Output feedback stabilization for a cascaded heat PDE‐ODE system subject to uncertain disturbance

Abstract

Summary In this paper, we are concerned with the output feedback exponential stabilization of a cascaded parabolic partial differential equation–ordinary differential equation system where the control end suffers from the external uncertain disturbance. We design an unknown input–type state observer with the disturbance estimator term, which is used to stabilize this cascaded system and compensate the external disturbance simultaneously. The stabilizing state feedback control is designed for the observer system by the three‐step backstepping transformation, which is the corresponding observer‐based output feedback stabilizing control for the original system. Finally, the closed‐loop system is shown to be exponentially stable.

Performance output tracking for cascaded heat partial differential equation‐ordinary differential equation systems subject to unmatched disturbance

“…In this paper, we take advantage of the method of trajectory planning to the performance output tracking for two cascades of one‐dimensional heat PDE‐ODE systems with Dirichlet and Neumann interconnection, respectively. Both systems are subject to the disturbance unmatched to the controller, and moreover, the disturbances come from exosystem, which is the main difference from the work of the current literatures 19,20 . By constructing the proper trajectory planning, the non‐collocated configurations can be converted into the collocated ones, so that the difficulties caused by these non‐collocated are overcome.…”

“…Both systems are subject to the disturbance unmatched to the controller, and moreover, the disturbances come from exosystem, which is the main difference from the work of the current literatures. 19,20 By constructing the proper trajectory planning, the non-collocated configurations can be converted into the collocated ones, so that the difficulties caused by these non-collocated are overcome. At last, exponential convergences of the regulation errors are proved.…”

“…It is noted that the reference signal is not specified due to disturbance in the above works, and moreover, the authors are only concerned with the performance output tracking for a single control system. Although there are some results about boundary control of PDE-ODE cascades, [19][20][21][22] how to deal with the output tracking problem of cascaded control system subject to unmatched disturbance is quite challenging. System control through actuator dynamics can usually be modeled as a cascaded control system.…”

Infinite‐dimensional extended state observer based stabilization for an ODE‐wave cascaded system with control matched disturbance

“…However, since it represents either inaccuracies in the control output or external excitation on the boundary, uncertain disturbance, usually entering the ODE-PDE cascade system from the input channel, is inevitable in many physical phenomena governed by PDE-PDE cascaded system [15,16]. In order to cope with the disturbance, several approaches are developed, including adaptive control for ODE-heat equation [17], sliding mode control (SMC, a variable structure control method) for ODE-heat equation [18], ODE-Schrödinger equation [19], ODE-wave equation [20] and active disturbance rejection control (ADRC) for ODE-heat equation [16,21,22], ODE-Schrödinger equation [15], ODE-wave equation [20,23], and so on. Among those, we have to mention the method ADRC, which was firstly proposed by Han [24].…”

Finite‐time parameter estimation based boundary control for a class of 2×2 hyperbolic partial differential equation systems with uncertain transport speeds

“…During the past few decades, much attentions have been paid to the control problem of systems described by partial differential equation (PDE) in References 1‐6. It is well recognized that PDE can be categorized into three groups: the parabolic PDE, the hyperbolic PDE, and the elliptic PDE.…”