robust control in presence of parametric uncertainties: observer-based feedback controller design

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Chemical Engineering Science 63 (2008) 1890 – 1900 www.elsevier.com/locate/ces Robust control in presence of parametric uncertainties: Observer-based feedback controller design Davide Fissore Dipartimento di Scienza dei Materiali e Ingegneria Chimica, Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino 10129, Italy Received 23 March 2007; received in revised form 20 September 2007; accepted 12 December 2007 Available online 17 December 2007 Abstract This paper proposes a complete procedure for the design of a robust controller for a nonlinear process, taking into account the various issues arising in the design and using the main theoretical results from the Literature about this topic. An extended model is set up, linking performance and robustness to the control law: the H norm of the extended system in closed loop measures the achievement of the objectives. The result is a state feedback control law which guarantees robust performance. The problem of the design of an observer to estimate the state of the system is also addressed, as the complete knowledge of the state is required to calculate the control action; moreover, the implications of the use of the observer in the design of the controller are pointed out. The methodology is illustrated via simulation of a regulation problem in a continuous stirred tank reactor (CSTR). The application of this methodology to more complex systems will be discussed. 2008 Elsevier Ltd. All rights reserved. Keywords: H control; Nonlinear control; Robust control; Robust performance; Observer design; Uncertain process 1. Introduction In the last decades there was a relevant effort to face with the design of controllers for nonlinear multivariable processes, par- ticularly in the chemical engineering field, where the dynamics of the systems can be a function of many parameters (such as physical properties, heat and mass transfer coefficients, kinetic constants) which can be nonlinear functions of the state vari- ables (temperature, pressure, composition). Thus, a controller designed around a nominal operating point, as in the classical approach, cannot guarantee satisfactory performance (distur- bance rejection and/or tracking control): the dynamic behaviour can be qualitatively different from one operating regime to an- other and instability can arise. Simple gain-scheduling schemes were proposed even in the recent past (Shinskey, 1996): linear process models valid within a “small” region around the linearisation point are derived and a local design is performed, thus resulting in a look-up table that interpolates controller gains as the process traverses the Tel.: +39 011 0904693; fax: +39 011 0904699. E-mail address: davide.fi[email protected]. 0009-2509/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2007.12.019 operating region. This procedure is time consuming and expen- sive, but is well accepted for many applications. Galan et al. (2000) discussed the construction of a multi-linear model ap- proximation for a single input single output (SISO) plant, but the question of how many models are required remains largely unanswered. Input/output (I/O) geometric linearisation (Singh and Rugh, 1972; Isidori and Ruberti, 1984; Isidori, 1989) can reduce a nonlinear system to a linear one using coordinate transforma- tion, thus avoiding the standard Jacobian linearisation. Using the I/O linearisation approach, Kravaris and Chung (1987) pro- posed the globally linearising controller, using a state feedback to make the I/O relationship linear and then using an exter- nal linear controller around the I/O linear system. Similarly, Sampath et al. (1998) proposed a multi-loop feedback config- uration: the inner loop is a state feedback law, based on a dif- ferential geometric method, meant for I/O linearisation, while the outer loop is designed for robust stability and nominal per- formance on the basis of the robust control theory for linear systems (Doyle, 1982). In principle, once the nonlinearities are cancelled, the outer loop can be designed to impose any desired stable dynamics

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Page 1: Robust control in presence of parametric uncertainties: Observer-based feedback controller design

Chemical Engineering Science 63 (2008) 1890–1900www.elsevier.com/locate/ces

Robust control in presence of parametric uncertainties: Observer-basedfeedback controller design

Davide Fissore∗

Dipartimento di Scienza dei Materiali e Ingegneria Chimica, Politecnico di Torino, C.so Duca degli Abruzzi 24, Torino 10129, Italy

Received 23 March 2007; received in revised form 20 September 2007; accepted 12 December 2007Available online 17 December 2007

Abstract

This paper proposes a complete procedure for the design of a robust controller for a nonlinear process, taking into account the variousissues arising in the design and using the main theoretical results from the Literature about this topic. An extended model is set up, linkingperformance and robustness to the control law: the H∞ norm of the extended system in closed loop measures the achievement of the objectives.The result is a state feedback control law which guarantees robust performance. The problem of the design of an observer to estimate the stateof the system is also addressed, as the complete knowledge of the state is required to calculate the control action; moreover, the implicationsof the use of the observer in the design of the controller are pointed out. The methodology is illustrated via simulation of a regulation problemin a continuous stirred tank reactor (CSTR). The application of this methodology to more complex systems will be discussed.� 2008 Elsevier Ltd. All rights reserved.

Keywords: H∞ control; Nonlinear control; Robust control; Robust performance; Observer design; Uncertain process

1. Introduction

In the last decades there was a relevant effort to face with thedesign of controllers for nonlinear multivariable processes, par-ticularly in the chemical engineering field, where the dynamicsof the systems can be a function of many parameters (such asphysical properties, heat and mass transfer coefficients, kineticconstants) which can be nonlinear functions of the state vari-ables (temperature, pressure, composition). Thus, a controllerdesigned around a nominal operating point, as in the classicalapproach, cannot guarantee satisfactory performance (distur-bance rejection and/or tracking control): the dynamic behaviourcan be qualitatively different from one operating regime to an-other and instability can arise.

