable and observable

Proof According to the Popov-Belevitch-Hautus (PBH) testthe following rank conditions should be verified for all 119904 isinspec(119860119886119906119892) ie

rank ([119904119868 minus 119860119886119906119892 119861119886119906119892]) = 119899 + 119898119897 (21)

rank([119904119868 minus 119860119886119906119892119862119886119906119892 ]) = 119899 + 119898119897 (22)

Controllability To assure the controllability of the augmentedsystem (19) the rank condition (21) should hold By expand-ing the corresponding matrix and after an appropriate rear-rangement of the block rows and columns the rank conditionis inherited over

rank((((

[[[[[[[[[[

119861 119904119868 minus 119860 0 0 00 minus119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 0 119904119868 minus119868 0 d0 0 0 119904119868 minus119868

]]]]]]]]]]))))

(23)

By the controllability condition of Assumption 1 the firstblock row has the rank of 119899 for all 119904 isin C Consequentlyowing to the identity matrices with different column indicesthe last 119897 minus 1 block rows have the rank 119898(119897 minus 1) Note that thesecond block row is independent of the other rows unlessfor 119904 isin spec(119878) Thereby to guarantee the controllabilitythe following rank condition needs to be satisfied for all 119904 isinspec(119878)

rank([119861 119904119868 minus 1198600 minus119862 ]) = 119899 + 119898 (24)

Clearly in view of Assumption 6 this condition is verified

Observability To show observability of the augmented system(17) the rank condition (22) needs to be verified Again weexpand the pertinent matrix rank condition as

rank((((

[[[[[[[[[[

119904119868 minus 119860 0 0 00 119904119868 minus119868 0 d minus119868119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 119868 0 0]]]]]]]]]]))))

(25)

Due to the observability condition of Assumption 1 the firstblock column has the rank 119899 for all 119904 isin C The last 119897 blockcolumns are independent due to the identity matrices withdifferent row indices Accordingly the overall rank of thematrix is 119899 + 119898119897 This completes the proof

In order to stabilize the augmented system (19) wepropose a dynamic output feedback controller of the form120581 = 1198651119909120581 + 1198652119890119906119905 = 1198653119909120581 (26)

where 119909120581 isin R119899120581 is the internal state of the controller and119865119894 119894 = 1 2 3 are real matrices of appropriate dimensions Thecontroller (26) in conjunction with the augmented system(19) yields a closed-loop system governed by the followingequation 119904 = 119860119904119909119904 (27)

6 Mathematical Problems in Engineering

Here 119909119904 = col(119909119886119906119892 119909120581) is the aggregated state vectorMoreover the system matrix is given as

119860 119904 = [ 119860119886119906119892 11986111988611990611989211986531198652119862119886119906119892 1198651 ] (28)

Owing to Proposition 8 there always exist matrices 11986511198652 and 1198653 such that the resultant closed-loop system isexponentially stable In view of this observation without lossof generality the following assumption is made

Assumption 9 The controller (26) is synthesized in such away that the matrix 119860119904 is Hurwitz stable

The proposed output regulation method uses the trans-formed control variable (15) in its feedback loop ie 119906119905 Con-sequently the primary control input is readily available byapplying the inverse of the operator Γ(119901) to the transformedvariable To this end the following state space realization isemployed 119906 = Λ119909119906 + 119864119906119905119906 = 119866119909119906 (29)

where 119909119906 = col(119906 119906(119897minus1)) and the system matrices are thesame as those of (17) The controller (26) together with thesystem (29) constitute the nominal output regulation loop ofthe overall control system

Remark 10 System (29) replicate a model of the exogenoussignal 119908 defined by (2) Therefore the proposed nominaloutput regulation has the internal model property which inturn guarantees asymptotic rejectiontracking of any signalbelonging to the solution space of (2) [7]

4 Extended State Observer

The output regulation technique presented in Section 3can handle the disturbances whose frequency spectraare embedded in the modes of system (2) For com-pensation of the disturbances without known frequencycharacteristicsmdashrepresented by the signal 120585(119905)mdashan ESO isdesigned Note that trivially based on the matched condi-tion the unstructured disturbance is assumed to belong tothe range space of 119861 otherwise the transfer function fromdisturbance to the output of interest cannot be set to null (seethe geometrical approach proposed in [27]) Considering thecomposite system (6) the following ESO is proposed119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 + 1198671120601 (119910 minus 119910) 119908 = 119878119908 + 1198672120601 (119910 minus 119910) 120585 = 1198673120601 (119910 minus 119910) (30)

where 119909 119908 and 120585 are the estimates of the states structuredand unstructured disturbances respectively Additionally theestimated system output is 119910 = 119862119909 In ESO dynamics 119867119894 119894 =1 2 3 are design matrices of appropriate dimensions Based

on the nonlinear observer proposed by Prasov and Khalil[28] for positive scalars 0 lt 120576 lt 1 and 119889 gt 0 the nonlineargain function of the observer ie 120601() is defined as120601 (119911) = [1206011 (1199111) 120601119898 (119911119898)]119879 120601119895 (119911119895) fl

120576119911119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 119889119911119895 + 119889 (120576 minus 1) sign (119911119895) 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 gt 119889119895 = 1 119898(31)

Remark 11 Thegain function (31) is a dead-zone nonlinearitydecreasing the observer gain when the estimation error fallsinside the zone [minus119889 119889] This allows adjusting a trade-offbetween fast state estimation and robustness to measurementnoises

In order to investigate convergence of the ESO (30)defining the error variables as 1205781 = 119909 minus 119909 1205782 = 119908 minus 119908 and1205783 fl 120585 minus 120585 we obtain the following differential equation forthe estimation error dynamics120578 = 1198600120578 + 119867120601 (120589) + 119876 120585120589 = 1198620120578 (32)

where 120578 = col(1205781 1205782 1205783) and the system matrices are givenby

1198600 = [[[119860 119862120590 1198610 119878 00 0 0]]]

