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On convergence analysis of feedback control with integral action and discontinuous relay perturbation

Michael Ruderman

TL;DR

This paper analyzes the convergence of a third-order system with integral action under a discontinuous Coulomb-friction perturbation modeled as a relay. Using a hybrid dynamics framework with Filippov solutions and sliding-mode concepts, it derives a global asymptotic stability condition and characterizes the stiction region where the system can temporarily halt (ddot{y}=0) while switching occurs. The main contributions are the GAS condition $ab>c$, a detailed description of the stiction region $S_0$, and a stick-slip convergence analysis that depends on the linear subsystem's root structure, supported by numerical and experimental evidence. The results inform robust design of integral-action controllers in friction-perturbed mechanical systems and clarify when stick-slip versus exponential convergence occurs.

Abstract

We consider third-order dynamic systems which have an integral feedback action and discontinuous relay disturbance. More specifically for the applications, the focus is on the integral plus state-feedback control of the motion systems with discontinuous Coulomb-type friction. We recall the stiction region is globally attractive where the resulting hybrid system has also solutions in Filippov sense, while the motion trajectories remain in that idle state (called in tribology as stiction) until the formulated sliding-mode condition is violated by the growing integral feedback quantity. We analyze the conditions for occurrence of the slowly converging stick-slip cycles. We also show that the hybrid system is globally but only asymptotically stable, and almost always not exponentially. A particular case of the exponential convergence can appear for some initial values, assuming the characteristic equation of the linear subsystem has dominant real roots. Illustrative numerical examples are provided alongside with the developed analysis. In addition, a laboratory example is shown with experimental evidence to support the convergence analysis provided.

On convergence analysis of feedback control with integral action and discontinuous relay perturbation

TL;DR

This paper analyzes the convergence of a third-order system with integral action under a discontinuous Coulomb-friction perturbation modeled as a relay. Using a hybrid dynamics framework with Filippov solutions and sliding-mode concepts, it derives a global asymptotic stability condition and characterizes the stiction region where the system can temporarily halt (ddot{y}=0) while switching occurs. The main contributions are the GAS condition , a detailed description of the stiction region , and a stick-slip convergence analysis that depends on the linear subsystem's root structure, supported by numerical and experimental evidence. The results inform robust design of integral-action controllers in friction-perturbed mechanical systems and clarify when stick-slip versus exponential convergence occurs.

Abstract

We consider third-order dynamic systems which have an integral feedback action and discontinuous relay disturbance. More specifically for the applications, the focus is on the integral plus state-feedback control of the motion systems with discontinuous Coulomb-type friction. We recall the stiction region is globally attractive where the resulting hybrid system has also solutions in Filippov sense, while the motion trajectories remain in that idle state (called in tribology as stiction) until the formulated sliding-mode condition is violated by the growing integral feedback quantity. We analyze the conditions for occurrence of the slowly converging stick-slip cycles. We also show that the hybrid system is globally but only asymptotically stable, and almost always not exponentially. A particular case of the exponential convergence can appear for some initial values, assuming the characteristic equation of the linear subsystem has dominant real roots. Illustrative numerical examples are provided alongside with the developed analysis. In addition, a laboratory example is shown with experimental evidence to support the convergence analysis provided.
Paper Structure (12 sections, 2 theorems, 41 equations, 9 figures, 1 table)

This paper contains 12 sections, 2 theorems, 41 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Global asymptotic stability
  4. Properties of stiction region
  5. Analysis of stick-slip convergence
  6. Numerical examples
  7. Numerical examples
  8. Three distinct real roots as in bisoffi2017
  9. One real and two conjugate complex roots
  10. Conjugate complex roots as in ruderman2022
  11. Experimental evidence
  12. Summary and discussion

Key Result

Theorem 1

The system eq:ode, eq:relay with $a,\,b,\,c,\,\gamma > 0$ is globally asymptotically stable (GAS) if

Figures (9)

  • Figure 1: Structure of the third-order system with feedback relay perturbation.
  • Figure 2: Stiction region $S_0$ in $\bigl(x_2, x_1\bigr)$ coordinates.
  • Figure 3: Slipping phases after leaving $S_0$.
  • Figure 4: Phase portraits in the $(x_2,x_1)$ coordinates.
  • Figure 5: Time series of the $x_2$ state.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2