Nash Game and Mixed H2/H? Control by H. de O. Florentino, R.M. Sales, 1997 and by D.J.N. Limebeer, B.D.O. Anderson, and Hendel, 1994

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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

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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

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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?

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de O. Florentino and Sales result
z2
(J.C. Geromel, P.L.D. Peres, S.R. Souza, 1992)

• Theorem (P1)
• For a given ? gt0,
• 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

• Theorem (P2)
• For a given ? gt0,
• 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

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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

J1 minimum
J2 minimum
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Limebeer, Anderson and Hendels result
w0
w0 white noise w1 signal of bounded power

• 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)

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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

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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

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Conclusion

• Mixed H2/H? control problem can be formulated as
  a two-player nonzero sum game
• De O. Florentino and Saless 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