Simple gain-scheduling schemes were proposed even in therecent past (Shinskey, 1996): linear process models valid withina “small” region around the linearisation point are derived anda local design is performed, thus resulting in a look-up tablethat interpolates controller gains as the process traverses the

∗ Tel.: +39 011 0904693; fax: +39 011 0904699.E-mail address: [email protected].

0009-2509/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.12.019

operating region. This procedure is time consuming and expen-sive, but is well accepted for many applications. Galan et al.(2000) discussed the construction of a multi-linear model ap-proximation for a single input single output (SISO) plant, butthe question of how many models are required remains largelyunanswered.

Input/output (I/O) geometric linearisation (Singh and Rugh,1972; Isidori and Ruberti, 1984; Isidori, 1989) can reduce anonlinear system to a linear one using coordinate transforma-tion, thus avoiding the standard Jacobian linearisation. Usingthe I/O linearisation approach, Kravaris and Chung (1987) pro-posed the globally linearising controller, using a state feedbackto make the I/O relationship linear and then using an exter-nal linear controller around the I/O linear system. Similarly,Sampath et al. (1998) proposed a multi-loop feedback config-uration: the inner loop is a state feedback law, based on a dif-ferential geometric method, meant for I/O linearisation, whilethe outer loop is designed for robust stability and nominal per-formance on the basis of the robust control theory for linearsystems (Doyle, 1982).

In principle, once the nonlinearities are cancelled, the outerloop can be designed to impose any desired stable dynamics

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D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900 1891

on the closed loop. The usual approach is to impose linear dy-namics with poles in the left half plane, thus resulting in sta-ble dynamics. However, the issue of where the poles should beplaced in the left half plane was not addressed in the Litera-ture. This is of outmost importance when there is uncertaintyin the model and/or the nonlinearities are not cancelled exactly.Uncertainty can have various origins, beside linearisation ofnonlinear dynamics:

• the process structure is known but some parameters are not,or they may change in time or they are known only in a rangeof approximation;

• the process is known but some of its dynamics are willinglyignored for simplicity;

• some dynamics of the process, especially at high frequencies,are unknown.

Moreover, even differential geometric techniques in presenceof uncertainties do not give perfectly linear models and thesenonlinearities require, for example, standard Jacobian lineari-sation around their steady-state values. This is different fromthe Jacobian linearisation of the original nonlinear system: onlythe perturbations due to uncertainties are linearised, but not thewhole model (Kolavennu et al., 2000). As a consequence, thecontroller in the outer loop must be designed not only for nom-inal stability and performance, but also for robustness in faceof uncertainties.

An alternative approach, based on the explicit constructionof a Lyapunov function, was proposed by Chen and Leitmann(1987): although their method opens an interesting way to thecontrol of nonlinear systems in presence of uncertainties, thereexists no general procedure to find an explicit Lyapunov func-tion for building a robust controller.

Recent papers gave extensive results on the control ofnonlinear systems that handle uncertainty and constraints.El-Farra and Christofides (2003) focused their attention oncontrol of multi-input multi-output nonlinear processes withuncertain dynamics and actuator constraints and proposed aLyapunov-based nonlinear controller design approach that ac-counts explicitly and simultaneously for process nonlinearities,plant–model mismatch, and input constraints. Mhaskar et al.(2005) investigated a robust hybrid model predictive controldesign that handles nonlinearity and uncertainty: the proposedmethod provides a safety net for the implementation of anyavailable MPC formulation, designed with or without takinguncertainty into account, and allows for an explicit character-isation of the set of initial conditions starting from where theclosed-loop system is guaranteed to be stable. Also in this casethe key idea is to use a Lyapunov-based robust controller, forwhich an explicit characterisation of the closed-loop stabilityregion can be obtained, to provide a stability region withinwhich MPC can be implemented. A set of switching lawsare designed that exploit the performance of MPC wheneverpossible, while using the bounded controller to provide thestability guarantees. Finally, Mhaskar (2006) investigated howit is possible to incorporate appropriate stability constraintsin the optimisation problem, thus guaranteeing feasibility

and closed-loop stability in the presence of constraints anduncertainty.

In the present paper a different approach will be shown, mod-elling the nonlinearities as uncertainties in a linearised model,thus allowing the use of the results of robust control theoryfor linear systems: most important results from the Literature(Doyle, 1982; Zhou et al., 1995; Colaneri et al., 1997) will berationalised and formalised in a full procedure for the design ofa robust controller. The focus of the paper is on the design of the“extended model” of the process which includes performancerequirements and parametric uncertainties. Moreover, it will beshown that when a feasible controller cannot be designed, i.e.,the objectives cannot be satisfied, the method allows to dis-criminate if the responsible of this is a too severe performancerequirement or a too large range of uncertainty for a certain pa-rameter, thus driving the design process to achieve the best re-sults. Moreover, the problem of the design of a robust observerfor the estimation of the state of the system in presence of un-certainties using the available measures will be addressed. Thisissue, which has been quite often neglected in the Literature, isof great importance as the calculation of the controller actionrequires the knowledge of the state of the system, which cannotbe completely available from the measures. Moreover, the useof the observer strongly affects the design of the controller andthe robust performance of the system. Few papers in the pastdealt with this issue; in particular El-Farra et al. (2005) synthe-sised a family of output feedback controllers using a combina-tion of bounded state feedback controllers, high-gain observers,and appropriate saturation filters to enforce asymptotic stabilityfor the individual closed-loop modes and provided an explicitcharacterisation of the corresponding output feedback stabilityregions in terms of the input constraints and the observer gain.In this paper a different approach is presented, based on the setup of an appropriate extended model which includes not onlythe control requirements but also the observer specifications.

