An exact solution to continuous-time mixed H2/H∞ control problems

The mixed H₂/H_∞ 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. While at the present time there exist efficient methods to solve a modified problem consisting of minimizing an upper bound of the H₂ cost subject to the H_∞ constraint, the original problem remains, to a large extent, still open. This paper contains a solution to general continuous-time mixed H₂/H_∞ problems, based upon constructing a family of approximating problems. Each of these approximations 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 us 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 (and thus physically implementable) controllers. Moreover, the computation of a controller yielding a performance ε-away from optimal requires the solution of a single optimization problem