An Application of QSR-Dissipativity to the Problem of Static Output Feedback Robust Stabilization of Nonlinear Systems

Abstract

In this work we deal with the asymptotic stabilization problem of polynomial (and rational) input-affine systems subject to parametric uncertainties. The problem of linear static output feedback (SOF) control synthesis is handled, having as a prerequisite a differential algebraic representation (DAR) of the plant. Using the property of strict QSR-dissipativity, theFinsler's Lemma and the notion of linear annihilators we introduce a new dissipativity-based strategy for robust stabilization which determines a static feedback gain by solving a simple linear semidenite program on a polytope. At the same time, an estimate of the closed-loop domain of attraction is given in terms of an ellipsoidal set. The novelty of the proposed approach consists in this combination of dissipativity theory and powerful semidenite programming(SDP) tools allowing for a simple solution of the challenging problem of static output feedback design for nonlinear systems. A numerical example allows the reader to verify the applicability of the proposed technique.

Necessary and Sufficient Dissipativity-Based Conditions for Feedback Stabilization

“…Theorem 2: The LTI system ( 36) is linear SOF stabilizable if and only if (40) is feasible with (R, P, N ) > 0 and ∆ = 0, where…”

“…Once a solution (P, N, Q, S, R) for ( 40) with ∆ = 0 is found, the combination (δP, δN, δQ, δS, δR) with any δ ∈ R + also solves (40) resulting in the same gain K and the same ∆. Then, without loss of generality, one can consider R = I.…”

“…subject to X d ≥ 0, (40), [36]. By solving optimization problem (54), one guarantees that the necessary condition ∆ ≤ 0 is fulfilled and, at the same time, tries to come as close as possible to the necessary and sufficient constraint that ∆ = 0.…”

“…A gain K is given by (42). Remark 2: Note that a necessary LMI condition for SOF stabilizability is given by simultaneously fulfilling X d ≥ 0 and (40).…”

“…An example in Section VI-B shows how that can be obtained simultaneously to (40) via semidefinite programming. Corollary 2: The LTI system (36) is linear SOF stabilizable if and only if (40) is feasible with (R, P, N ) > 0 and ∆ > 0, where ∆ is given by (41). A stabilizing gain is given by (42).…”