Stabilization Control Of Fractional-Order Reaction-Subdiffusion Systems

The fractional-order system has attracted extensive attention from researchers in science and engineering because of its good characterization of genetic effects and memory properties.Especially since 1970 s,the vigorous development of viscoelastic material science has laid the foundation for the application of fractional-order systems,and greatly promoted the study of various properties of fractional-order systems.Diffusion is a phenomenon or process that widely exists in nature or engineering fields,such as electrochemical processes in batteries,conduction and diffusion of heat,etc.When these phenomena or processes occur in homogeneous media,they follow Fick’s law and can be described by standard partial differential equations.However,when these phenomena occur in inhomogeneous media,such as porous media and fractal lattices,anomalous diffusion will occur,and Fick’s law is no longer applicable.The most powerful tool to describe anomalous diffusion is the time-fractional reaction-diffusion equation.Considering that stability is the basic property of the normal operation of the system,the stabilization control of fractional reaction-sub-diffusion has also been widely concerned.The existing control methods are mainly backstepping methods for the linear case,which require the full state information of the system to design the controller,and the expression of the controller needs to be obtained by solving a second-order hyperbolic partial differential equation.For the fractional-order nonlinear reaction-subdiffusion system,the spectral decomposition method is mainly used to reduce the order of the system,and then the controller is designed for the finite-dimensional system to achieve the stability of the system.Therefore,it is a challenging research topic that how to design a controller using finite local state information,and how to design a controller directly based on the system itself to achieve system stability instead of reducing the order of the nonlinear system.In this paper,we combine the distributed control and the boundary control to study the stabilization problem of time fractional reaction-sub-diffusion systems.The main work of this paper is as follows:1.For a class of uncertain linear fractional-order reaction-subdiffusion systems,based on point measurement and piecewise measurement,distributed robust point controllers and distributed robust piecewise controllers are designed,respectively.The MittagLeffler(M-L)stability of the closed-loop system is proved by using the fractionalorder Lyapunov theory,Lyapunov direct method,LMI theory and Wirtinger’s inequality,and the sufficient condition for the stability of the closed-loop system are given in terms of LMIs.2.The problem of distributed non-collocated feedback control for a class of linear fractionalorder reaction-subdiffusion systems with two kinds of measurements(point measurements and piecewise measurements)is studied for the case that the controller and sensor are non-collocated.For the case of non-collocated point measurements,a fractional-order Luenberger observer is designed to estimate the system states,and then a distributed feedback point controller is designed based on the observer.Based on LMI theory and Wirtinger’s inequality,the stability of the closed-loop system is analyzed in three cases(the number of sensors is greater than,equal to,and less than the number of controllers),and the sufficient conditions of M-L stability of the closedloop system are given in the form of LMIs.In the case of piecewise measurement,similar research methods are used and similar conclusions are drawn.Finally,the robustness results of the designed controller with respect to the system parameters(reaction coefficient and diffusion coefficient)are given.3.The problem of distributed robust stabilization for a class of semi-linear fractionalorder reaction-sub-diffusion systems is studied by using the output information of average measurements.Considering the placement of sensors and controllers,a distributed output feedback controller is designed directly for the case that the sensors and controllers are in the same position.When the controller and sensor are noncollocated,a fractional-order Luenberger-type observer is constructed according to the non-collocated average measurement output to estimate the system state,and then a distributed dynamic output feedback controller is designed based on the observer.Sufficient conditions for the M-L stability of the closed-loop system are given in the form of LMIs.4.For a class of nonlinear fractional-order reaction-subdiffusion systems,the boundary controllers are designed for three kinds of measurements: averaged measurements,boundary point measurements and non-collocated boundary point measurements.For the first two kinds of measurements,the static feedback control of the corresponding measurements is designed directly.For the non-collocated boundary point measurements,a fractional-order observer is constructed based on the non-collocated measurement information to estimate the measurements at the anti-boundary end,and then a boundary dynamic output feedback controller based on the observer is designed.For the above three cases,the sufficient conditions of M-L stability of the closed-loop system are given in the form of LMIs,respectively.5.The stabilization problem for a class of nonlinear fractional-order reaction-subdiffusion systems with time-varying delays is considered.First,for the case of average measured output,the boundary feedback hybrid event-triggered control is designed directly based on the output information to stabilize the closed-loop system.Then an observer is designed to estimate the system state information based on the boundary measurement information when only the boundary measurement information is available,and a dynamic event-triggered boundary controller based on the observer is designed.Based on the Lyapunov-Razumikhin theorem,sufficient conditions are obtained for the asymptotic stability of the closed-loop system in both cases,while Zeno behavior is excluded.