2. Robust control design

The aim of this section is to summarise the fundamentals ofthe theory of robust control for linear systems and to give asystematic procedure for the design of this kind of controllers.

2.1. Controller design

Let us consider a generic linear system described by thefollowing set of equations:{

x = Ax + B1w + B2u,

z = C1x + D11w + D12u,

y = C2x + D21w,

(1)

where x is the state of the system; u is the manipulated inputof the system, i.e., the control action; w represents the exter-nal (bounded) disturbances, i.e., unknown signals affecting thedynamics of the system; y represents the measured variables(which can be affected by external disturbances) available forthe control; z represents the objectives of the process, i.e., thesignals carrying requirements to be satisfied by the control.

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Within this framework, the system (1) corresponds to a dy-namical model with four ports constituted by two groups of in-puts (w and u), and two groups of outputs (z and y) signals; nolimitations about the dimension of x, u, w, and y are requiredand the approach is truly multivariable.

The control problem can be stated very preliminary as theselection of a control signal u to apply to the process to maintain(regulate) some output signals to a set point, or to force them tofollow (tracking) reference signals, in spite of the disturbancesacting on the plant. Thus, in a regulation problem the goalis to get insensitivity of the objectives z with respect to thedisturbances w. Also a tracking problem can be defined interms of insensitivity: if r is the reference trajectory for theobjective z, it is possible to introduce the tracking error er =z − r and try to achieve insensitivity of er with respect to r. Afurther performance requirement is on control activity due tomeasurement noises: excessive amplitude of the control actionin all frequency range should be avoided, i.e., a low sensitivityof the control action with respect to measurement noises.

To quantify the desired performance it is necessary to assigna measure to input and output signals. Various types of mea-sures can be defined, e.g. the energy of a signal, its power, itsmaximum value in frequency. These measures are called normsand generalise the classical concept of distance in the Euclideanspaces. The most important (and useful for our purposes) normfor an array (or a matrix) of signals U is the norm H∞:

‖U‖∞ = sup�

�[U(j�)], (2)

where U(j�) is the representation in the frequency (�) domainof the signals (which can be obtained, for example, from theLaplace transform of the function, followed by the substitutionof the Laplace variable s with j�) and �(�) is the maximumsingular value of the argument matrix for every value of �(the maximum singular value of a transfer function matrix hasintuitively the same interpretation of the gain of a scalarsystem).

The definition of a norm on the spaces of the input andoutput signals induces a norm on the dynamic operator whichestablishes a mapping between the input and the output signalspaces. Given two norms, one for the input signal u and one forthe output signal y, a bounded input bounded output (BIBO)operator is characterised by:

‖y‖ = ‖Gu‖�M‖u‖, ∀u. (3)

The norm of the operator G is defined as the lowest value ofM that verifies the previous inequality, thus:

‖G‖ = supu

‖y‖‖u‖ . (4)

This induced norm can be considered a generalisation of thegain as it is intended in electronic amplifiers.

Let us consider an SISO control problem where the aim is tocalculate the control u to make the objective z ∈ R insensitiveto the disturbance w ∈ R. Let Gw,z be the closed-loop transferfunction between w and z:

z(s) = Gw,zw(s) (5)

Fig. 1. Amplitude of |Ww,z|, |Ww,z|−1, and |Gw,z| as a function of thefrequency �.

and let Ww,z be a generic filter applied to the objective z:

z∗ = Ww,zz = Ww,zGw,zw. (6)

If it is possible to find a control u such that:

‖Ww,zGw,z‖∞ < 1 (7)

this means that:

|Gw,z| < |Ww,z|−1, ∀�. (8)

The filter Ww,z can thus be used to specify the performancerequirements for the controlled system. An example can beuseful to understand this key-point: let us assume that the goalof the controller is to reduce the disturbance effects of 1/100for all the frequencies till a crossover frequency of 10 rad s−1,with a maximum of 6 dB in the sensitivity at high frequencies.As a first guess it is possible to assume (see Fig. 1):

Ww,z = �(s + �)

s + �. (9)

Eq. (8) states that in the controlled system the module of theclosed-loop sensitivity function w → z (Gw,z) will be lowerthan |Ww,z|−1, thus, at low frequencies (� → 0, thus s → 0),|Ww,z|−1 = 10−2, i.e.,

|Ww,z(� → 0)| = ��

�= 102 = 40 dB (10)

while, at high frequencies (� → ∞, thus s → ∞), |Ww,z|−1 =6 dB, i.e.,

|Ww,z(� → ∞)| = � = 1

6 dB= −6 dB. (11)

The value of the pole � of the filter can be assumed equal to thecrossover frequency (10 rad s−1), so that, given the performancerequirements, the filter Ww,z is fully defined.

