Controller Design to Minimize a Composite Measure of the ℓ1, ℋ2 and ℓ∞ Norms

In this paper duality theory is employed to show that the problem of minimizing a given linear combination of the ℓ1 norm, the square of the ℋ2 norm, and the ℓ∞ norm of the closed loop over all stablizing controllers is equivalent to a finite dimensional convex optimization problem. It is shown that the dimension can be determined a priori. Relation to Pareto optimality is established and the continuity of the optimal solution with respect to changes in the coefficients of the linear combination is proven. The problem is studied for a single input single output, discrete time, linear time invariant system.