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Proceedings of the 35thConference on Decision and ControlKobe,Japan December 1996WP18 550

Identification of Linear Parameter-Varying Systemsvia L F T ~

Lawton N.Lee2 and Kames1iwa.rPoolla3Department of Mechanical Engineering, University of C;tlifornia, Berkeley CA 94720

AbstractThis paper considers the identification of LinearParameter-Varyiing (LPV) systems having linear-fractional parameter dependence. We present a naturalprediction error imethod, using gradient- and IIessian-based nonlinear optimization algorithms to minimizethe cost function,. Computing the gradients and (ail-proximate) Hessians is shown to reduce to siniiilatingLPV systems ancl computing inner products. Issues re-lating to initialization and identifiability are discussed.The algorithms are demonstrated on a numerical ex-ample.

1 IntroductioiiThis paper considers the identification of LinearParameter-Varyiing (LPV) systems. LPV models havereceived a great deal of recent attention (for example,see [ I , 4, 13, 161 and the references therein) in the con-text of developing systematic techniques for designinggain-scheduled controllers.An LPV system is a linear system that is dependent, onone or more time-varying parameters and hence repre-sents a family of LTV systems (one for each parametertrajectory). The parameter are considered measurablein real-time but not known in advance. Such modelshave been used effectively in missile [2, 141, aircraft[15], robotic [3], and process [5] control pr ol hn s.We consider here parametric identification methods forLPV systems; as is common [ I , 131, we deal wi th thecase where the time-varying paramrtrrs enter in linear-fractional fashion. Nemani et. al. [1 ] have been able toreduce simplified forms of the problem ( c g., exart ornoisy state measurement, eqnation error noise, scalartime-varying parameter block) to least squares. Still,investigation of this problem is still in relative infancy.

ric LTI system identification (Box-Jenkins and output-error models, for example [9]), consists of three steps:1, Form a cost function based on prediction error.2. Generate an initial parameter estimate.3 . Minirtrize the cost function via gradient- andHe~sian-based onlinear programming. These al-gorithms are sensitive to initial seeds, so the pre-cediiig step merits consitlerable at tention.

The reniaintler of this paper is organized as follows.Section 2 establishes some notation. In Section 3 weformulate the identificat,ion problem and form the costfunction. In Section 4 we present nonlinear optimiza-tion algorithms for parameter estimation and provideformulas for the gradient and (approximate) Hessian ofthe cost. In Section 5 we discuss identifiability and thegeneration of initial seeds. Section 6 gives a numericalexample. Conclusions are given in Section 7.

2 NotationDenote a discrete-time linear (LTI/LTV/LPV) systemt ( t f 1 ) = A( t ) t ( t )+ B ( t ) u ( t )

y ( t ) = C ( t ) X ( t ) D ( t ) u ( t )by M = [ and let S ( M ) := [ $ E ] denotethe corresponding state-space matrix. For properlypartitioned matrices or systems M = [ 2: 21and A , let F , , (M ,A) denote the familiarLFT intercon-nection F U (M , A )= A ~ Z Z M2tA(1- MllA)- 'MtZ.Given finit8e equences { ~ ( t ) ,( t ) E Rn}f',lr let (U, v ) ~denote the inner product

- L-1( U ,1)L := 1.-2T ( t ) V ( t )Y t =OThe approach considered here, standard in pnramet- Given matrices A E Rmxn nd B E Rp 'q , let A CB E

Rmpxnqdenote the J


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Finally, we should keep in mind that in this paper twodifferent ent ities are referred to as param etersJJ :hetime-varying parameters or scheduling variables thatare measured, and the unknouin structuml pornmetersthat are estimate d. It should be clear from the contextwhich of the two is being mentioned.

3 Problem FormulationConsider the nth-order discrete-time LPV plant

where 6 E R is the time-varying, measured parameterand 8 E RN s the fixed, unknown parameter. Assumethat P = F , ( M ( t ) ,A ) , a feedback interconnection ofthe nth-order LTI system

and the r x r time-varying parameter blockA := diag(&I,,,. ..,d31,.,) (3.3)

where r := r1 + .. + rs . Fig. 1 depicts the system,along with the measured output y E RP,measuredinput U E Rm,and noise e E RP.

Y e

Figure 1: The plantAs is standard [9], we assume the noise model to bestably invertible (hence D12 is nonsingular). Invert-ing the noise model produces the LPV system I; =Fu ( k ( z ) , ) , shown in Fig. 2 . Here P is a (weighted)one-stepahead prediction error and A?(%) is given by

The following result shows the equivalaiiceof t,he modelsets describing by P and via M ( r ) and M ( r ) .Lemma 3.5 The state-space data S ( M ) a nd S(A?)are related by the bijection 7,defirrerl as

0 0 0 0

Figure 2: The predictor and prediction error

The assochted model sets are therefore equivalent.

