Numerical Computation of the Minimal $\mathbfH_\infty$ Norm of the Discrete-Time Output Feedback Control Problem

Numerical computation of the minimal ${\bf H}_\infty$ norm for the discrete-time output feedback control problem is considered. First of all, a lower bound is established in terms of the ${\bf H}_\infty$ norm of certain stable transfer functions. Since the computational work in the evaluation of the ${\bf H}_\infty$ norm of a stable transfer function involves the determination of unimodular eigenvalues of the associated parameterized symplectic pencil of matrices, we discuss in detail how to get a numerically reliable solution when the pencil becomes singular as the parameter varies. Next, by exploiting the stable deflating subspaces of the two parameterized symplectic pencils derived by Iglesias and Glover in 1991, we characterize the critical points such that the corresponding two discrete-time Riccati equations (with parameter r) have stabilizing positive semidefinite solutions and satisfy certain inertia conditions. This characterization makes some kind of secant method applicable for finding these critical points. Finally, using the maximum of the two critical points as the starting point, we then devise an algorithm for computing the optimal (minimal) ${\bf H}_\infty$ norm by considering a secant method applied to the spectral radius (function of r) of the product of the corresponding two Riccati solutions. Numerical aspects are addressed throughout. In addition, some algebraic verifiable examples are given.