An Improved Error Estimate for Reduced-Order Models of Discrete-time Systems
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HINRICHSEN A N D A . J. PRITCHARD Abstract-In this note we derive, under weaker conditions, a discrete- time counterpart of Glover’s error estimate for reduced-order models. I. INTRODUCTION We consider finite-dimensional time-invariant discrete-time systems of the form x ( t + 1) = Ax(t) + Bu(t) Y ( 0 = (1.1) where (A,B,C)EKnX” KnXm KpX”, K = R o r C an d u(A) C D l , the unit disk in C . f ( A , B , C) is reachable, observable, and balanced, the two Lyapunov equations A M * C = -BB* ( 1 2) A*G1 - C = -C’C ( 1.3) Manuscript received October 3, 1988. D. Hinrichsen is with the Departme nt of Mathematics, University of Bremen, Bremen, A . J . Pritchard is with the Inst itute of Mathematics, U niversit y of Warwick, Cov entry IEEE Log Number 8933220. FRG. England. have a joint diagonal solution C > 0 (positive definite). Partitioning, in a com patible way, 0 .2 O],A=[:;; 3 , B = [z;] C =IC,, C21 (1.4) where C , , Al l E KrXr, < n one obtains an r-dimensional reduced- order model i ( t + 1) = Al1i(t) +BIu(t), y(t) = Cli(t) ( 1 5 ) of (1.1). The associated transfer functions are G(z) =C(ZZ, -A)-’B, G(z) =CI(ZZ, -AII)-IBI. Their difference E(z) = G(z) - G(z) ( 1.6) describ es the deviation of the input-output behavior of the reduced-ord er model (1.5) from that of the original system (1.1). If C = U1Im, 9 ’ %uo,Zm, ( 1.7) an d r = m, . .+mk, < 1 . AI-Saggaf [I] (see also [2]) has established the following bound on the H , -norm of the error function E : I IIEIIH== = og;Tl IE(~”)II 2 mIuI. j=k+l In the continuous-time case, G lover [4] has shown that a stronger estimate is valid Il G -GllH= =maxIIG(iu)-G(iu)ll 52x0,. (1.8) j=k-l w €R In [ 11 it has been pointed out that a transfer of this result to the discrete- time case via the standard bilinear transformation is not possible since the bilinear transformatio n introdu ces a direct input-output coupling into the reduced-order discrete-time model. In this note we will present an elementary direct proof of the stricter estimate for the discrete-time case under substantially weaker conditions. Following the same lines, a simple proof of G lover’s result (1.8) for continuous-time systems can be given. 11. ERROR STIMATE We will not need all the hypotheses made in [1]-[4] in our proof. In particular, we do not assume that ( A , B , C) is reachable, observable, and balanced. Su ppose that 5 denotes the usual order relation between Hermitian matrices P 5 Q( P < Q) H x*Px 5 x*Qx(x*Px < x*Qx) for all x E C”, x # 0 . Our analysis will proceed via the two Lyapunov inequalities ACA*-C< -BB* (2.1) A * M - C 5 -C*C. (2.2) In fact, we will show that if u(A) c DI and the inequalities (2.1), (2.2) have a joint solution C = C, @ & 0 with C, diagonal, then the truncated system is asymptotically stable and one obtains an error estimate of the type (1.8). The proof of the latter result proceeds- as in the continuous-t ime case- by successive truncation of the last n 4 , nq-, , ’ . nl state components. However, in contrast to the continuous-time case, the reduced-order models are not necessarily bal- 0018-9286/90/0300-0317$01 .OO 0 990 IEEE Authorized licensed use limited to: Technisc he Universiteit Delft. Downloade d on May 10,2010 at 10:01:03 UTC from IEEE Xplore. Restrictions apply.
