By Prof. Bruce A. Francis (eds.)
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Additional resources for A Course in H∞ Control Theory
Define the controllability and observability gramians oo Lc := I e-AtBB r e-art dt o (6) e,o L° := I e-artcTce-Atdt" 0 (7) It is routine to show that Lc and Lo are the unique solutions of the Lyapunov equations ALc +LcA T =BB T (8) A TLo +LoA = c T c . (9) Exercise 3. Prove that the matrix representations of WcW* and ~ W o are L c and L o respectively. Theorem 3.
In terms of a state-space model G is stabilizable iff its unstable modes are controllable from u (stabilizability) and observable from y (detectability). The next result is a stabilizability test in terms of left- and right-coprime factorizations G = N M -1 = ~ I - 1 N . Theorem 1. The following conditions are equivalent: (i) G is stabilizable, (it) M, [0 I ] N axe right-coprime and M , I ~ ] are left-coprime, (iii)  M, N I are left-coprime and M, [0 I] are right-coprime. The proof requires some preliminaries.
4 23 [A,B,C,D] stand for the transfer matrix D + C ( s - A )-IB . Now introduce state, input, and output vectors x, u, and y respectively so that y =Gu and =Ax + Bu (5a) y = Cx + D u . (5b) Next, choose a real matrix F such that AF :=A +BF is stable (all eigenvalues in Re s <0) and define the vector v : = u - F x and the matrix Ct~ :=C+DF. x = A FX + Bv u =Fx +v y = CFX + D v . Evidently from these equations the transfer matrix from v to u is m ( s ) := [AF, B, F, I] (6a) and that from v to y is N (s) :=[AF, B, CF,D ] .
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