A Linear Matrix Inequality Approach to Discrete Time Mixed ℓ1 - H∞ Problem

This paper presents a solution method to the general mixed l/sub 1///spl Hscr//sub /spl infin// problem for discrete time linear time invariant systems. The problem formulation involves finding a stabilizing feedback controller that minimizes the l/sub 1/ norm of a closed-loop transfer matrix subject to an /spl Hscr//sub /spl infin// norm constraint on another closed-loop transfer matrix. It is shown that for one-block problem the optimal solution can be approximated arbitrarily closely, in terms of the closed-loop l/sub 1/ norm, by solving a sequence of finite dimensional convex optimization problems over linear matrix inequalities. For multi-block problem we have also obtained superoptimal and suboptimal solutions which give lower bounds and upper bounds convergent to the optimal closed-loop l/sub 1/ norm. Numerical examples are provided to demonstrate the effectiveness of this approach.