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Non-overshooting continuous in convergence sliding mode control of second-order systems

Michael Ruderman, Denis Efimov

Abstract

This paper proposes a novel nonlinear sliding mode state feedback controller for perturbed second-order systems. In analogy to a linear proportional-derivative (PD) feedback control, the proposed nonlinear scheme uses the output of interest and its time derivative. The control has only one free design parameter, and the closed-loop system is shown to possess uniform boundedness and finite-time convergence of trajectories in the presence of matched disturbances. We derive a strict Lyapunov function for the closed-loop control system with a bounded exogenous perturbation, and use it for both, the control parameter tuning and analysis of the finite-time convergence. The essential features of the proposed new control law is non-overshooting despite the unknown dynamic disturbances and the continuous control action during the convergence to zero equilibrium. Apart from the numerical results, a revealing experimental example is also shown in favor of the proposed control and in comparison with PD and sub-optimal nonlinear damping regulators.

Non-overshooting continuous in convergence sliding mode control of second-order systems

Abstract

This paper proposes a novel nonlinear sliding mode state feedback controller for perturbed second-order systems. In analogy to a linear proportional-derivative (PD) feedback control, the proposed nonlinear scheme uses the output of interest and its time derivative. The control has only one free design parameter, and the closed-loop system is shown to possess uniform boundedness and finite-time convergence of trajectories in the presence of matched disturbances. We derive a strict Lyapunov function for the closed-loop control system with a bounded exogenous perturbation, and use it for both, the control parameter tuning and analysis of the finite-time convergence. The essential features of the proposed new control law is non-overshooting despite the unknown dynamic disturbances and the continuous control action during the convergence to zero equilibrium. Apart from the numerical results, a revealing experimental example is also shown in favor of the proposed control and in comparison with PD and sub-optimal nonlinear damping regulators.
Paper Structure (12 sections, 5 theorems, 29 equations, 7 figures, 1 table)

This paper contains 12 sections, 5 theorems, 29 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Novel nonlinear feedback control
  3. Problem statement
  4. Control law
  5. Exemplary
  6. Analysis
  7. Existence of solutions
  8. Analysis of stability
  9. Finite-time convergence and control boundedness
  10. Numerical simulations
  11. Experimental control results
  12. Summary

Key Result

Proposition 1

The system eq:CL_syst admits an unique solution for any $x(0)\in\mathbb{X}$ and $\|d\|_{\infty}\leq D$ well defined with $x(t)\in\mathbb{X}$ for all $t\geq0$.

Figures (7)

  • Figure 1: Phase portrait exemplary of the closed-loop system \ref{['eq:CL_syst']}.
  • Figure 2: Response of the dynamic state $x_{2}(t)$ in (a) and control signal $u$ in (b) of the proposed new nonlinear control \ref{['eq:CL_syst']}, sub-optimal nonlinear damping control ruderman2021a, and critically damped linear state feedback control.
  • Figure 3: Low bound of control gain depending on perturbation bound.
  • Figure 4: Output response of the perturbed control system \ref{['eq:CL_syst']} in (a), and the control signal versus disturbance value by using: thresholding scheme \ref{['eq:contFirst']} for zero in (b), and regularization scheme \ref{['eq:contSecond']} in (c).
  • Figure 5: Experimental setup of the actuator system.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • Lemma 7
  • ...and 1 more