SIAM Journal on Control and Optimization, Vol.53, No.4, 1734-1767, 2015
H-infinity PROBLEM WITH NONSTRICT INEQUALITY AND ALL SOLUTIONS: INTERPOLATION APPROACH
For given rational matrices V-a, U-a, V-b, U-b, we find necessary and sufficient conditions for existence of a stable rational matrix Phi satisfying parallel to Phi parallel to(infinity) <= 1, V-a Phi = U-a, and Phi V-b = U-b. A condition is the positive semidefiniteness of a matrix, denoted by R. We present a parametrization of all problem solutions. A property of the proposed algorithm is, as a first step, to reduce the problem to a minimal realization, by an orthogonal transformation. Another property is the ability to transform the problem into one with constant matrices, by another orthogonal transformation. A problem motivation is the optimal H-infinity control problem of descriptor systems. We show by an example that the existing numerical H-infinity control optimization algorithms, which solve the problem of obtaining stabilizing controllers such that parallel to Phi parallel to(infinity) <= gamma, where Phi is the closed loop transfer matrix, and gamma is slightly greater than the optimal one, compute controllers such that the closed loop system suffers from lack of stability robustness. Our proposed algorithm doesn't require gamma to be greater than the optimal one, and the closed loop system has the property of stability robustness. Consequences of the singularity of matrix R, which property generically appears for the optimal gamma, are (1) that parallel to Phi( j omega)parallel to = gamma for all real. and all optimal controllers, and (2) if the measured output is single, or the control input is single, then the H-infinity controller is unique. A numerical example is presented to illustrate the H-infinity control optimization algorithm.