Primal robustness and semidefinite cones

TL;DR

This paper presents a convex optimization framework for robust stability analysis of LTI systems using semidefinite programming, simplifying existing methods and enabling direct inclusion of structured uncertainties.

Contribution

It reformulates robust stability analysis as a primal SDP using semidefinite cones, streamlining the process and enhancing pedagogical clarity.

Findings

Robust stability can be analyzed via primal SDP formulations.

Structured uncertainties are directly incorporated into the SDP framework.

Standard results like the KYP lemma are derived through SDP duality.

Abstract

This paper reformulates and streamlines the core tools of robust stability and performance for LTI systems using now-standard methods in convex optimization. In particular, robustness analysis can be formulated directly as a primal convex (semidefinite program or SDP) optimization problem using sets of gramians whose closure is a semidefinite cone. This allows various constraints such as structured uncertainty to be included directly, and worst-case disturbances and perturbations constructed directly from the primal variables. Well known results such as the KYP lemma and various scaled small gain tests can also be obtained directly through standard SDP duality. To readers familiar with robustness and SDPs, the framework should appear obvious, if only in retrospect. But this is also part of its appeal and should enhance pedagogy, and we hope suggest new research. There is a key lemma proving closure of a grammian that is also obvious but our current proof appears unnecessarily cumbersome, and a final aim of this paper is to enlist the help of experts in robust control and convex optimization in finding simpler alternatives.