Optimal Control with Multiple Objectives: The H2 Case

Most control system design problems involve making tradeoffs among competing objectives. In this paper we consider the control system synthesis problem that arises when the objective is to find a controller such that seveal closed loop transfer functions are small in an appropriate sense. To be more precise, given a linear-time invariant plant, the objective is to find all stabilizing compensators that are noninferior (or Pareto-optimal) with respect to a vector-valued H2 performance criterion. It is shown that this multiple criteria Optimization problem can be reduced to a simultaneous model matching problem in a Hilbert space. Moreover, state-space formulas are given to solve this model matching problem. We also show that in some particular cases non-inferior controllers have the same order as that of the given plant, and in these cases the above state-space formulae can be further simplified.