Copyright IFAC System Identification Santa Barbara, California, USA, 2000 ON CONSISTENCY AND MODEL VALIDATION FOR SYSTEMS WITH PARAMETER UNCERTAINTY S. Gugercin*,l, A.C. Antoulas *,1 * Department of Electrical and Computer Engineering Rice University Houston, Texas 77251-1892, USA Abstract: In this note we discuss the issue of consistency and model validation for the identification framework introduced in (Antoulas, 1997) and (Antoulas and Anderson, 1999). Keywords: Identification, parameter uncertainty, model validation, consistency. 1. INTRODUCTION To apply robust control, the existence of a nominal model together with some description of the uncertainty is assumed. Usually, uncertainty in the system is taken to be norm bounded. In the past decade system identification has investigated the problem of obtaining an uncertain family of models, where the uncertainty is norm bounded. One of the well-known approaches for this purpose is set membership identification; this approach characterizes the membership set of the parameters consistent with the data, the model structure, and the perturbation characteristics (see e.g. (M. K. H. Fan et al., 1991), (A.C. Antoulas et al., 1999)). In (A.C. Antoulas et al., 1999), measured inputoutput data with some a priori knowledge on the norm of the perturbation, are assumed. A family of uncertain plants is thus obtained with realparametric uncertainty. The family is given in terms of a linear fraction, the uncertainty being a real contraction. In this note we address the question: What is the consistency problem in this framework? Does there always exist an input function which will invalidate a given model? 1 Partially supported by the NSF Grant DMS-9972591 For a given model how do the minimal energy input sequences required to invalidate it, look like both in the noise-free and noisy cases. The rest of this note is organized as follows. In Section 2, a summary of earlier results on the identification part is given. Section 3 summarizes the issue of consistency and model validation. We conclude with a numerical experiment in Section 4. 2. AN OVERVIEW OF THE IDENTIFICATION FRAMEWORK The results in this section are mainly based on (Antoulas, 1997), (Antoulas and Anderson, 1999), and (A.C. Antoulas et al., 1999). Let us consider the following input-output data records: &t= ~)t CII{2' t=o, 1,.-.,N generated by some discrete time dynamical system. We model the given input-output data as generated by a linear, time-invariant, discrete time system with an additive perturbation; consequently, the data is decomposable as follows: &~=wt+sjt = + Yt where wt comes from the nominal model (i.e. from the linear, time-invariant system) and &t is the 9t
perturbation. We introduce the input matrix U and output matrix Y: [ ] U0 ~1 ' " " Us--1?~1 U2 ' " " Us U :=.... E ]~(m+l) s ~m Urn+l "'" ~ra+s--1 YYl Yl " " " Ys-1 Yl Y~ Y~ N(~+i)x~ [ Y:=.. E Yn+l Yn+s-1 for appropriate indices m, n, s (n is the degree of the denominator, m the degree of the numerator, of the system generating the data, and s = N - n + 1). Define the data matrix Jv~: 34:= EN, r:=m+n+2 The data matrices 2Q and Ad are defined similarly. Assumption: throughout the paper it is assumed that some upper bound e on the 2-norm of the perturbation is a priori known. Thus = f14 + A4 with II.hdll2 <_ Consider a linear system with transfer function G(z) = p(a q(~), where degp(z) = m and degq(z) = n; this system will be described by means of its coefficient vector x = [-p0 - pl... pm qo ql --. q ]r (1) where Pi, qi are the coefficients of the monomiai z i in p(z), q(z) respectively. Given such a system x, its misfit with the data matrix 2~I is defined as follows:,(x, Jq)- IIxTJql]2 Ilxb Notice that as argued in (Antoulas, 1997), the true model x0 which generated the data satisfies #(xo, JQ) < e; we thus define the uncertain model family ~: F= { xe ]~; ~(x'yq)- ''xt2~4''2 - Our goal next is to give a characterization of the uncertain family 3 r. Assume that the data matrix has the following Singular Value Decomposition (SVD): J~ = W2V T : [Wr "'" Wl ] " V T where W E I~ ~x~, and V E R ~x~ are orthogonal matrices. A second assumption we will make is 0" 2 > ~: > 0"1; this assumption corresponds to the knowledge of the system order (e.g. by means of preprocessing of the data). Then x E ~ if and only if there exists a constant c such that x = cx~ and x~ = wl + a2(~2w2 + "'" Jr