119867 = [[[minus1198671minus1198672minus1198673]]]

119876 = [[[00119868]]]

1198620 = [119862 0 0]

(33)

Assumption 12 The unperturbed error dynamics given in120578 = 1198600120578 + 119867120601 (120589) 120589 = 1198620120578 (34)

has a globally exponentially stable equilibrium at the originThe feasibility of this assumption is addressed in Section 5

Theorem 13 Under Assumptions 3 and 12 the estimationerror of the ESO (30) is globally bounded and globally ulti-mately bounded in the sense that after a finite transient time1198790 gt 0 the following bound is valid10038171003817100381710038171205781003817100381710038171003817 le 1205880 + 1205731205850 (35)

where 1205880 gt 0 is arbitrarily small while 120573 gt 0 and 1205850 flsup119905ge0120585(119905)

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Mathematical Problems in Engineering 5

describing the dynamic relation between the distance coor-dinate and the reference tracking through applying Γ(119901) tothe tracking error (5)

119890(119897) + 119897minus1sum119894=0

120572119897minus119894minus1119890(119897minus119894minus1) = 119862119909119905 (16)

This differential equation is converted to the following statespace representation simply by considering 119909119905 as an input119909119890 = Λ119909119890 + 119864119862119909119905119890 = 119866119909119890 (17)

in which 119909119890 = col(119890 119890(119897minus1)) andΛ = [[[[[[[

0 119868 00 0 sdot sdot sdot 0 d 119868minus1205720119868 minus1205721119868 minus120572119897minus1119868]]]]]]]

119864 = [[[[[0119868]]]]]

119866 = [119868 0 0]

(18)

Setting 119909119886119906119892 = col(119909119905 119909119890) while selecting 119890 as the fictitiousoutput both (14) and (17) are integrated into the followingaugmented system119886119906119892 = 119860119886119906119892119909119886119906119892 + 119861119886119906119892119906119905119890 = 119862119886119906119892119909119886119906119892 (19)

where

119860119886119906119892 = [ 119860 0119864119862 Λ] 119861119886119906119892 = [1198610] 119862119886119906119892 = [0 119866]

(20)

Proposition 8 Under Assumptions 1 and 6 the augmentedsystem (19) is both controllable and observable

Proof According to the Popov-Belevitch-Hautus (PBH) testthe following rank conditions should be verified for all 119904 isinspec(119860119886119906119892) ie

rank ([119904119868 minus 119860119886119906119892 119861119886119906119892]) = 119899 + 119898119897 (21)

rank([119904119868 minus 119860119886119906119892119862119886119906119892 ]) = 119899 + 119898119897 (22)

Controllability To assure the controllability of the augmentedsystem (19) the rank condition (21) should hold By expand-ing the corresponding matrix and after an appropriate rear-rangement of the block rows and columns the rank conditionis inherited over

rank((((

[[[[[[[[[[

119861 119904119868 minus 119860 0 0 00 minus119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 0 119904119868 minus119868 0 d0 0 0 119904119868 minus119868

]]]]]]]]]]))))

(23)

By the controllability condition of Assumption 1 the firstblock row has the rank of 119899 for all 119904 isin C Consequentlyowing to the identity matrices with different column indicesthe last 119897 minus 1 block rows have the rank 119898(119897 minus 1) Note that thesecond block row is independent of the other rows unlessfor 119904 isin spec(119878) Thereby to guarantee the controllabilitythe following rank condition needs to be satisfied for all 119904 isinspec(119878)

rank([119861 119904119868 minus 1198600 minus119862 ]) = 119899 + 119898 (24)

Clearly in view of Assumption 6 this condition is verified

Observability To show observability of the augmented system(17) the rank condition (22) needs to be verified Again weexpand the pertinent matrix rank condition as

rank((((

[[[[[[[[[[

119904119868 minus 119860 0 0 00 119904119868 minus119868 0 d minus119868119862 1205720119868 1205721119868 (119904 + 120572119897minus1) 1198680 119868 0 0]]]]]]]]]]))))

(25)

Due to the observability condition of Assumption 1 the firstblock column has the rank 119899 for all 119904 isin C The last 119897 blockcolumns are independent due to the identity matrices withdifferent row indices Accordingly the overall rank of thematrix is 119899 + 119898119897 This completes the proof

In order to stabilize the augmented system (19) wepropose a dynamic output feedback controller of the form120581 = 1198651119909120581 + 1198652119890119906119905 = 1198653119909120581 (26)

where 119909120581 isin R119899120581 is the internal state of the controller and119865119894 119894 = 1 2 3 are real matrices of appropriate dimensions Thecontroller (26) in conjunction with the augmented system(19) yields a closed-loop system governed by the followingequation 119904 = 119860119904119909119904 (27)

6 Mathematical Problems in Engineering

Here 119909119904 = col(119909119886119906119892 119909120581) is the aggregated state vectorMoreover the system matrix is given as

119860 119904 = [ 119860119886119906119892 11986111988611990611989211986531198652119862119886119906119892 1198651 ] (28)

Owing to Proposition 8 there always exist matrices 11986511198652 and 1198653 such that the resultant closed-loop system isexponentially stable In view of this observation without lossof generality the following assumption is made

Assumption 9 The controller (26) is synthesized in such away that the matrix 119860119904 is Hurwitz stable

The proposed output regulation method uses the trans-formed control variable (15) in its feedback loop ie 119906119905 Con-sequently the primary control input is readily available byapplying the inverse of the operator Γ(119901) to the transformedvariable To this end the following state space realization isemployed 119906 = Λ119909119906 + 119864119906119905119906 = 119866119909119906 (29)

where 119909119906 = col(119906 119906(119897minus1)) and the system matrices are thesame as those of (17) The controller (26) together with thesystem (29) constitute the nominal output regulation loop ofthe overall control system

