Stability Analysis Of A Class Of Fractional-order Systems

In recent years, fractional calculus has attracted much attention. The so-called fractional-order is more accurately said as non-integer order. Also refers to the order of their calculus is not the traditional first, second, third and so on but arbitrary fraction. It is observed that the description of some systems is more accurate with the help of fractional derivatives and integrals. For example, signal processing and control, new electrical element, viscoelastic material, thermal diffusion process, pow-er fractal network are modelled as fractional-order equations. As for fractional-order singular systems, since the singular form not only contains the static information but also the dynamic constraints, we must consider not only stability, but also regular and impulse-free at the same time. The research of fractional-order singular sys-tems is more complicated than the traditional fractional-order system. At present, the study of fractional-order singular system is still at its beginning stage. As for fractional-order singular systems, efficient stability analysis and controller design methods are still absent. This thesis considers the stability analysis and controller design methods for fractional-order linear time invariant systems and fractional-order singular system. The main content of the thesis and the research results are as follows:(1)Some common basic functions, which are used to study fractional calculus system, such as Gamma function. Beta function, Mittag-Leffler function, etc. are studied. Then we introduce the three basic definitions of fractional calculus, which are the Grunwald definition, Riemann-Liouville definition and Caputo definition, and describe the relationship of the three definitions. Then the existence and u-niqueness of solutions of fractional-order differential equations is presented.(2)Through the analysis of pole distribution of the fractional-order linear time invariable system in the complex plane, using the decomposition of stability domain, a stability criterion of fractional-order linear time invariable system is derived in terms of linear matrix inequalities. Based on this result, the method for the design of a stabilizing controller is given.(3)Based on the method of linear matrix inequalities, a sufficient and necessary condition of admissibility for fractional-order singular systems with order 0< a< 1 is derived. Comparing with the existing results, this criterion don't need to use the decomposition of the original system with Weierstrass form. It has more concise formulation and fewer variables. Besides, the linear matrix inequalities criterion contains no complex matrix variable and is directly applicable to the design of a controller without bringing forth any conservatism. Finally, numerical examples illustrate the effectiveness of the proposed method.