It is well documented in any text about optimum controlthat with perfect knowledge of all states and with performance

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D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900 1893

measured by a square norm of the closed-loop operator the bestcontrol is an algebraic feedback from the states:

u = −Hx (12)

thus, in the design procedure, the goal is to find a controllergain H that satisfies Eq. (7). The available algorithms for thecalculation of the controller gain that minimises the norm ofan operator looks for the minimum of ‖Ww,zGw,z‖∞: if thisminimum is lower than 1, then the controller gain satisfiesEq. (8) and thus the performance requirements; otherwise, ifthe minimum of the norm is larger than 1, the controller doesnot satisfy the requirements. This may occur if the crossoverfrequency or the disturbance reduction at low frequencies or themaximum in the sensitivity at high frequencies are too severe:the method drives the design procedure to the best feasibleresult.

With the control law given by Eq. (12), the following equa-tions describe the dynamics of the controlled system:{

x = (A − B2H)x + B1w,

z = (C1 − D12H)x + D11w,

y = C2x + D21w.

(13)

Thus, beside the performance requirements, the controlled sys-tem has to be stable, i.e., the eigenvalues of the matrix (A −B2H), which depend on the gain H, must have negative realpart. Obviously system (13) has to be controllable, i.e., the polesof (A−B2H) should be placed in arbitrary positions varying H(this requirement can be relaxed in presence of poles that can-not be moved arbitrarily, but are stable; in this case the systemis said to be stabilisable).

In case of multiple objectives specification (on multiple dis-turbance rejection, tracking error, and control activity), the sameprocedure can be followed, defining a filter Wzi,wi

to describethe performance requirements for the closed-loop sensitivitybetween the ith objective and the ith disturbance (wi → zi)

and calculating a controller gain H that satisfies:

‖Wwi,ziGwi,zi

‖∞ < 1, ∀(wi, zi) (14)

being Gwi,zithe closed-loop transfer function between wi and

zi . Again, from the results of the minimisation of the norm,it is possible to verify if any of the performance requirementsmust be made less severe to satisfy Eq. (14). The various fil-ters Wzi,wi

can be designed as stated previously; for the con-trol activity, as the goal to guarantee that the control has notexcessive amplitude in all the frequency range, in particular athigh frequency, it is necessary that the weight of the control isa proper dynamical operator or a pure gain.

Let us consider now the presence of uncertainties in themodel, due, for example, to a parameter p that can assumevalues between pmin and pmax. In this case it is possible todefine:

Wp = pmax − pmin

pmax + pmin(15)

and then pose:

p = pmean(1 + �∗Wp) (16)

Fig. 2. Upper graph: block diagram corresponding to a process with uncer-tainties. Lower graph: block diagram where the uncertainty is described bymeans of the signals WG and zG.

where pmean is the mean value of the parameter p between pminand pmax and �∗ is a parameter whose value can vary in therange (−1; 1):

‖�∗‖∞ < 1. (17)

Nonlinearities can be treated in the same way. Let us consider,for example, a kinetic constant k varying with the temperatureaccording to an Arrhenius-type expression: if it is possible toassume that the temperature of the system varies in a certaininterval, thus also the kinetic constant will vary between kminand kmax and thus:

k = kmean(1 + �∗Wk), (18)

where

Wk = kmax − kmin

kmax + kmin(19)

and kmean is the mean value of the parameter k between kminand kmax. Eqs. (16) and (18) are two particular cases of themore general expression:

Gp = G(1 + �∗WG), (20)

where Gp is the true transfer function of the process, G is theapproximate transfer function, WG is the relative variation, and�∗ is such that ‖�∗‖∞ < 1. �∗ can be an arbitrary linear, proper,invariant, BIBO dynamic operator, with the unique constraintto have limited norm H∞: this operator parameterises all theelements in the region of the system uncertainty. Fig. 2 showsthe block diagram corresponding to Eq. (20) for an SISO sys-tem; the operator �∗ can be eliminated from the block diagram(and thus from the model of the system), thus giving origin to acouple of new signals: wG is a new input to the process, play-ing the role of a bounded disturbance (in fact wG is the outputof the operator �∗, which is unknown, even if bounded), whilezG plays the role of a new objective: the insensitivity between

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1894 D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900

the various wi and zi (specifying the performance towards ex-ternal disturbances) has now to be guaranteed also for all thecouples wG.zG, i.e., for all the values of the parameters �∗(robust performance).

This model is known as “extended system” and it embedsnominal model, process uncertainty, disturbances, and require-ments. Let us indicate with Geq the transfer matrix betweenw and z resulting from closing the loop of the extended sys-tem in nominal conditions: performances and stability mustbe preserved for any value of the uncertainty in the admissi-ble region. Robust stability is the property of the control thatguarantees stability not just for the nominal model, but forany model obtained from the uncertainty operator in the speci-fied region. The property is stated by the Small Gain Theorem(Zames, 1966a, b):

Let Geq be an asymptotically stable transfer matrix, and�∗ an arbitrary asymptotically stable operator with normH∞ < 1, then the feedback loop is stable if and only if theloop gain satisfies ‖�∗Geq‖∞ < 1. This implies that stabilityis guaranteed by ‖Geq‖∞ < 1.

