Boundary Control Of Parabolic Distributed Parameter Systems

In natural,physical phenomena,industrial production processes,and biochemical reac-tions,the state changes along with multiple independent variables.The distributed parameter system has infinite dimensions,which can describe the dynamic characteristics of the system more comprehensively compared with the lumped parameter system.Because it includes not only the characteristics of temporal evolution but also spatial distribution.Therefore,the dis-tributed parameter system has important theoretical significance and practical value.In this paper,for the open-loop instability problem of two types of orders of distributed parameter systems,the domain-averaged measurement and boundary-valued measurement are adopted,and the operator semi-group method,the Lyapunov direct method,the linear matrix inequality method,the backstepping method are employed.Considering the case of the diffusion coeffi-cient is constant or variable,and the situation with Robin boundary or mixed boundary condi-tions,the boundary controllers are designed and the effect of stabilizing the systems is achieved.1.For the boundary control problem of the semi-linear parabolic distributed parameter system,the boundary controllers are designed for the cases of the constant diffusion coefficient and space-dependent diffusion coefficient under the Robin boundary condition.The boundary controller whose feedback information comes from measured output directly and the observer-based output feedback boundary controller are designed by employing the domain-averaged measurement and boundary-valued measurement,and the goal of stabilizing the controlled sys-tem is achieved.Using the Lipschitz condition to deal with nonlinear term,the stability criteria of the controlled systems are obtained through the Lyapunov stability theory and the linear ma-trix inequality method.Simulation results verify the effectiveness of the boundary controllers.2.For the problem of external disturbance of the parabolic distributed parameter system,the H_∞control strategy is implemented to study the system with the space-dependent diffusion coefficient,and where the terminal boundary state has a great influence on the system under the mixed boundary condition.Construct the criterion in which the controlled system is H_∞stable under the observer-based output feedback H_∞boundary controller,by means of the Lyapunov stability theory and the linear matrix inequality method.Meanwhile,consider the noise when measuring with the domain-averaged and boundary-valued ways.The validity of the theoretical results is demonstrated by simulations.3.For the problem of external disturbance of the semi-linear parabolic distributed param-eter system,the H_∞control strategy is used to design the boundary controller for the system with the space-dependent diffusion coefficient under the Robin boundary condition.Domain-averaged and boundary-valued measurements are adopted and the noise in the process is con-sidered.Combining the performance index of H_∞with the Lyapunov stability criterion,the sufficient condition for H_∞stable of the controlled system is obtained through the linear ma-trix inequality method.Optimize the disturbance attention levelγof H_∞performance index by minimizingγto get the observer-based output feedback optimal H_∞boundary controller.Simulation results show the effectiveness of the boundary controllers.4.For the boundary control and state observation problem of the Caputo time-fractional distributed parameter systems,the backstepping method is utilized to study the system with the space-dependent reaction coefficient and the system with coupled boundaries.The bound-ary controller is designed and the state observer is constructed to estimate the system,so that the controlled system and the observer error system are Mittag-Leffler stable.Meanwhile,an algorithm for solving the numerical solution of the control gain is proposed.The feasibility of the algorithm is verified by simulations,which also show the effectiveness of the designed controller and observer.