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Robust control synthesis for uncertain linear systems with input saturation using mixed IQCs

Xu Zhang, Fen Wu

Abstract

This paper develops a robust control synthesis method for uncertain linear systems with input saturation in the framework of integral quadratic constraints (IQCs). The system is reformulated as a linear fractional representation (LFR) that captures both dead-zone nonlinearity and time-varying uncertainties. By combining mixed IQC-based dissipation inequalities with quadratic Lyapunov functions, sufficient conditions for robust stabilization are established. Compared with conventional approaches based on a single static sector condition for the dead-zone nonlinearity, the proposed method yields improved $\mathcal{L}_2$-gain performance through the use of scaled mixed IQCs. For systems subject to time-varying structured uncertainties, a new scaled bounded real lemma is further developed based on the IQC characterization. The resulting $\mathcal{H}_\infty$ synthesis conditions are expressed as linear matrix inequalities (LMIs), which are numerically tractable in all decision variables, including the scaling factors in the IQC multipliers. The proposed method is validated using a second-order uncertain system in linear fractional form, and its superiority over an anti-windup design is further illustrated by a cart-pendulum example.

Robust control synthesis for uncertain linear systems with input saturation using mixed IQCs

Abstract

This paper develops a robust control synthesis method for uncertain linear systems with input saturation in the framework of integral quadratic constraints (IQCs). The system is reformulated as a linear fractional representation (LFR) that captures both dead-zone nonlinearity and time-varying uncertainties. By combining mixed IQC-based dissipation inequalities with quadratic Lyapunov functions, sufficient conditions for robust stabilization are established. Compared with conventional approaches based on a single static sector condition for the dead-zone nonlinearity, the proposed method yields improved -gain performance through the use of scaled mixed IQCs. For systems subject to time-varying structured uncertainties, a new scaled bounded real lemma is further developed based on the IQC characterization. The resulting synthesis conditions are expressed as linear matrix inequalities (LMIs), which are numerically tractable in all decision variables, including the scaling factors in the IQC multipliers. The proposed method is validated using a second-order uncertain system in linear fractional form, and its superiority over an anti-windup design is further illustrated by a cart-pendulum example.
Paper Structure (13 sections, 4 theorems, 63 equations, 7 figures, 1 table)

This paper contains 13 sections, 4 theorems, 63 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. IQC preliminaries
  4. System description and control objective
  5. Main Results
  6. System Transformation
  7. IQC characterization
  8. Mixed IQC-Based State-Feedback Controller Synthesis
  9. Numerical Examples
  10. A Second-Order LFT Uncertain System
  11. Cart-spring pendulum system
  12. Conclusion
  13. $J$-Spectral Factorization

Key Result

Lemma 1

P2015 Consider a linear system $G\in\mathbb{RH}_\infty^{n_y\times n_u}$ and a bounded causal operator $\Delta\in\mathbb{R}^{n_q\times n_q}$. Assume that: Then, the feedback interconnection of $G$ and $\Delta$ is stable.

Figures (7)

  • Figure 1: Transformed uncertain LFT system.
  • Figure 2: Performance level $\gamma$ using different IQC's strategies as $\alpha$ increases.
  • Figure 3: State trajectory with the implementation of mixed IQC-based $\mathcal{H}_\infty$ controller.
  • Figure 4: Input saturation with the implementation of mixed IQC-based $\mathcal{H}_\infty$ controller.
  • Figure 5: Damped mass-spring-pendulum system.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Lemma 2
  • Remark 1
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Remark 2