O~r(~rWr where S-4 i= 1,2,...,r, +-.. _< 1 3. CONSISTENCY AND INVALIDATION The problems of consistency and model validation have been studied extensively, starting with Ljung's book ( L. Ljung., 1987); more recent contributions can be found in (K. Poola et hi., 1994), (R.S. Smith and J.C. Doyle, 1992), (R.S. Smith, 1996). However, in our identification framework, these issues need to be re-examined. The main result of our identification scheme asserts that given a level of misfit e and an input u, all models x satisfying #(x, A)I) < e, are not invalidated by the current input u and the corresponding output y. Furthermore all models x satisfying #(x, M) > e, are in fact invalidated. Hence the issue of consistency in this framework can be phrased as follows: Given a certain model described by the model parameter x, does there exist an input function u for which this parameter value is invalidated?. In other words what is the intersection of ellipses obtained by letting u vary through all possible input sequences of a given length and given energy? How does u look like? Below we will address these issues. As shown in (A.C. Antoulas et hi., 1999) and (S. Gugercin, 2000), when we have noisy output measurements, for a given model ~, ~TjQ can be written as xtj~-~u~(xo, fc) + Jkf,~n(Xo, x) where b/=[u(0) u(1)... u(g- n + m) 1' N" = [n(0) n(1)... n(n) ], (xo,~) E ]~,x8 and :,~(Xo, 5) E I~ (N+1) 8 depend only on Xo and wherel=n-n+m+l ands =N-n+l. It is assumed that the system is initially rest (zero initial conditions.) Therefore, the system 2 is invalidated by this set of measurements if t[// + HI:nil > e[[~[[. To obtain all systems which can be invalidated by using some appropriate input, we search for the points on the boundary of this set. Namely, given a parameter vector, we formulate the following optimization problem: min[lut[ s.t. [[UI:+Af ~ll=e[]21[ (2) Assuming that the noise sequence Af has known characteristics (e.g., Gaussian or fixed upper bound), for the given system 2, the problem in (2) is a nonhomogenous quadratic optimization problem on the ball. The minimization problem
2 is examined in details in (A.C. Antoulas et al., 1999) and in (S. Gugercin, 2000). Here we briefly review those results, The problem is solved following the method developed in (D. C. Sorensen, 1997). Thus for each parameter vector with the given noise sequence Af, we compute the minimum energy/.4 required to invalidate it. Therefore for a given noise sequence Af, using inputs of given energy, not all systems can be invalidated. The set of models which can be invalidated depends on e and the energy of the inputs u. For fixed e, this set of invalidatable models decreases as the energy of either the input sequence or the initial conditions decreases. The minimal energy input 5/* which solves (2) is unique. Moreover, the true model x0 can never be invalidated using an input of finite energy. Let 5/~f be the solution for the above minimization problem in the noise-free case. Then u,:r = ~ll:~llv~ 71" 1 where lrl is the largest singular value of : and vl is the corresponding left singular vector. Finally, it can be shown that even for infinite length measurements, the set of plants which cannot be invalidated does not shrink to the true model. In this note, we mainly examine the shape and norm of the minimal energy 5/ which invalidates a given model 2. It is easy to prove that for infinite length measurements Z: T is symmetric, positive definite and Toeplitz. Hence, in the noise-free case for infinite length measurements, the minimal input U. I is given by the eigenvector of a symmetric positive definite toeplitz matrix corresponding to the largest eigenvalue. Let A(z) be the filter designed as follows: A(z) = N - n-~-m Z k=0 u~f(k) z-k, As stated in (J. Makhoul, 1981), the eigenvectors of a symmetric positive definite toeplitz matrix are either symmetric or skew-symmetric with respect to middle. Also, the location of the zeros of the filters whose coefficients are the elements of the eigenvectors corresponding to the minimum or maximum eigenvalues lie on the unit circle. Moreover as S. S. Reddi (1984) proves these zeros are distinct. Hence it follows that in the noisefree case for infinite length measurements the minimal energy input 5/*f is either symmetric or skew-symmetric with respect to middle. Also A(z) above has distinct zeros on unit circle. 