# A cource in H Control Theory by Bruce A. Francis

By Bruce A. Francis

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Of [~]. (i): Choose, by right-coprimeness and Lemma 2, matrices X and Y 1Il RHoo such that Also, choose, by left-coprimeness, matrices Rand T in RHoo such that (1) Now define U:= TX (2a) eh. 4 34 V := Y - [0 I]NRX . (2b) Then we have from (1) and (2) that [I -RX] x] [[0 MI]N 0 Y I (3) v The two matrices on the left in (3) have inverses in RH()(), hence so does the matrix on the right in (3). The next step is to show that [ [0 MI]N [I [~]u1 V = all the matrices noting that ~ ~ G 21 G 22 in (4) at S We have [~]u [MO] V 0 [0 I] G [ Evaluate V is nonsingular.

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).

2 says that Laurent \\AF II = I\F by F. Obviously AF is linear. 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.