1. Nash Game and Mixed H 2 /H  Control by H. de O. Florentino, R.M. Sales, 1997 and by D.J.N. Limebeer, B.D.O. Anderson, and Hendel, 1994 Presensted by Hui-Hung Lin

2. Introduction Object in control system –Make some output behave in desired way by manipulating control input Want to determine what is (maximum) system gain Tow performance indexes –H 2 norms : well-motivated for performance –H  norms : measure of robust stability Consider both H 2 and H  norm in design controllers

3. Why Game Theory? Possible approaches –Minimize H 2 performance index under some H  constraints (P1) –Fix a priori H 2 performance level to optimize H  norms (P2) Motivation of Game Theory –Some performance level is lost if want a better disturbance rejection and vice-versa Idea of Game Theory –Two-player nonzero sum game –One for H 2 and the other for H 

4. de O. Florentino and Sale’s result Theorem (P1) For a given  >0, define the problem The following holds: 1. Above problem is convex 2. Being W* its optimal solution associated as  *, the gain is feasible solution for 3. If  , reduced to H 2 control problem z2z2 T K w ux zz Theorem (P2) For a given  >0, define the problem The following holds: 1.Above problem is convex 2.Being W* its optimal solution associated as  *, the gain is feasible solution for 3. If  , reduced to H  control problem (J.C. Geromel, P.L.D. Peres, S.R. Souza, 1992)

5. Two-player Nonzero Sum Game Apply Game Theory –Let  and  such that –Pay-off function for player are – Feasibility region (Theorem P1, P2) –Strategies set for the game,  –Nash equilibrium (  *,  *) , s.t. Existence of equilibrium point implies the existence of a controller K* such that Infinite NASH equilibrium points   ** ** ** ** mm ** ** mm (  * opt,  * opt ) J 1 minimum J 2 minimum

6. Limebeer, Anderson and Hendel’s result Goal –Find u * (t,x) s.t –u * (t,x) regulate x(t) to minimize the output energy when worst- case disturbance w * (t,x) is applied Minimize H 2 performance index over the set of feasible H  controllers (P1) w0w0 T K ux z w1w1 w 0 : white noise ; w 1 : signal of bounded power

7. Two-player Nonzero Sum Game One to reflect H  constraint, the other reflect H 2 optimality requirement Pay-off functions for players Nash equilibrium strategies u * (t,x), w * (t,x) satisfy –Iff exists P 1 (t)  0 and P 2 (t)  0 on [0,T] –u * (t,x) and w * (t,x) are specified by

8. Connection between H 2, H  and mixed H 2 /H  control problem Redefine pay-off function Solution is given by S 1 (t) and S 2 (t) satisfy H 2 (LQ) : set  = 0, and  H  : set  =  Mixed H 2 /H  : set  = 0 Infinite-horizon case (system is time- invariant): pair of cross-couple Riccati equations with (A,C) detectable or (A,B2) stabilizable for stability

9. Conclusion Mixed H2/H  control problem can be formulated as a two- player nonzero sum game De O. Florentino and Sales’s results can be sovled by convex optimization methods, but need croterion to choose optimal solution Results from Limebeer etc. need to solve a pair of cross-coupled (differential) Riccatti equations Overview paper by B. Vroemen and B. de Jager 1997