By C.T. Leondes (Eds.)

Compliment for earlier Volumes "This booklet may be an invaluable connection with keep watch over engineers and researchers. The papers contained conceal good the new advances within the box of contemporary regulate theory."-IEEE staff CORRESPONDENCE"This booklet may help all these researchers who valiantly attempt to continue abreast of what's new within the idea and perform of optimum control."-CONTROL

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**Extra resources for Discrete-Time-Control System Analysis and Design**

**Example text**

The sequence space g2(l+,/C), or simply g2(/C), is defined to be oo e2(Jc) . - { ~ ~ k ~/c, II~kll 2 < ~), k=O The norm for ~bk is the one on K; and the norm for g2(K;) is given by 1//2 11~112 - II~kll 2 The lifting operator W, mapping/22[0, co) to f2(~), is defined by - Wy~ ~k(t)- y(t+kh), We denote the lifted signal W y by/). 0<_ t < h. H - - OPTIMAL CONTROL Now we lift the system in the preceding figure t o get Here the lifted system G satisfies the discrete-time equations [14] [since w has support in [O,h), 6 is a pulse sequence in e2(IC)] ((k + 1) ik = Ad[(k) +8 + Bzdv(b), 1 ~ k Gk = Gobd(k) (16) = C1((k)+Dll~k$~12~(k), (17) where ((k) := x(kh) and the operators are given by The system G can be regarded as a linear time-invariant system in discrete time, with Zk and Gk being infinite-dimensional (functions in IC).

The following remarks are in order: + f1w ~ . H,- OPTIMAL CONTROL 27 1. N defined in (19) is a constant matrix pair because D ; ~ Cas~a 61 whole is a rnatrix (operator on C ) . Tlze formulas for 0 ~ ~ can be derived easily (see Section V). Similarly, F, though involving operators, is also a matrix. However, the feedforward gain FI is an operator mapping K to I ; its action on a fixed 27, can be determined a priori. 2. ';c(klr)]is indepe~ltle~rt ogenous input lo and can be realized by sampling x ( t ) a t the same rate as the hold operator.

34, No. 8, pp. 831-847, 1989. 34 TONGWEN CHEN AND BRUCE A . FRANCIS [ I l l 11. T. lity criterion," A ~ t o n z ~ t i cvol. a , 28, No. tems with an H 1, pp. 45-54, 1992. 1121 Y. Yamamoto, "A new approacll t o sampled-data control systems - a function space approach method," Proc. CDC, 1990. [13] B. Bamieh and J. B. Pearson, "A general framework for linear periodic systems with application t o H ' , sampled-data control," IEEE Trans. Autonzat. Control, vol. 37, pp. 418-435, 1992. [I41 B. Bamieh, J.