An Exact Solution to Continuous-Time Mixed 7-12/7-t , Control Problems

The mixed 3-12/3-1, control problem can be motivated as a nominal LQG optimal control problem subject to robust stability constraints, expressed in the form of an 3-1, norm bound. While at the present time there exist efficient methods to solve a modified problem consisting on minimizing an q p e r b o m d of the 2 2 cost subject to the IH, constraint, the original problem remains, to a large extent, still open. This paper contains a solution to general continuoustime mixed IHz/IH, problems, based upon constructing a family of approximating problems. Each of these approximations consists on a finite-dimensional convex optimization and an unconstrained standard IH, problem. The set oi solutions is such that in the limit the performance of the optimal controller is recovered, allowing 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, t h computation of a controller yielding a performance e-away from optimal requires the solution of a single optimization problem.