The identification problem is now the fol!owing: givenmeasurements {y( t ) , ~ ( i ) ,(t)}fz:,ind 0 to minimizethe meati-square prediction error as represented by thecost function14 0 ) = 5 MO), (e ) ) , (3 .6)

where e ( 0 ) i s as shown in Fig. 2. In general, suchpredictioh error methods may produce biased esti-mates [9], As is well known, this can be remedied bymaximum-likelihood estimation [6]. In our case, thesetwo estimates coincide and are asymptotically unbiasedif and only if D12(0 ) is known (i.e. , independent of e).

4 Gradient-Based Parameter EstimationIn this section we present an iterative gradient-basedscheme for minimizing the cost function J ( 0 ) . Thesetechniqiiea draw from existing identification methodsfor LT I sys kins (represented by the special case r = 0).The gradient .(I(@)E RN and [the positive semi-definiteapproximation of ] the Hessian H ( 0 ) E RNxN f J ( 0 )ar e given Ily

!Ik(O) =H i . i ( o ) =

Denoting the .Jacobiany(0 ) and 11(0 ) as of e ( 0 ) by E(O),we can write

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Given g(0) and , H ( B ) , we can estimate 0 using thesteepest-descent adgorithme(k + 1 ) = 6 ( k ) - p ( k ) cl(&)) (4.5)

(4.6)or the Gauss-Newton algorithm

i(k + 1) = e ( k ) - ( k )H - ' ( e ( k ) ) g ( e ( k ) )The damping factor p ( k ) can be held constant or cho-sen by a line search along the step direction.4.1 Computing Gradients & HessiansFor notational ewe, we will hereafter suppress the de-pendence on 0 where it can be readily inferred. Thefollowing result gives formulas for the sensitivity func-tions $ k = 1, I . .,N), from which the gradient8andapproximate Hessian can be obtained directly via theinner-product formulas (4.1)-(4.2) or (4 .3)-( .4).Lemma 4.7 Define the LPV systems

Q i .--

t.1elQ2 := 3u([vi. 1 D ,o 0 0 I p , A )

Given {u ( t ) ,y ( t ) ,s(t )} ;Zj l arid 0, ol)t-ain sequences{ z ( t )E R " , w ( t ) E R ' , e ( t ) E RP>fi; via the LPVsimulation

Each sensitivity function & of the error sigual e canthen be expressed as(4.9)

where v := [zT T U yTIT.The Jacobian E can be obtained either by executing(4.9) for each k := 1 , . . . N or v ia the more efficientbatch simulation

X ( t + l ) =: AX@)+&W( f )+ F,y(t) (4.10)Z ( t ) =: 6 o X ( t )+ i o " W ( / ) + F z ( t ) (4.11)E ( t ) =: C ' , X ( t ) + ~ i o I V ( l ) FE(^) (4.12)W( t ) =: A( t )Z( t ) (4.13)

whereFX 4F E t )

[ & ( t ) ] = [ W V ( t ) . . . * V ( l ) ](4.14)

Crilculoting the g n i d i m t and (approximate) Hessian ofJ ( B ) thcrefor-e reduces i o N + 1 L P V simulations andN(N +1)/2+ N inner products, where N is the numberof unknown pammrters.Let us mention two examples of special interest.

Example 4.15 Choose the vector 4 of estimated pa-rameters to be the elements of the state-space matrixS ( M ) ordered by columns (left to right). With thisparameterization, (4.14) assumes the form

Example 4.16 choose the vector 0 of estimated pa-rameters to be the dements of the state-space matrixS(n;r) ordered t)y rows (top taobottom). Then (4.14)asslimes the form0 0

F E (1) 0 0 Ipwhere v ;= [ wT U y']'.4.2 Gradient Computation Using AdjointsWe begin with a well-known result on adjoints of LTI,LTV, or LPV systems.Lemma 4.17 Consider the discrete-time linear sys-tem G governed by

z( t + 1) = A ( t ) z ( t )+ B(t)u(t)y(t) = C ( t ) z ( t ) D ( t ) u ( t )

for t = 0,. , , L - and z(0)= 0. The adjoint G' of Gsatisfiesfor all sequcnc*rs { ? t ( f ) , r( t) }f si and is governed by

{%, = (GI%,

{ ( t - 1) = A T ( t ) { ( t )+CT( t ) z ( t )U , ( / ) = B'( t ) ( ( t ) + DT( t ) z ( t )

for t = 1,- 1 , . . , O and ((IJ - 1) = 0.Snppose we only need to compute the gradient g ofJ (e.g., for the steepest,-descent estitnation algorithm(4 .6)) . Substituting (4.9) into (4 .1) and using the ad-joint property, we can express each element of 9 as

