research article closed-loop identification applied to a...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 519432 10 pageshttpdxdoiorg1011552013519432

Research ArticleClosed-Loop Identification Applied to a DC ServomechanismController Gains Analysis

Roger Miranda Colorado and Gamaliel Contreras Castro

Universidad Politecnica de Victoria Avenida Nuevas Tecnologıas 5902 Parque Cientıfico y Tecnologico de Tamaulipas87138 Ciudad Victoria TAMPS Mexico

Correspondence should be addressed to Roger Miranda Colorado rmirandacupvedumx

Received 13 June 2013 Accepted 11 October 2013

Academic Editor Wudhichai Assawinchaichote

Copyright copy 2013 R Miranda Colorado and G Contreras Castro This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Usually when parameter identification is applied there are some gains related to the identification algorithm whose value must becarefully adjusted in order to obtain a good performance of the algorithm However when performing closed loop identificationthere are some other constants that in general are not taken into account for the identification algorithm the controller gainswhichmay appear inside the identification algorithm specifically in the regressor vector which is very important for the parameterconvergence according to the persistence of excitation condition Therefore the effect of these gains on the estimated parametersshould be analyzed so that better estimates can be obtained This paper addresses the behavior of the parameter estimates for aclosed-loop identification methodology applied to a DC servomechanism with a bounded perturbation signal and a PD controllerIt is shown that with this perturbation the parameter estimates converge to a regionwhose size can bemodified not only by varyingthe identification algorithm gains but also by modifying the P and D controller gains in a suitable way

1 Introduction

It is well known that servomechanisms are fundamentalin modern robotic and mechatronic systems for industrialapplications where high speed and high precision are ofprime importance On the other hand in order to applymodel-based tuning methods it is necessary to apply anidentification algorithm to the servomechanism and if itis required to design a high performance controller wherea small steady state tracking error and high precision arenecessary Besides the system parameters are also importantto know if the system itself behaves as it should for exampleby evaluating its performancewith some specified test signalsThus parameter identification plays a very important role fordesigning a controller as pointed out from several points ofview [1ndash4]

For a DC servomechanism there are several techniquesthat may be applied for identifying its parameters whenthe servomechanism operates in open loop and in closedloop However it is important to note that if the variableof interest is the servo position then a linear model of

a servomechanism contains a pole on the imaginary axisthus making the system not BIBO stable that is a boundedinput applied to the servomechanism would not produce abounded position and this may lead to undesired effectsfor example in robotic applications Therefore for securityreasons parameter identification should be performed inclosed loop

One way to characterize the system identification algo-rithms is [5 6] (i) direct methods and (ii) indirect methodsMany of the identification procedures using a linear con-troller [7ndash9] fall into the category of direct methods that isthe parameter identification procedures are applied withoutregard to the controller being used to close the loop On theother hand if a technique takes into account the controllerstructure then it falls into the category of indirect methods

As mentioned before for security reason it is better toestimate the system parameters with the system workingin closed loop One way to perform closed loop parameteridentification of a servomechanism is with the relay-basedtechniques [10ndash12] which are widespread for servo identifi-cation The idea with this methodology is to close the loop

2 Mathematical Problems in Engineering

through a relay in order to obtain a sustained oscillationand then the values of the frequency and amplitude allowsidentifying a model of the systemThe relay based techniquesmayworkwell however the reference signal used for excitingthe system becomes constant in many time instants andthen no control is being applied in that time and then theeffect of the disturbances may affect the estimates obtainedwith this methodology Besides tuning of the relay controllercan be cumbersome and no one of the reviewed methodstakes explicitly into account the disturbances affecting theservomechanism

References [7ndash9 13ndash15] propose methods for closed loopidentification of position-controlled servos where the loopis closed using a linear controller The approach proposedin [13 14] uses a PD controller to close the loop and anon-line gradient algorithm allows estimating a linear andnonlinear model of a servomechanism Also in [15] a two-step identificationmethod for estimating the four parametersof a nonlinear model of a position control servomechanismis presented where the proposed identification scheme relieson the theory of operational calculus However the effect ofperturbations is considered only for constant values On theother hand a common technique used for identifying DCservomechanisms is based on the least squares algorithm In[16] an approach for identifying a DC motor is presentedconsidering the Coulomb friction and dead zone using arecursive discrete time least squares algorithmwith forgettingfactor using a Hammerstein model for the motor In [17]second-order sliding modes approach is applied for estimat-ing the parameters of aDCmotor operating in open-loop andclosed loop while in [18] the inertia of a second-order modelof a DC servomechanism driving a human assistive system isidentified using Bode plots obtained from input-output mea-surements Finally another interesting approach is presentedin [2 4] where an online algebraic identification method isproposed Besides [19 20] show how this methodology canbe used for parameter identification of a DC motor and alsothis methodology was used and modified in [21] introducinga batch least squares algorithm for estimating the parametersof an induction motor

All the previousworks consider in general the importanceof some gains related with the identification algorithm forclosed-loop identification However in some cases (as thatpresented in [13 14]) the controller gains get involved inwith some important elements for the identification schemefor example in the regressor vector In this case it has notbeen analyzed which effect actually has such gains in theparameter identification procedure and how the parameterestimates will behave when a disturbance is affecting thesystemThen this work presents the analysis of a closed loopidentification algorithm for a perturbed position-controlledDC servomechanism where a PD controller closes theloop and achieves stability without knowledge about theservomechanism parameters Theoretical results show thatwhen the perturbation signal is identically zero exponentialconvergence can be claimed and in the presence of abounded perturbation it is possible to obtain a region Ω120575

where the parameter estimates belong Whatrsquos more it willbe shown that this region can be made arbitrarily small if

the controller gains are tuned properly Some simulationsdepict the behavior of the region Ω120575 showing the effect thatthe controller gains have on the estimated parameters Thepaper is organized as follows Section 2 is devoted to presentpreliminary theoretical results of passivity based controladaptive control and the general description of the DC ser-vomechanism Section 3 presents the analysis for closed loopidentification of the servomechanism when no disturbancesare affecting the system and the exponential stability of theoverall identification scheme is shown Section 4 extendsthe analysis for the perturbed case and Section 5 deals withthe analysis of the results Finally Section 6 gives someconcluding remarks

2 Preliminary Results

The closed loop identification analysis that will be presentedis based on some results related with passivity theory andadaptive control and thus the following results are worthpresenting

21 Passivity and Adaptive Control As pointed out beforethe aim of this work is to show the convergence propertiesof the identification algorithm with respect to the value ofthe controller gains To this end it will be necessary to usethe observability property given in [22] and some results ofpassivity and stability that will be described in the following

Let us consider the linear time-varying system[119862(119905) 119860(119905)] defined by

= 119860 (119905) 119909 (119905)

119910 (119905) = 119862 (119905) 119909 (119905)

(1)

where 119909(119905) isin 119877119899 119910(119905) isin 119877

119898 while 119860(119905) isin 119877119899times119899 and

119862(119905) isin 119877119898times119899 are piecewise continuous functions A system

[119862(119905) 119860(119905)] is said to be uniformly completely observable(UCO) [22] if there exist strictly positive constants 1205731 1205732 120575such that for all 1199050 ge 0 1205731119868 le 119873(1199050 1199050 + 120575) le 1205732119868 where119873(1199050 1199050 + 120575) isin 119877

119899times119899 is the so-called observability Gramian as

119873(1199050 1199050 + 120575) = int

1199050+120575

1199050

Φ119879(120591 1199050) 119862

119879(120591) 119862 (120591)Φ (120591 1199050) 119889120591

(2)

where Φ(120591 1199050) is the state transition matrix associated with119860(119905) Now assume that for all 120575 gt 0 there exists 119896120575 gt 0 suchthat for all 1199050 ge 0 int

1199050+120575

1199050119870(120591)

2119889120591 le 119896120575 where the matrix 119870

corresponds to the system under output injection as

= 119860 (119905) + 119870 (119905) 119862 (119905) 119908 (119905)

119911 (119905) = 119862 (119905) 119908 (119905)

