Syrmos, Vassilis L.

Zagalak, Petr

English

Institute of Mathematics AS CR, v. v. i.

KybernetikaVassilis L. Syrmos; Petr ZagalakA computational method for reduced-order observers in linear systemsKybernetika, Vol. 32 (1996), No. 1, 95--100Persistent URL: http://dml.cz/dmlcz/125230Terms of use:© Institute of Information Theory and Automation AS CR, 1996Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A — V O L U M E 32 ( 1 9 9 6 ) , N U M B E R 1, P A G E S 9 5 - 1 0 0 A COMPUTATIONAL M E T H O D FOR R E D U C E D - O R D E R OBSERVERS IN LINEAR SYSTEMS 1 VASSILIS L. S Y R M O S AND P E T R Z A G A L A K 2 A computationally stable method for reduced-order observers of linear systems is pro-posed. This method is based on orthogonal transformations and adopts Diophantine matrix polynomial equations. 1. INTRODUCTION Reduced-order observer design has a long history spanning decades and involving various researchers in the control systems society. The first results on this problem were presented by Luenberger [1]. Thereafter, several papers have been presented examining the problems from different perspectives. One of these approaches is the Sylvester equation approach. This technique aeals mainly with the computational aspects of the problem. In particular, it was Van Dooren who presented some fun-damental concepts in this direction using block Hessenberg forms [8]. Later Tsui in [5] presented a parametrization using the aforementioned Sylvester equation. In this paper an alternative algorithm is presented for designing reduced-order observers, which is based on a Diophantine equation. Specifically, the problem of reduced-order observer is studied as a full-order observer problem on a reduced-order subspace using a Diophantine equation. The algorithm uses the Hessenberg form of the pair (C, A). The computation of an output injection is achieved in a computationally efficient way. 2. PRELIMINARIES Be given a linear time-invariant system (C, A, B) governed by x = Ax + Bu y = Cx 1 The original version of this paper was presented at the 2nd IFAC Workshop on System Structure and Control, Prague, 3 - 5 September, 1992. 2 P. Zagalak is supported by the Grant Agency of the Czech Republic through Grant No. 102/94/0294. 96 V. SYRMOS AND P. ZAGALAK where A G IRn x n , B G lRn x m and C G H p x n . In addition rankC = p and the observability of the pair (C, A) is further assumed. Let F(s), G(s) be p x p, p x n polynomial matrices over IR[s]. Then F(s), G(s) are said to form a (left) normal external description of (C, A) if [3] • [G(s), F(s)] is a minimal polynomial basis of the left kernel of si-A C • F(s) is noincreasingly row-degree ordered and row reduced; • G(s) is a minimal polynomial basis of the left kernel of (si — A) U where n is a matrix representation of the maximal annihilator of C, i.e. Cn = 0. Let P(s) be a p x (ki + k2 + row degrees k± — 1, k2 + kp) polynomial and row-reduced matrix with 1 , . . . , kp — 1 such that P(s)= block diag{[l,s,...,sk'-1]}l< where K G Mkxk is nonsingular, k = ]TP=1 fc,\ Then P(s) is said to be a polynomial basis of a k-dimensional and R-linear vector space [4]. Given an n x m polynomial matrix P(s), rankp(s) = k := min (n,m), we shall say that P(s) is irreducible if rank P(z) = k for every complex z. For instance, the \ si — A observability of the pair (C, A) is equivalent to the irreducibility of nn The concept of a (right) normal external description of the controllable pair (A, B) was established in [3] and used in [9] for compututing a state feedback F assigning a given pole structure to the system x = (A-\-BF)x. Here, we shall use the dual version of this algorithm. That is why we have introduced the concept of a (left) normal external description of the pair (C, A). In [1] it was shown that the problem of designing a reduced-order observer is equivalent to the problem of designing a full-order observer on a reduced-order sys-tem of order n — p. To get a description of the reduced-order system in a compu-tationaly stable