Existence of Continuous or Constant Finsler's Variables for Parameter-Dependent Systems

TL;DR

This paper provides conditions under which the auxiliary variables in parameter-dependent LMIs can be chosen as continuous or constant, simplifying analysis and reducing computational complexity in control and optimization problems.

Contribution

It introduces sufficient conditions ensuring that Finsler's lemma variables depend simply on parameters, avoiding unnecessary complexity in robust stability analysis.

Findings

Conditions for continuous dependence of variables on parameters

Conditions for constant variables in parameter-dependent LMIs

Reduction in computational complexity without loss of generality

Abstract

Finsler's lemma is a classic mathematical result with applications in control and optimization. When the lemma is applied to parameter-dependent LMIs, as such those that arise from problems of robust stability, the extra variables introduced by this lemma also become dependent on this parameter. This technical note presents some sufficient conditions which ensure, without losing generality, that these extra variables can assume a simple functional dependence on the parameters as continuity or even independence. The results allow avoiding an unnecessary use of a more functionally complicated parameter-dependent variable that increases the search computational burden without reducing the conservatism of the solution.