Remark 10 System (29) replicate a model of the exogenoussignal 119908 defined by (2) Therefore the proposed nominaloutput regulation has the internal model property which inturn guarantees asymptotic rejectiontracking of any signalbelonging to the solution space of (2) [7]

4 Extended State Observer

The output regulation technique presented in Section 3can handle the disturbances whose frequency spectraare embedded in the modes of system (2) For com-pensation of the disturbances without known frequencycharacteristicsmdashrepresented by the signal 120585(119905)mdashan ESO isdesigned Note that trivially based on the matched condi-tion the unstructured disturbance is assumed to belong tothe range space of 119861 otherwise the transfer function fromdisturbance to the output of interest cannot be set to null (seethe geometrical approach proposed in [27]) Considering thecomposite system (6) the following ESO is proposed119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 + 1198671120601 (119910 minus 119910) 119908 = 119878119908 + 1198672120601 (119910 minus 119910) 120585 = 1198673120601 (119910 minus 119910) (30)

where 119909 119908 and 120585 are the estimates of the states structuredand unstructured disturbances respectively Additionally theestimated system output is 119910 = 119862119909 In ESO dynamics 119867119894 119894 =1 2 3 are design matrices of appropriate dimensions Based

on the nonlinear observer proposed by Prasov and Khalil[28] for positive scalars 0 lt 120576 lt 1 and 119889 gt 0 the nonlineargain function of the observer ie 120601() is defined as120601 (119911) = [1206011 (1199111) 120601119898 (119911119898)]119879 120601119895 (119911119895) fl

120576119911119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 119889119911119895 + 119889 (120576 minus 1) sign (119911119895) 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 gt 119889119895 = 1 119898(31)

Remark 11 Thegain function (31) is a dead-zone nonlinearitydecreasing the observer gain when the estimation error fallsinside the zone [minus119889 119889] This allows adjusting a trade-offbetween fast state estimation and robustness to measurementnoises

In order to investigate convergence of the ESO (30)defining the error variables as 1205781 = 119909 minus 119909 1205782 = 119908 minus 119908 and1205783 fl 120585 minus 120585 we obtain the following differential equation forthe estimation error dynamics120578 = 1198600120578 + 119867120601 (120589) + 119876 120585120589 = 1198620120578 (32)

where 120578 = col(1205781 1205782 1205783) and the system matrices are givenby

1198600 = [[[119860 119862120590 1198610 119878 00 0 0]]]

119867 = [[[minus1198671minus1198672minus1198673]]]

119876 = [[[00119868]]]

1198620 = [119862 0 0]

(33)

Assumption 12 The unperturbed error dynamics given in120578 = 1198600120578 + 119867120601 (120589) 120589 = 1198620120578 (34)

has a globally exponentially stable equilibrium at the originThe feasibility of this assumption is addressed in Section 5

Theorem 13 Under Assumptions 3 and 12 the estimationerror of the ESO (30) is globally bounded and globally ulti-mately bounded in the sense that after a finite transient time1198790 gt 0 the following bound is valid10038171003817100381710038171205781003817100381710038171003817 le 1205880 + 1205731205850 (35)

where 1205880 gt 0 is arbitrarily small while 120573 gt 0 and 1205850 flsup119905ge0120585(119905)

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

6 Mathematical Problems in Engineering

Here 119909119904 = col(119909119886119906119892 119909120581) is the aggregated state vectorMoreover the system matrix is given as

119860 119904 = [ 119860119886119906119892 11986111988611990611989211986531198652119862119886119906119892 1198651 ] (28)

Owing to Proposition 8 there always exist matrices 11986511198652 and 1198653 such that the resultant closed-loop system isexponentially stable In view of this observation without lossof generality the following assumption is made

Assumption 9 The controller (26) is synthesized in such away that the matrix 119860119904 is Hurwitz stable

The proposed output regulation method uses the trans-formed control variable (15) in its feedback loop ie 119906119905 Con-sequently the primary control input is readily available byapplying the inverse of the operator Γ(119901) to the transformedvariable To this end the following state space realization isemployed 119906 = Λ119909119906 + 119864119906119905119906 = 119866119909119906 (29)

where 119909119906 = col(119906 119906(119897minus1)) and the system matrices are thesame as those of (17) The controller (26) together with thesystem (29) constitute the nominal output regulation loop ofthe overall control system

Remark 10 System (29) replicate a model of the exogenoussignal 119908 defined by (2) Therefore the proposed nominaloutput regulation has the internal model property which inturn guarantees asymptotic rejectiontracking of any signalbelonging to the solution space of (2) [7]

4 Extended State Observer

The output regulation technique presented in Section 3can handle the disturbances whose frequency spectraare embedded in the modes of system (2) For com-pensation of the disturbances without known frequencycharacteristicsmdashrepresented by the signal 120585(119905)mdashan ESO isdesigned Note that trivially based on the matched condi-tion the unstructured disturbance is assumed to belong tothe range space of 119861 otherwise the transfer function fromdisturbance to the output of interest cannot be set to null (seethe geometrical approach proposed in [27]) Considering thecomposite system (6) the following ESO is proposed119909 = 119860119909 + 119861119906 + 119862120590119908 + 119861120585 + 1198671120601 (119910 minus 119910) 119908 = 119878119908 + 1198672120601 (119910 minus 119910) 120585 = 1198673120601 (119910 minus 119910) (30)

where 119909 119908 and 120585 are the estimates of the states structuredand unstructured disturbances respectively Additionally theestimated system output is 119910 = 119862119909 In ESO dynamics 119867119894 119894 =1 2 3 are design matrices of appropriate dimensions Based

on the nonlinear observer proposed by Prasov and Khalil[28] for positive scalars 0 lt 120576 lt 1 and 119889 gt 0 the nonlineargain function of the observer ie 120601() is defined as120601 (119911) = [1206011 (1199111) 120601119898 (119911119898)]119879 120601119895 (119911119895) fl

