# Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis

## TL;DR

This paper derives formulas for the analytic center of the solution set of LMIs in passivity analysis, relating it to Riccati equations, and proposes numerical methods with robustness benefits for continuous- and discrete-time systems.

## Contribution

It introduces a novel characterization of the analytic center of LMI solution sets in passivity analysis and develops numerical algorithms to compute it.

## Key findings

- Analytic center described by related matrix equations.
- Numerical methods for computing the analytic center are developed.
- The analytic center exhibits robustness properties in passive system representation.

## 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 ascent and Newton-like 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.