Elsevier

Automatica

Volume 42, Issue 10, October 2006, Pages 1697-1704
Automatica

Brief paper
A novel numerical method for exact model matching problem with stability

https://doi.org/10.1016/j.automatica.2006.04.024Get rights and content

Abstract

In system design the exact model matching problem with stability consists of compensating a given system, using a realizable control law of a specified structure, in order to ensure the stability of the compensated system and achieve a target closed-loop transfer function. In this paper we develop a novel numerical method to verify the solvability of the problem for left invertible systems and further construct a desired solution. Our method has a complexity which is cubic in the state dimension of the system and the desired model and can be implemented in a numerically reliable way.

Introduction

In this paper we study the solution of the exact model matching problem with stability for a system (A,B,C) characterized byx˙=Ax+Bu,y=Cx,where ARn×n, BRn×q, CRp×n, and where xRn is the state, uRq is the input and yRp is the output of the system. We consider a linear state feedback control law of the formu=Fx+Gv,where FRq×n, GRq×q and vRq is the new input. Then the closed-loop system of (1)–(2) isx˙=(A+BF)x+BGv,y=Cx.The transfer function matrix of the closed-loop system (3) from the new input v to the output y is then given byMF,G(s)C(sIn-A-BF)-1BG.In this paper we state the exact model matching problem with stability as follows:

Definition 1

Given system (1) and a strictly proper transfer matrix M(s), the exact model matching problem with stability consists of finding matrices FRq×n and GRq×q with G nonsingular such thatMF,G(s)=M(s),and furthermore the matrix A+BF is stable (i.e. has all its eigenvalues in the open left half plane).

The exact model matching problem is of both theoretical and practical importance since a number of control problems can be linked to it (Ichikawa, 1985). Among those are the decoupling problem, the model tracking problem and model reference adaptive control. The exact model matching problem was originally presented without stability requirement in Wolovich (1972), where a solution was given for the case of invertible systems, using feedback invariants, a coordinate transformation and a set of polynomial matrix equations. In Wang and Desoer (1972) and Wang and Davison (1972) the restriction of invertibility was eliminated and the problem was reduced to that of solving a system of linear algebraic equations. The problem was treated in Rutman and Shamash (1975) by means of polynomial matrix equations under the assumption that the open-loop system is invertible. A time-domain solution was developed in Tzafestas and Paraskevopoulos (1976), which led to a set of nonlinear matrix equations yielding the feedback law. In Paraskevopoulos (1977) a solution was given in the frequency domain which reduced the problem to that of solving a large set of linear equations. In Kaczorek (1982) and Kucera (1981), the problem was studied via a polynomial approximation approach. Further related results were given in Ichikawa (1997), Kase and Mutoh (2000), Kase et al. (1998), Vardulakis and Karcanias (1985), Yamanaka et al. (1997) and Torres and Malabre (2003).

The exact model matching problem has thus been studied extensively in the last three decades but several issues have not been addressed appropriately:

(i) Numerically verifiable necessary and sufficient solvability conditions for the exact model matching problem for a general linear time-invariant system and a given general proper model are still not available in the literature: existing results transform the underlying problem to another kind of problem such as nonlinear equations or rational/polynomial matrix equations.

(ii) Solutions based on rational/polynomial matrix equations (Kaczorek, 1982, Kimura et al., 1982, Kucera, 1981, Rutman and Shamash, 1975, Torres and Malabre, 2003, Wolovich, 1972) or nonlinear matrix equations (Tzafestas and Paraskevopoulos, 1976) with constant unknowns, do not lead to numerically reliable methods for solving the exact model matching.

(iii) Results in Paraskevopoulos (1977), Wang and Desoer (1972) and Wang and Davison (1972) are delicate from a numerical point of view because the resulting linear systems of equations are much larger than the state dimension of the models M0(s)C(sIm-A)-1B and M(s) and the computations of the involved coefficient matrices are poorly conditioned.

(iv) Although the model matching problem in H control can be solved using algebraic Riccati equations, this does not appear to be the case—as far as we know—for the exact model matching problem.

Therefore, we believe there is still a lack of numerically reliable methods for solving it. In this paper we thus revisit the exact model matching problem with stability from a numerical point of view.

We first point out that if the original system (1) is unstabilizable then the problem has no solution. On the other hand, if its uncontrollable modes are stable, then they play no role in the problem. We can therefore assume without loss of generality that the system (1) is controllable (but not necessarily observable). We will denote its transfer function by M0(s)=C(sIn-A)-1B. If the given realization of the desired transfer matrix M(s) is not minimal, we can always obtain one via a numerically stable procedure (Van Dooren, 1981). We can therefore assume without loss of generality that we are given a minimal realization of the desired model M(s): M(s)=C(sIm-A)-1B, with ARm×m, BRm×q and CRp×m. Notice that if M(s) is left invertible then M0(s) must also be left invertible since this property is not affected by the matrices F and G in (5), and that mn is required since M(s) is also given by (4).

The main purpose of this paper is to develop a new method to verify the solvability and compute a desired solution for the exact model matching problem with stability for left invertible systems. In contrast to Torres and Malabre (2003), we do not require the computation of the zero structure of the systems M0(s), M(s) and [M0(s)M(s)]. Furthermore, our method has a computational complexity which is cubic in the state dimensions of M0(s) and M(s) and can be implemented using orthogonal transformations only.

