On optimal performance for linear time-varying systems

In this paper we consider the optimal disturbance attenuation problem and robustness for linear time-varying (LTV) systems. This problem corresponds to the standard optimal H^∞ problem for LTI systems. The problem is analyzed in the context of nest algebra of causal and bounded linear operators. In particular, using operator inner-outer factorization it is shown that the optimal disturbance attenuation problem reduces to a shortest distance minimization between a certain operators to the nest algebra in question. Banach space duality theory is then used to characterize optimal time-varying controllers. Alignment conditions in the dual are derived under certain conditions, and the optimum is shown to satisfy an all pass condition for LTV systems, therefore, generalizing a similar concept known to hold for LTI systems. The optimum is also shown to be equal to the norm of a time-varying Hankel operator analogous to the Hankel operator, which solves the optimal standard H^∞ problem. Duality theory leads to a pair of finite dimensional convex optimizations which approach the true optimum from both directions not only producing estimates within desired tolerances, but also allow the computation of optimal time varying controllers.