Computation of the Analytic Center of the Solution Set of the Linear Matrix Inequality Arising in Continuous- and Discrete-Time Passivity Analysis

Daniel Bankmann; Volker Mehrmann; Yurii Nesterov; Paul Van Dooren

doi:10.1007/s10013-020-00427-x

Computation of the Analytic Center of the Solution Set of the Linear Matrix Inequality Arising in Continuous- and Discrete-Time Passivity Analysis

Vietnam J Math. 2020;48(4):633-659. doi: 10.1007/s10013-020-00427-x. Epub 2020 Jul 23.

Authors

Daniel Bankmann¹, Volker Mehrmann¹, Yurii Nesterov², Paul Van Dooren²

Affiliations

¹ Institut für Mathematik MA 4-5, TU Berlin, Str. des 17. Juni 136, D-10623 Berlin, Germany.
² Department of Mathematical Engineering, Université catholique de Louvain, Louvain-La-Neuve, Belgium.

Abstract

In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous- and discrete-time systems. Numerical methods are derived to solve these equations via steepest descent and Newton methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.

Keywords: Algebraic Riccati equation; Analytic center; Linear matrix inequality; Passivity; Positive real system; Robustness.