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where the adjoint operator QZ is as in Lemma 4.17.This clearly reduces the computation of 9 to only tw oLPV simulations (one forward in time, one backward)and N inner products. The following example illus-trates this point.Example 4.19 Suppose the parameter vector consistsof the elements of S ( M ) . Given 0 E RN and measure-ments { ~ ( t ) ,(t) , 6 ( t ) } f l / , obtain { z ( t ) ,w( t ) , ( t ) } t : /via (4.8). Also compute the signals [ = & ;e andv = [zTwT uT y TIT . Then the gradient vector is de-termined by

whereE= [((O)...t(L-l)]and V = v(O).+.v(L-l)].4.3 Recursive AlgorithmsThe Gauss-Newton algorithm (4.6) has a natural re -cursive form 191 for on-line identification:

H ( t + 1) = H ( t )+ E T ( t ) E ( f )R ( t + 1) = H - y t + 1 )b( t + 1) = b ( t )- ( t ) ~ ( tl ) E T ( t ) e ( t )

Applying the matrix inversion lemma, we can imple-ment the algorithm more efficiently asG(t+ 1) = R ( t ) E T ( t ) [ l + (t)R(t)ET(t)]-lR (t+ 1) = R ( t )-G (t+ l ) E ( t ) R ( t )e ( t + 1) = e @ )- ( t ) G ( t+ l)e(t)

At each time t these calculations are preceded byV( t )= A(t)[I- &o(t)A(t)]- 'w ( t ) = V(t)[Co(t)t(t)+ bo1 ( i )u ( t )+ f ioz(f)g( t)Ie ( t ) = Cl ( t ) t ( t ) +~ l o ( t )w ( t ) +D11t )7d( t )+D12( t ) ,Y( t )W(t)= V ( t ) [ C O ( t ) X ( t ) F z ( t ) lE( t )= G ( t ) X ( t )+B)lo(t)W(t) F E @ )z(t+1) = A(t z ( t )+Bo ( t )w ( i ) i 1 ( t ) 7 l ( )+62( t ) g ( )X(t+l) = A ( t ) X ( t )+ B o ( t ) W ( t )+ F x ( t )

where A( t ) = A ( g ( t ) ) ,etc. This algorithm can I xinitialized at z(0) = 0, X ( 0 ) = 0, d ( 0 ) = 8 0 , andR(0)= c l , where c is a large scalar.

5 DiscussionWe now address various supporting issues. I n particu-lar , we discuss an approach to gmerating init,ial secds,methods fo r dealing with equivalent state-space rcaliza-tions, and normalization procedures fo r avoiding spu-rious solutions.

5.1 Initial EstimatesHecause of the existence of local minima in the costfunction J ( O ) , the success of these iterative predictionerror methods depends OIL providing reasonable initialseeds.One (naive) approach would be to generate anLTI estimate ( A ,B1 ,C1, D11) using subspace-basedmethods, choose the block dimensions T I , . . ,n priori, mid initialize the LPV algorithm using(Ba, 0 2 , D12) = (O,O, ) and random matrices for(Bo,CO,Do01 Dol, &o ) .Although such random seeds may be sufficient in a fewcmes, more systematic methods are preferred. Moreintelligently, WP cnn use the LTI estimate to generatean initial LPV model of the form (3 .1 ) that dependsafinely on 6. Such a model can be obtained via LTIsubspace-bwecl methods and/or modified versions ofthe gradient-based algori thms [8]. For example, theaffine LPV model

z(t + 1) = Az(t )+ [BoB, . . B,]G(t)y ( t ) = Cc ( l )+ [DOD1 .. D , ] G ( t )whereq t ) := [I dT(t)]T @ U ( t )can be written (and estimated) as an LTI system.We can then apply the following result to express theaffinely dependent, atate-space data as an LFT (withDo0 = 0):d(6) &(a ) &(d) -C(6) DI(6) DZ(6) 1 -

Proposition 5.1 Consider the 11 x n matrix function

Assume that rank(Ilfk) = Tk for IC = I , . . .,s. Using asingular value tlecomposition, write Mk = UkVk, hereU, is ti x vk arid vk is r k x t i, and define U = [U, . . U s ]and V = [VT . . . Then

where A is as definad in (3 .3 )

Notje that th e siiigtilar values obtained by applyingProposition 5.1 to t,hP estimated state-space dat a givesus N p o s t c i w r r information that we can use t o selectblock dirnc~risiorisor A

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5.2 Identifiability & NormalizationThe LPV plant E is obviously input-output equivalentto any system of the form