We will denote the complex plane by C, the closed right half plane by Cunst, the spectrum of a square matrix A by σ(A) and the spectrum of a regular pencil sE-A by σ(E,A). We will call KRn×pσ(A+KC) the unobservable spectrum of the pair (A,C) and rankg(R(s))rank(R(s))foralmostallsCthe generic rank of a rational matrix R(s).

Lemma 2

Let{A,B,C}be a realization of dimension n for which the pair(A,B)is controllable. Let{A22,B2,C2}be a minimal realization of dimensionn2ofC(sIn-A)-1B. Then

  • (i)

    σ(A)is the union ofσ(A22)and the unobservable spectrum of the pair(A,C),

  • (ii)

    the invariant zeros of the system{A,B,C}are the union of the invariant zeros of the system{A22,B2,C2}and the unobservable spectrum of the pair(A,C).

Proof

The proof is easy and thus omitted here. 

Section snippets

Reduction to invertible subblocks

The exact model matching problem with stability is better understood for square invertible systems than for left invertible ones (see e.g. Torres and Malabre, 2003). This observation motivates us to reduce the underlying problem to one for an invertible system. For this purpose, we show in this section that there always exists an orthogonal transformation W such that the top q×q block of WM(s) is invertible. Rephrasing the original problem in this coordinate system will eventually lead to a

Conditions for F and G

In this section we derive solvability conditions for F and G related to rank condition (16), under the assumption that condition (12) already holds.

Lemma 7

With notation of Theorem6and assuming that(12)holds, denote byΦ-sΘthe pencilA11-sE11A12-sE1200(A13-sE13)Z120A22-sE2200(A23-sE23)Z1200A11-sE11A12-sE12(A13-sE13)Z22000A22-sE22(A23-sE23)Z22.Then there exist orthogonal matrices P and Q such thatP(Φ-sΘ)Q=Φ11-sΘ11Φ12Φ13-sΘ1300Φ23-sΘ23,whereΦ11Rτ1×τ1, Φ12Rτ1×q,rank[Φ11-sΘ11Φ12]=rank(Θ11)=τ1sC,rankg(Φ23

Stability condition of A+BF

Up to now, we have not included the condition of stability of A+BF, which we will discuss in this section. In the following theorem we show that only the anti-stable eigenspaces of the pencils sE11-A11 and sE11-A11 play a role in the stability requirement of the exact model matching problem for the system (1) and the desired model M(s). Our proof is new and different from that in Torres and Malabre (2003). It clarifies the role played by the anti-stable invariant zeros in the exact model

Numerical method and example

As a consequence of Theorem 10 we have the following algorithmic implementation.

Algorithm 1

Input: ARn×n, BRn×q, CRp×n, ARm×m, BRm×q, CRp×m such that the system (1) and the model M(s) are left invertible.

Output: FRq×n and GRq×q such that (5) holds and A+BF is stable, if such a solution exists.

Step 1: (a) Compute orthogonal matrices W, U,Vb,g(KerC), U and V such thatAnew-sEnewU(A-sIm)Vb,g(KerC)=m1m2qm1A11-sE110A31-sE31m2A12-sE12A22-sE22A32-sE32qA13-sE13A23-sE23A33-sE33,Bnew=UB=m1m2q00B3,Cnew=WCVb,

Concluding remarks

In this paper we have studied the exact model matching problem with stability for left invertible systems. Based on matrix pencil theory, we have developed a new numerical method to verify the solvability of the underlying problem and further construct a desired solution. Our new method can be implemented via a numerically reliable manner, and its computational complexity is cubic in the sizes of the system (1) and the desired model M(s). The results trivially extend to the discrete-time case

Delin Chu is working in Department of Mathematics, National University of Singapore, as an Associate Professor. He got his doctoral degree from Department of Applied Mathematics, Tsinghua University, PR China, in 1991. His research interests include numerical linear algebra and its applications in systems and control, numerical analysis and scientific computing.

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Delin Chu is working in Department of Mathematics, National University of Singapore, as an Associate Professor. He got his doctoral degree from Department of Applied Mathematics, Tsinghua University, PR China, in 1991. His research interests include numerical linear algebra and its applications in systems and control, numerical analysis and scientific computing.

Paul M. Van Dooren received the engineering degree in computer science and the doctoral degree in applied sciences, both from the Katholieke Universiteit te Leuven, Belgium, in 1974 and 1979, respectively. He held research and teaching positions at the Katholieke Universiteit te Leuven (1974–1979), the University of Southern California (1978–1979), Stanford University (1979–1980), the Australian National University (1984), Philips Research Laboratory Belgium (1980–1991), the University of Illinois at Urbana-Champaign (1991–1994), Florida State University (1998) and the Universite Catholique de Louvain (1994–now) where he is currently a Professor of Mathematical Engineering.

Dr. Van Dooren received the Householder Award in 1981 and the Wilkinson Prize of Numerical Analysis and Scientific Computing in 1989. He became an IEEE Fellow in 2006. He is an Associate Editor of several journals in numerical analysis and systems and control theory. His main interests lie in the areas of numerical linear algebra and systems and control theory.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Siep Weiland under the direction of Editor Roberto Tempo. This paper presents research supported by the Belgian Programme on Inter-University Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture. The first author was also supported by the NUS Research Grant R-146-000-016-112 and the second author by the NSF Research Grant CCR-20003050.

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