[11] robust identification and control with time-varying parameter perturbations_2004

7/29/2019 [11] Robust Identification and Control With Time-Varying Parameter Perturbations_2004

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Robust identification and control with time-varyingparameter perturbationsTarek A hmed-Ali", Fabienne Floretb', Franqoise Lamnabhi-Lagarrigueb.

a ENSIETA,2 RueFranqois Verny, 29 806 Brest Cedex 9, FRANCEL2S, SUPELEC - C.N.R.S.- UniversitC Paris-Sud,3 Rue Joliot Curie, 91190 Gif-sur-Y vette, FRANCE.

AbstractThis paper is dedicated to the identification of param-eters for a reasonably large class of nonlinear systems.We propose to design adirect parameter identificationmethod in open loop and an indirect one in closed loopusing the Variable Structure Theory. The exponentialconvergence of parameter identifiers in both cases isstudied. Finally, we highlight some robustness proper-ties of the method with respect to bounded parameterperturbations.Key-words: Parameter Identification, Variable Struc-ture Control, Robustness Analysis, Time-varying per-turbations.

1 IntroductionThis contribution is dedicated to on-line parameteridentification for a reasonably large class of uncer-tain nonlinear systems. Here, the parameter identi-fication is studied in the continuous-time domain be-cause parameters andequations of the model arephys-ically meaningful in this domain. As a matter of fact,the classical recursive identification (Recursive LeastSquares Algorithms, Maximum Likelihood, ...) is nottreated here. In the continuous-time domain, the no-tion of parameter identification is widespread in theliterature. Nevertheless, there are two major trends inthis field. The first one concerns the issue of parameteridentification as a part of a state-observer. Here, thechosen observer restores the successive derivatives ofthe state and has to be robust with respect to noisymeasurements and nonlinearities. One can cite thework of Niethammer, Menold and Allgower [7] usinga high-gain observer combined with the Least-SquaresMethod or the work of Floret-Pontet and Lamnabhi-Lagarrigue [2], [3] dedicated to parameter identifica-tion as a part of a Variable Structure Observer. Thesecond trend of parameter identification is much morewidespread. It concerns the parameter identificationas apart of a controller. Implicitly, it seems thereforeinteresting to focus on parameter identification in open-loop and closed-loop operations. The main advantage

of algorithms in closed loop lies in the fact that theidentification in the presence of unstable open-loop sys-tems is feasible. In this field, one can cite the work ofXu and Hashimoto [lo]based on the Variable StructureTheory (VST) and extended to MIMO systems in [ll]and to time-varying parameters in [12]. Furthermore,one can cite the work of Landau, Anderson and DeBruyne [6] devoted to open-loop and closed-loop iden-tification, based on the notion of passivity. Algorithmsintroduced in [6] are designed for systems with nonlin-ear parameterisation and present some very interestingrobustness properties with respect to additive measure-ment noise and to significant parameter perturbations.In this paper, the authors study the SISO case (singleinput, single output). In our contribution, we wouldlike to extend the work of [6] to the SIMO case (sin-gle input, multiple outputs). Then, using some usualand simple assumptions, we propose, in this paper, aparameter identification algorithm based on the VST,keeping and improving the parameter robustness prop-erties.In section 2, we introduce a direct identification al-gorithm based on the VST. Thanks to the invarianceproperties, inherent into the VST, one can prove theexponential convergence of the parameter identifier.Furthermore, in section 3, we propose to combine theparameter identifier with a VST controller. In fact,we need an indirect identification method in order toachieve the design of the closed loop. As a matter offact, we assume that the system could be stabilizablewhen the vector of parameters 6 is known. In otherwords, this means that there exits an input U =~ ( z ,)achieving the tracking goal (z is the vector of states).From ageneral point of view, the parameter values arenot well-known. Then, we introduce an indirec: pa-rameter identification computing the estimates 6 andone can prove, thanks to Lyapunov's _arguments, theconvergence of the controller U =~ ( z ,) . Finally, sec-tion 4 is dedicated to the robustness analysis. Themain contributionof this paper consists in the study ofthe robustness properties with respect to time-varyingparameter perturbations using some notions on identi-fiability in combination with properties of the VariableStructure Theory.