way, we shall exploit the properties of the block-upper Hessenberg form of the matrix A [7, 8]. To this end, let x = Ux' be a state-space similarity transformation, where U is orthogonal, such that [7,8] A' = UTAU Ã A C Ã ' Aц Aí2 • • J-i.jь-i Aik A2i A22 • • -4i,ł_i A2k (0) • ^jfc-i.jt-i Ak-i,k . (0) Ak^k-i Akk A Computational Method for Reduced-Order Observers in Linear Systems 97 and C' = CU = [ 0 • • • 0 Cp ], B' = UTB, i.e. the matrix A is brought into the block-upper Hessenberg form [7] a n d Cp £ 1RP p is nonsingular. w Let further x' = T where Ln-p u 0 C" 1 be another similarity transformation that brings the system (C',A',B') into the form A\ A2 Ci A3 y = [0 IP] ' w ' Bi ' V + . 5 2 . w = V V and denote A \ Ai A2 [ Ci A3 B Bi в2 Õ := 0 /„ Luenberger's theory of observers [1] now impLes that an observer for the system w = A\w + A2y + B\u yw : = Cxw = y - A3y - B2u can be constructed using the reduced-order subsystem (C\,A\) and is of the form k = (Ax-LC\)i + (A2 - LA3) y + (B\-LB2)u + Ly. (I) The relationship (1) reveals that the most crucial point in designing the observer is to find an output injection L such that the observation process £ will converge to the state w, i.e. the eigenstructure of the matrix A\ — LC\ must be set up properly. 3. THE STRUCTURE OF THE OBSERVER In this section we exploit the structure of the system in order to build a reduced-order observer using a reduced-order and square Diophantine equation. Let G\(s), G2(s) and F(s) form a left normal external description of the system (C,A), i .e. ' [ d l У ) G2(s) F(s) } sín-p — A\ -Å2 - C i slp - A3 0 ~h . = o, (2) V. SYRMOS AND P. ZAGALAK and let k\ > k2 >•••> kp be the row indices of [G\(s) G2(s) F(s)]. Clearly, ki, i = 1,2,... ,p are the observability indices of the pair (C, A) and hence those of (C>A\-It is natural to explore the relationship between the observability indices of the pairs (C,A) and (C\,A\). The following lemma provides this relationship. Lemma 1. Let((7, A) be an observable pair with observability indices and k\, k2,... . . ., kp. Then the pair (C\, A\) defined above is also observable and its observability indices are k\ — 1, k2 — 1 , . . . , kp — 1. P r o o f . As (C,A) is observable, (C,A) is also observable and has the same ob-servability indices. This implies that sln-p ~~ A\ -A2 -C\ slp - A3 0 -IP _ is an irreducible matrix. In view of the fact that the second column of the above matrix forms an ireducible submatrix, we have that sln-p — A\ -C\ is also irreducible and hence, (C\,A\) is observable. Let K(s) := [G\(s) C?2(s) F(s)] form a left normal external description of (C, A) and let k\,k2,... ,kp be the row degrees of K(s). Since C(sln — A)~- is a strictly proper rational function, F~1(s) [G\(s) G2(s)] is strictly proper, too. Then, F(s) is row reduced with the row degrees k\,k2,... ,kp and [Gi(s) G2(s)] is also row reduced with the row degrees k\ — l,k2 — 1,... ,kp — 1. As [G\(s), G2(s)] forms a left normal external description of (Ci,.Ai), we can repeat the above considerations and the claim follows. • The effect of an output injection L on the system (C, A, B) can be described by [ G\(s) G2(s) F(s) ]MM - ì sln-p — A\ —A2 where M = ln—p 0 0 -C7i 0 -L It follows from the above equation that [ G\(s) G2(s) + G\(s)L ] sln-p — A\-\- LC\ -C\ = 0 (3) A Computational Method for Reduced-Order Observers in Linear Systems 99 and hence, the output injection gain L can be obtained from a constant solution pair X, Y ( i .e. X nad Y have entries in IR) with X nonsingular [2] of the equation G2(s)X +G\(s)Y = D(s) (4) where D(s) is an (n — p) x (n — p) polynomial and row-reduced matr ix having the same row indices as G2(s) and reflecting the desired pole structure of the observer (1). Indeed, it follows from (3) that D(s) reflects the pole structure of the observer. Thus, if we find a constant solution pair X, Y with X nonsingular to equation (4), we get the desired output injection L on putt ing L = YX~l. It should be noted that there exists an efficient and