Finally, the design of a robust controller reduces to the calcu-lation of the gain H of the controller which satisfies Eq. (14)for all the couples (wi, zi) representing performance specifica-tions (through the filters Wzi,wi

) and for all those representingthe uncertainties (through WG).

The algorithms that are used to calculate the controller gainlook for a value of H that minimises the H∞ norm of all thedynamic system (and thus of the multivariable operator GW)and not of all the input–output couples; ‖GW‖∞ < 1 ensuresrobust performance, but this is not a necessary condition toguarantee robust performance. Without going into details, theproblem can be modified introducing a matrix D and the al-gorithms, beside the gain H, calculate the values of D and of‖D−1(GW)D‖∞; if it is possible to find a gain H and a matrixD that verify:

‖D−1(GW)D‖∞ < 1 (21)

then robust performance is guaranteed.

2.2. Observer design

The use of the control law (12) requires the knowledge ofall the states x; some components of this array can be accessi-ble through measurements, but some others not. Moreover, themeasure of all the states x can be expensive or time consuming(thus introducing a delay in the control loop) or even not fea-sible. This justifies the use of an observer to get a quick andreliable estimation of the state array x, indicated as x in the fol-lowing. Fig. 3 depicts the block diagram of the observer: giventhe value of the input u and of the measurements y (and also ofthe objectives z), which are obviously available, the observeris a dynamic system returning the estimates of the states andalso of the objectives (z) and of the measured variables (y):the difference between the values of the measured variables yand their estimates y quantifies the adequacy of the observer.

Fig. 3. Block diagram of an extended system with its observer.

Given a linear system such as that of Eq. (1), the followingset of equations constitutes the observer:{ ˙x = Ax + K(y − y) + B2u,

z = C1x + D12u,

y = C2x.

(22)

If Eq. (22) is subtracted from Eq. (1) the equations describingthe dynamics of the errors on the estimation of the states andof the objectives are obtained:{

ex = (A − KC2)ex + (B1 − KD21)w,

ez = C1ex + D11w.(23)

Obviously, the observer has to be stable, i.e., the eigenvaluesof the matrix (A − KC2), which are a function of the observergain K, must have negative real part; moreover, the couple ofmatrices (A, C2) must be observable, i.e., it should be possibleto place the poles of (A − KC2) in arbitrary positions varyingK (this requirement can be relaxed in presence of poles thatcannot be moved arbitrarily, but are stable; in this case thesystem is said to be detectable).

The goal is now to calculate the observer gain K so that theestimation error on the objectives ez is insensitive with respectto the process disturbances w, i.e., to verify:

‖Gw,ez‖∞ < 1, (24)

where Gw,ez is the transfer function between the disturbancesw and the estimation errors ez. The same algorithms previouslydescribed for the calculation of the controller gain can be usedto calculate K: the only difference is in the extended systemthat is used.

2.3. Observer-based state feedback

The control design is affected by the presence of the observer,as now:

u = −Hx. (25)

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D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900 1895

The separation principle states that for a linear system con-trolled with (25) the poles of the closed-loop system are givenby the poles of the controller and those of the observer, so thatthe two can be designed independently and then put together.Even if robust performance is guaranteed for the controller andfor the observer following the previously described procedures,robust performance is not guaranteed for the closed-loop systemif the observer and the controller are designed independently.

Two issues should be taken into account for the design ofthe robust controller:

1. The first is that if the best performance that can be obtainedby estimating the objectives with an observer is measuredby ‖Gw,ez‖∞ ��o, when a full-state feedback control isused the controller performance (measured by ‖Gey,z‖∞),cannot be better than the estimations obtained.

2. The second intuition is that the observer model adopted fordesigning state feedback, cannot be any, but it must resultfrom the best observer in the presence of disturbances in theworst conditions (Zhou et al., 1995; Colaneri et al., 1997),i.e.,

w = �x. (26)

These two intuitions integrate each other in the following result(Zhou et al., 1995; Colaneri et al., 1997):

If the best objective estimator, designed in the worst dis-turbance conditions for a given bound ‖Gw,ez‖∞ ��o existsand a state feedback on the model obtained from this esti-mator with control performance ‖Gey,z‖∞ ��c exists, thenthe output feedback control obtained integrating estimatorand state feedback guarantees internal stability and perfor-mances given by ‖Gw,z‖∞ ��o if and only if �c < �o.

In conclusion, the observer model offers guaranteed perfor-mances for the original process only if it is derived from aspecific class of observers and the model in closed loop has anorm that outperforms the error of estimating the objectives.

From previous considerations the steps constituting the de-sign process are:

1. For a given �o < 1, the feedback K of an optimum observerin the worst disturbance condition is determined. If a solu-tion exists, �o represents a guaranteed upper bound of thenorm of the operator disturbances-objective estimation er-rors. For this observer the matrix � corresponds to the worstdisturbances. With K and � the model for feedback designis built (see Fig. 4, block diagram A).

2. H, the feedback from the estimated states, is computed inthe second step solving a state feedback problem, assumingas new disturbances the measure reconstruction errors and,as new objectives, the objective estimates. The feedbackdesign results in a closed-loop operator ey → z, with anorm bounded by �c (see Fig. 4, block diagram B).

