By Bruce A. Francis
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1. The following conditions are equivalent: (i) G is stabilizable, (ii) M, [0 I] N are right-coprime and M, (iii) [~] are left-coprime, __  M, N 1 'M, are left-coprime and [0 I] are right-coprime. 4 31 The proof requires some preliminaries. fact. The reader will recall the following For each real matrix F there exist real matrices The matrices G and H may be obtained G and H such that by elementary row and column opera- tions, and the size of the identity matrix equals the rank of F. analogous result for RHoo-matrices is stated without proof.
1100'so AF is bounded. This operator Theorem is called a operator and F is called its symbol; so AF is the Laurent operator with symbol F . A related operator is AF I Hz, the restriction to Lz. ) Example 4. This is the time-domain lution in the time-domain Suppose F (s) containing analog of the previous example. corresponds to multiplication is a matrix-valued the imaginary function Recall that convo- in the frequency-domain. which is analytic axis and which belongs to Loo' in a vertical strip Taking the region of I'll.
1. Theorem parametrized The of all (proper real-rational) = -MQ )(X G is (XQNt1(y t (2) -QM) (3) -NQ 1 Let's first prove equality (3). Let Q ERHoo' [oI K 's stabilizing by the formulas K = (Y Proof. QM)] [M Y -MQ] __ [ Equating -N M N X -NQ -- 1 the (1,2)-blocks on each side in (4) gives (X -QN)( Y -MQ ) = Or -QM )(X -NQ ) , which is equivalent to (3). 4 39 Next, we show that if K is given by (2), it stabilizes U := Y-MQ, V:= - ~ Define X --NQ .... U := Y -QM, G. 2). Also from (5) [~~r E RHoo So from Lemma 1 K stabilizes G.
A cource in H Control Theory by Bruce A. Francis