120576119911119895 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 le 119889119911119895 + 119889 (120576 minus 1) sign (119911119895) 1003816100381610038161003816100381611991111989510038161003816100381610038161003816 gt 119889119895 = 1 119898(31)

Remark 11 Thegain function (31) is a dead-zone nonlinearitydecreasing the observer gain when the estimation error fallsinside the zone [minus119889 119889] This allows adjusting a trade-offbetween fast state estimation and robustness to measurementnoises

In order to investigate convergence of the ESO (30)defining the error variables as 1205781 = 119909 minus 119909 1205782 = 119908 minus 119908 and1205783 fl 120585 minus 120585 we obtain the following differential equation forthe estimation error dynamics120578 = 1198600120578 + 119867120601 (120589) + 119876 120585120589 = 1198620120578 (32)

where 120578 = col(1205781 1205782 1205783) and the system matrices are givenby

1198600 = [[[119860 119862120590 1198610 119878 00 0 0]]]

119867 = [[[minus1198671minus1198672minus1198673]]]

119876 = [[[00119868]]]

1198620 = [119862 0 0]

(33)

Assumption 12 The unperturbed error dynamics given in120578 = 1198600120578 + 119867120601 (120589) 120589 = 1198620120578 (34)

has a globally exponentially stable equilibrium at the originThe feasibility of this assumption is addressed in Section 5

Theorem 13 Under Assumptions 3 and 12 the estimationerror of the ESO (30) is globally bounded and globally ulti-mately bounded in the sense that after a finite transient time1198790 gt 0 the following bound is valid10038171003817100381710038171205781003817100381710038171003817 le 1205880 + 1205731205850 (35)

where 1205880 gt 0 is arbitrarily small while 120573 gt 0 and 1205850 flsup119905ge0120585(119905)

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Mathematical Problems in Engineering 7

Proof By Assumption 12 there exist a positive definitefunction 119881(120578) and positive scalars 1205730 1205731 and 1205732 satisfying(see Theorem 517 of [29])

d119881 (120578)d119905 100381610038161003816100381610038161003816100381610038161003816(32) le minus1205730119881 (120578) 100381710038171003817100381710038171003817100381710038171003817120597119881 (120578)120597120578 100381710038171003817100381710038171003817100381710038171003817 le 1205731radic119881(120578)

10038171003817100381710038171205781003817100381710038171003817 le 1205732radic119881(120578)(36)

Calculating the time derivative of 119881(120578) along the solu-tion of (32) gives (d119881(120578)d119905)|(31) = (d119881(120578)d119905)|(32) +(120597119881(120578)120597120578)119876 120585 le minus1205730119881(120578) + 12057311205850radic119881(120578) By dividing bothsides of the last inequality by 2radic119881(120578) invoking the compar-ison lemma and using (36) we obtain 120578 le 1205732(radic119881(120578) minus(12057311205730)1205850) exp(minus12057301199052) + 1205731120573212058501205730 which clearly showsthe global boundedness of the estimation error Moreoverconsidering the decaying exponential term after the time1198790 = (21205730)Log(1205732|radic119881(120578) minus (12057311205730)1205850|1205880) the inequality(35) holds with 120573 fl 1205731120573212057305 ESO Design via LMIs

The key to the convergence result of Theorem 13 is Assump-tion 12 which requires the existence of a positive definitefunction 119881(120578) satisfying (36) In this regard we propose aconvex programming-based method to construct the func-tion 119881(120578) and also to design the gain matrices 119867119894 119894 = 1 2 3in (29) Instrumental for the development of our methodthe unperturbed error dynamics (34) is considered as a Lurrsquoesystem in terms of the gain function 120601(120589) (which is assumedto be a sector nonlinearity)

Remark 14 Thegain function 120601() belongs to the sector [1 120576]That is for all 120589 isin R119898

120576120589119879120589 le 120589119879120601 (120589) le 120589119879120589 (37)

Using (34) the sector condition (37) is equivalent to thefollowing quadratic inequality (see [30])

12057612057811987911986211987901198620120578 + 120601119879 (120589) 120601 (120589) minus (1 + 120576) 1205781198791198621198790120601 (120589) le 0 (38)

In the framework of quadratic stabilization we search for apositive definite function of the form 119881(120578) = 120578119879119875120578 whichguarantees Assumption 12 by satisfying the first requirementof (36) Thereby

120578119879 (1198751198600 + 1198601198790119875 + 1205730119875) 120578 + 2120578119879119875119867120601 (120589) le 0 (39)

We should note that since 119881(120578) is a quadratic function theother two requirements of (36) are trivially satisfied accordingto Rayleighrsquos principle

Theorem 15 For a given constant 1205730 gt 0 assume thereexists a positive definite matrix 119875 a matrix 119884 with appropriatedimension and a scalar 120599 ge 0 satisfying the LMI

[[[1198600 + 1198601198790119875 + 1205730119875 minus 12057612059911986211987901198620 119884 + 120599(1 + 1205762 )1198621198790119884119879 + 120599 (1 + 1205762 )1198620 minus120599119868 ]]]le 0 (40)

Then the function119881(120578) along with the gainmatrix119867 = 119875minus1119884guarantees the exponential stability of the origin of (34)