3. If �c < �o, the control law with guaranteed bound �o hasbeen found (see Fig. 4, block diagram C), otherwise a largervalue to �o has to be assigned and the observer-feedbackdesign repeated. Again, if �o becomes larger than 1 some

Fig. 4. Extended systems used to design the controller in presence of anobserver.

of the requirements (the range of uncertainties of the pa-rameters and/or the performance requirements) have to bemade less severe.

3. Illustrative example

The purpose of this section is to apply the proposed designprocedure and to evaluate its performance through an examplein the chemical engineering field. Let us consider a continu-ous stirred tank reactor (CSTR) in which an isothermal, liquidphase, multi-component chemical reaction is carried out:

A�2C → B. (27)

The objective is to keep the concentration of B at a desiredset point by manipulating the molar feed rate of species C (in-dicated as u). The (nonlinear) balance equations are (Kravarisand Palanki, 1988):⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

cA = −k1cA + F

V(cA,feed − cA) + k2c

2C,

cB = −F

VcB + k3c

2C,

cC = k1cA − F

VcC − (k2 + k3)c

2C + u,

y = cA.

(28)

The state variables are the concentrations cA, cB , and cC ,while the measured concentration is cA. The system param-eters are: k1 = 1 s−1, k3 = 5 m3 mol−1 s−1, F = 3 m3 s−1,

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1896 D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900

V =3 m3, cA,feed=2 mol m−3; at the desired steady state cA,S =2.18 mol m−3, cB,S = 3.93 mol m−3 and cC,S = 0.87 mol m−3,and uS = 5 mol s−1.

There is uncertainty in the parameter k2, which can vary be-tween 0.1 and 5.9 m3 mol−1 s−1, thus, using the representationgiven by Eq. (16):

k2 = k2,mean(1 + �∗Wk2) (29)

with k2,mean = 3 m3 mol−1 s−1 and Wk2 = 0.97. Moreover, thesystem is nonlinear in the state variable cC ; standard Jacobilinearisation around the steady state gives:

c2C = (cC,S)2 + 2cC,S(cC − cC,S). (30)

Thus, the system of equations (28) takes the form:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

cA = −k1cA + F

V(cA,feed − cA) + k2[(cC,S)2

+ 2cC,S(cC − cC,S)],cB = −F

VcB + k3[(cC,S)2 + 2cC,S(cC − cC,S)],

cC = k1cA − F

VcC − (k2 + k3)[(cC,S)2

+ 2cC,S(cC − cC,S)] + u,

y = cA.

(31)

At steady state, Eqs. (31) yield:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 = −k1cA,S + F

V(cA,feed,S − cA,S) + k2(cC,S)2,

0 = −F

VcB,S + k3(cC,S)2,

0 = k1cA,S − F

VcC,S − (k2 + k3)(cC,S)2 + uS,

y = cA,S.

(32)

Subtracting Eq. (32) from Eq. (31) the dynamics of the systemis described in terms of deviations from the steady-state values:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

˙cA = −k1cA + F

V(cA,feed − cA) + 2k2cC,S cC,

˙cB = −F

VcB + 2k3cC,S cC,

˙cC = k1cA − F

VcC − 2(k2 + k3)cC,S cC + u,

y = cA,

(33)

where cA =cA −cA,S , cB =cB −cB,S , cC =cC −cC,S , cA,feed =cA,feed − cA,feed,S , u = u − uS . In nominal conditions cC,S =0.887 mol m−3 and this value can be used for the calculation ofthe controller with classical techniques (PID, LQR, etc.); herecC,S can be assumed to vary between 0 and 1.774 mol m−3,i.e., a variation of ±100% around the steady-state value, thus:

cC = 0.887(1 + �∗WcC) mol m−3 (34)

with WcC= 1.

As a first guess, a reduction of the disturbance effects of1/100 till a crossover frequency of 10 rad s−1 is required, witha maximum of 6 dB in the sensitivity at high frequencies, thusresulting in the following performance filter:

Ww,z = 0.5(s + 2000)

s + 10. (35)

Fig. 5. Block diagram of the extended system corresponding to the CSTR ofthe example (the circles correspond to summation nodes).

The state representation of the filter (35) is:

z = −10z + u,

y = 995z + 0.5u. (36)

The representation (36) is required because when the filter (35)is used, a further state (z) is introduced in the system. Withrespect to the control activity a constant filter is generally usedand, as no requirements are given about this performance, avalue of 0.1 is assumed. Fig. 5 shows the block diagram of theextended system corresponding to this control problem: eachblock identifies an operator that acts over the input and gives, asoutput, the input multiplied by the function of the block (whichcan be either a constant, or an operator); the block that performsthe integration of the input signal is indicated as “1/s”; theothers are indicated with the notation used in Eqs. (33)–(35).The summation nodes are indicated as circles and the signalsarriving to each node are summed, apart from those that havea minus sign, that are subtracted from the summation. Variouscouples of input–output signals emerge:

1) cA,feed.z represents the channel disturbance-objective (Wis the performance filter corresponding to Eq. (36));

2) n.zn represents the channel measurement noise-control ac-tivity (the performance filter is a constant equal to 0.1);

3) wk2.zk2 represents the uncertainty channel on the kineticconstant k2 (0.97 is the value of Wk2 );

4) wCc.zCc represents the uncertainty channel due to cC,S ;5) y.u represents the channel measure-control action.

