An exact solution to general four-block discrete-time mixed ℋ 2/ℋ∞ problems via convex optimization

The mixed ℋ₂ℋ_∞ control problem can be motivated as a nominal LQG optimal control problem subject to robust stability constraints, expressed in the form of an ℋ_∞ norm bound. This paper contains a solution to a general four-block mixed ℋ₂/ℋ_∞ problem, based upon constructing a family of approximating problems. Each one of these problems consists of a finite-dimensional convex optimization and an unconstrained standard ℋ_∞ problem. The set of solutions is such that in the limit the performance of the optimal controller is recovered, allowing one to establish the existence of an optimal solution. Although the optimal controller is not necessarily finite-dimensional, it is shown that a performance arbitrarily close to the optimal can be achieved with rational controllers. Moreover, the computation of a controller yielding a performance ε-away from optimal requires the solution of a single optimization problem, a task that can be accomplished in polynomial time