(3)

with119908(119905) isin 119877119899119870(119905) isin 119877119899times119898 and 119911(119905) isin 119877119898Then the system[119862 119860] is UCO if and only if [119862 119860 + 119870119862] is UCO where theterm 119870119862 stands for the output injection term as indicated in[22] Moreover if the observability Gramian of the system[119862 119860] satisfies the previous upper and lower bounds on 119873

Mathematical Problems in Engineering 3

then the observability Gramian of the system [119862 119860 + 119870119862]

satisfies these inequalities with identical 120575 and

1205731015840

1=

1205731

(1 + radic1198961205751205732)

1205731015840

2= 1205732 exp (1198961205751205732) (4)

Now assume that 119891(119905 119909) 119877+ times 119877119899 rarr 119877119899 has continuous

and bounded first partial derivatives in 119909(119905) and is piecewisecontinuous in 119905 for all 119909 isin 119861ℎ 119905 ge 0 Then the followingstatements are equivalent [22]

(1) 119909 = 0 is an exponentially stable equilibrium point of = 119891(119905 119909) 119909(1199050) = 1199090

(2) There exists a function 119881(119905 119909) and some strictlypositive constants ℎ1015840 1205721 1205722 1205723 and 1205724 such that forall 119909 isin 119861ℎ1015840 and 119905 ge 0 1205721119909

2le 119881(119905 119909) le 1205722119909

2(119905 119909)| 119909=119891(119905119909) le minus1205723119909

2 and |120597119881(119905 119909)120597119909| le 1205724119909

Now if there exists a function119881(119905 119909) and strictly positiveconstants 1205721 1205722 1205723 and 120575 such that for all 119909 isin 119861ℎ119905 ge 0 1205721119909

2le 119881(119905 119909) le 1205722119909

2 (119905 119909)| 119909=119891(119905119909) le

0 and int119905+120575119905

119889119881(120591 119909(120591))119889120591|119909=119891(119905119909)

119889120591 le minus12057231199092 then 119909(119905)

converges exponentially to zeroFinally [23] the state equation [119860 119861 119862119863] is a minimal

realization of a proper rational function 119892(119904) if and only if(119860 119861) is controllable (119860 119862) is observable or if and onlyif dim(119860) = deg(119892(119904)) where 119892(119904) = 119873(119904)119863(119904) anddeg(119892(119904)) = deg(119863(119904))

In the next definitions let us consider the system Π

defined as

Π

(119905) = 119891 (119909 119906)

119909 (0) = 1199090 isin 119877119899

119910 (119905) = 119867 (119909 119906)

(5)

where 119909 isin 119877119899 is the state vector 119906 isin 119877119898 the input and 119910 isin

119877119898 the system output Consider the set Ξ of 119899 dimensional

real valued functions 119891(119905) 119877+ rarr 119877119899 and define the set

1198712 = 119909 isin Ξ 1198912

2= intinfin

0119891(119905)

2119889119905 lt infin with sdot 2 the

Euclidean norm This set constitutes a normed vector spacewith the field119877 and norm sdot 2 Let us introduce the extendedspace 1198712119890 as

1198712119890 = 119909 isin Ξ 10038171003817100381710038171198911003817100381710038171003817

2

2119879= int

119879

0

1003817100381710038171003817119891 (119905)1003817100381710038171003817

2119889119905 lt infin forall119879 (6)

where 1198712 sub 1198712119890 In the same way let us introduce the innerproduct and the truncated inner product of functions 119906 and119910 as

(119906 119910) = int

infin

0

119906119879(119905) 119910 (119905) 119889119905

(119906 119910)119879= int

119879

0

119906119879(119905) 119910 (119905) 119889119905

(7)

The system (5) is dissipative [24] with respect to the supplyrate 120596(119906 119910) 119877119898 times 119877119898 rarr 119877 if and only if there exists a

storage function 119877119899 rarr 119877ge0 such that119867(119909(119879)) le 119867(119909(0)) +int119879

0120596(119906(119905) 119910(119905))119889119905 for all 119906(119905) all 119879 ge 0 and all 1199090 isin 119877

119899 Thesystem is passive if its supply rate is given by (119906 119910) = 119906119879119910 Itis input strictly passive (ISP) if there exists a positive constant120575119894 such that the supply rate can be expressed as (119906 119910) =

119906119879119910 minus 120575119894119906

2 120575119894 gt 0 Finally it is output strictly passive(OSP) if there exists a positive constant1205750 such that the supplyrate can be expressed as (119906 119910) = 119906

119879119910 minus 120575119900119910

2 120575119900 gt 0The system (5) is called 1198712 finite gain stable if there exists apositive constant 120574 such that for any initial condition 1199090 thereexists a finite constant 120573(1199090)such that 1199102119879 le 1205741199062119879+120573(1199090)and one important result from [24] is that if Π 119906 rarr 119910 isOSP then it is 1198712 stable

A state space system (119905) = 119891(119909) 119909 isin 119877119899 is zero state

observable from the output 119910(119905) = ℎ(119909) if for all initialconditions 119909(0) isin 119877

119899 the output 119910(119905) equiv 0 implies that119909(119905) equiv 0 It is zero state detectable if 119910(119905) equiv 0 implies thatlim119909(119905) = 0 as 119905 rarr infin

Regarding the identification algorithms the persistenceexcitation (PE) condition is very important and its definitionis given now according to [22] a vector 120601 119877

+rarr 119877

2119899 ispersistently exciting (PE) if there exist constants 1205721 1205722 120575 gt0 such that

1205721119868 le int

1199050+120575

1199050

120601 (120591) 120601119879(120591) 119889120591 le 1205722119868 forall1199050 ge 0 (8)

Besides let 119908 119877+rarr 119877

2119899 If 119908 is PE the signals 119908 belong to the space 119871infin and 119880 is a rational stable strictlyproper minimum phase transfer function then 119880(119908) is PE

Theorem 1 (small signal IO stability [22]) Consider theperturbed system = 119891(119905 119909 119906) 119909(0) = 1199090 and theunperturbed system = 119891(119905 119909 0) 119909(0) = 1199090 where 119905 ge 0119909 isin 119877

119899 and 119906 isin 119877119898 Let 119909 = 0 be an equilibrium point of

the unperturbed system that is 119891(119905 0 0) = 0 for all 119905 ge 0Let 119891 be piecewise continuous in 119905 and have continuous andbounded first partial derivatives in 119909 for all 119905 ge 0 119909 isin 119861ℎ119906 isin 119861119888 Let 119891 be Lipchitz in u with Lipchitz constant 119897119906 forall 119905 ge 0 119909 isin 119861ℎ 119906 isin 119861119888 Let 119906 isin 119871infin If 119909 = 0 isan exponentially stable equilibrium point of the unperturbedsystem then the perturbed system is small-signal 119871infin stablethat is there exist 120574infin 119888infin gt 0 such that 119906infin lt 119888infin impliesthat 119909infin le 120574infin119906infin lt ℎ where 119909 is the solution of 119891(119905 119909 119906)starting at 1199090 = 0 There exists 119898 ge 1 such that for all1199090 lt ℎ119898 0 lt 119906infin lt cinfin implies that 119909(119905) converges toa 119861120575 ball of radius 120575 = 120574infin119906infin lt ℎ that is for all 120593 gt 0 thereexists 119879 ge 0 such that 119909(119905) le (1 + 120593)120575 for all 119905 ge 119879 along thesolutions of 119891(119905 119909 119906) starting at 1199090 Also for 119905 ge 0 119909(119905) lt ℎ

22 Model of the DC Servomechanism The closed loopidentification analysis that will be presented considers themodel of a perturbed DC servomechanism described by thefollowing equation

119869 119902 + 119891 119902 (119905) = 119896119906 (119905) + ]1 (119905) = 120591 (119905) + ]1 (119905) (9)

where 119869 119891 119896 119906 ]1 and 120591(119905) = 119896119906(119905) are the inertia viscousfriction coefficient amplifier gain input voltage perturbation


signal and torque input respectivelyThe perturbation signalmay be due to possibly constant disturbances or parasiticvoltages appearing inside the amplifier The model (9) can berewritten as

119902V (119905) = minus119886 119902 (119905) + 119887119906 (119905) + ] (119905) (10)

where 119886 = 119891119869 and 119887 = 119896119869 are positive constants and] = ]1119869 and 119902(119905) 119902(119905) are the servo angular position andvelocity respectively It is also assumed that the perturbationsignal is bounded that is ]1(119905) le 120573 120573 isin 119877