A simplified block diagram of the extended system is given inFig. 6, where only the input (disturbances and control actions)and output (objectives and measures) signals are evidenced;the control loop, as well as the loop due to the uncertainties,

Page 8: Robust control in presence of parametric uncertainties: Observer-based feedback controller design

D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900 1897

Fig. 6. Simplified block diagram of the extended system where only the inputsignals are evidenced.

are also evidenced. The transfer functions between the variousinput and output of the system are given explicitly in Table 1 .

Let us assume, for the moment, that all the states of the sys-tem (i.e., the values of cA, cB , cC , and z) are available forthe control. Various softwares, e.g. the Robust Control Tool-box of MatLab, can be used to minimise the H∞ norm of theinput–output operator according to Eq. (21) giving a value of0.416 for the ‖D−1(GW)D‖∞: this means that the requiredrobust performance can be achieved and the controller gain is:

H = [1.03737 6199.42 303641 3206.43]. (37)

Fig. 7 shows the variations of cB when cA,feed changes from 2to 7 mol m−3: the time evolution (from the steady-state value)without any control and with the previously calculated con-troller gain are compared, so that its effectiveness is pointedout. In order to verify the robust performance, the same testhas been repeated for three values of the parameter k2: the re-sults are shown in Fig. 8 and demonstrate that the performancerequirements are fulfilled for all the values of k2 considered.

The designed controller requires the knowledge of the statesof the system; actually, only cA is measured, thus an observeris required to get the estimation of the other states. The ex-tended system used for the design of the observer is the samedepicted in Figs. 5 and 6, i.e., the same uncertainties due to k2and to the nonlinearity and the same performance specificationare considered. When the calculations are performed, Eq. (24)cannot be satisfied: the norm of the operator is bounded by avalue of 1.52 which does not guarantee anything. The norm ofthe single channels is thus analysed and it is found that the re-sponsible for the high value of the norm of the operator is theuncertainty channel due to the linearisation; thus, a lower valueof WcC

is assumed, passing from 1 to 0.8. The calculations arerepeated and the norm of the operator can now be lowered to0.861, giving an observer gain equal to:

K = [29823.4 2.73 − 5.32 − 0.19]. (38) Tabl

e1

Tra

nsfe

rfu

nctio

nsbe

twee

nth

eva

riou

sin

put

and

outp

utsi

gnal

sof

the

exte

nded

syst

emco

rres

pond

ing

toth

eC

STR

ofth

eex

ampl

e

Inpu

t→

c A,f

eed

nw

k2

wC

Cu

outp

ut↓

z4.

35(s

+20

00)

(s+

10)(

s+

1)(s

+15

.312

1)(s

+1.

6078

8)0

−4.3

5(s

+1)

(s+

2000

)

(s+

10)(

s+

1)(s

+15

.312

1)(s

+1.

6078

8)

2.5(

s+

1)(s

+2)

(s+

2000

)

(s+

10)(

s+

1)(s

+15

.312

1)(s

+1.

6078

8)

4.35

(s+

2)(s

+20

00)

(s+

10)(

s+

1)(s

+15

.312

1)(s

+1.

6078

8)zn

00

00

0.1

zk

2

5.06

34

(s+

15.3

121)

(s+

1.60

788)

0−5

.063

4(s

+1)

(s+

15.3

121)

(s+

1.60

788)

2.91

(s+

1)(s

+2)

(s+

15.3

121)

(s+

1.60

788)

5.06

34(s

+2)

(s+

15.3

121)

(s+

1.60

788)

zC

C

1.74

(s+

15.3

121)

(s+

1.60

788)

0−1

.74(

s+

1)

(s+

15.3

121)

(s+

1.60

788)

−13.

92(s

+1.

625)

(s+

15.3

121)

(s+

1.60

788)

1.74

(s+

2)

(s+

15.3

121)

(s+

1.60

788)

ys

+14

.92

(s+

15.3

121)

(s+

1.60

788)

1s

+9.

7

(s+

15.3

121)

(s+

1.60

788)

3(s

+1)

(s+

15.3

121)

(s+

1.60

788)

5.22

(s+

15.3

121)

(s+

1.60

788)

Page 9: Robust control in presence of parametric uncertainties: Observer-based feedback controller design

1898 D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900

Fig. 7. Time evolution (from the steady-state value) of the variation of cB

when cA,feed changes from 2 to 7 mol m−3: upper graph: no control action,lower graph: state feedback control (the symbols indicate the value of thecontrol action).

Fig. 8. Time evolution (from the steady-state value) of the variation ofcB when cA,feed changes from 2 to 7 mol m−3 for three values of k2

(©: 0.1 m3 mol−1 s−1, : 3 m3 mol−1 s−1, �: 5.9 m3 mol−1 s−1).

Fig. 9. Comparison between the evolution of molar concentration of species Band C from the steady-state calculated by means of simulation of the systemof nonlinear equations (solid lines) and by the observer (©: cA, : cB , ♦: cC ).

