Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

TL;DR

This paper introduces a novel Lagrange basis formulation for Hermite matrix in polynomial matrix inequalities, significantly improving scaling and solver performance in static output feedback control design.

Contribution

It presents a new Lagrange basis approach for Hermite PMIs, enhancing problem scaling and solver efficiency in static output feedback control.

Findings

Improved problem scaling and convergence with the Lagrange basis approach

Reduced number of iterations for solver convergence

Enhanced stability and performance in benchmark tests

Abstract

Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the substantial improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behavior of PENNON.