Read e-book online A Course in H∞ Control Theory PDF

# Read e-book online A Course in H∞ Control Theory PDF

By Prof. Bruce A. Francis (eds.)

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Extra info for A Course in H∞ Control Theory

Example text

Exercise 1. In Figure 4 suppose G (s) = - . Consider a controller of the form s (s-l) K= -Q 1-GQ where Q is real-rational. Find necessary and sufficient conditions on Q in order that K 41 Ch. 4 stabilize G. 5 Closed-Loop Transfer Matrices Now we return to the standard set-up of Figure 1, Chapter 3. 1 gives every stabilizing K as a transformation of a free parameter Q in RH**. The objective in this section is to find the transfer matrix from w to z in terms of Q. In the previous section we dropped the subscripts on G22; now we must restore them.

Then = = = because II2Fg-l_h = = because g _I_FflF-h. Ch. 5 55 We conclude that r; =l-I2A}IH}. For example if F (s)= ( s - l ) -1 , then F} maps h (s) in H~- to - h (-1)/(s+1) in H2. ) Exercise 2. Show that the adjoints of Wc and tPo are as follows: tlJ* : cn---)L2[O, oo) (tlL*X)(t ) =-B r e-A rtX , t_>O; W~ : L2 (--~,0]---~C n 0 ~oY= I eArtcTy(t)dt" The rank of an operator q) : X-+Y is the dimension of the closure of its image space ~X. Our interest is in Hankel operators with real-rational symbols, Example 7 being a special case.

2 says that its norm also equals IIF[I~. 2, AFH 2 c H 2 . The converse is true too: if AFH2CH2, then F ~ H~. ) Example 4. This is the time-domain analog of the previous example. Recall that convolution in the time-domain corresponds to multiplication in the frequency-domain. Suppose F (s) is a matrix-valued function which is analytic in a vertical strip containing the imaginary axis and which belongs to L , . Taking the region of convergence to be this strip, let f (t) denote the inverse bilateral Laplace transform of F(s).