[Paper Review] Output-Positive Adaptive Control of Hyperbolic PDE-ODE Cascades

Ji Wang, Miroslav Krstić|arXiv (Cornell University)|Sep 11, 2023

Vibration and Dynamic Analysis4 citations

TL;DR

This paper presents a novel safe adaptive control framework for hyperbolic PDE-ODE cascades with unknown parameters, using finite-time batch least-squares identification (BaLSI) and a backstepping-based safety filter. The method ensures exact parameter identification in finite time, maintains state safety throughout the entire original safe set, and achieves exponential regulation of the system to zero, enabling collision-free control in applications like UAV-cargo delivery systems.

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: $2 imes2$ 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.

Motivation & Objective

• To address the challenge of safe adaptive control in hyperbolic PDE-ODE cascades with unknown parameters, particularly when the control barrier function (CBF) is of high relative degree.
• To overcome the limitation of conventional adaptive control, which restricts initial conditions to a shrinking subset of the safe set due to parametric uncertainty.
• To develop a controller that guarantees finite-time exact parameter identification, state safety for the distal ODE, and exponential regulation of the entire system to zero.
• To extend the application of the regulation-triggered BaLSI method to PDE-embedded systems, enabling safe control in real-time engineering contexts such as UAV-cargo delivery.

Proposed method

• The method employs batch least-squares identification (BaLSI) to achieve finite-time exact identification of unknown parameters in the PDE and distal ODE, avoiding the inverse dependence of safe set size on adaptation gain.
• A backstepping-based safety filter is designed to enforce the non-negativity of both the PDE Control Barrier Function (CBF) and the ODE CBF, ensuring state safety throughout the entire original safe set.
• The controller is explicitly constructed to operate over the full initial safe set, eliminating the need for conservative initial condition restrictions common in continuous adaptation schemes.
• The system is modeled as a 2×2 hyperbolic PDE sandwiched between a nonlinear ODE (UAV dynamics) and a linear ODE (payload dynamics), with unknown coefficients in the PDE coupling terms and the distal ODE input.
• Finite-time convergence of parameter estimates is proven via contradiction, showing that persistent excitation is maintained and the parameter identification cannot fail due to signal degeneracy.
• The stability and safety guarantees are derived using Lyapunov-based analysis and Riemann transformation techniques to convert wave PDEs into heterodirectional transport PDEs for tractable design.

Experimental results

Research questions

• RQ1 Can finite-time parameter identification via BaLSI be leveraged to eliminate the need for restricting initial conditions to a shrinking subset of the safe set in adaptive control of PDE-ODE cascades?
• RQ2 How can a safety filter be designed to ensure non-negativity of both PDE and ODE Control Barrier Functions (CBFs) in systems with high relative degree?
• RQ3 Is it possible to achieve exponential regulation of the entire PDE-ODE cascade while maintaining state safety and exact parameter identification in finite time?
• RQ4 What are the conditions under which the BaLSI estimator avoids signal degeneracy (e.g., zero excitation) that would prevent finite-time convergence?

Key findings

• The proposed controller achieves finite-time exact identification of all unknown parameters in the PDE and distal ODE, with convergence guaranteed in finite time regardless of initial uncertainty.
• The entire original safe set is preserved for control operation, eliminating the need to restrict initial conditions to a shrinking subset as in conventional adaptive control.
• The state of the distal ODE (e.g., the payload in a UAV-cargo system) remains safely bounded within the safe region at all times, due to the non-negativity of the ODE CBF enforced by the safety filter.
• The PDE CBF is explicitly defined and its non-negativity is ensured via the backstepping-based safety filter, marking the first such design for PDE systems.
• The overall system state is exponentially regulated to zero, ensuring asymptotic stability and fast convergence.
• Numerical simulations confirm the effectiveness of the controller in achieving safe, stable, and adaptive control of the PDE-ODE cascade under parametric uncertainty.