Reduction of SDPs in H∞ control of SISO systems and performance limitations analysis

In SDP-based H_∞ control, we often encounter numerical difficulties when solving SDPs by various pieces of software. It is empirically known that such numerical difficulty occurs if an SDP at hand or its dual has no interior point feasible solutions, and this is indeed the case of some SDPs in H_∞ control. To conceive a way for getting around such numerical difficulties in a concrete problem setting, in this paper, we focus on the dual SDP for the H_∞ control problem of the transfer function (1 + PK)-1P and simplify it. More precisely, by actively using the information of unstable zeros (non-minimum phase zeros) of the plant P, we reduce the original dual SDP into a set of simplified SDPs each of which and its dual have interior point feasible solutions. In this way, we show by numerical experiments that reliable numerical computation can be done by SDP software. On the other hand, once we have obtained simplified SDPs, it becomes possible to further reduce them into the computation of maximum singular values of matrices determined by unstable zeros. In this way, if the number of unstable zeros is moderate, we can obtain analytical expressions of the best achievable H_∞ performance or its lower bounds in terms of the unstable zeros. Keywords: H_∞ control, SDP, numerical reliability, best achievable performance.