Finite-dimensional characterizations of H∞ control for linear systems with delays in control input and controlled output

Abstract

We formulate an H∞ control problem for linear systems with delays in control input and controlled output and discuss possibility of finite-dimensional characterizations of the solution. First we derive a state feedback H∞ control formula constructed by using any solution to an infinite-dimensional Riccati matrix equation or an infinite-dimensional Riccati matrix inequality. Second, we show that if the controlled output is chosen such that it satisfies the “prediction condition”, the solution to the infinite-dimensional Riccati equation can be calculated by solving a finite-dimensional Riccati equation and similarly the infinite-dimensional Riccati inequality can be solved with a solution to a finite-dimensional Riccati inequality. We provide a system theoretic interpretation for the prediction condition, and show finally that, if the prediction condition is satisfied, there is an H∞ control problem for finite-dimensional linear systems which is equivalent to the problem formulated in this paper for linear systems with delays in control input and controlled output.