Author to whom all correspondence should be addressed

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2 Parameter identification in open loopIn this section, we propose a parameter identifyingmethod based on the Variable Structure Theory (VST).2.1 Description of studied systemsLet us consider the SIMO class of nonlinear systems,linearly parameterized, described as follows

where x E R" is the vector of states, U E R is themeasurable scalar input and 8 E RP is the vector ofunknown constant parameters with 1< p 5 n. Weassume that 9 E Re which is acompact set of Rp. Weconsider the complete measure of the vector of statesx. We suppose that the system (S)described by (1)is identifiable in the sense of the work of Lecourtier,Lamnabhi-Lagarrigue and Walter [5]. I t means (seethe book of Walter [13]) that there exists a matrix7 E RnXnwhich is symmetric and non negative defi-nite. This matrix 7 s chosen such that the inverse ofF(x )TF(x ) * exists2.2 Direct identification methodLet us introduce the notation P depending on e, theestimated value of the parameter vector 6. Based ontheVST, the adaptive observerP satisfies the followingequation

d=fo(x)+F *(2)8+go(x)u+v (2)where v is the input of the identifier and is definedby v = Kwsign x- 2) . By defining e = P - x andeo =8- 6 and by taking into account (1)and (2), onecan obtain

e = FT(x)ee+UThe equivalent control weq can be obtained frome=0,see Slotine and Li [9]

U = ueq=-FT(x)es (3)

Kw > I ~T( ~)eel maz (4)if the gain K , is chosen such

if IFT(x)eol is bounded (meaning that F T ( x ) isbounded and 8E 0 0 ) .Now: we introduce the direct identifier

e = Be =.AoF(x)veq with e(0) E Cl0 (5)where Xo is a strictly positive constant. Due to the in-variance properties (see Eq.(3)), the indirect identifierbecomes4 = 80= -XoF(x)FT(x)eo (6)

We are now able to state the main result of this part.Theorem Let us consider a nonlinear system ( p>1)described by ( I ) and let us assume that an ideal slid-ing mode could be generated (condition (4) satisfied).Then, the estimated parameters 8given by (5 ) convergeexponentially to their true values.Proof of the exponential convergence To provethe convergenceof the identifying algorithm, let us con-sider the non-negative quadratic Lyapunov function

'See section 4.2 for more detail s on this condition.

1909

By taking its time derivative and replacing de by itsexpression ( 6 ) , one can finally obtainV(es) = eFeg =-Xoe;fF(x)F(z)Tee

Then, V ( g ) is negative semi-definite. If F ( z ) E 0 issatisfied, then the system is not globally identifiablein the sense of the work of Lecourtier, Lamnabhi andWalter [5]. Indeed, if F ( x ) E 0, the state xn is inde-pendent of 8. A s a consequence, any parameter vector8 satisfies this equation which is in contradiction withthe definition of the global identifiability.Then, ee is bounded and is always decreasing. How-ever, the convergence of the estimated parameters totheir nominal values depends on the excitation of theinput signal. Indeed, eg converges exponentially to zeroif the well-known condition of persistence of excitationis verified, in other words if there exist some positiveconstants T and p suchV t 2 0

where I p xp epresents the identity matrix of dimensionP x P.

3 Parameter identification in a closed loopIn this section, we propose an indirect parameter iden-tification method, required in the study of a closedloop. The control problem is to achieve the vec-tor of states x to track a specific time varying vec-tor of states xd(t). Let us note the tracking errorez(t)= z(t)- xd(t). The vector x(t) is still the so-lution of the differential equation (1).Remark We highlight the fact that we do not impose aspecific method to realize the achievement of the track-ing. Nevertheless, we could choose the Variable Struc-ture Control in order to be in adequacy wi th the methodused in section 2.3.1 8 is considered to be knownFirst of all, letus assume that the vector of parameters0 is known and satisfies the following natural hypothe-sis

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4 Robustness analysis4.1 'T ime-varying parameter perturbationsFromapractical point of view, physical parameters inreal plants are not constant. Their values depend onthe conditions of the environment. Therefore, the ro-bustness behaviour of the proposed method with re-spect to bounded time-varying parameter perturba-tions'has to be analyzed. Let us now consider the sys-tem

Because, each expression in the right-hand side of theprevious equation is known, one can conclude thatsign(e0) is known.Finally, f or the implementations, the estimated param-eter is given by8 = x O F ( X ) F ~ ( ~ ) ( F ( x ) T F ( ~ )) F(z)Tveq

\ w /ee_

+ Asign [(F(z)TF(x)T)-l(z)Tveq]= O ( 2 ) +FT(4en,, +A@(t>)+go(x)u

The new value to be identified is 6 =enom+Ao(t)whereOn,, corresponds to the constant nominal valueof the parameter vector (sections2 and 3) and A@ sabounded time-varying parameter uncertainty such thatits time derivative verifies

With these notations, the parameter error eo is nowdefined byeo = 8- e

or eo = e -e,,, - A@(t) (13)4.2 RobustnessIn order to confirm the robustness of the method pre-sented in section 2, wesuggest to introduce the follow-ing.parameter adaptation law

8 = -XoF(x)FT(z)es - Asign(e0) (14)Remark From a practical point of view, it is not pos-sible to implement es and sign(ee) because the vectorof parameters 8 i s assumed to be unknown. Neverthe-less, eo and sign(e0) could be determined regarding theinvariance properties. Indeed, let us first consider theequation (3) again,

