Algorithms for solving multidisk problems in H∞ optimization

The article concerns the H^∞ multidisk problem which in other coordinates would be called an integral quadratic constraint (IQC) problem. We are given m×m matrix valued functions K^p(e^iθ) for p=1,···ν, from which we form matrix valued performance functions Γ^p(e^iθ,f)=(K ^p(e^1θ)-f)^T(K^p(e ^iθ)-f), (1) Our objective is to (MDISK) find γ*⩾0 and continuous f* in H_m×m^∞ . In “projective coordinates” this is the same as trying to satisfy ν IQCs simultaneously. The well known H^∞ problem of control is a one disk case. H^∞ 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