+ In thefollowing it will be shown that there exists a regionΩ120575 wherethe parameter estimates converge even in the presence ofthe perturbation signal ](119905) Furthermore it will be shownthat the size of this region can be arbitrarily reduced byincreasing the controller gains employed for stabilizing theservomechanism This analysis will consider two cases

(i) Closed loop identification of the unperturbed DCservomechanism where the main objective will beto get a formula to estimate the system parametersand show the condition under which the system isexponentially stable

(ii) Closed loop identification of the perturbed DC ser-vomechanism where the existence of the region Ω120575will be shown where the parameter estimates con-verge when a perturbation signal affects the system

3 Closed-Loop Identification AlgorithmNonperturbed Case

This section describes the closed loop identification approachand the way the parameter updating law is constructedBesides it is proved that under some assumptions the overallsystem is exponentially stableThen first the stability analysiswill be presented to ensure the asymptotic stability of theclosed loop system and the parameter error dynamics Laterthe system exponential stability property will be established

31 Stability Analysis and Parameter Convergence The closedloop identification analysis to be described is similar to thatpresented in [14] and depicted in Figure 1 For this (unper-turbed) case it will be considered that ](119905) is identically zeroleading to the following unperturbed servo model

119902 (119905) = minus119886 119902 (119905) + 119887119906 (119905) (11)

The closed loop identificationmethodology consists of select-ing a model of (11) with the following dynamics

119902119890 (119905) = minus119886 119902119890 (119905) + 119906119890 (119905) (12)

where 119902119890(119905) is the estimated servo position 119902119890(119905) the estimatedvelocity 119886 are the estimates of 119886 119887 and 119906119890 the controllaw for the estimated model Then the loop is closed aroundthe real servomechanism and its model by using two PDcontrollers where the same gains are used for both of themNow let us define the output error 120576(119905) = 119902(119905) minus 119902119890(119905)and then this error and its time derivative are employed to

feed an identification algorithm which estimates the systemparameters and updates them in the estimatedmodel All thisprocedure is now theoretically summarized

For closing the loop around the unperturbed system andits model let us consider the PD controllers u(t) and 119906119890(119905) asfollows 119906(119905) = 119896119901119890(119905) minus 119896119889 119902(119905) and 119906119890(119905) = 119896119901119890119890(119905) minus 119896119889 119902119890(119905)where 119890(119905) = (119902119889(119905) minus 119902(119905)) 119890119890(119905) = (119902119889(119905) minus 119902119890(119905)) 119902119889(119905) isthe reference signal and 119896119901 gt 0 119896119889 gt 0 the proportional andderivative controller gains which are used in both controllersThen the system (11) in closed loop with 119906(119905) leads to the nextclosed-loop dynamics (for the sake of simplicity fromnowonthe time argument will be omitted) as

Σ1 119902 = minus (119886 + 119887119896119889) 119902 + 119887119896119901119890 (13)

while the system (12) in closed loop with 119906119890(119905) leads to thefollowing dynamics

Σ2 119902119890 = minus (119886 + 119896119889) 119902119890 + 119896119901119890119890 (14)

By using the Routh-Hurwitz criterion it is easy to showthat the control law 119906(119905) stabilizes the unperturbed servomodel (11) However even when the system (13) is stablethe same conclusion cannot be drawn for (14) because thecoefficients of this estimated model (ie 119886 ) are updated byone identification algorithm to be described later and thenthese parameters are time varying therefore it is necessary toanalyze the stability of (14) From the definition of 120576(119905) it ispossible to evaluate its second time derivative and employing(13) and (14) the error dynamics is established as follows

120576 +119888 120576 + 119887119896119901120576 = 120579119879120601 (15)

where 119888 = (119886 + 119887119896119889) gt 0 120579(119905) is the parameter error vectorand 120601(119905) is the so-called regressor vector where 120579 = 120579 minus 120579 =(119886 minus 119886 minus 119887)

119879 120601(119905) = ( 119902119890 minus119906119890)119879 with 120579 being the estimate

of 120579 In order to analyze the behavior of the signals involvedin the error dynamics (15) passivity based arguments will beused [24 25] To this end let 119909 = (120576 120576)119879 be the state vector of(15) and consider the following storage function

1198671 (119909) =1

2119909119879(119887119896119901 120583

120583 1)119909 = 119909

119879119872119909 (16)

where 120583 isin 119877+ The choice of 1198671 will lead to concluding theOSP property of (15) Then taking the time derivative of1198671along the trajectories of (15) leads to

1 = 120579119879120601 (120583120576 + 120576) minus

119888

2(120583120576 + 120576)

2minus (

119888

2minus 120583) 120576

2

minus 120583(119887119896119901 minus120583119888

2) 1205762

(17)

It is easy to prove that 1198671 gt 0 if 120583 lt radic119887119896119901 Besides if120583 lt min1198882 2119887119896119901119888 then the time derivative of 1198671 alongthe trajectories of (15) yields 1 le 120579

119879120601(120583120576+ 120576)minus119888(120583120576 + 120576)

22

that is (15) defines an OSP operator 120579119879120601 rarr (120583120576 + 120576)Moreover it is well known that the feedback interconnection


Figure 1 Blocks diagram for the identification algorithm

of passive subsystems is passive [24] thusmaking an intuitiveto consider the parameter error dynamics as follows Σ3

120579 =

minusΓ120601(120583120576+ 120576) withmatrix Γ = Γ119879 gt 0Then by considering thestorage function1198672(120579) = 120579

119879Γminus11205792 its time derivative along

the trajectories of Σ3 leads to 2 = 120579119879Γminus1 120579 = (120583120576 + 120576)(minus120579

119879120601)

so that Σ3 defines the passive operator (120583120576 + 120576) rarr (minus120579119879120601)

Finally let us consider the feedback interconnection of (15)with Σ3 given by

Σ

120576 (119905) + 119888 120576 (119905) + 119887119896119901120576 (119905) = 120579119879120601 (119905)

120579 = minusΓ120601 (120583120576 + 120576)

(18)

Then by considering the storage function for (18) as the sumof 1198671 and 1198672 it is easy to prove that Σ is still OSP thus itfollows that (120583120576+ 120576) isin 1198712 Let us define the signal 119910(119905) = (120583120576+120576) as the output for the interconnected system (18)Then notethat 120576(119905) corresponds to the output of an exponentially stablefilterwhose input belongs to the1198712 space therefore 120576(119905) tendsto zero as 119905 rarr infin [26] Therefore it has been proved that119902(119905) isin 119871infin and 120576(119905) rarr 0 as 119905 rarr infin and thus 119902119890(119905) rarr 119902(119905)

as 119905 rarr infin that is 119902119890(119905) isin 119871infin and then it turns out thatthe time varying system (14) is stable as desired although itremains to prove that the parametric error converges to zerothat is that 120579(119905) converges to the real parameter vector 120579 Tothis end let us consider the state-space description of (15) as

= 119860119909 + 119861119880

119910 = 119862119909

(19)

with

119860 = (0 1

minus119887119896119901 minus119888) 119861 = (

0

1)

119862119879= (

120583

1) 119880 = 120579

119879120601

(20)

As described in [22] a necessary and sufficient condition forparameter convergence in linear systems is the PE conditionof the regressor vector 120601(119905) However the results presentedin [22] assume that all the signals in 120601(119905) come from a lineartime invariant system which is not the case here because herethe regressor 120601(119905) has signals from the estimated model (14)which is a time varying system To overcome this technicaldifficulty it is possible to consider the real regressor vector120601119879

119903(119905) = ( 119902 minus119906)

119879 which consists of signals from the realservomechanism (13) Following the same procedure as thatpresented in [22] it is not a difficult task to show that if 119888 gt 120583then 120601119903(119905) will be PE Now let us consider the difference

