Optimal Disturbance Rejection and Robustness for Infinite Dimensional LTV Systems

TL;DR

This paper develops a novel operator algebra framework for optimal disturbance rejection in infinite-dimensional linear time-varying systems, enabling the computation of robust controllers through finite-dimensional convex optimization.

Contribution

It introduces a new operator-theoretic approach that does not rely on state space models, providing a systematic way to compute optimal TV controllers using convex optimization.

Findings

Existence of optimal solutions shown via duality theory.

Optimal TV controllers are essentially unique under mild conditions.

Provides a practical method for computing controllers through finite-dimensional convex problems.

Abstract

In this paper, we consider the optimal disturbance rejection problem for possibly infinite dimensional linear time-varying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. This approach does not assume any state space representation and views LTV systems as causal operators. After reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, which satisfy a "time-varying" allpass or flatness condition. Under mild assumptions the optimal TV controller is shown to be essentially unique. Next, the concept of M-ideals of operators is used to show that the computation of time-varying (TV) controllers reduces to a search over compact TV Youla parameters. This involves the norm of a TV compact Hankel operator defined on the space of causal trace-class 2 operators and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is provided. These results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz. The final outcome of the approach developed here is that it leads to two tractable finite dimensional convex optimizations producing estimates to the optimum within desired tolerances, and a method to compute optimal time-varying controllers.