Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Safe Adaptive Control of Parabolic PDE-ODE Cascades

Yun Jiang, Ji Wang

TL;DR

This work addresses safety-critical control of parabolic PDE-ODE cascades under parametric uncertainty by deploying a safe adaptive boundary controller built on an adaptive Control Barrier Function (aCBF) and Batch Least-Squares Identification (BaLSI). The method combines a distal-ODE transform and PDE backstepping to produce a stable target system, while an update mechanism with a triggering rule identifies the unknown parameters $\lambda$ and $b$ in finite time, ensuring both safety via barrier functions and convergence to zero of all states. Key contributions include extending safe adaptive control to parabolic PDE-ODE cascades with in-domain instabilities, proving safety guarantees and finite-time parameter identification, and validating the approach through numerical simulations. The practical impact lies in enabling robust, safety-guaranteed control for diffusion-dominant systems in engineering where both PDE and ODE subsystems exhibit uncertainty and safety constraints must be maintained during transients.

Abstract

In this paper, we propose a safe adaptive boundary control strategy for a class of parabolic partial differential equation-ordinary differential equation (PDE-ODE) cascaded systems with parametric uncertainties in both the PDE and ODE subsystems. The proposed design is built upon an adaptive Control Barrier Function (aCBF) framework that incorporates high-relative-degree CBFs together with a batch least-squares identification (BaLSI)-based adaptive control that guarantees exact parameter identification in finite time. The proposed control law ensures that: (i) if the system output state initially lies within a prescribed safe set, safety is maintained for all time; otherwise, the output is driven back into the safe region within a preassigned finite time; and (ii) convergence to zero of all plant states is achieved. Numerical simulations are provided to demonstrate the effectiveness of the proposed approach.

Safe Adaptive Control of Parabolic PDE-ODE Cascades

TL;DR

This work addresses safety-critical control of parabolic PDE-ODE cascades under parametric uncertainty by deploying a safe adaptive boundary controller built on an adaptive Control Barrier Function (aCBF) and Batch Least-Squares Identification (BaLSI). The method combines a distal-ODE transform and PDE backstepping to produce a stable target system, while an update mechanism with a triggering rule identifies the unknown parameters and in finite time, ensuring both safety via barrier functions and convergence to zero of all states. Key contributions include extending safe adaptive control to parabolic PDE-ODE cascades with in-domain instabilities, proving safety guarantees and finite-time parameter identification, and validating the approach through numerical simulations. The practical impact lies in enabling robust, safety-guaranteed control for diffusion-dominant systems in engineering where both PDE and ODE subsystems exhibit uncertainty and safety constraints must be maintained during transients.

Abstract

In this paper, we propose a safe adaptive boundary control strategy for a class of parabolic partial differential equation-ordinary differential equation (PDE-ODE) cascaded systems with parametric uncertainties in both the PDE and ODE subsystems. The proposed design is built upon an adaptive Control Barrier Function (aCBF) framework that incorporates high-relative-degree CBFs together with a batch least-squares identification (BaLSI)-based adaptive control that guarantees exact parameter identification in finite time. The proposed control law ensures that: (i) if the system output state initially lies within a prescribed safe set, safety is maintained for all time; otherwise, the output is driven back into the safe region within a preassigned finite time; and (ii) convergence to zero of all plant states is achieved. Numerical simulations are provided to demonstrate the effectiveness of the proposed approach.
Paper Structure (22 sections, 4 theorems, 95 equations, 7 figures)

This paper contains 22 sections, 4 theorems, 95 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Boundary Control of Parabolic PDEs
  3. Safe Adaptive Control
  4. Main Contribution
  5. Notation
  6. Problem Formulation
  7. Nominal Safe Control
  8. Transformations for the distal ODE
  9. PDE Backstepping Transformation
  10. Selection of safe design parameters
  11. Result of the nominal safe control
  12. Safe adaptive control design
  13. Batch least-squares identifier
  14. Safe Adaptive Controller
  15. Results of safe adaptive control
  16. ...and 7 more sections

Key Result

Lemma 1

With the design parameters $\kappa_i$, $i=1,2,\cdots,n-1$ satisfying kappa1, the high-relative-degree ODE CBFs is initialized positive, i.e., $h_i(\underline{z}_i(0),0) > 0$ for $i=1,2,\cdots,n$.

Figures (7)

  • Figure 1: Results of $u(x,t)$ and $y_1(t)$ in open-loop system
  • Figure 2: The trajectory of $y_1(t)$ when the initial condition is safe
  • Figure 3: The trajectory of $y_2(t)$ when the initial condition is safe
  • Figure 4: Estimation of parameters when the initial condition is safe
  • Figure 5: Results of $u(x,t)$ when the initial condition is safe
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Lemma 2
  • Theorem 2
  • Claim 1