Control Analysis And Design For Uncertain Hyperbolic Distributed Parameter Systems

In practical engineering, hyperbolic distributed parameter systems are widely used to model many dynamic processes, such as the vibration of string, the de-velopment of population, the motion of overhead crane, the flow of fluid, the exploitation and transport in pipelines of oil and gas. Compared with lumped pa-rameter systems(described by ordinary differential equations), distributed param-eter systems(described by partial differential equations) can describe the dynamic of the practical systems more accurately and comprehensively. However, mainly due to the property of infinite dimensional state spaces and the complexity of dis-tributed parameter systems, many control problems which have been completely solved in lumped parameter systems field are still very challenging so far in dis-tributed parameter systems field. Moreover, in practice, uncertainties/unknowns are inevitable due to the limitation in understanding how the controlled objects works, the measurement errors or the influence of external disturbances, which makes the control methods proposed in the existing literature inapplicable. Con-sequently, the study of control the hyperbolic distributed parameter systems with uncertainties/unknowns has both theoretical and practical significance.This dissertation focuses on the state-feedback/output-feedback boundary stabilization for several classes of uncertain hyperbolic distributed parameter sys-tems. By using the adaptive technique, time varying technique, backstepping method, LaSalle invariance principle, semigroups and Lyapunov stability theories, several boundary control schemes and methods for analyzing the well-posedness and stability are developed. Compared with the existing results, more serious uncertainties/unknowns are contained in the considered systems in this disserta-tion, and consequently relax the restrictions of the systems, extend the application scope of the existing methods and improve the control theory of distributed pa-rameter systems. Moreover, the developed boundary control schemes can also provide reference and guidance for other control problems. The main contents of the dissertation consist of the following four parts:(I) Adaptive boundary stabilization for first-order hyperbolic PDE systems with unknown spatially varying parameterThis part is Chapter 3 of the dissertation, and studies the adaptive boundary stabilization for a class of systems described by first-order hyperbolic PDE with unknown spatially varying parameter. Different from the existing relative works which are not allowed to have unknown parameters or unknown spatially vary-ing parameter, the considered systems have more serious uncertainties/unknowns, which makes the existing methods inapplicable. To solve the problem, a dynamic compensation is first given to deal with the system serous unknowns by using infinite-dimensional backstepping method, adaptive technique and projection op-erator. Then, an adaptive state-feedback boundary controller is successfully con-structed by certainty equivalence principle, which can stabilize the original system.(II) Adaptive state-feedback stabilization for PDE-ODE cascade systems with harmonic disturbanceThis part is Chapter 4 of the dissertation, and studies the adaptive state-feedback stabilization for a class of systems described by second-order hyperbolic PDE and ODE with uncertain input harmonic disturbance. The essential differ-ence between this chapter and the existing related hterature is the presence of the uncertain disturbance belonging to an unknown interval, which makes the problem more practical. To solve the problem, an adaptive boundary feedback controller is first constructed in two steps by adaptive technique and backstepping method developed for ODE in the finite dimensional framework. Then, it is shown that the resulting closed-loop system is well-posed and asymptotically stable by the semigroup approach and LaSalle invariance principle, respectively. Moreover, the parameter estimates involved in the designed controller are shown to ultimately converge to their own real values.(III) Adaptive output-feedback stabilization for uncertain PDE-ODE cascade systemsThis part is Chapter 5 and 6 of the dissertation. Chapter 5 investigates the adaptive out-feedback stabilization for a class of PDE-ODE cascade systems with uncertain input disturbance. Compared with Chapter 4 in which all states are required available for feedback, the considered systems are more practical and the developed boundary control schemes are more applicable. To solve the problem, an observer is first introduced to estimate the unmeasured states of the original system. Then, by adaptive technique and backstepping method devel- oped for ODE in the finite dimensional framework, an output-feedback controller is successfully constructed, which guarantees that the resulting closed-loop sys-tem is asymptotically stable and the parameter estimates ultimately converge to their own real values. Moreover, by semigroup approach, the well-posedness of the resulting closed-loop system is achieved. Chapter 6 investigates the adaptive out-feedback stabilization for a class of PDE-ODE cascade systems with both un-certain parameter and input disturbance. Essentially different from Chapter 5, not only input disturbance but also unknown parameter belonging to an unknown interval are contained in the considered systems, and moreover less information are available for feedback in this chapter. By constructing a state observer and using adaptive technique and backstepping method, an output-feedback bound-ary controller is successfully designed which guarantees the well-posedness and stability of the resulting closed-loop system. Moreover, it can be proved that the parameter estimates ultimately converge to their own real values.(IV) Output-feedback stabilization for PDE-ODE cascade systems with non-periodic time-varying input disturbanceThis part is Chapter 7 of the dissertation, and studies the out-feedback stabi-lization for a class of PDE-ODE cascade systems with non-periodic time-varying input disturbance. Essentially different from Chapter 6, the disturbance consid-ered in this chapter is only required to be bounded and the bounds are unknown. This makes the control design more complex, and moreover the methods and ideas used in the existing literature inapplicable. To solve the problem, an time-varying observer is first introduced to estimate the unmeasured states of the original sys-tem. Then, by time-varying technique and backstepping method developed for ODE in the finite dimensional framework, an output-feedback boundary controller is successfully constructed, which can stabilize the original system.