Fu([ ,A)where T, and Tl , are invertible matrices and TAA =ATA. For this model structure to be i n any way iden-tifiable, we must restrict our identified models to somecanonical form. Choosing a particular form is still anopen problem, though. The controllable and observ-able canonical forms and their corresponding "overlap-ping" extensions [g, IO, 121 to multivariable systemstend to be ill-conditioned, and indeed no continuouscanonical form exists for MIMO systems [7]. Nnmeri-cally attrac tive alternatives may, for example, involveputting A into modal, triadiagonal, or real-St*hur formand extracting maximally controlla1)le subspares.Note that the problem of realizing an LPV system withlinear-fractional parameter dependence is analogous tothat of realizing a multidimensional (M D ) systeni withs + 1 transform variables. Given a df.sirctl canonicalform, this suggests a heuristic method for obtainingsimilar forms for LPV models. For exarnple, considerthe LPV realization

where A = 61, consists of a single block. A ssume that ( A ,B1) is controllable, ( A ,Cl) is observalde,( D o o ,CO o l ] ) s, controllable, and (Doo,B,TDTolT)isobservable. Genlerate the transformation matrices T,and TA as follows:

can take one of two approaches to incorporating theseconstraints into tfhcestimation algorithm:1. Fix redutidant state-space entries at zero, one,etc. ifs nreded (thereby reducing the dimensionof the pararnetc,r vector e ) , and optimize over theremaining parameters.2. Retain some or all of the redundant entries inthe parameter vector and optimize over the wholevariable space while transforming and normaliz-ing the system after each iteration.

While the first, approach has obvious numerical advan-tages, it, dcm not nerrssarily have better convergenceproperties. I n fact,, for some problems the second ap-proach niay produce superior estimates.

6 Example

The iterative algorithms were applied to a SISO numer-ical example. This LPV system has dimensions n = 2and m = p = r = 1. We have chosen A = 6 E R andM ( r ) o be the 2nd-order LTI systemr 0.000 1.000 I 0.000 1.073 o 1-0.100 0.700 0.816 1.075 0I 0.524 -0.625 I -0.100 0.500 0 I( z ) = 1 0.443 0.060 I 0.000 0.500 1 ]A Axed, outpilt-error noise model is assumed (i.e.,l?z = Dog = 0,Dlz = 1). The system was excitedfor L = 400 iime steps by a unit-magnitude pseudo-random binary input U, a sinusoidal parameter trajec-tory d(t) = sin(27rt/100), and Gaussian white noise ewith variance 0.03. An initial LPV model with affine

1.

2.

3.

dependenre was obtained using an LTI subspace-basedLet T, E R"'" be the niat,rix transforming algoritlim.( A ,B1,Cl) into the canonical form ( A I ,&, cl).The N = 16 unknown parameters were estimated usingthe st,eepest-descent and Gauss-Newton optimizationfiles written for MATLAB v4.2 running on a DEC 3000

Let TA E RrX' be the matrix transformingical form (1300, [CO Do l ] ,BOT D;"] ).(DOO,[COT;-'Dol], [BrTTDT ') "he techniques, The example was implemented in MEX-O-r TIf necessary, scale T, and/or T~ to rlormalize Hand/or C. forms).

Alpha witchOSF/1 V2.0. The adjoint equations wereused to calculate gradients for the steepest-descent al-gorithm, while the Jacobian equations were used to cal-culate the Gauss-Newton step direction. Step sizes forboth cwes were chosen by simple biscction algor ithms.The A matrix was transformed to controllable canoni-cal form after each iteration. A total of fifty itera tionsof the steepest-descent method were completed in 260seconds; thc Gauss-Newton method converged muchmore quickly, needing only nine iterations and 65 sec-onds, and produced a superior estimate. Fig. 3 plotsthe cost function VS. iteration number for each method,and Fig. 4 shows the results of a validation experimentusing the f i n d estimates.

For various reasons, we should also normalize the ma-trix 012(8) of the plant P. For exarnplr, J ( 8 ) can bemade arbi trari ly small simply by making DlZ(8) arbi-trarily large; J( t?) is also invariant under orthogonaltransformations on e . One form of this normalizationis to set D12 = Zp . A less stringent alternative is toconstrain it to be triangular with oncs 011 t he diagonalRemark 5.2 Given that we need certain constraintson the LPV mo dd structure to achieve nniqiieness, one

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Figure 3: J ( 0 ) vs. iteration number for steepest-descentand Gauss-Newton estimation algorithms

2.5r I

Figure 4: Validation output (. . ) and prediction errors fo rsteepest-descent (- -) and Gauss-Newton (-)

7 CouclusioiisIn this paper we examine the problem of parameterestimation for linear parameter-varying systems hav-ing linear-fractional parameter dependence. We ex-amine iterative prediction error methods and presentgradient- and Hessian-based parametric optimizationalgorithms, for which gradient and (approximate) Hes-sian computation reduces to simulating LPV systemsand taking inner products. The algorithms are demon-strated on a simple example. We also discuss theissues of initialization, identifiability, and normaliza-tion; other issues meriting investigation inrlude exper-iment design, model s tructu re determinat,ion, conver-gence properties, and optimality.

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