Backstepping Control Of Infinite Dimensional Coupled Systems With Disturbances

Infinite dimensional coupled systems are usually used to model many phenomena that happened in engineering practice,such as road traffic,blood circulation,and catalytic reactor etc.The dynamics of each subsystem that coupled together as well as their complicate con-nections increase the complexity of the coupled system;On the other hand,due to changes of work environment,and aging or damage of components etc,the coupled systems are in-evitably subject to the external disturbances,which may induce complex behavior,e.g.,oscil lations,instability etc.Therefore,the research on the control of infinite dimensional coupled systems with disturbances is of theoretical challenge and application valueBased on the backstepping control approach,the thesis is devoted to the control design of infinite dimensional coupled systems with disturbances such that the coupled system is transformed to a target system with some desirable stability properties.Moreover,we study two kinds of control issues for the target system:(1)when the disturbance enters at the controlled end,we consider stabilization problem,i.e.,the state trajectory is forced by the designed controller to arrive at a suitable sliding surface in a finite time and becomes stable on the surface;(2)when the disturbances act at the uncontrolled end and the body equations,we consider output regulation,i.e.,the controller is designed for rejection of disturbances and asymptotic tracking of time-varying signals,which are generated by an exosystem.Moreover,the tracking error decays exponentially at a prescribed rate.The thesis is organized as follows:In Chapter 1,we introduce the engineering background,development situation on back-stepping control and analysis of infinite dimensional coupled systems.Some preliminaries relevant to basic concepts,main theorems,and sliding mode control(SMC)and output regu-lation are then presented.Finally,the main results of the thesis are briefly summarizedChapter 2 is devoted to the stabilization of the Orr-Sommerfeld equation cascaded by both Squire equation and ODE,in which Orr-Sommerfeld and Squire equations are subject to boundary controls matched disturbances respectively.A modified backstepping transfor-mation with matrix kernel is integrated with SMC to reject the bounded disturbances.The well-posedness of the solution for the closed-loop system is obtained based on the Riesz basis method and the finite time "reaching condition" is verifiedIn Chapter 3,we are 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 time-varying signal,which is generated by an exosystem.Moreover,the tracking error decays exponentially at a prescribed rate.The design is based on a two-step 2-dimensional backstepping approach and relies on solving SFRE.The solvability condition of SFRE is characterized by the transfer matrix of the coupled wave equations and eigenvalues of the exosystem.An output-feedback regula-tor is then constructed by developing an observer and solving the output-feedback regulator equationsChapter 4 investigates the output regulation of an unstable reaction-diffusion PDE with regulator delay and external disturbances in a suitable Hilbert space.The unstable reaction-diffusion PDE with time delay is first written as a reaction-diffusion PDE cascaded with a transport equation.The systematic design procedure of a backstepping regulator is then pre-sented by mapping the cascaded system to an error system,which is shown to be exponen-tially stable with a prescribed rate.The output regulation is then verified by solving cascaded state feedback regulator equations(SFRE).The solvability condition of the cascaded SFRE is characterized by a transfer function and eigenvalues of the exosystemIn Chapter 5,we consider the backstepping-based output regulation of ODE cascaded by wave equation.Using backstepping transformation,the orginal systems is first mapped into a target system with some desirable features:the state matrix of ODE becomes a Hurwitz,and a damping term appears in the equation body of wave-part.A backstepping regulator is then designed to force the output to track time-varying signal.Moreover,the tracking error decays exponentially at a prescribed rate.The design is based on solving cascaded regulator equations(SFRE).The solvability condition of SFRE is finally characterized in terms of a transfer function and the eigenvalues of the exosystem.An observer-based output-feedback regulator is then designed to solve the output regulation problem.At last,a summary of the thesis and some unsolved problems are addressed.