Input-to-state stabilization of an ODE-wave system with disturbances
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Input-to-state stabilization of an ODE-wave system with disturbances Yu-Long Zhang1 · Jun-Min Wang1
· Donghai Li2
Received: 18 September 2019 / Accepted: 11 September 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, we consider the input-to-state stabilization of an ODE-wave feedbackconnection system with Neumann boundary control, where the left end displacement of the wave equation enters the ODE, while the output of the ODE is fluxed into boundary of the wave equation. The disturbance is appeared as a nonhomogeneous term in the ODE. Based on the backstepping approach, a state feedback control law is designed to guarantee the exponential input-to-state stability of the closed-loop system. The resulting closed-loop system has been shown to be well-posed by the semigroup approach. Moreover, we construct an exponentially convergent state observer based on which an output feedback control law is obtained, and the closed-loop system is proved to be input-to-state stable. Keywords Input-to-state stabilization · Backstepping · Boundary control · Feedback-connection systems · Output feedback
Published in the topical collection Input-to-state stability for infinite-dimensional systems. This work is partly supported by the National Natural Science Foundation of China with Grant/Award Number: 61673061.
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Jun-Min Wang [email protected] Yu-Long Zhang [email protected] Donghai Li [email protected]
1
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2
China State Key Laboratory of Power Systems and Generation Equipment, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China
123
Mathematics of Control, Signals, and Systems
1 Introduction Ordinary differential equation (ODE)–partial differential equation (PDE) coupled systems have attracted a lot of attention in different engineering aspects such as electromagnetic coupling, mechanical coupling and coupled chemical reactions. The backstepping method is one of the powerful methods to design feedback control laws for ODE–PDE systems [23]. In [32], the stabilization of an ODE-Schrödinger cascade system is considered. By using the backstepping method, the system is converted into an equivalent target system which is stable under a state feedback boundary control. When only the boundaries of PDEs are available for measurement, observer-based output feedback control laws are designed to stabilize ODE–PDE systems by the backstepping method in [1,2,15,36]. In [34,35], the stabilizations of wave equations with nonlocal terms are considered by adopting the backstepping method and converting the wave equations into coupled hyperbolic PDE–ODE systems. In recent years, some effective methods have been applied to deal with ODE– PDE systems subject to external disturbances. The active disturbance rejection control (ADRC) approach is one of the effective control methods with advantages such as simplicity in implementation and low dependence on mathematical model
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