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Get A cource in H Control Theory PDF

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By Bruce A. Francis

ISBN-10: 0387170693

ISBN-13: 9780387170695

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A preliminary 1, split apart the factors, result. Insert and introduce the factoriza- two new signals ~ to get Figure 2. Lemma 1. The nine transfer belong of (i) and (ii) is simply matrices to RHoo iff the six transfer in Figure matrices 1 from w, in Figure 2 from v l' V 2 to z, w, v V 2 l' u, y to ~, 'Yf belong to RHoo' Proof. (If) This direction follows immediately from the equations which in turn follow from Figure 2. matrices X and Y such that YN = I. Hence ~= XM E + YN ~. (2) 28 Ch. = Substitution [ y -v z 2 ] .

Example 2. 3 the direct sum Each function f in Lz(-oo,oo) f lELz(-oo,O] and f zELz[O,oo): has a unique decomposition f = f 1 +- f z with Ch. 5 49 f l(t) = f (t), fl(t)=O, The function f nal projection f z(t) = 0, fz(t)=f(t), --+ f 1 t>O. from Lz(-oo,oo) of Lz(-oo,oo) t <0 to Lz(-oo,O] is an operator, onto L2(-oo,0]. the orthogo- It's easy to prove that its norm equals 1. In the same way we have The orthogonal projection from Lz onto Hi will be denoted ITI and from Lz onto Hz by ITz. Example 3. Let F ELoo and define the function AF from Lz to Lz via AF g := Fg .

Exercise 1. In Figure 4 suppose G 1 (s ) = s (s -1)' Consider a controller of the form J( -Q = --- 1-GQ where Q is real-rational. Find necessary and sufficient conditions on Q in order that K stabilize G . 5 Closed-Loop Transfer Matrices Now we return to the standard gives every stabilizing set-up of Figure 1, Chapter 3. 1 K as a transformation of a free parameter The objective in this section is to find the transfer matrix from w Q in RHoo' to z in terms of Q. In the previous section we dropped restore them.

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A cource in H Control Theory by Bruce A. Francis


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