By Bruce A. Francis
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4 24 with (A ,B) stabilizable new data structure: [A,B, and (C ,A) detectable. It's convenient to introduce a let C,D] stand for the transfer matrix Now introduce state, input, and output vectors x, y=Gu 11" and y respectively so that and x = Ax + Bu y = Cx (5a) + D11, (5b) Next, choose a real matrix F such that AF := in Re s <0) and define the vector v := 11,~Fx A +BF is stable (all eigenvalues and the matrix CF := C +DF . e. G =NM-1. 4 25 (6d) we get G =M-1 N. ) (This can be derived as above by starting with G (s )T Thus we've obtained four matrices in RHoo satisfying (2).
But by (10) and (12) those from w to v Hence those from w to ~1' r] conclude from (9) that the transfer matrix from Exercise w belong to RHoo. l' Finally, we to ~ belongs to RHoo. [J 2. Suppose G 11= G 12= G 21= G 22. Prove that G is stabilizable. 4 Parametrization This section contains a parametrization simplify notation of all K's slightly, in this section the subscripts which stabilize G 22. To 22 on G 22 are dropped. The relevant block diagram is Figure 1. Bring in a doubly-coprime G = NM-1 factorization of G , = M-1jy (1) ('Ii.
Lemma 1 is due to Doyle and Stein (1981) and Chen and Desoer (1982). There are several other examples of the standard weighted sensitivity problem problem, for example, the (Zames (1981)) and the mixed sensitivity (Verma and Jonckheere (1984), Kwakernaak (1985)). problem CHAPTER 4 STABILITY THEORY In this chapter model-matching matrix it is shown how the standard problem. The procedure problem can be reduced to the is to parametrize, via a parameter Q in RHoo, all K 's which stabilize G . 1 Coprime Factorization f (8) and Recall that two polynomials said to be coprime if their greatest 9 (8 ), with, say, real coefficients, are common divisor is 1 (equivalently, no common zeros).
A cource in H Control Theory by Bruce A. Francis