Research On Control Of Two Types Of Orders Of Distributed Parameter Systems

Generally speaking,integer-order distributed parameter systems can describe diffusion processes,fractional-order distributed parameter systems can describe anomalous diffusion processes and fractional reaction diffusion processes.These two types of orders of systems are widely used in fields of engineering,ecology,society and environment.Thus they have been attracted much attention.This dissertation mainly focuses on the control problems of two types of orders of distributed parameter systems,including integer-order distributed parameters systems(usually referred to as distributed parameter systems)and fractional-order distributed parameter systems,whose contributions will be illustrated as follows:1.With the help of static mesh sensor networks and PI controllers,the control problem of a diffusion system with a static or moving pollution source,is investigated.By constructing a optimal actuators location objective function and centroidal Voronoi tessellations(CVTs),the actuator motion planning is obtained.Based on the Lyapunov stability theory,convergence of the actuators location is proved,i.e.,actuators can converge to the mass centroids of Voronoi cells under control input with the PI controller.Moreover,a optimal spraying control objective function is constructed to determine the PI controller for neutralizing control,which makes the difference of spraying and pollution amounts minimal and avoids excessive spraying to cause pollution again.Finally,the modified integer-order simulation platform(Diff-MAS2D-PID simulation platform)is established.By it,the fact that control effect of PI controllers outperforms the one of P controllers is verified.2.For the control problem of a tempered anomalous diffusion system,a fractionalorder PI controller is introduced for mobile actuators motion control and spraying control.Using the Lyapunov stability theory,convergence analysis of the actuators location is firstly presented under control input with the fractional-order PI controller.Moreover,a new CVT algorithm based on fractional-order PI controllers is provided together with a modified fractional-order simulation platform(FO-Diff-MAS2D-FOPI simulation platform).Lastly,extensive numerical simulations for the anomalous diffusion process are presented to verify the effectiveness of our proposed fractional-order PI controllers.3.The backstepping method is introduced into a fractional-order reaction diffusion system with mixed or Robin boundary conditions.Then a boundary feedback control problem of this system is discussed.Here,the diffusivity of this system is spaceindependent,i.e.the diffusion coefficient is constant.Dirichlet,Neumann and Robin boundary feedback controllers are designed by the backstepping method.By the integral transformation,the system with designed controllers is mapped into a Mittag-Leffler stable target system.Then,the control problem is transformed into the problem of solving the gain kernel.The fractional-order Lyapunov method is used to prove that the designed boundary feedback controllers can make the closed-loop system Mittag-Leffler stable.Extensive numerical simulations are presented to demonstrate the validness of the proposed method.4.An output feedback control problem of a fractional-order reaction diffusion system with mixed boundary conditions is considered.This system is endowed with only boundary sensing available.In cases of collocated and anti-collocated sensors and actuators(i.e.sensors and actuators are in the same ends and in the different ends),the observers are designed under Dirichlet actuation respectively.The combination of it and above backstepping-based boundary feedback controller results in an output feedback controller.In terms of the fractional-order Lyapunov method,it is proved that this output feedback controller can make the closed-loop system Mittag-Leffler stable.Numerical simulations are further given to test the theoretical results.5.The boundary stabilization problem of a fractional-order distributed parameter system(original system)with space-dependent(nonconstant)diffusivity is addressed.This problem can be viewed as a generalization of the case of constant diffusivity,which is more suitable to practice.By a change of variables,this original system is converted into a general fractional-order reaction diffusion system(new system).A boundary feedback controller is designed for the new system by the backsteping method and an integral transformation.Then,the boundary feedback controller is obtained for the original system via the given change of variables.Based on the fractional Lyapunov method,the sufficient conditions are obtained for Mittag-Leffler stability of the original system under the boundary feedback controller.Numerical simulations are shown to verify the Mittag-Leffler stability of the closed-loop system.