Research On Stability And Control For Several Classes Of Fractional Systems

Recently,due to its extensive applications to many fields of science and technology,many researchers have been applying the fractional order operator to solve the dynamical and control.And the research topics involves the existence and uniqueness of solution for the fractional order differential equations or systems,and the stability for the fractional order differential equations or systems.These theory results provide the basis for the control of the systems.Recently,control of the fractional order chaotic systems have attracted attention from various scientific fields.For the pioneering contributions on fractional order control methodologies,many scholars have proposed technical methods for control of fractional order nonlinear chaotic systems.When using the fractional order stability theories to guarantee the stability of the system,the sliding mode control approach is a well-known control technique.And currently,the study of sliding mode control has been extended to the variable order fractional systems and the infinite dimensional distributed parameter systems.The dissertation is divided into the following parts.We briefly introduce the basic theory of fractional calculus and the variable order fractional calculus,mainly including three common definitions,important basic properties and several commonly used functions.This chapter also simply introduces the implementation of the numerical methods.At the same time,the Lyapunov's sense stability is introduced.In addition,we give the simple description of the basic theory of fractional order sliding mode control,which provides the basis for the design of controllers for later chapters.By two type of approximate methods,we have obtained the existence and uniqueness of solution for Cauchy problem of a fractional differential equation with causal operator in Banach spaces.An example is given to illustrate the results.In addition,the fixed point theorem is employed to study the Hyers-Ulam-Rassias stability for fractional differential equations with causal operators on both bounded and unbounded time intervals.Moreover,a theorem related to Hyers-Ulam stability for fractional differential equation with causal operator is proposed on bounded time interval.Finally,two typical examples are given to demonstrate the applications of theoretical results proposed.Based on the Volterra integral form that the initial value problem is transformed into,the condition of solution existence is obtained for the variable order fractional order differential systems by adopting Arela-Ascoli theorem.Based on the constant order fractional order Lypunov's stability theorem,variable order fractional stability criteria in the sense of Mittag-Leffler is provided and proved in terms of the Fractional Comparison Principle.For the unperturbed system,the first control law is focused on the constructing of a variable order fractional integral sliding mode surface and results in a free of chattering signal.And the second one introduces a variable order fractional derivative sliding mode surface,which is also adapted for the variable order fractional system with uncertainty and external disturbance.And the sign function in the switching control law is transferred into the fractional derivative of the control input to avoid the undesirable chattering.In addition,in the presence of uncertainty and external disturbance,an adaptive sliding mode control law is designed for the variable order fractional chaotic system.Based on the fractional order Barbalat's Lemma,which is extended from the integer order one,the asymptotical stability is proved for the controlled uncertain system.At last,numerical simulations are presented to verify the validity and efficiency of the proposed fractional order controllers.For the fractional order uncertain wave equation subject to persistent external disturbances in Hilbert spaces,the adaptive twisting fractional order sliding mode controllers are designed for the fractional order perturbed wave equation by introducing the adaptive control law to the twisting controller and the bound of the external disturbances which is unknown is dealt with.When the bound is a known constant,the super-twisting sliding mode controller is designed,and a fractional order sliding mode manifold is utilized which results in a continuous input control and a chattering free signal.In addition,the relative theorem involved in the paper for the proof of the stability is proved by choosing the appropriate Lyapunov functional.Then,the control algorithms are extended to globally asymptotically stabilize the fractional order uncertain wave equation.And both of the controllers are realized stabilized.Finally,numerical simulations are presented to verify the viability and efficiency of the proposed fractional order controllers.For the boundary control of the fractional order wave equation with uncertainties,by using the sliding mode control method,we design the boundary controller.Based on fractional order global asymptotic stability theorem,the effectiveness of the controller is proved.The results open the research field for the boundary control of distributed parameter systems with the fractional order derivative by sliding mode control.By extending the recursive algorithm,which determines the generalized frequency response functions for a nonlinear differential system to determine the generalized frequency response functions of a nonlinear fractional order differential equation,the generalized frequency response functions are determined for the system with fractional order damping.Based on the generalized frequency response functions,the output spectra can be represented by the output frequency response function.Then,for a single-time sinusoidal input,the analytical relationships among the force transmissibility,nonlinear terms,namely nonlinear stiffness and nonlinear fractional order damping coefficients and fractional order parameters are established for the frequency domain analysis by using the concept of output frequency response function.The theoretical analysis is conducted about the effects of nonlinear fractional order damping on the dynamical responses of the single degree of freedom oscillator over different frequency ranges and the benefits of applying these nonlinearities with fractional order terms.