H∞ synthesis with unstable weighting filters: An LMI solution

Synthesis of a controller is considered for achieving a desired level of guaranteed worst-case energy gain performance in the case of unstable (input and output) weighting filters. The solution of this problem is shown to require a proper replication of the unstable filter dynamics accompanied by a controller of free structure. By merging the replicated filter dynamics with the plant, the problem is reformulated as the design of the free part of the controller in a way to stabilize an extended plant and ensure the considered performance objective. A solution is then provided based on LMI conditions accompanied by affine equation constraints. A procedure is also outlined for the synthesis of a controller whose order is equal to the order of the plant plus the orders of the filters.