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Optimal and Robust Multivariable Reaching Time Sliding Mode Control Design

J. C. Geromel, L. Hsu, E. V. L. Nunes

TL;DR

The paper develops LMI-based methods to design optimal, robust state-feedback gains for two finite-time control strategies, Variable Structure Control (VSC) and Unit Vector Control (UVC), ensuring global finite-time convergence of uncertain multivariable systems under convex parameter uncertainty and norm-bounded disturbances. By using a Persidiskii-type Lyapunov function for VSC and a quadratic Lyapunov function for UVC, it derives reach-time bounds and solves the synthesis problems via convex LMIs over vertex representations of the uncertainty set, including a line search on scalar variables. Two practical demonstrations—visual robotics servoing and an underwater ROV—show that UVC provides tighter reach-time estimates while VSC can be more sensitive to uncertainty. The framework is computationally efficient and compatible with standard LMI solvers, offering a practical path for robust minimum-time control of multivariable systems.

Abstract

This paper addresses two minimum reaching time control problems within the context of finite stable systems. The well-known Variable Structure Control (VSC) and Unity Vector Control (UVC) strategies are analyzed, with the primary objective of designing optimal and robust state feedback gains that ensure minimum finite time convergence to the origin. This is achieved in the presence of convex bounded parameter uncertainty and norm-bounded exogenous disturbances. In both cases, the optimality conditions are expressed through Linear Matrix Inequalities (LMIs), which are solved efficiently within the framework of multivariable systems using existing numerical tools. The theoretical results are demonstrated with two practically motivated examples.

Optimal and Robust Multivariable Reaching Time Sliding Mode Control Design

TL;DR

The paper develops LMI-based methods to design optimal, robust state-feedback gains for two finite-time control strategies, Variable Structure Control (VSC) and Unit Vector Control (UVC), ensuring global finite-time convergence of uncertain multivariable systems under convex parameter uncertainty and norm-bounded disturbances. By using a Persidiskii-type Lyapunov function for VSC and a quadratic Lyapunov function for UVC, it derives reach-time bounds and solves the synthesis problems via convex LMIs over vertex representations of the uncertainty set, including a line search on scalar variables. Two practical demonstrations—visual robotics servoing and an underwater ROV—show that UVC provides tighter reach-time estimates while VSC can be more sensitive to uncertainty. The framework is computationally efficient and compatible with standard LMI solvers, offering a practical path for robust minimum-time control of multivariable systems.

Abstract

This paper addresses two minimum reaching time control problems within the context of finite stable systems. The well-known Variable Structure Control (VSC) and Unity Vector Control (UVC) strategies are analyzed, with the primary objective of designing optimal and robust state feedback gains that ensure minimum finite time convergence to the origin. This is achieved in the presence of convex bounded parameter uncertainty and norm-bounded exogenous disturbances. In both cases, the optimality conditions are expressed through Linear Matrix Inequalities (LMIs), which are solved efficiently within the framework of multivariable systems using existing numerical tools. The theoretical results are demonstrated with two practically motivated examples.

Paper Structure

This paper contains 9 sections, 2 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Statement
  4. Variable Structure Control (VSC) Design
  5. Unit Vector Control (UVC) Design
  6. Illustrative examples
  7. Visual robotics servoing
  8. Underwater Remotely Operated Vehicle
  9. Conclusions

Key Result

Theorem 1

Let $B \in \mathbb{B}$ be given. If there exist a diagonal matrix $0 < P_d \in {\mathbb R}^{n \times n}$ and positive scalars $\omega \mu > \delta^2$ such that the inequality holds, then the origin $\sigma = 0$ of the closed-loop system is globally finite-time stable for all exogenous disturbances satisfying (eq03). In the affirmative case, the reaching time satisfies $T_r(\sigma_0) \leq 2\mu v(\

Figures (1)

  • Figure 1: The plot of $\theta$ against $\rho$ for UVC design with the closed-loop system free of exogenous disturbance ($\delta =0$) and subject to exogenous norm bounded disturbance with ($\delta = 2.0$).

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2