As an example of the various tests made to verify the effec-tiveness of the proposed observer, Fig. 9 shows the results ob-tained in presence of a change from 2 to 4 mol m−3 of cA,feedand from 5 to 3 of u: the evolution of molar concentration ofspecies A, B, and C from the steady state has been calculated bymeans of simulation of the system of nonlinear equations (28)and the results obtained with an observer, using the previouslycalculated gain and using the value of cA as measured input,are compared, evidencing how the observer is able to followthe dynamics of the system and to give a precise estimation ofthe states.

This result can be useful if the aim of the observer is tomonitor the process, i.e., to get an estimation of some variablesthat cannot be measured, but if the goal is to use the observerin a state feedback loop, the optimum observer in the worstdisturbance condition has to be calculated. The calculationsevidenced that the condition �c < �o can be fulfilled, but theminimum value of �o that can be obtained is about 4. Again,the norm of the single channels is analysed and it is found thatthe responsible for the high value of the norm of the operator isthe uncertainty channel due to the linearisation; thus, the valueof WcC

is lowered to 0.5 (now the observer has to estimate the

Page 10: Robust control in presence of parametric uncertainties: Observer-based feedback controller design

D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900 1899

Fig. 10. Time evolution (from the steady-state value) of the varia-tions of cB when cA,feed changes from 2 to 7 mol m−3 in presence ofobserver + full-state feedback controller (the symbols indicate the value ofthe control action).

state variables in a controlled system, so it is reasonable thatcC,S varies in a narrow range). The best results are obtainedwith �c < �o=0.784. Fig. 10 (upper graph) shows the variationsof cB when cA,feed changes from 2 to 7 mol m−3 (the timeevolution from the steady-state value without any control wasgiven in Fig. 7).

4. Conclusions and further remarks

The construction of an extended model for the design of arobust controller has been discussed in the paper: this modelallows for the specification of the regulation and tracking per-formance requirements as well as of the control activity re-quirements; also parametric uncertainties can be included inthe extended model. Performance and robustness are linked tothe control law, which consists of a simple state feedback.

Moreover, this procedure can guide the designer towards thebest feasible controller: consider each requirement or sourceof uncertainty, e.g. disturbance rejection or steady-state gainchanges, and the restriction of the closed-loop operator refer-ring to the corresponding pair disturbance-objective; if the H∞norm of this restriction is less than 1, the performance is sat-isfied in nominal conditions (or robust stability is guaranteedwith respect to one source of uncertainty), while if the value isgreater than 1 it means that the specified performance cannotbe satisfied by any control gain (or that the range of uncertaintyis too large to allow for robust performance).

Nonlinear system can be easily handled in this frameworkby describing it as deviations of parameters from a nominalvalue, and then utilising linear system based control methods.This approach obviously implies that the range of the uncer-tainty could be very large and, thus, yield very conservativeresults, in particular when uncertainty is also considered. As aresult, the robust controller could be unfeasible only because ofthe nonlinearity has been managed in a too conservative way.Anyway, in case the controller is unfeasible, i.e., it is not pos-

sible to get a value of the norm H∞ of the extended modellower than 1, the proposed procedure allows to point out if theresponsible of this is a too severe performance requirement ora too large range of uncertainty for a certain parameter (or fora certain nonlinear variable). This is the main limitation of theproposed approach, which is in turn compensated by its sim-plicity (according to Luyben’s philosophy of keeping a controlsystem as simple as possible).

If the state of the system is not fully measured, then anobserver is required: also in this case an extended model hasto be set up and again the H∞ norm of this extended systemensures the performance of the observer. The observer modeloffers guaranteed performances for the original process only ifit is derived from a specific class of observers and the model inclosed loop has a norm that outperforms the error of estimatingthe objectives.

Notation

A, B, C chemical speciesc molar concentration, mol m−3

D matrix to be used in Eq. (21)e estimation errorer tracking errorF feed flow rate, m3 s−1

G approximate transfer functionGeq equivalent transfer functionGp true transfer function of the processGw,ez transfer function between the disturbances w and

the estimation errors ez

Gw,z transfer function between w and zGW matrix transfer function of the extended systemH controller gainj imaginary unit (j2 = −1)

k, p parameterK observer gainr reference trajectorys Laplace variableu control actionU array (or matrix) of signalsV reactor volume, m3

w disturbancesWG relative variation of a parameterWw,z filter used to specify the requirements on the sen-

sitivity of the output z with respect to the input w

x state of the systemy measured variablesz objectives of the controlled systemz∗ filtered objectives of the controlled system

Greek letters

�, �, � parameter of a generic filter Ww,z

�∗ bounded operator with norm H∞lower than 1� threshold value of the norm� maximum singular value

Page 11: Robust control in presence of parametric uncertainties: Observer-based feedback controller design

1900 D. Fissore / Chemical Engineering Science 63 (2008) 1890–1900

� frequency, rad s−1

� array used to calculate the worst disturbance

Subscripts and superscripts

∧ estimated value− deviation from the steady-state value1, 2, 3 reaction identifierc controllerfeed feeding valuemax maximum valuemean mean valuemin minimum valueo observers steady-state value

Abbreviations

BIBO bounded input bounded outputCSTR continuous stirred tank reactorI/O input/outputLQR linear quadratic regulatorMPC model predictive controlPID proportional integral derivativeSISO single input single output

Acknowledgement

The author would like to acknowledge Prof. Giuseppe Mengaand Prof. A. Barresi (Politecnico di Torino) for their valuablesuggestions.

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