numerically stable way to compute the matrices G\(s), G2(s) and D(s); see [9] for more details. In [2] it has been shown tha t there exists a constant solution to (4), where X is nonsingular, if and only if D(s) is a row reduced matrix with the same row indices as G2(s). In order to calculate a constant solution X, Y of the equation (4), we can proceed as follows. Let a = k{ — 1, i = 1, 2 , . . . ,p and let S(s) := block diag {[1, s,..., s^'1]} . Then the equation (4) can be written in the form diag [ s c >, . . . , sc»-»] G2lrX + S(s) G2\Y = = diag [ s C l , . . . , sc»-'] D\r + S(s) D\ where G2\r and D\r are the leading coefficient matrices of G2(s) and D(s), respec-tively, and G2\ and D\ are constant matrices (i.e. having elements in IR). In fact, we are to solve the following two systems of linear equations G2\rX = D\r G2lY = D\, which can be done for instance using some of the well-known methods exploiting orthogonal transformations to achieve high numerical stability. The output injection L is then given by L = YX~l. To sum up, the proposed algorithm for finding L can be described as follows: into the block-A C 1. Using orthogonal transformations, bring the matrix upper Hessenberg form and put C\ = CPC, A\ = A. 2. Find a left normal external description of (C\, A\). 3. Construct a row-reduced matr ix D(s) having the desired invariant factors and the same row degrees as G2(s). 100 V. SYRMOS AND P. ZAGALAK 4. Find a constant solution pair X, Y with X nonsingular of the equation (4). 5. Put L = Y x - 1 . It is to be noted tha t the solution given by the above algorithm is by no means unique [2]. The particular form of L depends mainly on the particular form of D(s). 4. CONCLUSIONS In this paper we presented a computationally efficient method for reduced-order observers in linear systems. The proposed technique is based on the Hessenberg form of the pair (C, A) and a square reduced-order Diophantine equation (4), which enables us to modify the dynamics of observer, i.e. the zero structure of s / n _ p — A\ + LC\, in the limits given by the Rosenbrock theorem; see [4] for details. The method was tested using MATLAB. (Received February 23, 1993.) REFERENCES [1] D.G. Luenberger: An introduction to observers. IEEE Trans. Automat. Control AC-16 (1971), 596-602. [2] V. Kucera and P. Zagalak: Constant solutions of polynomial equations. Internat. J. Control 53 (1991), 495-502. [3] M. Malabre, V. Kucera and P. Zagalak: Reachability and controllability indices for linear descriptor systems. Systems Control Lett. 15 (1990), 119-123. [4] P. Zagalak and J. J. Loiseau: Invariant factors assignment in linear systems. In: Proc. SINS'92, ARRI, Ft. Worth 1992, pp. 197-204. [5] C.C. Tsui: A complete analytical solution to the equation TA — FT = LC and its applications. IEEE Trans. Automat. Control AC-32 (1987), 3, 742-744. [6] P. Van Dooren: The generalized eigenstructure problem in linear system theory. IEEE Trans. Automat. Control AC-26 (1981), 111-129. [7] P. Van Dooren: The computation of Kronecker's canonical form of a singular system. Linear Algebra Appl. £7(1979), 103-140. [8] P. Van Dooren: Reduced-order observers: A new algorithm and proof. Systems Con-trol Lett. 4 (1984), 243-250. [9] V. L. Syrmos and P. Zagalak: Computing normal external descriptions and feedback design. Linear Algebra Appl. 188, 189 (1993), 613-639. Dr. Vassilis L. Syrmos, Department of Electrical Engineering, University of Hawaii at Manoa, 2540 Dole St., Holmes Hall 483, Honolulu, HI 96822. Hawaii. Ing. Petr Zagalak, CSc, Ustav teorie informace a automatizace AV CR (Institute of Information Theory and Automation - Academy of Sciences of the Czech Republic), Pod vodárenskou věží 4, 18208 Praha 8. Czech Republic. 

A computational method for reduced-order observers in linear systems

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A computational method for reduced-order observers in linear systems

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Institute of Mathematics AS CR, v. v. i.