Let us consider a weighting matrix 7 symmetric andnon negative definite. If the matrix F ( x) T F (x) T 2 sinversible, we obtaineo = - F T F T ) - ~Tw,,

F inally, sign(e0) is given bysign(e0) = -sign [ (F TF T)- ' FTweq] (15)

'The matrix to inverse F ( z ) T F ( z ) ~s symmetric andF ( z ) ' T F ( z ) ~ RpXp. One can highlight that the dimensionof F(z)TF(z)* does not depend on the number of the data. Ex-cept if the system is not identifiable or if the. inputs are poor,this matrix would be inversible. This condition is also requiredin the well-known Least-Squares Method.

=-sign(ee)In order to prove the robustness properties, letusagainconsider the non-negative quadratic Lyapunov functionv(es> 1 ,V(es)= -eg esBy taking its time derivative and by replacingeo by itsexpression form (13), one can easily obtain

2

V ( e s ) = ee (6 - A'@(t))Now, let us consider the equation (14). The previousequation can be rewritten asV = e; ( XoF(x)FT(z)es -Asign(e0)- A@(t))We already know that the first term in the right-handside of the previous equation is negative definite (seesection 2 with the persistence of excitation).To conclude, we. only need to prove thateT(-Asign(es) - A@(t))s also negative definite.In fact, from (12)

i = we;f(-Asign(es) - A'@(t))= -A lesi I - eFA'@(t)i =l

i=l i =lThen, by considering the previous inequality, weobtain

5 -XoeTF (x>F (xc>Tes- A leg, +po lee,a=p i=pi=l i = 1

Therefore, if A is chosen such thatA 2 CL0

then, V(8) s negative definite. Then, eo is boundedand is always decreasing. eo converges exponentiallyto zero if the well-known condition of persistence ofexcitation is verified, in other words if there exist somepositive constants T and p such M _> 0

t+T1 F ( 4 F ( 4 T 2 PIPPXP (16)where I pxp epresents the identity matrix of dimensionp x p. Finally, one can attest the robustness of thesuggested identification algorithm.

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5 Numerical exampleIn order to illustrate the theoretical aspects and toshow the advantages of our algorithm, we propose toapply the previous direct identifier to a nonlinear sys-tem that was already studied in the literature.5.1 System to identifyConsider the open-loop unstable second-order nonlin-ear system with two unknown parameters 81 and 62 tobe identified

r = lms. Finally, the identification gains are chosensuch that K,,, =8 and K,,, =3 from (19) and XO =3from (20). The indirect identification consists in keep-ing the equations (19) and (20) in the design of theclosed loop. Here, we suggest a closed loop designedthanks to the Variable Structure Control (see remark3). Then, by a classical way (we choose the VariableStructure Control), one can obtain the following inputU

i 1=x2+elx: +e2x2k2= +e2x2 whereG is the unique solutionof d- =0with for instancer n =yl(x1- d l ) +(x2- d,). Therefore, i i satisfies17)where the value of the unknown parameters are as- U = -y1(x2 +el%: +62x2 - d l ) - 2x2+x&2)sumed to be chosen as The gain of the VST controller is K, =4 and y1 =

5. The desired trajectories, xd l and xd,, are chosenconstant.el = -0.5 e2=-0.25 (18)

The matrices f a, F and go are defined by

In this case, we choose the matrix 7 such that 7=1because & ~( F F ~)~ 9 ( ~ ~ ~ ~ ) ~is different from zero if X I and x2 are also different fromzero.5.2 Simulations without parameter perturba-tions in closed loopWe assume that x = ( X I , x2)* is measurable. Consid-ering the equation (2), one can obtain for the example(17), the following systemconcerning the adaptive ob-server

d1 = x2+61,; +&zt. ,+K,,sign (x1- 21>22 = U +62x2 +K,,,sign 2 2 - 2) (19)where the estimated values 61 and 62 are designedthanks to Eq.(5) by

81 =XOx:211,,

6 2 =Xo[X2(211,, +212J (20)2r l =&,sign (z1- 51)1:2 =K,,,sign ( 2 2 - 22)From a theoretical point of view, simulations have tobe realized with the discontinuous sign function. Froma practical point of view, this discontinuity deterio-rates the sharpness of the identifier because of thewell-known chattering behaviour. As a matter of fact,the sign(X) function is advantageously replaced by asaturation function &, (Slotine and Li [9]) where6 = 0.01 in our simulation. Moreover, vl,, and vz, ,are used through a first-order filter (fi ltering the high-frequency components of 211 and 212 respectively) with

