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

We formulate H^∞ control problems for linear systems with delays in input and output, and discuss possibility of finite-dimensional characterizations of solutions. In the case when delay exists in control input and controlled output, first, we derive an output feedback H^∞ control formula of the central solution type, which is given by using solutions of finite- and infinite-dimensional Riccati matrix inequalities. Second, we show that, if the controlled output is chosen such that it satisfies the 'prediction condition', the solution to the infinite-dimensional Riccati inequality can be calculated by solving a finite-dimensional Riccati inequality. We provide a system theoretic interpretation for the prediction condition, and show that, if the prediction condition is satisfied, there is an equivalent H^∞ control problem for finite-dimensional linear systems with no delay. Finally, the equivalence result is extended to the case when delay exists also in measurement output.