We therefore obtain the following algorithm. ALGORITHM I1

Abs6~1c-A new promlure for s t a b i i an unstable system by stable feedback is presented. The procedure is based on using a lower order model for the given system. Stabilization conditions for tbe original system are given in terms of the e m in the approximation. The proeednre is extended to largescale systents whereby stabilkation coaditions with decentralized feedback are presented whereuponp(A)=[fi')l~)I... I f i q. requires 3nZ storage. A carefid operation count for s> 1 shows that w2(s)n3 multiplications are required where Since arrays are needed for only A, A S, and p(A), Algorithm I1 where 2 ( s- l) + [ q / s]- 2 ifsdividesq 2(s- 1) + [ q / s] - 1 otherwise. W Z ( 4 = This work count is approximately minimized by setting s =[w] whereupon w2(s)=2 @-about 1.5 times the work associated with At the expense of one extra n X n array, A S can be computed via its Algorithm I. binary expansion A J = ( A B q ( A 812')... (A BmZm) s = 2 Bkzk m k=O in alog2(s)n3 multiplications where aE[l], [2] depends upon the number of nonzero &'s. (If s is an exact power of two, the extra storage is not required.) When this feature is incorporated in Algorithm 11, the work count becomes i2(s)n3 where G2(s)=aiog2(s)-~+wl(s). These ideas have been incorporated in a subroutine for approximating the matrix exponential e A with a truncated Taylor series For reasons based on the machine precision used (10- 15), the norm of A, and the desired accuracy, q was chosen to be 16. (See [ I] for an analysis of Taylor approximation to eA.) Setting s = 4 and using binary squaring to compute A4, Algorithm I1 requires 7n3 multiplications and 3n2 storage to evaluate Tl,(A). This is less than half the work associated with traditional Homer evaluation and only marginally slower than the 6n3 multiplications needed by Algorithm I which requires 5n2 storage for this problem