${\mathscr H}_\infty$ Problem with Nonstrict Inequality and All Solutions: Interpolation Approach

Abstract

For given rational matrices ${\boldsymbol V}_a$, ${\boldsymbol U}_a$, ${\boldsymbol V}_b$, ${\boldsymbol U}_b$, we find necessary and sufficient conditions for existence of a stable rational matrix ${\boldsymbol\Phi}$ satisfying $\Vert{\boldsymbol\Phi}\Vert_\infty \leq 1$, ${\boldsymbol V}_a{\boldsymbol\Phi} = {\boldsymbol U}_a$, and ${\boldsymbol\Phi} {\boldsymbol V}_b={\boldsymbol U}_b$. A condition is the positive semidefiniteness of a matrix, denoted by ${\cal R}$. 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 ${\mathscr H}_\infty$ control problem of descriptor systems. We show by an example that the existing numerical ${\mathscr H}_\infty$ control optimization algorithms, which solve the proble...