5.3 Simulations with time-varying parameterperturbations in closed loopHere, weagain study the previous example, eq. (17), inthe context of aclosed-loop operation in order to pointout the robustness behaviour of the algorithm with re-spect to time-varying parameter uncertainties. The in-direct identification consists in keeping the equations(19) concerning the adaptive observer and equations(21) and (22) in the design of the input U for the closedloop.We suppose that the parameter vector 8 undergo somebounded perturbations such A01(t)= 0.2 sin(2t) forthe first parameter and A&(t) =0.2sin(lt) for thesecond parameter. As previously and in order to ro-bustify the parameter identifier for 0, we suggest thenew parameter identification described by (14).However, sign B - =sign(e8) s unknown. Never-theless, from the remark 5, i t is possible to determinesign(e0). Indeed, due to invariance properties of theVST adaptive observer, we have obtained Eq.(15). Byconsidering Eq.(15) with 7=1,wecan deduce that

(* )

Then, the parameter identifier for the parameter isgiven by

where each term is known. The conditions of simula-tions are the sameas the ones proposed in the previoussection (with indirect identifier). A is chosen equal to

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Figure 1: Closed loop - State restoring errors and stabi-lizing errors1. Figure (1) shows simulations in the closed loop forthe tracking operations. One can note that the staterestoring errorsel =21- 1 (21and22 are representedby 21, and 22, in Figure (1)) and e2 =22 - 2, andthe tracking errors e,, =21 - dl and esa=z2 - d 2go towards zero in aquite short finite time. Figure (2)shows simulations in the closed loop for the parame-ter identification operations designed with (23) underparameter perturbations. Figure (2) validates the ro-

i21

-' \ *3-40 1 2 3 1 s 8 7 8 9 10

Figure 2: Closed loop - Parameter identifier under time-varying parameter perturbationsbustness of the method with respect to significant timevarying perturbations.

6 ConclusionIn this paper, a continuous-time parameter identifica-tion is introduced for open-loop and closed-loop op-erations of a general class of nonlinear systems. Theadvantages of the closed-loop operations are better per-formances. I t is also worth noting that due to the use

of the Variable Structure Theory, the method presentssome inherent robustness properties with respect totime-varying perturbations.

References[l] T . Basar, G. Didinsky and Z. Pan, A new classof identifiers for robust parameter identification andcontrol in uncertain systems, Robust Control via Vari-able Structure and Lyapunov techniques Eds F. Garo-folo and L. Glielmo 149-173, 1996 .[2] F . Floret and F. Lamnabhi-Lagarrigue, Param-eter identification using sliding regimes, InternationalJournal of Control, Vol. 74, N 18, 1743-1753, 2001.[3] F. Floret, F. Lamnabhi-Lagarrigue and H.Nkwawo, Parameter identification for nonlinear uncer-tain systems partially measurable, Proceedings of the5th IFAC Symposium NOLCOS'OI Saint-Petersburg,Russia, 2001.[4] K . Khalil, Nonlinear systems Macmillan Publish-ing Company, New-York, 1992.[5] Y . Lecourtier, F . Lamnabhi-Lagarrigue, E. Wal-ter, Volterra and generating power series approachesto identifiability testing, Eds. by E. Walter, PergamonPress, 50-66,1987.[6] I.D. Landau, B.D.O. Anderson and F. DeBruyne, Algorithms for identification of continuous-time nonlinear systems: a passivity approach, Non-linear Control in the Year 2000 Eds A . Isidori, F.Lamnabhi-Lagarrigue and W. Respondek Springer-Verlag Vol. 2 13-44, 2000.[7] M. Niethammer, P.H. Menold and F. Allgo-wer, Parameter and derivative estimation for nonlin-ear continuous-time system identification, Proceedingsof the 5th I FAC Symposium Nonlinear Control Systems(NOLCOS'O1) Saint-Petersburg, Russia, 2001.[8] H. Sira-Ramirez, Differential geometric methodin variable structure control, International J ournal ofControl Vol48 1359-1390,1988.[9] J .J .E Slotine andW. Li, Applied Nonlinear Con-trol Prentice-Hall International Editions, EnglewoodCliffs, 1991.[lo] J .X Xu and H. Hashimoto, Parameter identifica-tion methodologies based on variable structure control,International J ournal of Control Vol 57 1207-1220,1993.[ll] J .X Xu and H. Hashimoto, VSS theory-based pa-rameter identification scheme for MIMO systems, Au-tomatica Vol 32, N 2 279-284, 1996.[12] J .X Xu, Y .J Pan and T.H Lee, VSS identificationscheme for time-varying parameters, Proceedings of the15th World Congress of IFAC Code 1160, 2002.[13] E . Walter and L. Pronzato, Identification demodhles parametriques, Masson, 1994.

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[11] robust identification and control with time-varying parameter perturbations_2004

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