Proof The origin of the system (34) is exponentially stableif the inequality (39) holds for all 120578 and 120601(120589) satisfying thesector condition (38) According to the S-procedure Lemma[30] this statement is equivalent to the existence of a scalar120599 ge 0 such that 120578119879(1198751198600 + 1198601198790119875 + 1205730119875)120578 + 2120578119879119875119867120601(120589) minus120599(12057612057811987911986211987901198620120578+120601119879(120589)120601(120589)minus(1+120576)1205781198791198621198790120601(120589)) le 0 Convexifyingthe inequality by the transformation 119884 = 119875119867 and using theSchur complement Lemma LMI (40) is achieved Thus theproof is complete

6 Closed-Loop Stability

The modular structure of the proposed control systemenables an independent design of the output regulation anddisturbance rejection loops However due to the interactionof the components the overall closed-loop system stabilitystill needs to be investigated The stability of the disturbancerejection loop is established by the ESO convergence results ofTheorem 13 and the design method presented inTheorem 15Therefore we consider the effect of the ESO-based distur-bance rejection on the performance of the nominal regulatorThe nominal closed-loop system operating in conjunctionwith the disturbance rejection loop is described by theequations of the form = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 + 119876119888119897 (120585 minus 120585) (41)

where 119909119888119897 = col(119909 119909119906 119909120581) and119860119888119897 = [[[

119860 119861119866 00 Λ 11986411986531198652119862 0 1198651 ]]] 119861119888119897 = [[[

1198621205900minus1198652119862119898]]] 119876119888119897 = [[[

11986100]]] (42)

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

8 Mathematical Problems in Engineering

Additionally to examine the effect of disturbance rejectionon regulating the distance coordinate 119909119905 we consider system(27) perturbed by the disturbance cancellation error119904 = 119860119904119909119904 + 119876119904120585119905 (43)

where 120585119905 fl Γ(119901)(120585 minus 120585) is the transformed perturbation and119876119904 = [119861119879119886119906119892 0]119879Assumption 16 The signal 120585119905() is Lebesgue measurable andbounded

The next lemma is instrumental for the closed-loopstability analysis

Lemma 17 We have spec(119860 119904) = 119904119901119890119888(119860119888119897)Proof We show that the characteristic polynomials of119860119904 and119860119888119897 coincide For this it suffices to show that for all 119904 isin C

det (119860 119904 minus 119904119868) = det (119860119888119897 minus 119904119868) (44)

First regarding the block components of 119860119904 we havedet (119860 119904 minus 119904119868) = det([[[

119860 minus 119904119868 0 1198611198653119864119862 Λ minus 119904119868 00 1198652119866 1198651 minus 119904119868]]]) = det (119860 minus 119904119868)det([Λ minus 119904119868 minus119864119862 (119860 minus 119904119868)minus1 11986111986531198652119866 1198651 minus 119904119868 ])= det (119860 minus 119904119868) det (Λ minus 119904119868)det (1198651 minus 119904119868 + 1198652119866 (Λ minus 119904119868)minus1 119864119862 (119860 minus 119904119868)minus1 1198611198653)

(45)

Following the same procedure for 119860119888119897 we obtaindet (119860119888119897 minus 119904119868) = det (119860 minus 119904119868)det (Λ minus 119904119868) det (1198651 minus 119904119868 + 1198652119862 (119860 minus 119904119868)minus1 119861119866 (Λ minus 119904119868)minus1 1198641198653) (46)

Since 119866(Λ minus 119904119868)minus1119864 = minus(1Γ(119904))119868 the matrix multiplicationof 119866(Λ minus 119904119868)minus1119864 commutes with 119862(119860 minus 119904119868)minus1119861 and therebyequality (44) holds for all 119904 isin C

Theorem 18 Consider the system (41) Assume that Assump-tions 1ndash16 are satisfied 1205820 = minusmax119904isin119904119901119890119888(119860119888119897)Re(119904) and 1205850119905 =sup120591ge0120585119905(120591) Then

(1) The state trajectories 119909 119909119906 and 119909120581 are all globallybounded

(2) In the state space of the composite system (6) the statetrajectory 119909 converges practically to the manifoldM inthe sense that119909 minus 119909 le 12058310158400 + 12058310158401 exp (minus1205820119905) (47)

for some positive scalars 12058310158400 and 12058310158401(3) The error variable 119890 and its first 119897 minus 1 derivatives

are globally bounded and globally ultimately boundedThat is there is a finite time 119879101584010158400 gt 0 after which thefollowing inequalities holds10038171003817100381710038171003817119890(119894) (119905)10038171003817100381710038171003817 le 120588101584010158400 + 120583101584010158401 12058501199051205820 119894 = 0 119897 minus 1 (48)

where 120588101584010158400 gt 0 is arbitrarily small and 1205820 gt 0 is apertinent constant

Proof The proof is separated into three parts(I) We consider the closed-loop equation (41) without

perturbation = 119878119908119888119897 = 119860119888119897119909119888119897 + 119861119888119897119908 (49)

In view of Lemma 17 and Assumption 9 the matrix 119860119888119897 isHurwitz stable Therefore according to the center manifoldtheorem [29] there exists a stable invariant manifold M1015840attracting all trajectories of (49) The motion of the systemtrajectories on M1015840 is equivalent to the immersion of thesystem (49) into (2) Hence there is a linear map of theform 119909119888119897 = Π119908 in which the matrix Π satisfies the Sylvesterequation Π119878 = 119860119888119897Π + 119861119888119897 (50)We note that existence of Π is guaranteed since spec(119878) capspec(119860119888119897) = 0 For the perturbed closed-loop dynamics (41)an attractivity index with respect to M1015840 is defined as 119902 =119909119888119897 minus Π119908 Then it is straightforward to obtain119902 = 119860119888119897119902 + 119876119888119897 (120585 minus 120585) (51)

