A linear matrix inequality approach to the discrete time mixed l 1/ℋ∞ control problem

This paper presents a solution method to the general mixed l₁ /ℋ_∞ problem for discrete time linear time invariant systems. The problem formulation involves finding a stabilizing feedback controller that minimizes the l₁ norm of a closed-loop transfer matrix subject to an ℋ_∞ 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 l₁ 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 l₁ norm. Numerical examples are provided to demonstrate the effectiveness of this approach