On the solutions to complex parameter-dependent LMIs involved in the stability analysis of 2D discrete models

TL;DR

This paper adapts a known approximation result for parameter-dependent LMIs to the stability analysis of 2D discrete models, focusing on complex parameters and their impact on LMI solutions.

Contribution

It extends Bliman's real-parameter LMI approximation results to complex parameters in the context of 2D Roesser model stability analysis.

Findings

LMI solutions can polynomially depend on a complex parameter on the unit circle.

The stability condition for 2D Roesser models can be relaxed into a parameter-dependent LMI.

Precautions are necessary when extending real-parameter results to complex parameters.

Abstract

The aim of this short communique is to adapt a result established by Bliman, related to the possible approximation of the solutions to real-parameter-dependent linear matrix inequalities (LMIs), to the special context of stability analysis of 2D discrete Roesser models. While Bliman considered the case of LMIs involving several real parameters, which is especially crucial for the analysis of linear systems against parametric deflections, the stability of Roesser models leads to consider LMIs with only one single complex parameter. Extending the results from real parameters to complex ones is not straightforward in our opinion. This is why the present note discusses precautions to be taken concerning this case before applying the results in a 2D context. Actually, it is shown that a well-known condition for structural stability of a 2D discrete Roesser can be relaxed into an LMI system whose solution polynomially depends on a single complex parameter over the unit circle.