By Theorem 13 and following the same notation the dis-turbance estimation error is bounded as 120585 minus 120585 le 1205731205850 +1205780 exp(minus12057301199052) for some 1205780 gt 0 As a result the solution of(51) satisfies an inequality of the form10038171003817100381710038171199021003817100381710038171003817 le 12057312058501205830 + 1205831 exp (minus1205820119905) + 1205832 exp(minus12057301199052 ) (52)

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Mathematical Problems in Engineering 9

where 1205830 1205831 and 1205832 are appropriate positive numbersInequality (52) implies practical convergence of the closed-loop trajectories to the manifold M1015840 Using the triangleinequality 119909119888119897 le 119902 + Π119908 boundedness of the trajectory119909119888119897 is deduced

(II) Apportioning matrix Π as

[[[119909119909119906119909120581]]] = [[[

Π119909Π119906Π120581]]]119908 (53)

the following matrix equation is obtained from (50)Π119909119878 = 119860Π119909 + 119861119866Π119906 + 119862120590 (54)

Comparing (54) with (8) it follows from the uniquenessresult ofTheorem 5 that119883 = Π119909 and119880 = 119866Π119906TherebyM isa submanifold ofM1015840 and finally it is implied from inequality(52) that 119909minus119909 le 12058310158400+12058310158401 exp(minus1205820119905) in which12058310158400 = 12057312058501205830+1205832and 12058310158401 = 1205831

(III) The solution of the state equation (43) satisfies119909119904(119905) le (119909119904(0) minus 119876119904(1205850119905 1205820)) exp(minus1205820119905) + 119876119904(1205850119905 1205820)

Given 120588101584010158400 gt 0 after the transient time 119879101584010158400 =(11205820)Log(|(119909119904(0) minus 1198761199041205850119905 1205820)120588101584010158400 |) the inequality119909119904(119905) le 120588101584010158400 + 120583101584010158401 (1205850119905 1205820) holds with 120583101584010158401 = 119876119904 Thiscompletes the proof

7 Simulation Example

In order to illustrate the efficacy of the proposed controlmethod we explain its application to the plotter systemwhich is a computer printer used for vector graphics [15]The schematic model of the plotter is shown in Figure 2 andthe corresponding symbols and numerical values are given inTable 1 The basic components of the plotter are a DC motorand three disks (referred to as disks 1-3)The state variables ofthe system 1199091 1199092 and 1199093 are the angular displacement of disk1 the angular velocity of disk 1 and the current in the motorrespectively The control variable is 119906 in the input voltage ofthe motor Through a first principle modeling we obtain thestate equation of the plotter as follows = 119860119909 + 119861119906 + 120574 (119909 119905) (55)

where

119860 = [[[[[[[[0 1 0minus 1198701 + (11990311199032)211987021198691 + (11990311199032)2 (1198692 + (12)11987211990322) minus 1198611 + (11990311199032)2 11986121198691 + (11990311199032)2 (1198692 + (12)11987211990322) 1198701198941198691 + (11990311199032)2 (1198692 + (12)11987211990322)0 minus 119870V119871119898 minus119877119898119871119898

]]]]]]]]

119861 = [[[[[001119871119898

]]]]] (56)

The total disturbance comprises two terms in such a waythat 120574(119909 119905) = 1205741(119905) + 1198611205742(119909) The first which has aknown frequency spectrum affects both motor dynam-ics and mechanical part of the system and is given by1205741(119905) = [0 001 sin(20119905) 02 sin(20119905)]⊤ The second term isa matched state-dependent uncertainty of the form 1205742(119909) =minus01tanh(101199092) which represents frictional effects in themotor Taking the disk 1 angle as the output 119910 = 1199091 weassume that the control objective is to track a reference signalof the form 119910119898(119905) = 001 sin(119905) According to the availablefrequency data of the referencedisturbance signals system(2) is characterized by the following matrices

119878 = [[[[[0 1 0 0minus1 0 0 00 0 0 10 0 minus400 0

]]]]] 119862119898 = [1 0 0 0]

(57)

In view of Assumption 3 and Remark 4 we consider thedisturbance model (4) with

119862120590 = [[[0 0 0 01 0 1 01 0 1 0]]] (58)

The controller (26) is designed using linear quadratic regu-lation in such a way that all eigenvalues of the matrix 119860119904 areplaced on the left of the lineR(119904) = minus3 in the complex planeParameters of the nonlinear gain function (31) are selected as120576 = 08 and 119889 = 01 To obtain the matrices 119867119894 119894 = 1 2 3taking 1205730 = 001 the LMI (40) is solved per CVX [31] Weobtained

1198671 = [[[32598626663805471 ]]]

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

10 Mathematical Problems in Engineering

Table 1 Symbols of the plotter system model with the numerical values [15]

Symbol Description Value119870119894 Motor magnetic flux 10minus3 kgm2s2119870V Motor back electromotive force 045 Vsrad119877119898 Motor resistance 1 Ω119871119898 Motor inductance 10minus2 H1198691 Disk 1 moment of inertia 001213 kgm21198611 Disk 1 viscous friction 01 Nmsrad1198701 Disk 1 elastic constant 10minus3 Nmrad1199031 Disk 1 radii 01 m1198692 Disk 2 moment of inertia 001213 kgm21198612 Disk 2 viscous friction 01 Nmsrad1198702 Disk 2 elastic constant 10minus3 Nmrad1199032 Disk 2 radii 01 m119872 Disk 3 (the pen nib) mass 05 kg

Figure 2 Schematics of the plotter system [15]

1198672 = [[[[[[140456minus41513minus79373minus4428252

]]]]]]

1198673 = 356611(59)

