By Prof. Bruce A. Francis (eds.)
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Extra info for A Course in H∞ Control Theory
E. ) More generally, two matrices F and G in RH~. are right-coprime (over RH~) if they have equal number of columns and there exist matrices X and Y in RH~ such that [X y ] [ F ] =XF+YG=I. This is equivalent to saying that the matrix [ G] is left-invertible in R H , . Similarly, two matrices F and G in RH~ are left-coprime (over RH,~) if they have equal number of rows and there exist X and Y in RH,, such that 22 Ch. 4 equivalently, [F G ] is right-invertible in RH~. Now let G be a proper real-rational matrix.
2). Also from (5) So from Lemma 1 K stabilizes G. 38 Ch. 4 Finally, suppose K stabilizes G. ~. Let K=UV -1 be a right-coprime factorizafion. From (1) and defining D :=~IV-NU we have The two matrices on the left in (6) have inverses in RH~, the second by Lemma 1. Hence D-1 e RH~. Define Q :=-(XU-I'V)D-I, so that (6) becomes [_~ ~ ] IN U] =[10 -QDD] . (7) Pre-mulfiply (7) by [::] and use (1) to get Therefore (X -NQ )DJ " Substitute this into K= UV-I to get (2). e. Ge RH~. Then in (1) we may take N=~' =G X= r--1 y=0, in which case the formulas in the theorem become simply Ch.
4 37 Theorem 1. The set of all (proper real-rational) K's stabilizing G is parametrized by the formulas K = (Y-MQ)(X-NQ)-I (2) Q~ RH•. Proof. Let's first prove equality (3). Let Q~RH~. From (1) we have I'o ;] [io-,O]--, so that X-NQJ= I . - (4) Equating the (1,2)-blocks on each side in (4) gives ( 2 - Q ~ X r - M Q ) = ( f ' - Q ~ t X X - N Q ~, which is equivalent to (3). Next, we show that if K is given by (2), it stabilizes G. 2). Also from (5) So from Lemma 1 K stabilizes G. 38 Ch. 4 Finally, suppose K stabilizes G.
A Course in H∞ Control Theory by Prof. Bruce A. Francis (eds.)