4. EXAMPLES We consider a LTI first order dynamical system described by two parameters a and b: a(z) - b z--a The numerical values are assumed to be b = 1, a = 0.5, and number of data points N=100. The noise is additive white Gaussian Yt "~ N(0,0.05); e is chosen to be 0.05. 4.1 Minimal Norm Inputs We solve the minimization problem defined in (2) using the tools developed in (D. C. Sorensen, 1997). Experiments were run 100 times and then averaged. Let the solution be denoted by H*. Figure I shows the average minimum energy of H on a logarithmic scale, i.e. log10(115/*]1), required to invalidate a given model represented as a deviation from the true model. On that figure (~ stands for the deviation from the nominal pole and/~ for the deviation from the nominal gain. Notice that as the deviation from true model becomes smaller, the minimum energy ]lh*]l increases, and finally as 2-~xo, Ilu*ll-~cc. 05. ~- - -t. mm nocm O v dare,an from the mac mo~l I..-. : :-..! 15 i 2 a~pha Fig. 1. Minimal energy IlU*ll (log scale) necessary to invalidate given model Figure 2 shows the cross section of the 3 dimensional plot in Figure 1 for 115/11=1. It shows that for any choice of a unit energy input function, all models inside the area shown cannot be invalidated. Moreover, all the models 2, on the boundary of this area need unit energy to be invalidated. Now we examine the shape of the input sequences required to invalidate models &, which are on the boundary in Figure 2. We choose 4 models (see the figure) : &T = [--1.5 -- 0.31 1] 2 T = [--1.25 -- 0.66 1] 2[ = [--0.8 -- 0.4 1] 24 ~ = [--0.75 -- 0.71 1] beta
The ~t o( (amnl~t s Ihat c~,nol be,nvahdate<l by unit energy ' =*! O m,~alidati~ inputs for x2 0,,,,,,,,, --0.05................................ -o,i 0.~ t xt x3 0115 2: 20 10 ~0 3O 40 50 60 70 80 90 1 O0 ~0 -O2 X2 X4 o' / invahdaling inputs for x3 0. 3 ~ 02 --.- Nobly ~.3 01 ~8 ~DI6 J i I I -04 ~Q2 0 02 0.4 06 t~ca 0 Fig. 2. Cross section of Figure I for ItUll = 1: Set of parameter values which cannot be invalidated For each of these chosen models, we compute the minimal energy input sequences L/* and L/*f required to invalidate them in the noisy and noisefree cases. The resulting plots are shown in Figure 3 for xl, x4 and in Figure 4 for x2, x3. It should be noticed that for each case the noisefree solutions L/~I (i) are given by either ± sin (~ i), or -4-(-1) -i sin(~i). Figure 3 and Figure 4 reveal that the noisy solution is a perturbation around the noise-free one; and when the noise-free solution alternates in sign, the resulting noisy solution has a complex structure. Fig. 4. Input sequences that invalidate x2 (above) and x3 (below). result illustrates the fact that even with infinite length measurements, not all the models can be invalidated. 10( 1 os 1.04 ~1.03 I.o2 ~n ~ U vs Oata L~gZh ~e~a~ ~put* for xl ~'i i : :! i : : i ~o 40 6O ao ice 120 140 I 1 0 200 Oala Leng~ (N) kwldk~at]ng inpots for x4 ] o,s.... i i i/ Fig. 5. Minimal energy IlU~N[I required to invalidate the given model x3 vs N, the length of the data. o -o.5 5. CONCLUSION ~t0 10 20 30 4(1 50 60 70 80 9o 100 Fig. 3. Input sequences that invalidate xl (above) and x4 (below). 4.2 Norm of the Input vs Data Length In this example we examine the minimum energy of 5/, as a function of the length of the measurements. Let X3 be the model given in the previous example. We solve the minimization problem (2) for x3 as N varies from N=10,.-., 200. Let L/~N be the solution for a given N. Figure 5 shows IIL/~Nll corresponding to the chosen model x3 as N changes. Figure 5 indicates that even as N goes to infinity, ]IL/~NII does not go to zero; it stays constant with a finite nonzero value. This The set of models which can be invalidated using some measurement, depends on the misfit ( and the energy of the inputs u. For fixed ~, this set of invalidatable models decreases as the energy of either the input sequence or the initial conditions decreases. In the noise-free case for the infinite length measurements the minimal norm input L/~, I which is the eigenvector of a symmetric positive definite Toeplitz matrix is symmetric or skew symmetric with respect to middle. Moreover simulations illustrate that the minimal norm input N* for the noisy problem is a perturbation around ~f. 6. REFERENCES Antoulas, A.C. (1997). On modeling for robust control. Proc. SysId'97 pp. 1707-1709.
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