We also compare the tracking response of the controller withthe hybrid controller [15] which comprises a sliding modeobserver and a sliding mode output feedback controllerSince only the stabilization problem is considered in [15] weperform the following state and control transformation1199091 = 1199091 minus 1199101198981199092 = 1199092 minus 1199101198981199093 = 1199093 minus 120579119898119906 = 119906119871119898 + 11988632119898 + 11988633120579119898 minus 120579119898

120579119898 = minus111988623 (11988621119910119898 + 11988622 119910119898 minus 119910119898) (60)

where 119886119894119895 are the entries of the state matrix 119860 According tothe transformation (60) the following system is obtained forthe stabilization problem = 119860119909 + 119861 (119906 + 1205742 (119905)) + 1205741 (119905) 119890 = 119862119909 (61)

where 119909 = [1199091 1199092 1199093]⊤ and 119862 = [1 0 0] The hybridcontroller is given by119909 = 119860 + 119861119906 + 119870ℎ119910119887 (119890 minus 119862)+ 119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 1205742 = 119861+119870ℎ119910119887 (119890 minus 119862)+ 119861+119873ℎ119910119887sign (119872ℎ119910119887 (119890 minus 119862)) 119906 = minus1205742 minus 119861+119870ℎ119910119887119890 + 119861+119873ℎ119910119887sign (119872ℎ119910119887119890)

(62)

where and 1205742 are estimates of 119909 and 1205742 respectively 119870ℎ119910119887119873ℎ119910119887 and 119872ℎ119910119887 are design matrices of suitable dimensions(119872ℎ119910119887119862 should be positive semidefinite) and 119861+ is thepseudoinverse of 119861 and is the standard signum function [15]We note that in the design of the hybrid controller onlymatched disturbances are considered [15] Using eigenvalueplacement the design matrices of the hybrid controller aretuned as follows119870ℎ119910119887 = [00675 08936 14613]⊤ times 103119873ℎ119910119887 = 001119872ℎ119910119887 = [1 1 1]⊤ (63)

The tracking responses of both controllers are shown inFigure 3The settling times of the proposed controller and thehybrid controller are 4119904 and 25119904 respectively Our controller

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40

minus001minus0005

00005

0010015

ReferenceHybrid controllerProposed controller

0 5 10 15 20 25 30 35 40minus4

minus2

0

2

Time (s)

36 38 40minus2

0

2

e(r

ad)

yy

m(r

ad)

times10minus3

times10minus4

Figure 3 Tracking responses of the proposed controller and thehybrid controller

shows a much faster transient response but its overshootis larger The approximate values of the absolute steady-state tracking errors of the controllers are 1567 times 10minus7 radfor the proposed controller and 1219 times 10minus4 rad for thehybrid controller That is our controller outperforms thehybrid controller in terms of transient time and steady-statetracking This is mainly due to the output regulation prop-erty of our controller which enables guaranteed asymptotictracking as well as mismatched disturbance rejection Thecontrol signals generated by the controllers are shown inFigure 4 Both control signals are approximately the samehowever there is a high-frequency component in the controlinput of our controller This component is caused by theinternal model property of our controller which embeds theknown frequencies of the referencedisturbances to learntheir behavior [8] In order to assess the performance of theESO the estimated state variables of the plotter system aregiven in Figure 5 According to the ESO differential equations(30) state reconstruction is a part of disturbance estimationThe ESO successfully tracks the state variables of the systemwhich in turn confirms its disturbance estimation capability

8 Conclusion

This paper presented a control structure combining outputregulation and disturbance rejection control techniquesThis method extends the range of the disturbances con-sidered in the conventional output regulation theory Inthe output regulation aspect the internal model principleenables the controller to handle themismatched disturbanceswith known frequency characteristics In the disturbancerejection aspect an ESO is employed to compensate forthe disturbances without known dynamical characteristicswhich for example may arise from nonlinear effects and

0 5 10 15 20 25 30 35 40minus3

minus2

minus1

0

1

2

3

Time (s)

Proposed controllerHybrid controller

u(V

)

Figure 4 Time trajectories of the control inputs of the proposedcontroller and the hybrid controller

0 05 1 15 2 25 3 35 4 45 5minus001

0001002

TrueEstimated

0 05 1 15 2 25 3 35 4 45 5minus002

0

002

0 05 1 15 2 25 3 35 4 45 5minus5

0

5

Time (s)

x1 x1

(rad

)x2 x2

(rads

)x3 x3

(A)

Figure 5 Estimation of the plotter state variables using the ESO

structural variations of the underlying plant Generalizationand application of the method to a class of mechanicalsystems will be pursued in future works

Data Availability

Theauthors are willing to share the implementation scripts inthe form of someMATLABm-files with the interested reader

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

12 Mathematical Problems in Engineering

Supplementary Materials(1)MATLABlowastm file ldquoPlotter systemmrdquowhich represents thedesign and implementation of output regulation via the proposedNESO for a plotter system It should be noted that theexecution of this file requires the installation of CVX pro-gram which is a free disciplined convex programming tool(2)MATLAB lowastm file ldquoPlotter system Hybrid Controllermrdquowhich represents the design and implementation of thehybrid control system on the plotter benchmark problem (3)SIMULINK lowastmdl files ldquoHybrid control systemmdlrdquo andldquoPlotter system controlmdlrdquo which are the realizations ofclosed-loop systems as SIMULINK models (SupplementaryMaterials)

References

[1] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[2] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization An International Journal withApplications to Stochastics vol 1 no 1 pp 64ndash86 197475

[3] A A Adegbege andW P Heath ldquoInternalmodel control designfor input-constrained multivariable processesrdquo AIChE Journalvol 57 no 12 pp 3459ndash3472 2011

[4] E J Davison and A Goldenberg ldquoRobust control of a generalservomechanismproblem the servo compensatorrdquoAutomaticavol 11 no 5 pp 461ndash471 1975

