State-space formulae for all the stabilizing discrete controllers that satisfy anH∞norm bound Part 1. Main results and the full information problem

Abstract

Given a linear discrete time system, necessary and sufficient conditions for the existence of a linear or nonlinear controller, which satisfies a specified closed-loop H ∞ norm bound, are stated in terms of the solutions to two uncoupled algebraic discrete Riccati equations of the order of the system. State-space formulae for all the stabilizing controllers, some of the order of the system, which satisfy an achievable bound are also given. In this first part of the paper, new growth and convergence properties of the solution to a time-varying discrete Riccati equation are used to derive the necessary and sufficient conditions for the existence of a solution in the full information case. These properties generalize those of the corresponding discrete Riccati equation related to the LQR problem. In the second part of the paper, a non-standard estimation problem is posed and solved. This problem arises as a novel reinterpretation of the H ∞ norm objective. Its solution can be used to reduce a partial information to a full information control problem. Our approach to solving the H ∞ output feedback control problem is based on this result. This approach, the dual of that typically used in H ∞ control theory, yields significant new insights into the H ∞regulation problem. Minimax difference game theory is central to the developments of this paper and it is shown that minimax, not saddle point game theory, provides the proper framework to solve the H ∞ regulation problem.