120601119903 minus 120601 = (119902

minus119906) minus (

119902119890

minus119906119890) = (

120576

119896119901120576 + 119896119889 120576) (21)

and consider the Lyapunov function candidate 1198811 = 1198671 +

1198672 Clearly 1198811 gt 0 if 120583 lt radic119887119896119901 and 1 le minus1205731205762 120573 =

(2120572)(1205831198871198961199011205722 minus 120583211988828) with 120572 = (119888 minus 120583) gt 0 Therefore

if 120583 le 4119887119896119901119888(4119887119896119901 + 1198882) then 120576(119905) isin 1198712 and considering

the fact that 119910(119905) = (120583120576 + 120576) isin 1198712 permits concluding that120576(119905) isin 1198712 which makes clear that (120601119903 minus 120601) isin 1198712 Now givena PE signal 120596(119905) and a signal 119911(119905) isin 1198712 the sum (120596 + 119911) isstill PE [22] Thus 120601 = 120601119903 minus (120601119903 minus 120601) is PE as desired andthe convergence of 120579(119905) to zero can be claimed that is the


estimated parameters converge to the real ones Then basedon the previous analysis the following proposition follows

Proposition 2 Consider the system (13) and assume that120583 le minradic119887119896119901 2119887119896119901119888 4119887119896119901119888(4119887119896119901 + 119888

2) Then 120576 120576 and the

parameter error vector 120579(t) converge to zero as time tends toinfinity

From the previous result asymptotic stability has beenproved for the closed loop system and identificationapproach However in the identification framework usuallyit is preferable to have a more robust stability conditionthat is exponential stability to guarantee that the parameteridentification scheme will be robust against disturbanceswhich may affect the system such as unmodeled dynamicsand measurement noise Therefore the next analysis willestablish the necessary conditions for the servomechanism tobe an exponentially stable system

32 Exponential Convergence As mentioned above oneappealing and important property of the parameter identifi-cation schemes is the exponential stability because it guaran-tees robustness of the identification algorithm in presence ofexternal disturbances or unmodeled dynamics Thus in thefollowing it will be proved which conditions must be satisfiedin order to make the system to be exponentially stable Firstrecall the state space description given by (19) Clearly thesystem (119860 119861 119862) is observable and controllable [23] that isit is a minimal realization if 120583 lt 1198882 Let us consider thesystems

Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)

The last analysis proved that 120601(119905) is PE which turns out to beequivalent to the UCO condition on the system Σ4 and so doΣ5 Then the main result of this section can be established asfollows

Theorem 3 Given the system (18) assume that 120583 fulfills thebound conditions of Proposition 2 Let w = (119909 120579) be anequilibrium point of Σ Then w is an exponentially stableequilibrium point

Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)

119879 be the state of (18) andconsider the following Lyapunov function candidate

1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)

with1198672 as previously mentioned and1198673 defined as follows

1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)

Clearly1198673 gt 0 if the inequality 120583 lt (119888+radic1198882 + 4119887119896119901)2 holdsBesides it is possible to show that

1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)

Taking the time derivative of 1198812 along the trajectories of (18)yields

2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)

Let us define the diagonal matrix119876 = diag120583119887119896119901 119888 minus 120583 thenwe have that 2 le minus119909

119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888

Now assume that there exist strictly positive constants 1205723 120575such that

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)

First note from (18) and the time derivative of 1198812 that

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)

and then (27) will be valid if for 1205723 gt 0 the followinginequality holds

120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)

for all 1199050 ge 0 and w(1199050) Now consider the following system

= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)

Let us consider the UCO property of Σ5 with119870 = minusΓ(120583120576 + 120576)It is clear that for the UCO condition

119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)

Also we know that (120583120576 + 120576) isin 1198712 thus there exists a positiveconstant 120581 gt 0 such that 119896120575 le 120581120582

2

max(Γ) Whatrsquos more theoutput of (19) is given by

119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)

because from (30) we note that the vector 120579 is constant Letus define the signals 1199111(119905) and 1199112(119905) as follows

1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)


From the stability analysis of Section 3 it was concluded that120601(119905) 120601(119905) isin 119871infin and that 120601(119905) is PE Thus it is clear that[22] 120601119891(119905) = int

119905

1199050119862119879119890119860(119905minus120591)

119861120601(120591)119889120591 is PE Therefore there

exist positive constants 1205721 1205722 120590 such that 1205721120579(1199050)2

le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2

for all 1199051 ge 1199050 ge 0 and 120579(1199050)Moreover because matrix 119860 is stable there exist positiveconstants 1205741 1205742 such that int

infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590

for all 1199050 ge 0 119909(1199050) and an integer 119898 gt 0 to be defined laterBecause (119860 119862) is observable there exists 1205743(119898120590) gt 0 with1205743(119898120590) increasing such thatint

1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2

for all 1199050 ge 0 119909(1199050) and 119898 gt 0 Let 119899 gt 0 be another integerto be defined and 120575 = (119898 + 119899)120590 Then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)

Let both119898 and 119899 be large enough such that 1198991205721minus1198981205722 ge 1205721 gt0 and 1205743(119898120590) minus 1205741119890

minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)

Similarly it is possible to conclude that

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(36)Now we define 1205731 = min(1205721 1205743(119898120590)2) and 1205732 =

max(21205741 2(119898 + 119899)1205722) Then it is possible to obtain thefollowing bounds

1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)

where 1205721 = min(120582min (1198722) 120582min(Γ))2 1205722 = max(120582max =(1198722) 120582max(Γ))2 and 1205723 = 120582min(119876)min(1205721 1205743(119898120590)2)(120583

2+

1) Then we conclude that the system (18) is exponentiallystable

The last result established the exponential stability of(18) for the unperturbed case However in practice there areseveral disturbance sources (eg unmodeled dynamics noisefrom the environment or from the measuring devices etc)which may affect the system performance of both the systemand the identification algorithm Therefore the next analysisshows the extension of the last analysis to those scenarios

4 Closed-Loop Identification AlgorithmPerturbed Case

Till now exponential stability of the equilibrium of (18) hasbeen established Now it will be considered the perturbedsystem (10) in closed loop with the PD controller 119906(119905) andit will be proved that this closed loop system is 119871infin stablein the sense of Theorem 1 Let us consider the system (10) inclosed loop with 119906(119905) the system Σ2 and the output error 120576(119905)defined as before Then it is possible to obtain the followingerror dynamics

120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)

For the system (39) it is important to know if it is still stablein the presence of the perturbation signal ](119905) If it is thecase then the system will be robust even in the presence ofperturbations which is desirable in the identification contextbecause reliable parameter estimates can be obtained even inthe perturbed caseThe next result obtained from [22] allowsconcluding that the perturbed system (39) is 119871infin stable

Theorem 4 Consider the perturbed system (39) and theunperturbed system (18) If the equilibrium w0 of (18) isexponentially stable then one has the following

(1) The perturbed system (39) is small signal 119871infin-stablethat is there exists 120574infin such that w(119905) le 120574infin120573 lt ℎwhere w(119905) is the solution of (39) starting at w0

(2) There exists 119898 ge 1 such that w0 lt ℎ119898 implies thatw(119905) converges to a ballΩ120575 of radius 120575 = 120574infin120573 lt ℎ thatis for all 120576 gt 0 there exists 119879 ge 0 such that w(119905) le120575(1+120576) for all 119905 ge 119879 along the solutions of (39) startingat w0 Also for all 119905 ge 0 w0 lt ℎ

Proof Consider again the Lyapunov function 1198812(w(119905))Assuming that inequalities for 120583 given in Proposition 2 itis possible to obtain 2 le minus120582min(119876)119909

2+ 120573radic1205832 + 1119909

Let us define the constants 1205723 = 120582min(119876) 1205724 = radic1205832 + 1120574infin = 12057241205723radic12057221205721 120575 = 120574infin120573 and 119898 = radic12057221205721 ge 1Then the time derivative of1198812(w(119905)) along the trajectories of(39) yields 2 le minus1205723119909[119909 minus 120575119898] and two cases may beconsidered

(1) First it will be proved that w(119905) le 120574infin120573 lt ℎ Tothis end consider the case where 1199090 le 120575119898 Note


that 120575119898 le 120575 because of 119898 ge 1 which implies thatw(119905) isin Ω120575 for all 119905 ge 0 Suppose that it is not trueand then by continuity of the solutions there exists1198790 and 1198791 with 1198791 gt 1198790 ge 0 such that 119909(1198790) = 120575119898119909(1198791) gt 120575 and for all 119905 isin [1198790 1198791] w(119905) ge 120575119898and then from the time derivative of1198812 it is clear thatin [1198790 1198791] 2 le 0 but in this case 1198812(1198790 119909(1198790)) le1205722(120575119898)

