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Multivariable Generalized Super-Twisting Algorithm Robust Control of Linear Time-Invariant Systems

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

TL;DR

This work addresses robust control of uncertain MIMO linear plants by formulating a Multivariable Generalized Super-Twisting Algorithm ($MGSTA$) within an $LMIs$ framework. The plant is decomposed into linear and nonlinear subsystems, each analyzed with dedicated Lyapunov functions, and a max-type non-differentiable Lyapunov function is used to ensure global stability and sliding-mode finite-time convergence. Exogenous Lipschitz disturbances and convex bounded parameter uncertainty in all system matrices are handled via convex LMIs, with a one-to-one variable change enabling tractable gain synthesis and a 2D search over design parameters. A fault-tolerant MGSTA controller is demonstrated on a 3-DOF mechanical chain, illustrating robust performance and practical applicability. The methodology provides a systematic, solvable route to compute robust gains and guaranteed performance in the presence of internal dynamics and actuator faults, with potential extensions to full-order output feedback and time-optimal reaching.

Abstract

This paper presents a novel procedure for robust control design of linear time-invariant systems using a Multivariable Generalized Super-Twisting Algorithm (MGSTA). The proposed approach addresses robust stability and performance conditions, considering convex bounded parameter uncertainty in all matrices of the plant state-space realization and Lipschitz exogenous disturbances. The primary characteristic of the closed-loop system, sliding mode finite-time convergence, is thoroughly examined and evaluated. The design conditions, obtained through the proposal of a novel max-type non-differentiable piecewise-continuous Lyapunov function are formulated as Linear Matrix Inequalities (LMIs), which can be efficiently solved using existing computational tools. A fault-tolerant MGSTA control is designed for a mechanical system with three degrees of freedom, illustrating the efficacy of the proposed LMI approach.

Multivariable Generalized Super-Twisting Algorithm Robust Control of Linear Time-Invariant Systems

TL;DR

This work addresses robust control of uncertain MIMO linear plants by formulating a Multivariable Generalized Super-Twisting Algorithm () within an framework. The plant is decomposed into linear and nonlinear subsystems, each analyzed with dedicated Lyapunov functions, and a max-type non-differentiable Lyapunov function is used to ensure global stability and sliding-mode finite-time convergence. Exogenous Lipschitz disturbances and convex bounded parameter uncertainty in all system matrices are handled via convex LMIs, with a one-to-one variable change enabling tractable gain synthesis and a 2D search over design parameters. A fault-tolerant MGSTA controller is demonstrated on a 3-DOF mechanical chain, illustrating robust performance and practical applicability. The methodology provides a systematic, solvable route to compute robust gains and guaranteed performance in the presence of internal dynamics and actuator faults, with potential extensions to full-order output feedback and time-optimal reaching.

Abstract

This paper presents a novel procedure for robust control design of linear time-invariant systems using a Multivariable Generalized Super-Twisting Algorithm (MGSTA). The proposed approach addresses robust stability and performance conditions, considering convex bounded parameter uncertainty in all matrices of the plant state-space realization and Lipschitz exogenous disturbances. The primary characteristic of the closed-loop system, sliding mode finite-time convergence, is thoroughly examined and evaluated. The design conditions, obtained through the proposal of a novel max-type non-differentiable piecewise-continuous Lyapunov function are formulated as Linear Matrix Inequalities (LMIs), which can be efficiently solved using existing computational tools. A fault-tolerant MGSTA control is designed for a mechanical system with three degrees of freedom, illustrating the efficacy of the proposed LMI approach.

Paper Structure

This paper contains 12 sections, 6 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem statement
  4. Robust MGSTA Lyapunov control design
  5. Lyapunov function candidates
  6. The linear subsystem $\mathcal{S}_L$
  7. The nonlinear subsystem $\mathcal{S}_N$
  8. Robust performance analysis
  9. LMI-based conditions
  10. The realization of second-order sliding modes
  11. Control of a chain of trailers
  12. Conclusion

Key Result

Lemma 1

If the matrix inequality is satisfied for scalars $\alpha>0$, $\rho>0$, and matrices $S>0$, $P>0$ of appropriate dimensions, then the inequality holds, for all $0 \neq \zeta \in {\mathbb R}^r$ and $0 \neq x \in {\mathbb R}^{2n}$.

Figures (3)

  • Figure 1: Chain of two active trailers and one passive trailer controlled by three actuators
  • Figure 2: Plot of the sliding variables $\sigma(t)$ and $\bar{z}(t)$, and of the control input $u(t)$ for all vertices.
  • Figure 3: Time evolution for all vertices of: (a) $q_a(t)$ (solid) and $q_d(t)$ (dashed); (b) $\dot{q}_a(t)$ (solid) and $\dot{q}_d(t)$ (dashed); (c) $e_y(t)$; (d) $e_v(t)$; (e) $e_p(t)$

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3