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Discontinuous integro-differential equations and sliding mode control

Andrey Polyakov

TL;DR

The paper addresses robust control for systems described by discontinuous integro-differential equations (IDEs), where standard Filippov regularization is not directly applicable to infinite-dimensional dynamics. It extends the Filippov framework to a class of IDEs with distributed delays and develops an affine-in-control equivalent-control method (Utkin-like) for IDEs, enabling finite-time sliding-mode control (SMC) design. The authors establish well-posedness, continuous dependence, and uniqueness results for Filippov IDEs, and demonstrate SMC design for systems with distributed input delay as well as IDEs representing PDEs in Banach spaces, notably applying to a heat equation example. This IDE-based approach provides a practical regularization and design toolkit for SMC in infinite-dimensional settings, bridging IDE modeling with robust sliding-mode strategies and enabling finite-time stabilization in PDE-controlled systems.

Abstract

The paper deals with analysis and design sliding mode control systems modeled by integro-differential equations. Filippov method and equivalent control approach are extended to a class of nonlinear discontinuous integro-differential equations. Sliding mode control algorithm is designed for a control system with distributed input delay. The obtained results are illustrated by numerical example.

Discontinuous integro-differential equations and sliding mode control

TL;DR

The paper addresses robust control for systems described by discontinuous integro-differential equations (IDEs), where standard Filippov regularization is not directly applicable to infinite-dimensional dynamics. It extends the Filippov framework to a class of IDEs with distributed delays and develops an affine-in-control equivalent-control method (Utkin-like) for IDEs, enabling finite-time sliding-mode control (SMC) design. The authors establish well-posedness, continuous dependence, and uniqueness results for Filippov IDEs, and demonstrate SMC design for systems with distributed input delay as well as IDEs representing PDEs in Banach spaces, notably applying to a heat equation example. This IDE-based approach provides a practical regularization and design toolkit for SMC in infinite-dimensional settings, bridging IDE modeling with robust sliding-mode strategies and enabling finite-time stabilization in PDE-controlled systems.

Abstract

The paper deals with analysis and design sliding mode control systems modeled by integro-differential equations. Filippov method and equivalent control approach are extended to a class of nonlinear discontinuous integro-differential equations. Sliding mode control algorithm is designed for a control system with distributed input delay. The obtained results are illustrated by numerical example.
Paper Structure (20 sections, 28 theorems, 129 equations, 10 figures, 1 table)

This paper contains 20 sections, 28 theorems, 129 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. State of the art
  3. Motivation example: SMC design for a heat system
  4. Contributions of the paper
  5. Organization of the paper
  6. Preliminaries: Discontinuous ODEs
  7. Some classes of functions
  8. Well-posed ODEs
  9. Regularization of ill-posed ODE by Filippov method
  10. Uniqueness of Filippov solutions of ODEs
  11. Equivalent control method for systems modeled by ODEs
  12. Discontinuous Integro-differential equations
  13. Well-posed IDEs
  14. Regularization of ill-posed IDEs by Filippov method
  15. Uniqueness of Filippov solutions of IDEs
  16. ...and 5 more sections

Key Result

Theorem 1

If $f$ is continuous then the IVP eq:ch2_generalsystem, eq:ch2_initialcondition has a classical solution defined, at least, locally in time.

Figures (10)

  • Figure 1: The states of the closed-loop system \ref{['eq:LTI_IDE']}-\ref{['eq:SMC_LTI']}
  • Figure 2: The output of the closed-loop system \ref{['eq:LTI_IDE']}-\ref{['eq:SMC_LTI']}
  • Figure 3: The control input of the closed-loop system \ref{['eq:LTI_IDE']}-\ref{['eq:SMC_LTI']}
  • Figure 4: The sliding mode indicator $\delta$ of the closed-loop system \ref{['eq:LTI_IDE']}- \ref{['eq:SMC_LTI']}
  • Figure 5: The states of the closed-loop system \ref{['eq:LTI_IDE']}-\ref{['eq:SMC_LTI']} with $\Phi=\mathbf{0}$
  • ...and 5 more figures

Theorems & Definitions (61)

  • Definition 1: Classical solution of ODE
  • Theorem 1: Peano Existence Theorem, 1890
  • Definition 2: Carathéodory solution of ODE
  • Theorem 2: Carathéodory Existence Theorem, 1918
  • Theorem 3: Continuity of solutions on initial data CoddingtonLevinson1955:Book, page 58
  • Theorem 4: Uniqueness of solutions
  • Example 1
  • Definition 3: Filippov solution
  • Example 1: continued
  • Theorem 5: Filippov Existence Theorem, 1960
  • ...and 51 more