2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which

is a contradiction therefore for all w(119905) with initialcondition w0 a solution for (39) remains onΩ120575

(2) Assume that w0 gt 120575119898 and that for all 120576 gt 0

there exists 119879 ge 0 such that w(119879) le 120575(1 + 120576)119898

and suppose that it is not true then for some 120576 gt 0

and for all 119905 ge 0 w(119905) gt 120575(1 + 120576)119898 and from thetime derivative of 1198812 given previously it is possible toobtain 2 le minus1205723(120575119898)

2120576(1 + 120576) which is a strictly

negative constant However this contradicts with thefact that 1198812(0 1199090) le 1205722x0

2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge

0 for all 119905 ge 0 because the inequality must be strictOn the other hand let us assume that for all 119905 ge 119879

the following inequality holds 119909(119905) le 120575(1 + 120576) andit is possible to probe this affirmation in the sameway that the first step of this proofThus we concludethat w(119905) converges toΩ120575 as desired and the proof iscompleted

5 Analysis of the Results

The previous analysis showed that even in the presence of abounded perturbation signal the parameter estimates remainin a region Ω120575 = 120574infin120573 Now it is important to analyze thebehavior of this region and to see that how the controller gainsinfluence the way this region behaves To this end recall fromTheorem 4 that the state of (39) is bounded by

Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)

Besides it is easy to obtain the eigenvalues 120582119894 of1198722 as 120582119894 =(119887119896119901 +120583119888+ 1plusmnradic(119887119896119901 + 120583119888 minus 1)

2+ 41205832)2 Thus the minimal

(120582min) and maximum (120582max) eigenvalues for1198722 are given by

120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)

Besides for the diagonal matrix 119876 note that 120582min(119876) =

min120583119887119896119901 119888 minus 120583 120582max(119876) = max120583119887119896119901 119888 minus 120583Then notethat the region Ω120575 depends on several parameters amongwhich are the controller gains 119896119901 and 119896119889 In order tosee the effect of the controller gains let us consider theminimal and maximum eigenvalues of 1198722 From (40) it

is possible to note that the region Ω120575 can be reduced byminimizing the value of 120582max(1198722) and maximizing that of120582min(1198722) that is the value of radic(119887119896119901 + 120583119888 minus 1)

2+ 41205832 must

be the minimum possible Then let us consider the function119891119898(120583 119896119901 119896119889) = (119887119896119901 + 120583119888 minus 1)

2+ 41205832 and its first and second

partial derivatives with respect to 120583 119896119901 and 119896119889 as follows

120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)

If we set the first partial derivatives to zero we obtain theequilibrium points 120583lowast 119896lowast

119901 119896lowast

119889 with the following expres-

sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)

Now let us analyze the qualitative behavior of the region Ω120575for different values of 120582min(119876) 120582min(Γ) 120582max(119876) 120582max(Γ) 119896119901119896119889 and 120583 Figure 2 depicts the behavior of region Ω120575 for dif-ferent values of 119896119901 and 119896119889 (the controller gains) and Figure 3shows also theΩ120575 behavior for different values of 120582min(Γ) and120582max(Γ) (thematrix relatedwith the identification algorithm)From this set of graphics and (40) note the following

(i) For a given constant value of 119896119889 and 120583 if 119896119889 is bigthe value of 119896lowast

119901is reduced and the region Ω119889 is also

reduced For a given constant value of 119896119901 if 119896119901 is bigthe values of 119896 lowast

119889and 120583lowast are reduced which turns out

to reduce again the size of the regionΩ120575(ii) The region Ω120575 can be made arbitrarily small if the

values for 120582min(119876) and min(120582min(1198722) 120582min(Γ)) areas large as possible and the value for 120582min(119876) can beincreased if the controller gains 119896119889 and 119896119901 have largevalues which is the case if a high gain PD controlleris used Then the size of Ω120575 may be reduced andtherefore the quality of the parameter estimates maybe improved by means of the controller gains

(iii) By increasing the value of 119896119901 and 119896119889 the value of120582min(1198722) is increased which may reduce the size ofΩ120575 and by increasing the values of the gain matrix Γthe value of 120582min(Γ) is also increased reducing thenthe size ofΩ120575

(iv) As it can be seen from Figure 4 for the controllergains the effect of the gain 119896119901 is more importantfor reducing the size of region Ω120575 than that ofgain 119896119889 because large values of 119896119889 do not influencesignificantly in the reduction of the regionΩ120575 Whatrsquosmore the effect of gain 119896119889 is almost imperceptible ifthe gain 119896119901 is large enough


12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd

Figure 2 Behavior of the region Ω119889when the values of 119896

119901and 119896

119889

are taken as variables

Figure 3 Behavior of the regionΩ119889 when the values of 120582min(Γ) and120582max (Γ) are taken as variables

(v) Finally as expected if the values of the maximum orminimumeigenvalues for the gainmatrix Γdominatethe size ofΩ120575 will also be reduced

From the last analysis it is possible to conclude that forrobustness of the identification algorithm it is important topay attention not only to the setup of the gain matrix Γ and120583 from the identification algorithm but also to the valueof the controller gains 119896119901 and 119896119889 Then the region Ω119889 iseffectively reduced if a high gain 119896119901 is employed and also iflarge eigenvalues are assigned to matrix Γ

Another subject to be considered is the controller struc-ture because it is possible to note that not all the gainshave the same effect of reducing region Ω120575 Thereforewhen the parameter identification algorithm is designedit is important to note that the structure of the controllershould be an important issue that has been underestimatedand that will be important in real applications where noisymeasurements or unmodeled dynamics may be affecting thesystem

6 Conclusion

This paper analyzes a method for on-line closed loop identi-fication of a linear DC servomechanism with a bounded per-turbation signal and a PD controllerTheoretical results showthat without perturbation signals the system is exponentially

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)

Figure 4 Behavior of the regionΩ119889 for different cases (a) variationof 119896119901 (b) variation of 119896119889 (c) variation of 120583

stable and the parameter error converges to zero but whena perturbation signal is present the parameter estimatesconverge to a region Ω120575 and the overall system is 119871infin stablethat is the system is robust in face of perturbation signalsBesides by increasing the controller and adaptation gains theregion Ω119889 can be arbitrarily reduced which means that theaccuracy of the parameter estimates can be increased whichgives some insight into the importance of the controller


structure in the identification process Future work will bedevoted to design more complicated controllers and verifythe controller structure needed to obtain a better estimateof the system parameters that is a controller which reducesΩ120575 as much as possible and ensures the overall exponentialstability

References

[1] H Garnier M Mensler and A Richard ldquoContinuous-timemodel identification from sampled data implementation issuesand performance evaluationrdquo International Journal of Controlvol 76 no 13 pp 1337ndash1357 2003

[2] M Fliess andH Sira-Ramırez ldquoAn algebraic framework for lin-ear identificationrdquo ESAIM Control Optimisation and Calculusof Variations no 9 pp 151ndash168 2003

[3] G P Rao andHUnbehauen ldquoIdentification of continuous-timesystemsrdquo IEE ProceedingsmdashControl Theory and Applicationsvol 153 no 2 pp 185ndash220 2006

[4] M Fliess and H Sira Ramırez ldquoClosed loop parametric iden-tification for continuous time linear systems via new algebraictechniquesrdquo in Continuous Time Model Identification fromSampled Data H Garnier and LWang Eds Springer LondonUK 2007

[5] L Ljung System Identification Prentice Hall 1987[6] U Forssell and L Ljung ldquoClosed-loop identification revisitedrdquo

Automatica vol 35 no 7 pp 1215ndash1241 1999[7] T Iwasaki T Sato A Morita and H Maruyama ldquoAutotuning

of two-degree-of-freedom motor control for high-accuracytrajectory motionrdquo Control Engineering Practice vol 4 no 4pp 537ndash544 1996

[8] E J AdamandEDGuestrin ldquoIdentification and robust controlof an experimental servo motorrdquo ISA Transactions vol 41 no2 pp 225ndash234 2002

