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Dissipativity-based $\mathcal{L}_2$ gain-scheduled static output feedback design for rational LPV systems

Valessa V. Viana, Diego de S. Madeira, Thiago Alves Lima

TL;DR

The proposed approach uses the definition of strict QSR-dissipativity, Finsler’s Lemma, and the notion of linear annihilators to formulate conditions expressed in the form of polytopic linear matrix inequalities for determining the gain-scheduled static output feedback control for system stabilization.

Abstract

This paper proposes the design of gain-scheduled static output feedback controllers for the stabilization of continuous-time linear parameter-varying systems with $\mathcal{L}_2$-gain performance. The system is transformed into the form of a differential-algebraic representation which allows dealing with the broad class of systems whose matrices can present rational or polynomial dependence on the parameter. The proposed approach uses the definition of strict QSR dissipativity, Finsler's Lemma, and the notion of linear annihilators to formulate conditions expressed in the form of polytopic linear matrix inequalities for determining the gain-scheduled static output feedback control for system stabilization. One of the main advantages of the strategy is that it provides a simple design solution in a non-interactive manner. Furthermore, no restriction on the plant output matrix is imposed. Numerical examples highlight the effectiveness of the proposed method.

Dissipativity-based $\mathcal{L}_2$ gain-scheduled static output feedback design for rational LPV systems

TL;DR

The proposed approach uses the definition of strict QSR-dissipativity, Finsler’s Lemma, and the notion of linear annihilators to formulate conditions expressed in the form of polytopic linear matrix inequalities for determining the gain-scheduled static output feedback control for system stabilization.

Abstract

This paper proposes the design of gain-scheduled static output feedback controllers for the stabilization of continuous-time linear parameter-varying systems with -gain performance. The system is transformed into the form of a differential-algebraic representation which allows dealing with the broad class of systems whose matrices can present rational or polynomial dependence on the parameter. The proposed approach uses the definition of strict QSR dissipativity, Finsler's Lemma, and the notion of linear annihilators to formulate conditions expressed in the form of polytopic linear matrix inequalities for determining the gain-scheduled static output feedback control for system stabilization. One of the main advantages of the strategy is that it provides a simple design solution in a non-interactive manner. Furthermore, no restriction on the plant output matrix is imposed. Numerical examples highlight the effectiveness of the proposed method.

Paper Structure

This paper contains 13 sections, 3 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. LPV systems
  4. Differential-Algebraic Representation - DAR
  5. Finsler's Lemma
  6. Dissipativity
  7. GS-SOF Stabilization
  8. $\mathcal{L}_2$-gain Performance
  9. Optimization Problem
  10. Numerical Examples
  11. Example 1
  12. Example 2
  13. Conclusion

Key Result

Lemma 1

Consider $\mathcal{W} \subseteq \mathbb{R}^{n_s}$ a given polytopic set, and let $Q_d:\mathcal{W} \rightarrow \mathbb{R}^{n_q \times n_q}$ and $C_d:\mathcal{W} \rightarrow \mathbb{R}^{n_r \times n_q}$ be given matrix functions, with $Q_d$ symmetric. Then, the following statements are equivalent

Figures (1)

  • Figure 1: Example 1: States trajectories of the closed-loop system.

Theorems & Definitions (7)

  • Remark 1
  • Lemma 1
  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof