Presensted by Hui-Hung Lin

1. Nash Game and Mixed H2/H Controlby H. de O. Florentino, R.M. Sales, 1997and byD.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 • H2 norms : well-motivated for performance • H norms : measure of robust stability • Consider both H2 and H norm in design controllers

3. Why Game Theory? • Possible approaches • Minimize H2 performance index under some H constraints (P1) • Fix a priori H2 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 H2 and the other for H

4. w T z u x K de O. Florentino and Sale’s result z2 Theorem (P1) For a given  >0, define the problem The following holds: • Above problem is convex • Being W* its optimal solution associated as *, the gain is feasible solution for 3. If  , reduced to H2 control problem (J.C. Geromel, P.L.D. Peres, S.R. Souza, 1992) Theorem (P2) For a given  >0, define the problem The following holds: • Above problem is convex • Being W* its optimal solution associated as *, the gain is feasible solution for 3. If  , reduced to H control problem

5. * * (*opt, *opt)  m * * m *  * 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 J1 minimum J2 minimum

6. z w1 T u x K Limebeer, Anderson and Hendel’s result w0 • 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 H2 performance index over the set of feasible H controllers (P1) w0: white noise ; w1: signal of bounded power

7. Two-player Nonzero Sum Game • One to reflect H constraint , the other reflect H2 optimality requirement • Pay-off functions for players • Nash equilibrium strategies u*(t,x), w*(t,x) satisfy • Iff exists P1(t)  0 and P2(t)0 on [0,T] • u*(t,x) and w*(t,x) are specified by

8. Connection between H2, H and mixed H2/Hcontrol problem • Redefine pay-off function • Solution is given by • S1(t) and S2(t) satisfy • H2(LQ) : set  = 0, and  • H: set  =  • Mixed H2/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