[Paper Review] Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs

Jean Auriol, Ulf Jakob F. Aarsnes+2 more|arXiv (Cornell University)|Sep 13, 2017

Stability and Controllability of Differential Equations43 references81 citations

TL;DR

The paper shows that finite-time stabilization for two coupled hyperbolic PDEs by canceling proximal reflections can destroy delay robustness, and proposes a delay-robust control design that preserves a small reflection. It also reformulates the system as a distributed-delay neutral equation to analyze robustness.

ABSTRACT

We detail in this article the necessity of a change of paradigm for the delay-robust control of systems composed of two linear first order hyperbolic equations. One must go back to the classical trade-off between convergence rate and delay-robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay-robustness. Indeed, for such systems, using a backstepping-controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.

Motivation & Objective

• Motivate the need to balance convergence rate with delay robustness in two-heterodirectional linear hyperbolic PDEs.
• Show that canceling proximal reflections can eliminate delay margins for small delays.
• Develop a delay-robust backstepping-based control design that preserves a small amount of proximal reflection.
• Reformulate the two-PDE system as a neutral system with distributed delay to analyze robustness.

Proposed method

• Analyze open-loop transfer functions for boundary-coupled transport PDEs to derive delay robustness conditions.
• Use backstepping to map the original system to a target system with a distributed-delay neutral form.
• Derive conditions on reflection gains (proximal and distal) that determine delay-robust stabilizability.
• Propose adjusted control laws that trade convergence speed for delay robustness by preserving part of the proximal reflection.
• Establish exponential stability of the adjusted target system under small delays and provide a constructive controller.
• Illustrate the approach with a tutorial two-transport-equation example and extend to general coupled hyperbolic PDEs.

Experimental results

Research questions

• RQ1 Can a backstepping-based controller stabilize two heterodirectional linear hyperbolic PDEs robustly to small delays?
• RQ2 What role do proximal and distal reflection gains play in delay-robust stabilization?
• RQ3 How can the target system be designed to trade off finite-time convergence for delay robustness?
• RQ4 Can the two-PDE system be reformulated as a distributed-delay neutral system to facilitate analysis of delays?

Key findings

• Finite-time stabilization by completely canceling proximal reflections can yield zero delay robustness in certain systems.
• If the open-loop gain |ρq| ≥ 1, the system cannot be delay-robustly stabilized.
• For |ρq| < 1, delay-robust stabilization is possible by modifying the control law to preserve a small amount of proximal reflection.
• A backstepping-based reformulation maps the two-PDE system to a distributed-delay neutral system, enabling delay analysis.
• A modified control law with a tunable parameter K can achieve delay-robust exponential stabilization when |ρq| < 1, allowing a trade-off between convergence rate and delay robustness.
• The framework extends from the simple two-transport example to the general two-coupled hyperbolic PDE case.