H_∞ control of discrete-time systems with multiple input delays

This paper is concerned with the finite horizon H_∞ full-information control for discrete-time systems with multiple control and exogenous input delays. We first establish a duality between the H_∞ full-information control and the H₂ smoothing of a stochastic backward system in Krein space. Like the duality between the llinear quadratic regulation (LQR) of linear systems without delays and the Kalman filtering, the established duality allows us to address complicated multiple input delay problems in a simple way. Indeed, by applying innovation analysis and standard projection in Krein space, in this paper we derive conditions under which the H_∞ full-information control is solvable. An explicit controller is constructed in terms of two standard Riccati difference equations of the same order as the original plant (ignoring the delays). As special cases, solutions to the H_∞ state feedback control problem for systems with delays only in control inputs and the H_∞ control with preview are obtained. An example is given to demonstrate the effectiveness of the proposed H_∞ control design.