[9] Y Zhou A Han S Yan and X Chen ldquoA fast method foronline closed-loop system identificationrdquo International Journalof Advanced Manufacturing Technology vol 31 no 1-2 pp 78ndash84 2006

[10] A Besancon-Voda and G Besancon ldquoAnalysis of a two-relaysystem configuration with application to Coulomb frictionidentificationrdquo Automatica vol 35 no 8 pp 1391ndash1399 1999

[11] K K Tan T H Lee S N Huang and X Jiang ldquoFrictionmodeling and adaptive compensation using a relay feedbackapproachrdquo IEEE Transactions on Industrial Electronics vol 48no 1 pp 169ndash176 2001

[12] S-L Chen K K Tan and S Huang ldquoFriction modelingand compensation of servomechanical systems with dual-relayfeedback approachrdquo IEEE Transactions on Control SystemsTechnology vol 17 no 6 pp 1295ndash1305 2009

[13] R Garrido and R Miranda ldquoAutotuning of a DC servomecha-nismusing closed loop identificationrdquo inProceedings of the IEEE32nd Annual Conference on Industrial Electronics (IECON rsquo06)Paris France October 2006

[14] R Garrido and R Miranda ldquoDC servomechanism parameteridentification a closed loop input error approachrdquo ISA Trans-actions vol 51 no 1 pp 42ndash49 2012

[15] R Garrido and A Concha ldquoAn algebraic recursive methodfor parameter identification of a servo modelrdquo IEEEASMETransactions onMechatronics vol 18 no 5 pp 1572ndash1580 2012

[16] T Kara and I Eker ldquoNonlinear closed-loop direct identificationof a DC motor with load for low speed two-directional opera-tionrdquo Electrical Engineering vol 86 no 2 pp 87ndash96 2004

[17] J Davila L Fridman and A Poznyak ldquoObservation andidentification of mechanical systems via second order slidingmodesrdquo International Journal of Control vol 79 no 10 pp 1251ndash1262 2006

[18] K Kong J Bae and M Tomizuka ldquoA compact rotary serieselastic actuator for human assistive systemsrdquo IEEEASMETrans-actions on Mechatronics vol 17 no 2 pp 288ndash297 2012

[19] G Mamani J Becedas V Feliu-Batlle and H Sira-RamırezldquoOpen- and closed-loop algebraic identification method foradaptive control of DC motorsrdquo International Journal of Adap-tive Control and Signal Processing vol 23 no 12 pp 1097ndash11032009

[20] J Becedas G Mamani and V Feliu ldquoAlgebraic parametersidentification of DC motors methodology and analysisrdquo Inter-national Journal of Systems Science vol 41 no 10 pp 1241ndash12552010

[21] J C Romero C G Rodrıguez A L Juarez and H S RamırezldquoAlgebraic parameter identification for induction motorsrdquo inProceedings of the 37th Annual Conference of the IEEE IndustrialElectronics Society (IECON rsquo11) pp 1734ndash1740 MelbourneAustralia November 2011

[22] S Sastry and M Bodson Adaptive Control Stability Conver-gence and Robustness Prentice Hall EnglewoodCliffs NJ USA1989

[23] C-T Chen Linear SystemTheory andDesign OxfordUniversityPress Oxford UK 4th edition 2012

[24] R Ortega A Loria P J Nicklasson and H Sira-RamırezPassivity-Based Control of Euler-Lagrange Systems SpringerLondon UK 3rd edition 1998

[25] R Sepulchre M Jankovic and P Kokotovic ConstructiveNonlinear Control Springer New York NY USA 1996

[26] C A Desoer and M Vidyasagar Feedback Systems Input-Output Properties Academic Press New York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Differential EquationsInternational Journal of

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International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


through a relay in order to obtain a sustained oscillationand then the values of the frequency and amplitude allowsidentifying a model of the systemThe relay based techniquesmayworkwell however the reference signal used for excitingthe system becomes constant in many time instants andthen no control is being applied in that time and then theeffect of the disturbances may affect the estimates obtainedwith this methodology Besides tuning of the relay controllercan be cumbersome and no one of the reviewed methodstakes explicitly into account the disturbances affecting theservomechanism

References [7ndash9 13ndash15] propose methods for closed loopidentification of position-controlled servos where the loopis closed using a linear controller The approach proposedin [13 14] uses a PD controller to close the loop and anon-line gradient algorithm allows estimating a linear andnonlinear model of a servomechanism Also in [15] a two-step identificationmethod for estimating the four parametersof a nonlinear model of a position control servomechanismis presented where the proposed identification scheme relieson the theory of operational calculus However the effect ofperturbations is considered only for constant values On theother hand a common technique used for identifying DCservomechanisms is based on the least squares algorithm In[16] an approach for identifying a DC motor is presentedconsidering the Coulomb friction and dead zone using arecursive discrete time least squares algorithmwith forgettingfactor using a Hammerstein model for the motor In [17]second-order sliding modes approach is applied for estimat-ing the parameters of aDCmotor operating in open-loop andclosed loop while in [18] the inertia of a second-order modelof a DC servomechanism driving a human assistive system isidentified using Bode plots obtained from input-output mea-surements Finally another interesting approach is presentedin [2 4] where an online algebraic identification method isproposed Besides [19 20] show how this methodology canbe used for parameter identification of a DC motor and alsothis methodology was used and modified in [21] introducinga batch least squares algorithm for estimating the parametersof an induction motor

All the previousworks consider in general the importanceof some gains related with the identification algorithm forclosed-loop identification However in some cases (as thatpresented in [13 14]) the controller gains get involved inwith some important elements for the identification schemefor example in the regressor vector In this case it has notbeen analyzed which effect actually has such gains in theparameter identification procedure and how the parameterestimates will behave when a disturbance is affecting thesystemThen this work presents the analysis of a closed loopidentification algorithm for a perturbed position-controlledDC servomechanism where a PD controller closes theloop and achieves stability without knowledge about theservomechanism parameters Theoretical results show thatwhen the perturbation signal is identically zero exponentialconvergence can be claimed and in the presence of abounded perturbation it is possible to obtain a region Ω120575

where the parameter estimates belong Whatrsquos more it willbe shown that this region can be made arbitrarily small if

the controller gains are tuned properly Some simulationsdepict the behavior of the region Ω120575 showing the effect thatthe controller gains have on the estimated parameters Thepaper is organized as follows Section 2 is devoted to presentpreliminary theoretical results of passivity based controladaptive control and the general description of the DC ser-vomechanism Section 3 presents the analysis for closed loopidentification of the servomechanism when no disturbancesare affecting the system and the exponential stability of theoverall identification scheme is shown Section 4 extendsthe analysis for the perturbed case and Section 5 deals withthe analysis of the results Finally Section 6 gives someconcluding remarks

2 Preliminary Results

The closed loop identification analysis that will be presentedis based on some results related with passivity theory andadaptive control and thus the following results are worthpresenting

21 Passivity and Adaptive Control As pointed out beforethe aim of this work is to show the convergence propertiesof the identification algorithm with respect to the value ofthe controller gains To this end it will be necessary to usethe observability property given in [22] and some results ofpassivity and stability that will be described in the following

Let us consider the linear time-varying system[119862(119905) 119860(119905)] defined by

= 119860 (119905) 119909 (119905)

119910 (119905) = 119862 (119905) 119909 (119905)

(1)

where 119909(119905) isin 119877119899 119910(119905) isin 119877

119898 while 119860(119905) isin 119877119899times119899 and

119862(119905) isin 119877119898times119899 are piecewise continuous functions A system

[119862(119905) 119860(119905)] is said to be uniformly completely observable(UCO) [22] if there exist strictly positive constants 1205731 1205732 120575such that for all 1199050 ge 0 1205731119868 le 119873(1199050 1199050 + 120575) le 1205732119868 where119873(1199050 1199050 + 120575) isin 119877

119899times119899 is the so-called observability Gramian as

119873(1199050 1199050 + 120575) = int

1199050+120575

1199050

Φ119879(120591 1199050) 119862

119879(120591) 119862 (120591)Φ (120591 1199050) 119889120591

(2)

where Φ(120591 1199050) is the state transition matrix associated with119860(119905) Now assume that for all 120575 gt 0 there exists 119896120575 gt 0 suchthat for all 1199050 ge 0 int

