On optimal performance for linear time-varying systems

Abstract

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 operator 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 allpass 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.