[Paper Review] Output feedback exponential stabilization of a nonlinear 1-D wave equation with boundary input

Hua‐Cheng Zhou, George H. Weiss|arXiv (Cornell University)|Jun 7, 2017

Stability and Controllability of Differential Equations6 references3 citations

TL;DR

This paper presents an output feedback exponential stabilization method for a nonlinear 1-D wave equation with boundary input, where both internal uncertainty and external disturbance are present. Using only two displacement measurements, it designs a high-gain-free disturbance estimator and a state observer to enable disturbance rejection and exponential stability via backstepping and active disturbance rejection control (ADRC). The key contribution is exponential stability of the closed-loop system with minimal output measurements.

ABSTRACT

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in $L^2(0,\infty)$ but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded.

Motivation & Objective

• To address the output feedback exponential stabilization of a 1-D unstable/anti-stable wave equation with both internal nonlinear uncertainty and external disturbance.
• To design a disturbance estimator that is free of high-gain and estimates the total disturbance (unknown function f and external d) in the L²(0,∞) sense.
• To construct a state observer based on the disturbance estimator that converges exponentially to the true system state using only two output signals.
• To develop an observer-based output feedback control law that cancels the estimated disturbance and ensures exponential stability of the closed-loop system.
• To minimize the number of measurements by using only two displacement signals (w(0,t) and w(1,t)), which is shown to be nearly minimal.

Proposed method

• A dynamic compensator is designed using a PDE-based disturbance estimator that avoids high-gain structures, ensuring robustness and avoiding peaking in the estimation error.
• The disturbance estimator is constructed using the measured outputs w(0,t) and w(1,t), and it estimates the total disturbance F(t) = f(w(·,t),w_t(·,t)) + d(t) in the L²(0,∞) sense.
• A state observer is designed based on the disturbance estimator, using the measured outputs and the estimated disturbance to reconstruct the full state (w, w_t) exponentially.
• The control law is designed via backstepping, using the observer state to form a feedback control that cancels the estimated disturbance in real time.
• The closed-loop system is analyzed in a Hilbert state space, and exponential stability is proven using Lyapunov functionals and energy estimates.
• The method is extended to an alternative anti-stable wave equation with a negative damper at x=0, achieving the same exponential stability with the same minimal measurement set.

Experimental results

Research questions

• RQ1 Can exponential stabilization be achieved for a nonlinear 1-D wave equation with boundary input under both internal uncertainty and external disturbance using only two output signals?
• RQ2 Can a disturbance estimator be designed without relying on high-gain mechanisms while still ensuring L²(0,∞) estimation error for the total disturbance?
• RQ3 Is it possible to design a state observer that converges exponentially to the true system state using only displacement measurements and the estimated disturbance?
• RQ4 Does the combination of disturbance estimation, state observation, and backstepping control lead to exponential stability of the closed-loop system?
• RQ5 Can the proposed method be extended to anti-stable wave equations with negative damping instead of negative spring at the boundary?

Key findings

• The closed-loop system is exponentially stable with all internal signals uniformly bounded, under the proposed output feedback control law.
• The disturbance estimator successfully estimates the total disturbance F(t) in the L²(0,∞) sense without using high-gain structures.
• The state observer converges exponentially to the true state of the wave system, even in the presence of unknown nonlinearities and disturbances.
• The control law achieves exponential stability using only two displacement measurements, which is shown to be nearly minimal in the number of required sensors.
• For the case f ≡ d ≡ 0, the transient states (v, v_t, z, z_t, W) decay exponentially with rate μ′ > 0.
• The method is extended to the anti-stable wave equation with negative damping at x=0, and exponential stability is still achieved with the same minimal measurement set.