Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Disturbance-to-state stabilization by output feedback of nonlinear ODE cascaded with a reaction-diffusion equation

Abdallah Ben Abdallah, Mohsen Dlala

TL;DR

This work addresses the problem of stabilizing a cascaded nonlinear ODE coupled with a 1-D heat equation in the presence of in-domain and boundary disturbances, using only partial state measurements. The authors develop an observer-based dynamic output feedback that integrates a high-gain observer with backstepping for the coupled ODE-PDE system, ensuring global disturbance-to-state stabilization (DISS). They establish well-posedness via $C_0$-semigroup theory and Bari's theorem, and prove global exponential DISS through a composite Lyapunov analysis that combines separate Lyapunov functions for the observer and the plant. The results advance ISS theory for infinite-dimensional systems under output feedback and nonlinear ODE terms, with potential implications for robust control of PDE-ODE couplings in engineering applications.

Abstract

In this paper, we analyze the output stabilization problem for cascaded nonlinear ODE with $1-d$ heat diffusion equation affected by both in-domain and boundary perturbations. We assume that the only available part of states is the first components of the ODE-subsystem and one boundary of the heat-subsystem. The particularity of this system is two folds i) it contains a nonlinear additive term in the ODE-subsystem, and ii) it is affected by both boundary and in-domain perturbations signals. For such a system, and unlike the existing works, we succeeded to design an output observer-based feedback that guarantees not only asymptotic stabilization result but also a globally {\it disturbance-to-state stabilization} for our cascaded system. The output feedback is designed using an adequate backstepping transformation recently introduced for coupled ODE-heat equations combined with high-gain observer and high-gain controller.

Disturbance-to-state stabilization by output feedback of nonlinear ODE cascaded with a reaction-diffusion equation

TL;DR

This work addresses the problem of stabilizing a cascaded nonlinear ODE coupled with a 1-D heat equation in the presence of in-domain and boundary disturbances, using only partial state measurements. The authors develop an observer-based dynamic output feedback that integrates a high-gain observer with backstepping for the coupled ODE-PDE system, ensuring global disturbance-to-state stabilization (DISS). They establish well-posedness via -semigroup theory and Bari's theorem, and prove global exponential DISS through a composite Lyapunov analysis that combines separate Lyapunov functions for the observer and the plant. The results advance ISS theory for infinite-dimensional systems under output feedback and nonlinear ODE terms, with potential implications for robust control of PDE-ODE couplings in engineering applications.

Abstract

In this paper, we analyze the output stabilization problem for cascaded nonlinear ODE with heat diffusion equation affected by both in-domain and boundary perturbations. We assume that the only available part of states is the first components of the ODE-subsystem and one boundary of the heat-subsystem. The particularity of this system is two folds i) it contains a nonlinear additive term in the ODE-subsystem, and ii) it is affected by both boundary and in-domain perturbations signals. For such a system, and unlike the existing works, we succeeded to design an output observer-based feedback that guarantees not only asymptotic stabilization result but also a globally {\it disturbance-to-state stabilization} for our cascaded system. The output feedback is designed using an adequate backstepping transformation recently introduced for coupled ODE-heat equations combined with high-gain observer and high-gain controller.

Paper Structure

This paper contains 16 sections, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement and controller design
  3. Problem formulation
  4. Assumptions and discussion
  5. Well-posedness
  6. Global exponential disturbance-to-state stabilization
  7. Observer design
  8. Backstepping controller design
  9. Target closed-loop system
  10. Stability analysis of the closed-loop system
  11. Conclusion
  12. Proof of Theorem \ref{['Th1']}
  13. Proof o Lemma \ref{['invtransuw']}
  14. Proof of Lemma \ref{['lemk']}
  15. Proof of Lemma \ref{['V1']}
  16. ...and 1 more sections

Figures (1)

  • Figure 1: The block diagram of the cascaded PDE-ODE system (\ref{['sysdimfini']})-(\ref{['boundlcont']}) to boundary control disturbance with output $(CX(t), u(1, t))$.