1199050+120575

1199050119870(120591)

2119889120591 le 119896120575 where the matrix 119870

corresponds to the system under output injection as

= 119860 (119905) + 119870 (119905) 119862 (119905) 119908 (119905)

119911 (119905) = 119862 (119905) 119908 (119905)

(3)

with119908(119905) isin 119877119899119870(119905) isin 119877119899times119898 and 119911(119905) isin 119877119898Then the system[119862 119860] is UCO if and only if [119862 119860 + 119870119862] is UCO where theterm 119870119862 stands for the output injection term as indicated in[22] Moreover if the observability Gramian of the system[119862 119860] satisfies the previous upper and lower bounds on 119873




1205731015840

1=

1205731

(1 + radic1198961205751205732)

1205731015840

2= 1205732 exp (1198961205751205732) (4)





2le 119881(119905 119909) le 1205722119909

2(119905 119909)| 119909=119891(119905119909) le minus1205723119909

2 and |120597119881(119905 119909)120597119909| le 1205724119909


2le 119881(119905 119909) le 1205722119909

2 (119905 119909)| 119909=119891(119905119909) le

0 and int119905+120575119905

119889119881(120591 119909(120591))119889120591|119909=119891(119905119909)

119889120591 le minus12057231199092 then 119909(119905)




defined as

Π

(119905) = 119891 (119909 119906)

119909 (0) = 1199090 isin 119877119899

119910 (119905) = 119867 (119909 119906)

(5)




1198712 = 119909 isin Ξ 1198912

2= intinfin

0119891(119905)



1198712119890 = 119909 isin Ξ 10038171003817100381710038171198911003817100381710038171003817

2

2119879= int

119879

0

1003817100381710038171003817119891 (119905)1003817100381710038171003817

2119889119905 lt infin forall119879 (6)


(119906 119910) = int

infin

0

119906119879(119905) 119910 (119905) 119889119905

(119906 119910)119879= int

119879

0

119906119879(119905) 119910 (119905) 119889119905

(7)





119906119879119910 minus 120575119894119906


119879119910 minus 120575119900119910






+rarr 119877


1205721119868 le int

1199050+120575

1199050

120601 (120591) 120601119879(120591) 119889120591 le 1205722119868 forall1199050 ge 0 (8)

Besides let 119908 119877+rarr 119877






119869 119902 + 119891 119902 (119905) = 119896119906 (119905) + ]1 (119905) = 120591 (119905) + ]1 (119905) (9)




119902V (119905) = minus119886 119902 (119905) + 119887119906 (119905) + ] (119905) (10)








119902 (119905) = minus119886 119902 (119905) + 119887119906 (119905) (11)


119902119890 (119905) = minus119886 119902119890 (119905) + 119906119890 (119905) (12)




Σ1 119902 = minus (119886 + 119887119896119889) 119902 + 119887119896119901119890 (13)


Σ2 119902119890 = minus (119886 + 119896119889) 119902119890 + 119896119901119890119890 (14)


120576 +119888 120576 + 119887119896119901120576 = 120579119879120601 (15)




1198671 (119909) =1

2119909119879(119887119896119901 120583

120583 1)119909 = 119909

119879119872119909 (16)


1 = 120579119879120601 (120583120576 + 120576) minus

119888

2(120583120576 + 120576)

2minus (

119888

2minus 120583) 120576

2

minus 120583(119887119896119901 minus120583119888

2) 1205762

(17)


119879120601(120583120576+ 120576)minus119888(120583120576 + 120576)

22





120579 =




119879120601)



Σ

120576 (119905) + 119888 120576 (119905) + 119887119896119901120576 (119905) = 120579119879120601 (119905)

120579 = minusΓ120601 (120583120576 + 120576)

(18)



= 119860119909 + 119861119880

119910 = 119862119909

(19)

with

119860 = (0 1

minus119887119896119901 minus119888) 119861 = (

0

1)

119862119879= (

120583

1) 119880 = 120579

119879120601

(20)


119903(119905) = ( 119902 minus119906)


120601119903 minus 120601 = (119902


119902119890

minus119906119890) = (

120576

119896119901120576 + 119896119889 120576) (21)









2) Then 120576 120576 and the




Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)



Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)


1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)


1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)


1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)


2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)


119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)


120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)


= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)


119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)


2


119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)


1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)



119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1205731015840

1=

1205731

(1 + radic1198961205751205732)

1205731015840

2= 1205732 exp (1198961205751205732) (4)





2le 119881(119905 119909) le 1205722119909

2(119905 119909)| 119909=119891(119905119909) le minus1205723119909

2 and |120597119881(119905 119909)120597119909| le 1205724119909


2le 119881(119905 119909) le 1205722119909

2 (119905 119909)| 119909=119891(119905119909) le

0 and int119905+120575119905

119889119881(120591 119909(120591))119889120591|119909=119891(119905119909)

119889120591 le minus12057231199092 then 119909(119905)




defined as

Π

(119905) = 119891 (119909 119906)

119909 (0) = 1199090 isin 119877119899

119910 (119905) = 119867 (119909 119906)

(5)




1198712 = 119909 isin Ξ 1198912

2= intinfin

0119891(119905)



1198712119890 = 119909 isin Ξ 10038171003817100381710038171198911003817100381710038171003817

2

2119879= int

119879

0

1003817100381710038171003817119891 (119905)1003817100381710038171003817

2119889119905 lt infin forall119879 (6)


(119906 119910) = int

infin

0

119906119879(119905) 119910 (119905) 119889119905

(119906 119910)119879= int

119879

0

119906119879(119905) 119910 (119905) 119889119905

(7)





119906119879119910 minus 120575119894119906


119879119910 minus 120575119900119910






+rarr 119877


1205721119868 le int

1199050+120575

1199050

120601 (120591) 120601119879(120591) 119889120591 le 1205722119868 forall1199050 ge 0 (8)

Besides let 119908 119877+rarr 119877






119869 119902 + 119891 119902 (119905) = 119896119906 (119905) + ]1 (119905) = 120591 (119905) + ]1 (119905) (9)




119902V (119905) = minus119886 119902 (119905) + 119887119906 (119905) + ] (119905) (10)








119902 (119905) = minus119886 119902 (119905) + 119887119906 (119905) (11)


119902119890 (119905) = minus119886 119902119890 (119905) + 119906119890 (119905) (12)




Σ1 119902 = minus (119886 + 119887119896119889) 119902 + 119887119896119901119890 (13)


Σ2 119902119890 = minus (119886 + 119896119889) 119902119890 + 119896119901119890119890 (14)


120576 +119888 120576 + 119887119896119901120576 = 120579119879120601 (15)




1198671 (119909) =1

2119909119879(119887119896119901 120583

120583 1)119909 = 119909

119879119872119909 (16)


1 = 120579119879120601 (120583120576 + 120576) minus

119888

2(120583120576 + 120576)

2minus (

119888

2minus 120583) 120576

2

minus 120583(119887119896119901 minus120583119888

2) 1205762

(17)


119879120601(120583120576+ 120576)minus119888(120583120576 + 120576)

22





120579 =




119879120601)



Σ

120576 (119905) + 119888 120576 (119905) + 119887119896119901120576 (119905) = 120579119879120601 (119905)

120579 = minusΓ120601 (120583120576 + 120576)

(18)



= 119860119909 + 119861119880

119910 = 119862119909

(19)

with

119860 = (0 1

minus119887119896119901 minus119888) 119861 = (

0

1)

119862119879= (

120583

1) 119880 = 120579

119879120601

(20)


119903(119905) = ( 119902 minus119906)


120601119903 minus 120601 = (119902


119902119890

minus119906119890) = (

120576

119896119901120576 + 119896119889 120576) (21)









2) Then 120576 120576 and the




Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)



Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)


1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)


1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)


1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)


2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)


119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)


120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)


= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)


119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)


2


119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)


1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)



119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902V (119905) = minus119886 119902 (119905) + 119887119906 (119905) + ] (119905) (10)








119902 (119905) = minus119886 119902 (119905) + 119887119906 (119905) (11)


