By Bruce A. Francis
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Extra resources for A Course in H_infinity Control Theory
This system is linear, but not necessarily causal because f (t) may not equal zero for negative time. e. "El leaves the future invariant". The operators Ef and A F are intimately related via the Fourier transform. The relationship is exhibited in the commutative diagram of Figure 1. Example 5. Again let F a L~. The Toeplitz operator with symbol F, denoted OF, maps H 2 to H z and is defined as follows: for each g in It 2, ®Fg equals the orthogonal projection of Fg onto H2. Thus OF = FIzAF IH2. The relevant commutative diagram is Figure 2.
Also, since G22 is strictly proper, we have that 0 I G21 G22 1 is nonsingular when evaluated at s =~,: its determinant equals 1 at s =~. Thus both matrices on the right-hand side of (3) are nonsingular. The equations corresponding to Figure 2 are ] -EO, N Thus by Lemma 1 K stabilizes G iff -[0 I]N V RH~. 29 Ch. 4 But this is equivalent to (it).  Exercise 1. Prove equivalence of (i) and (iii) in Theorem 1. 3 Stabilizability Let's say that G is stabilizable if there exists a (proper real-rational) K which stabilizes it.