Positive Trigonometric Polynomials on the Stability of Spatially Interconnected Systems

TL;DR

This paper introduces a novel stability analysis method for spatially interconnected systems using sum-of-squares decomposition of positive trigonometric polynomials, applicable to various topologies.

Contribution

It develops necessary and sufficient stability conditions for SISs with different topologies using SOS decomposition and semidefinite programming, considering global and local positivity.

Findings

The methods are effective for multiple interconnected structures.

Numerical examples demonstrate the approach's efficiency.

The approach simplifies stability analysis via SDP formulations.

Abstract

This paper is devoted to the stability analysis of spatially interconnected systems (SISs) via the sum-of-squares (SOS) decomposition of positive trigonometric polynomials. For each spatial direction of SISs, three types of interconnected structures are considered. Inspired by the idea of rational parameterization and robust stabilizability function, necessary and sufficient conditions are derived for establishing the stability of SISs with two different combined topologies respectively. For these results, the primary issue concerns the global or local positivity of trigonometric polynomials. SOS decomposition and generalized trace parameterization of positive trigonometric polynomials are utilized so that the addressed problems can be quantified by two semidefinite programs (SDPs). The proposed methods are applicable to all possible interconnected structures due to the assumption of spatial reversibility. Numerical examples are given to illustrate the efficiency of the derived theoretical results.