Game theoretic approach to H^∞ control for time-varying systems

A representation formula for all controllers that satisfy an L^∞-type constraint is derived for time-varying systems. It is now known that a formula based on two indefinite algebraic Riccati equations may be found for time-invariant systems over an infinite time support (see [J.C. Doyle et al., IEEE Trans. Automat, Control, AC-34 (1989), pp. 831-847]; [K. Glover and J.C. Doyle, Systems Control Lett., 11 (1988), pp. 167-172]; [K. Glover et al., SIAM J. Control Optim., 29 (1991), pp. 283-324]; [M. Green et al, SIAM J. Control Optim., 28 (1990), pp. 1350-1371]; [D.J.N. Limebeer et al., in Proc. IEEE conf. on Decision and Control, Austin, TX, 1988]; [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301-324]). In the time-varying case, two indefinite Riccati differential equations are required. A solution to the design problem exists if these equations have a solution on the optimization interval. The derivation of the representation formula illustrated in this paper makes explicit use of linear quadratic differential game theory and extends the work in [J.C. Doyle et al., IEEE Trans. Automat. Control, AC-34 (1989), pp. 831-847] and [G. Tadmor, Math. Control Systems Signal Processing, 3 (1990), pp. 301-324]. The game theoretic approach is particularly simple, in that the background mathematics required for the sufficient conditions is little more than standard arguments based on 'completing the square.'