Output regulation of anti‐stable coupled wave equations via the backstepping technique

Abstract

This study is concerned with the output regulation of an anti-stable system of coupled wave equations with external disturbances. A state-feedback regulator is designed to force the output of the coupled wave equations to track the reference signal, which is generated by an exosystem. Moreover, the tracking error decays exponentially at a prescribed rate. The design is based on backstepping approach and relies on solving the regulator equations. The solvability condition of the regulator equations is characterised by the transfer matrix of the coupled wave equations and eigenvalues of the exosystem. An outputfeedback regulator is then constructed by developing an observer. Finally, the numerical simulations are demonstrated for the effectiveness of the theoretical results.

Sliding mode control for N‐coupled reaction‐diffusion PDEs with boundary input disturbances

“…where W(x, t) is governed by (14) (or (16)). Then, the error system is obtained by subtracting (52) from (14)…”

“…Hence, the control laws need to be designed to stabilize the systems in the time of rejecting the disturbances. Several control approaches have been developed for the disturbance rejection, including output regulation, [11][12][13][14] active disturbance rejection control, 15,16 disturbance observer-based control, 17,18 and sliding mode control (SMC). [19][20][21][22][23] The SMC approach is based on designing a unit control, 24,25 the norm of which is equal to one when beyond a suitable sliding manifold.…”

Error feedback regulation for 1D anti-stable wave equation

“…Roughly speaking, the main strategy is to map the closed-loop system to a stable target system by constructing invertible transformations. To the best of our knowledge, this method was first introduced into output regulation problem for an unstable heat equation in Deutscher (2015) and then extended to the anti-stable wave equation with external disturbances in Gu et al (2018, 2019). An exponential tracking result for an anti-stable wave equation was reported in Jin and Guo (2019) by using backstepping transformation and constructing the special solution for regulator equations.…”

Output Feedback Regulation for 1-D Anti-Stable Wave Equation with External System Disturbance

“…Output regulation of 1-D wave equation with both internal and external uncertainties is considered in [12] and finally achieves exponential tracking. In [9], the tracking problem of coupled wave equations with external disturbance is solved through the backstepping method. F.F.…”