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A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers

Pedro Henrique Silva Coutinho, Iury Bessa, Victor Hugo Pereira Rodrigues, Tiago Roux Oliveira

Abstract

This paper deals with sliding mode control for multivariable polytopic uncertain systems. We provide systematic procedures to design variable structure controllers (VSCs) and unit-vector controllers (UVCs). Based on suitable representations for the closed-loop system, we derive sufficient conditions in the form of linear matrix inequalities (LMIs) to design the robust sliding mode controllers such that the origin of the closed-loop system is globally stable in finite time. Moreover, by noticing that the reaching time depends on the initial condition and the decay rate, we provide convex optimization problems to design robust controllers by considering the minimization of the reaching time associated with a given set of initial conditions. Two examples illustrate the effectiveness of the proposed approaches.

A Systematic LMI Approach to Design Multivariable Sliding Mode Controllers

Abstract

This paper deals with sliding mode control for multivariable polytopic uncertain systems. We provide systematic procedures to design variable structure controllers (VSCs) and unit-vector controllers (UVCs). Based on suitable representations for the closed-loop system, we derive sufficient conditions in the form of linear matrix inequalities (LMIs) to design the robust sliding mode controllers such that the origin of the closed-loop system is globally stable in finite time. Moreover, by noticing that the reaching time depends on the initial condition and the decay rate, we provide convex optimization problems to design robust controllers by considering the minimization of the reaching time associated with a given set of initial conditions. Two examples illustrate the effectiveness of the proposed approaches.

Paper Structure

This paper contains 11 sections, 2 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Robust Variable Structure Control Design
  3. The proposed robust VSC design condition
  4. Optimization issues related to the reaching time of VSC
  5. Robust Unit Vector Control Design
  6. The proposed robust UVC design condition
  7. Optimization issues related to the reaching time of UVC
  8. Numerical Results
  9. Example 1: Robotics visual servo system
  10. Example 2: Underwater ROV system
  11. Conclusion

Key Result

Theorem 1

Consider the uncertain system eq:plant and the sliding-mode controller eq:controller. Given $\xi > 0$, if there exist diagonal matrices $W \in \mathbb{R}^{n \times n}$ and $X \in \mathbb{R}^{n \times n}$, a symmetric matrix $R \in \mathbb{R}^{n \times n}$, and a full matrix $Z \in \mathbb{R}^{m \tim then, the origin of the closed-loop system eq:closed-loop with $K = Z X^{-1}$ is globally asymptoti

Figures (4)

  • Figure 1: Regions $\mathcal{B}_{\mathrm{vsc}}$ and $\mathcal{B}_{\mathrm{uvc}}$ and closed-loop trajectories with the VSC (in blue) and with the UVC (in red) -- Example 1.
  • Figure 2: States of the closed-loop robotics visual servo system with the robust (a) VSC and (b) UVC -- Example 1.
  • Figure 3: Upper-bound on the reaching times for the (a) VSC and (b) UVC -- Example 2.
  • Figure 4: States of the closed-loop over-actuated ROV system with the robust (a) VSC and (b) UVC -- Example 2.

Theorems & Definitions (4)

  • Remark 1
  • Definition 1: see hsu2000matrix
  • Theorem 1
  • Theorem 2