[5] J Huang and Z Chen ldquoA general framework for tackling theoutput regulation problemrdquo Institute of Electrical and ElectronicsEngineers Transactions on Automatic Control vol 49 no 12 pp2203ndash2218 2004

[6] A Isidori ldquoRegulation and Tracking in Linear Systemsrdquo inLectures in feedback design for multivariable systems pp 83ndash133Springer New York NY USA 2016

[7] Z Chen and J Huang ldquoRobust Output Regulation A Frame-workrdquo in Stabilization and Regulation of Nonlinear Systems ZChen and J Huang Eds pp 197ndash237 Springer InternationalPublishing Cham Switzerland 2015

[8] A Oveisi M Hosseini-Pishrobat T Nestorovic and JKeighobadi ldquoObserver-based repetitive model predictive con-trol in active vibration suppressionrdquo Structural Control andHealth Monitoring vol 13 no 4 p e2149 2018

[9] M Hosseini-Pishrobat and J Keighobadi ldquoRobust VibrationControl and Angular Velocity Estimation of a Single-AxisMEMS Gyroscope Using Perturbation Compensationrdquo Journalof Intelligent Robotic Systems vol 32 no 10 p 94 2018

[10] Z Gao ldquoOn the centrality of disturbance rejection in automaticcontrolrdquo ISA Transactions vol 53 no 4 pp 850ndash857 2014

[11] A Oveisi and T Nestorovic ldquoRobust observer-based adaptivefuzzy sliding mode controllerrdquo Mechanical Systems and SignalProcessing vol 76-77 pp 58ndash71 2016

[12] Z Gao ldquoActive disturbance rejection control A paradigmshift in feedback control system designrdquo in Proceedings of theAmerican Control Conference 7 pages IEEE June 2006

[13] B-Z Guo and Z-L Zhao Active Disturbance Rejection Controlfor Nonlinear Systems John Wiley amp Sons Singapore 2016

[14] Y Huang and W Xue ldquoActive disturbance rejection controlMethodology and theoretical analysisrdquo ISA Transactions vol53 no 4 pp 963ndash976 2014

[15] J D J Rubio ldquoHybrid controller with observer for the estima-tion and rejection of disturbancesrdquo ISA Transactions vol 65pp 445ndash455 2016

[16] Z Chen and D Xu ldquoOutput regulation and active disturbancerejection control unified formulation and comparisonrdquo AsianJournal of Control vol 18 no 5 pp 1668ndash1678 2016

[17] M Hosseini-Pishrobat and J Keighobadi ldquoRobust output reg-ulation of a triaxial MEMS gyroscope via nonlinear activedisturbance rejectionrdquo International Journal of Robust andNonlinear Control vol 28 no 5 pp 1830ndash1851 2018

[18] Z Zuo CWangWYang et al ldquoRobust disturbance attenuationfor a class of uncertain Lipschitz nonlinear systems with inputdelayrdquo International Journal of Control vol 6 no 2 pp 1ndash7 2017

[19] M Li and M Deng ldquoOperator-based external disturbancerejection of perturbed nonlinear systems by using robust rightcoprime factorizationrdquo Transactions of the Institute of Measure-ment and Control vol 57 2018

[20] X-J Wei Z-J Wu and H R Karimi ldquoDisturbance observer-based disturbance attenuation control for a class of stochasticsystemsrdquo Automatica vol 63 pp 21ndash25 2016

[21] F Gao M Wu J She and W Cao ldquoDisturbance rejectionin nonlinear systems based on equivalent-input-disturbanceapproachrdquo Applied Mathematics and Computation vol 282 pp244ndash253 2016

[22] Y Zhang L Wang J Zhang and J Su ldquoRobust observerbased disturbance rejection control for Euler-Lagrange sys-temsrdquoMathematical Problems in Engineering Art ID 383950513 pages 2016

[23] A Isidori L Marconi and A Serrani ldquoFundamentals ofInternal-Model-Based Control Theoryrdquo in Robust AutonomousGuidance An Internal Model Approach A Isidori L Marconiand A Serrani Eds pp 1ndash85 Springer London UK 2003

[24] C Coleman ldquoLocal trajectory equivalence of differential sys-temsrdquoProceedings of the AmericanMathematical Society vol 16pp 890ndash892 1965

[25] H Sira-Ramırez A Luviano-Juarez M Ramırez-Neria et alldquoGeneralities of ADRCrdquo inActive Disturbance Rejection Controlof Dynamic Systems pp 13ndash50 Elsevier 2017

[26] J Huang ldquoLinear Output Regulationrdquo in Nonlinear output reg-ulation theory and applications pp 1ndash34 SIAM PhiladelphiaPa USA 2004

[27] AOveisi andTNestorovic ldquoReliability of disturbance rejectioncontrol based on the geometrical disturbance decouplingrdquoPAMM vol 16 no 1 pp 823-824 2016

[28] A A Prasov and H K Khalil ldquoA nonlinear high-gain observerfor systems with measurement noise in a feedback controlframeworkrdquo Institute of Electrical and Electronics EngineersTransactions on Automatic Control vol 58 no 3 pp 569ndash5802013

[29] S Sastry Nonlinear Systems Analysis Stability and ControlSpringer New York NY USA 1999

[30] S Boyd L El Ghaoui E Feron et al Linear matrix inequalitiesin system and control theory society for industrial and appliedmathematics SIAM Philadelphia Pa USA 1994

[31] M C Grant and S P Boyd ldquoGraph implementations fornonsmooth convex programsrdquo in Recent Advances in Learningand Control V Blondel S Boyd and H Kimura Eds vol 371of Lecture Notes in Control and Information Sciences pp 95ndash110Springer Berlin Germany 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom