MIMO optimal control design: The interplay between the h₂ and the ℓ₁ norms

In this paper we consider controller design methods which can address directly the interplay between the H₂ and ℓ₁ performance of the closed loop. The development is devoted to multi-input/multi-output (MIMO) systems. Two relevant multiobjective performance problems are considered each being of interest in its own right. In the first, termed as the combination problem, a weighted sum of the ℓ₁ norms and the square of the H₂ norms of a given set of input-output transfer functions constituting the closed loop is minimized. It is shown that, in the one-block case, the solution can be obtained via a finite-dimensional quadratic optimization problem which has an a priori known dimension. In the four-block case, a method of computing approximate solutions within any a priori given tolerance is provided. In the second, termed as the mixed problem, the H₂ performance of the closed loop is minimized subject to an ℓ₁ constraint. It is shown that approximating solutions within any a priori given tolerance can be obtained via the solution to a related combination problem.