By Bruce A. Francis
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Additional resources for A cource in H Control Theory
Again let F ELoo. The Toeplitz operator Hz to Hz and is defined as follows: for each of Fg onto Hz. projection The relevant commutative the scalar-valued Fg EHzl, eF diagram g zEHz 1 g l(s ) = g (1)/(8 -1) = [g (s )-g (l)]j(s maps g to g z. Hz, eF , maps g equals the orthogonal is Figure 2. As a concrete example consider gz g gz(s) Thus + g in eF Thus function F (s) = l/(s = g 1 with symbol F, denoted -1) . ) ApIH:! 2. 3. 5 53 Example 6. For F in Lao the Hankel Hi and The operator with symbol F, denoted f p, maps H2 to is defined as corresponding commutative diagram is Figure 3.
A + U 2F , := A j\ Il Then the fOl'lJllJins 2( S ) = [AF , B 2, F , I] N 2( S ) AI 2( S ) = [AF , B = [A Il , H, C 2' I] C 2' 2 0] N2(s) = [All' B2, X 2(S) = [AF, ~H, C 2' I] Y 2(S ) = [AF' ~H, F, 0] C2, 0] X2(s) = [All' ~B2' F, I] Y 2(s ) = [All' ~H, F, Finally, substitution HC are as follows: M 2' + 0] . into (3) yields the following realizations: [A~ -:;F] [Bl:;D 21 ] Ch. 4 45 Q = [C1+D12F --D12F] T 2(8) = [AF,B 2'C l+D 12F ,D u] T 3(8) = [AH,B l+HD It can be observed that T, ERHoo [A , B 2' (i =1-3), as claimed in Theorem 1.
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. I ~][~ ] ~ (4) . V =00; then take determinants of both sides G 22 is strictly proper and the matrix on the left-hand side of (4) is invertible in RH()().