By Eitan Altman, Tamer Başar, Naira Hovakimyan (auth.), Jerzy A. Filar, Vladimir Gaitsgory, Koichi Mizukami (eds.)

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**Example text**

Step 1. Set a domain *~ 1 5; lRn that contains the origin. For all ~ matrix Jl. )+D,1 =0 Eqn2(Y) + D,2 = 0. (30) (31) At this time, the obtained S, Y give a controller (r) in the form of (20), (21). Step 2. Find 0'1 such that Q1 = {(x,~) I ~TS~ + y2(x - ~)Ty-I(X -~) ~ ad 5; <1>1. (32) Q1 is a domain in which the closed-loop system (1:, r) is internally stable, where <1>1 is an admissible domain that contains QI, defined as <1>1 := (lRn x ~ I) n <1>12. (33) Here <1>12 is defined as <1>12 := {(x,~) I -e[S(B(~)BT (~) - BBT)S + D,tl~ + y2(x _ ~)Ty-I[Y{Q(~)M}T + {(Q~)M}Y - D,2]y- I (X -~) < OJ. *

W) < y. , xo = 0), so that Fp. would be associated with a linear gain from W to Z, if J-L is chosen to be linear. Another point to mention is that, as shown in [8]\ the study of such H oo -optimal control problems is equivalent to the study of a differential game with dynamics (1) and kernel function Jy(to. xo; u, w)= t(IIClx(t)II~+lIu(t)lIt -y2I1w(t)II~)dt+(Qfx(tf ), x(tf )h. lto (7) which is to be minimized with respect to the controller uO = J-L(', x('», and maximized with respect to w. This equivalence is as far as the upper value of this differential game goes, though one can show that under some appropriate conditions, for all y > y*, this differential game has a saddle point, while for y < y*, its upper value is infinite.

S +a p s+I The model M (s) does not reflect on any particular actual problem. We consider an external input W3 only for the orthongonality condition to hold. Correspondingly, we add a gain block K2 = 10 to the nominal plant in order to attenuate an influence of W3 on constructing a controller. 1 given in Section 4. The following numerical calculations use MATLAB/LMITOOL [4]/Optimization Toolbox [5]. 1. Step 1. We shall construct a controller. The bilinear term of (45) gives where ~ denotes a state of the controller.