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Output-Positive Adaptive Control of Hyperbolic PDE-ODE Cascades

Ji Wang, Miroslav Krstic

TL;DR

The paper tackles safe control of a cascade system consisting of $2\times2$ hyperbolic PDEs sandwiched by a nonlinear ODE and a linear ODE under unknown parameters. It introduces a novel aCBF-based adaptive control framework that uses Batch Least-Squares Identification (BaLSI) to achieve finite-time parameter convergence, enabling the safety set to be preserved within its original boundary after learning. The nominal safe control via backstepping yields exponential stability and safety guarantees; the adaptive extension combines a CE-like controller with a safety-filtered BaLSI to maintain safety during learning and achieves asymptotic alignment with the nominal safe control once parameters are identified. Numerical simulations on UAV-cable-payload-inspired models validate safety (nonnegativity of the distal output) and exponential regulation, demonstrating practical applicability of the approach in uncertain PDE-ODE cascades.

Abstract

In this paper, we propose a new adaptive Control Barrier Function (aCBF) method to design the output-positive adaptive control law for a hyperbolic PDE-ODE cascade with parametric uncertainties. This method employs the recent adaptive control approach with batch least-squares identification (BaLSI, pronounced "ballsy") that completes perfect parameter identification in finite time and offers a previously unforeseen advantage in safe control design with aCBF, which we elucidate in this paper. Since the true challenge is exhibited for CBF of a high relative degree, we undertake a control design in this paper for a class of systems that possess a particularly extreme relative degree: $2\times2$ hyperbolic PDEs sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the unknown coefficients are associated with the PDE in-domain coupling terms and with the input signal of the distal ODE. The designed output-positive adaptive controller guarantees the positivity of the output signal that is the furthermost state from the control input as well as the exponential regulation of the overall plant state to zero. The effectiveness of the proposed method is illustrated by numerical simulation.

Output-Positive Adaptive Control of Hyperbolic PDE-ODE Cascades

TL;DR

The paper tackles safe control of a cascade system consisting of hyperbolic PDEs sandwiched by a nonlinear ODE and a linear ODE under unknown parameters. It introduces a novel aCBF-based adaptive control framework that uses Batch Least-Squares Identification (BaLSI) to achieve finite-time parameter convergence, enabling the safety set to be preserved within its original boundary after learning. The nominal safe control via backstepping yields exponential stability and safety guarantees; the adaptive extension combines a CE-like controller with a safety-filtered BaLSI to maintain safety during learning and achieves asymptotic alignment with the nominal safe control once parameters are identified. Numerical simulations on UAV-cable-payload-inspired models validate safety (nonnegativity of the distal output) and exponential regulation, demonstrating practical applicability of the approach in uncertain PDE-ODE cascades.

Abstract

In this paper, we propose a new adaptive Control Barrier Function (aCBF) method to design the output-positive adaptive control law for a hyperbolic PDE-ODE cascade with parametric uncertainties. This method employs the recent adaptive control approach with batch least-squares identification (BaLSI, pronounced "ballsy") that completes perfect parameter identification in finite time and offers a previously unforeseen advantage in safe control design with aCBF, which we elucidate in this paper. Since the true challenge is exhibited for CBF of a high relative degree, we undertake a control design in this paper for a class of systems that possess a particularly extreme relative degree: hyperbolic PDEs sandwiched by a strict-feedback nonlinear ODE and a linear ODE, where the unknown coefficients are associated with the PDE in-domain coupling terms and with the input signal of the distal ODE. The designed output-positive adaptive controller guarantees the positivity of the output signal that is the furthermost state from the control input as well as the exponential regulation of the overall plant state to zero. The effectiveness of the proposed method is illustrated by numerical simulation.
Paper Structure (30 sections, 7 theorems, 95 equations, 5 figures)

This paper contains 30 sections, 7 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. Boundary control of coupled hyperbolic PDE systems
  4. CBF Safe Control
  5. Main Contribution
  6. Notation
  7. Problem Formulation
  8. Nominal Safe Control design
  9. First (ODE) nonundershooting backstepping transformation
  10. Second PDE backstepping transformation
  11. Third (ODE) nonundershooting backstepping transformation
  12. Selection of nonundershooting design parameters
  13. Result with nominal safe control
  14. Safe adaptive control design
  15. Adaptive controller
  16. ...and 15 more sections

Key Result

Lemma 1

For the time period no control action reaches the $Y$-ODE, $y_1(t)$ is kept in the safe region, i.e., under Assumption as:initial regarding the initial data.

Figures (5)

  • Figure 1: The diagram of the proposed safe-adaptive control.
  • Figure 2: Responses of the state to be safely regulated.
  • Figure 3: Responses of $y_2(t)$ and the parameter estimates.
  • Figure 4: Responses of $w(x,t),z(x,t)$ under the nominal safe and safe adaptive controllers.
  • Figure 5: Responses of $x_1(t), x_2(t)$ under the nominal safe and safe adaptive controllers.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • ...and 14 more