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Stability Analysis And Stabilizing Controller Design For Several Classes Of Fractional Order Systems

Posted on:2017-04-23Degree:DoctorType:Dissertation
Country:ChinaCandidate:Y G ZhaoFull Text:PDF
GTID:1108330485980150Subject:Control theory and control engineering
Abstract/Summary:PDF Full Text Request
Fractional calculus is an extension and expansion of integer-order calculus. The development of fractional calculus is almost together with that of integer-order cal-culus. Fractional calculus plays an important role in various fields. Comparing to integer-order models, fractional order models are used for a better description of the natural phenomena and a better simulation of the physical phenomena and dynam-ic processes in nature. In view of the applications of fractional calculus in various fields, it is particularly urgent for considering the extensive theoretical significance or practical research values of fractional calculus. Therefore, it is a wide range of theoretical significance and practical value to study fractional differential equations and fractional order systems. The researches on fractional differential equations and fractional order systems have aroused attention of the domestic and foreign scholars and gradually been in a hot issue.In this paper, the stability analysis and the stabilizing controller design problem-s for several classes of fractional order systems, and the existences of solutions for two classes of boundary value problems of nonlinear fractional differential equation-s are investigated. Some new stability criteria and the stabilizing controller design methods of fractional order systems, and some new existence theorems of bound-ary value problems of nonlinear fractional differential equations are given. Some illustrative examples are provided to illustrate the effectiveness of the main results, respectively. The main contents of this paper are listed as follows:1. Based on the existing properties for Caputo fractional derivative, some new properties for Caputo fractional derivative are presented, which allow finding a quad-ratic Lyapunov candidate function for many given fractional order systems.2. The stability and the stabilization for several classes of fractional order sys-tems are discussed. First, a sufficient condition of asymptotical stability for frac- tional order linear systems is obtained by the fractional Lyapunov function method. By designing state feedback controller for fractional order linear controlled system-s, asymptotical stabilization for closed-loop systems is considered. Then, using the fractional Razumikhin theorem, a sufficient condition of asymptotical stability for fractional order linear time-delay systems is given. Asymptotical stabilization for closed-loop systems is studied by designing state feedback controller for fractional order linear time-delay controlled systems. Finally, based on the fractional Lyapunov function method, a sufficient condition of asymptotical stability for fractional order nonlinear systems is established. The state feedback controller design problem for fractional order nonlinear systems in the triangular form is investigated by using the Backstepping technique.3. The feedback stabilizing controller design problems for fractional order non-linear triangular systems are considered. The feedback stabilizing controller design problems are converted into the determination of finding some parameters by intro-ducing appropriate transformations of coordinates. Based on the static gain control method and the fractional Lyapunov function method, both state and output feedback controller design problems for fractional order nonlinear systems in both the lower and the upper triangular form are studied.4. The feedback stabilizing controller design problems for fractional order non-linear time-delay triangular systems are investigated. The feedback stabilizing con-troller design problems are converted into the determination of finding some parame-ters by introducing appropriate transformations of coordinates. Using the static gain control method and the fractional Razumikhin theorem, both state and output feed-back controller design problems for fractional order nonlinear time-delay systems in both the lower and the upper triangular form are discussed.5. The existences of solutions for two classes of boundary value problems of nonlinear fractional differential equations are studied. By upper and lower solu-tion method, Shauder fixed point theorem and Leggett-Williams fixed point theorem, some sufficient conditions for the existence of at least one or three positive solutions for a class of boundary value problems of nonlinear fractional differential equations are given. By a fixed point theorem in Banach algebra due to Dhage, an existence theorem for boundary value problems of hybrid fractional differential equations is established.
Keywords/Search Tags:Fractional order system, Triangular system, Time-delay system, Stabil- ity, Stabilizing controller design, Fractional Lyapunov function method, Fractional differential equation, Boundary value problem
PDF Full Text Request
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