Algorithms for solving multidisk problems in H/sup ∞/ optimization

Abstract

The article concerns the H/sup /spl infin// multidisk problem which in other would be called an integral quadratic constraint (IQC) problem. We are given m/spl times/m matrix valued functions K/sup p/(e/sup i/spl theta//) for p=1,/spl middot//spl middot//spl middot//spl nu/, from which we form matrix valued performance functions /spl Gamma//sup p/(e/sup i/spl theta//,f)=(K/sup p/(e/sup 1/spl theta//)-f)/sup T/(K/sup p/(e/sup i/spl theta//)-f), (1) Our objective is to (MDISK) find /spl gamma/*/spl ges/0 and continuous f* in H/sub m/spl times/m//sup /spl infin//. In projective coordinates this is the same as trying to satisfy /spl nu/ IQCs simultaneously. The well known H/sup /spl infin// problem of control is a one disk case. H/sup /spl infin// multidisk problems occur whenever there are classical performance constraints which compete. LMI state space numerical solutions are typically extremely conservative compromises. The paper develops the mathematics needed to understand and develop numerical algorithms based on writing the equations that an optimum must satisfy and then invoking a Newton algorithm (or something similar) to solve these equations.