119902119890 (119905) = minus119886 119902119890 (119905) + 119906119890 (119905) (12)




Σ1 119902 = minus (119886 + 119887119896119889) 119902 + 119887119896119901119890 (13)


Σ2 119902119890 = minus (119886 + 119896119889) 119902119890 + 119896119901119890119890 (14)


120576 +119888 120576 + 119887119896119901120576 = 120579119879120601 (15)




1198671 (119909) =1

2119909119879(119887119896119901 120583

120583 1)119909 = 119909

119879119872119909 (16)


1 = 120579119879120601 (120583120576 + 120576) minus

119888

2(120583120576 + 120576)

2minus (

119888

2minus 120583) 120576

2

minus 120583(119887119896119901 minus120583119888

2) 1205762

(17)


119879120601(120583120576+ 120576)minus119888(120583120576 + 120576)

22





120579 =




119879120601)



Σ

120576 (119905) + 119888 120576 (119905) + 119887119896119901120576 (119905) = 120579119879120601 (119905)

120579 = minusΓ120601 (120583120576 + 120576)

(18)



= 119860119909 + 119861119880

119910 = 119862119909

(19)

with

119860 = (0 1

minus119887119896119901 minus119888) 119861 = (

0

1)

119862119879= (

120583

1) 119880 = 120579

119879120601

(20)


119903(119905) = ( 119902 minus119906)


120601119903 minus 120601 = (119902


119902119890

minus119906119890) = (

120576

119896119901120576 + 119896119889 120576) (21)









2) Then 120576 120576 and the




Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)



Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)


1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)


1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)


1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)


2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)


119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)


120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)


= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)


119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)


2


119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)


1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)



119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120579 =




119879120601)



Σ

120576 (119905) + 119888 120576 (119905) + 119887119896119901120576 (119905) = 120579119879120601 (119905)

120579 = minusΓ120601 (120583120576 + 120576)

(18)



= 119860119909 + 119861119880

119910 = 119862119909

(19)

with

119860 = (0 1

minus119887119896119901 minus119888) 119861 = (

0

1)

119862119879= (

120583

1) 119880 = 120579

119879120601

(20)


119903(119905) = ( 119902 minus119906)


120601119903 minus 120601 = (119902


119902119890

minus119906119890) = (

120576

119896119901120576 + 119896119889 120576) (21)









2) Then 120576 120576 and the




Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)



Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)


1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)


1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)


1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)


2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)


119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)


120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)


= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)


119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)


2


119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)


1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)



119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2) Then 120576 120576 and the




Σ4

120579 = 0

1199102 = 120601119879(119905) 120579 (119905)

Σ5

120579 = minusΓ120601 (120583120576 + 120576)

1199102 = 120601119879(119905) 120579 (119905)

(22)



Proof Let w(119905) = (119909 120579)119879= (120576 120576120579)


1198812 (w (119905)) = 1198672 (120579 (119905)) + 1198673 (119909 (119905)) (23)


1198673 (119909 (119905)) =1

21199091198791198722119909 1198722 = (

119887119896119901 + 120583119888 120583

120583 1) (24)


1

2120582min (1198722) 119909

2+1

2120582min (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

le 1198812 (w)

le1

2120582max (1198722) 119909

2+1

2120582max (Γ)

1003817100381710038171003817100381712057910038171003817100381710038171003817

2

(25)


2 = minus119909119879(120583119887119896119901 0

0 119888 minus 120583)119909 (26)


119879119876119909 le 0 where 119876 = 119876

119879gt 0 if 120583 lt 119888


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

10038161003816100381610038161003816100381610038161003816(18)

119889120591 le minus1205723w (119905)2 (27)


int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 1int

119905+120575

119905

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 (28)


120582min (119876)

1205832 + 1int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge 1205723 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(29)


= 119860119909 + 119861119880120579 = 0 119910 = 119862119909 (30)


119896120575 = int

1199050+120575

1199050

Γ119879(120583120576 + 120576)

2Γ 119889120591 le int

1199050+120575

1199050

1205822

max (Γ) (120583120576 + 120576)2119889120591

(31)


2


119910 (119905) = 119862119879119890119860(119905minus1199050)119909 (1199050) + int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050) (32)


1199111 (119905) = 119862119879119890119860(119905minus1199050)x (1199050)

1199112 (119905) = int

119905

1199050

119862119879119890119860(119905minus120591)

119861120601 (120591) 119889120591120579 (1199050)

(33)



119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119905

1199050119862119879119890119860(119905minus120591)



le

int1199051+120590

11990511199112

2(120591)119889120591 le 1205722120579(1199050)

2


infin

1199050+1198981205901199112

1(120591)119889120591 le 1205741119909(1199050)

2119890minus1205742119898120590


1199050+119898120590

11990501199112

1(120591)119889120591 le 1205743(119898120590)119909(1199050)

2


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591

ge int

1199050+119898120590

1199050

1199112

1(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

1(120591) 119889120591

+ int

1199050+119898120590

1199050

1199112

2(120591) 119889120591 minus int

1199050+120575

1199050+119898120590

1199112

2(120591) 119889120591

ge 1205743 (119898120590)1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2minus 1205741119890minus12057421198981205901003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2

+ 1198991205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

minus 1198981205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

=1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2[1205743 (119898120590) minus 1205741119890

minus1205742119898120590]

+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

[1198991205721 minus 1198981205722]

(34)


minus1205742119898120590 ge 1205743(119898120590)2 gt 0 then

int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 ge

1205743 (119898120590)

2

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+ 1205721

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

(35)


int

1199050+120575

1199050

1003816100381610038161003816119910 (120591)1003816100381610038161003816

2119889120591 le 21205741

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2119890minus1205742119898120590 + 2 (119898 + 119899) 1205722

10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2



1205731 (1003817100381710038171003817119909 (1199050)

1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le int

1199050+120575

1199050

1003817100381710038171003817119910 (120591)1003817100381710038171003817

2119889120591 le 1205732 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

(37)

and therefore

int

119905+120575

119905

119889

1198891205911198812 (w (120591))

100381610038161003816100381610038161003816100381610038161003816(18)

le minus120582min (119876)

1205832 + 11205731 (

1003817100381710038171003817119909 (1199050)1003817100381710038171003817

2+10038171003817100381710038171003817120579 (1199050)

10038171003817100381710038171003817

2

)

le minus12057231003817100381710038171003817w0

1003817100381710038171003817

(38)


2+





120576 (119905) = minus119888 120576 (119905) minus 119887119896119901120576 (119905) + 120579119879120601 (119905) + ] (119905) (39)






2+ 120573radic1205832 + 1119909





2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2= 1205721120575

2 and 1198812(1198791 119909(1198791)) gt 12057211205752 which








2lt 1205722ℎ

21198982 1198812(119905 x(119905)) ge





Ω120575 = 120574infin120573 =radic(1205832+ 1) 120573

2max (120582max (1198722) 120582max (Γ))

(120582min (Q))2min (120582min (1198722) 120582min (Γ))

(40)




120582min (1198722) =119887119896119901 + 120583119888 + 1 minus

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

120582max (1198722) =119887119896119901 + 120583119888 + 1 +

radic(119887119896119901 + 120583119888 minus 1)2

+ 41205832

2

(41)




2+ 41205832 must




120597119891119898

120597120583= 2119888 (119887119896119901 + 120583119888 minus 1) + 8120583

1205972119891119898

1205971205832= 21198882+ 8

120597119891119898

120597119896119901

= 2119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962119901

= 21198872

120597119891119898

120597119896119889

= 2120583119887 (119887119896119901 + 120583119888 minus 1) 1205972119891119898

1205971198962

119889

= 21205832119888

(42)


119901 119896lowast


sions

119896lowast

119901=1 minus 120583119888

119887 119896lowast

119889=

1 minus (119887119896119901 + 120583119886)

120583119887 120583lowast=

119888 (1 minus 119887119896119901)

1198882 + 4

(43)











12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










12345678910

0506

0708

0907

08

09

1

kd

kp

Ωd


119901and 119896

119889






6 Conclusion


0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kp

(a)

0 1 2 3 4 5 6 7 80

05

1

15

2

25

3

Ωd

kd

(b)

0 1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

Ωd

120583

(c)





References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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