Study On Some Problems In Analysis And Control Of Fractional-order Systems

Fractional calculus as an extension of ordinary calculus has a history ofmore than 300 years. Fractional-order systems are represented by di?eren-tial equations with non-integer order. It has been found that the behaviorof many physical systems can be more properly described by fractional-order systems than those of integer-order models. At present, the studyof fractional-order systems is still at its beginning stage. As for a generalfractional-order system, e?cient stability analysis and controller designmethods are still absent. This paper considers two classes of system inthe fractional-order system theory, i.e. the fractional generalized systemand the fractional delayed system. To the best knowledge of the author,it is the first time that the fractional generalized system is studied in ourcountry. For the fractional linear time-invariant generalized system, wediscuss its stability problem using linear matrix inequalities(LMI) method,and then address the robust stability of the fractional generalized systemwith parametric uncertainties and its robust controller design. For thefractional-order system with constant delays, we analyze the relation be- tween its characteristic roots and its stability, and investigate its frequencydomain analysis via complex Lyapunov function and matrix measure the-ory. The main contributions and achievements of this paper are detailedbelow:1. The regularity and the impulsiveness of fractional generalized sys-tems are investigated. On the basis of the presented state-space descriptionof the fractional generalized system, its state response is obtained by usingLaplace transform. The problem of the existence and the uniqueness ofthe fractional generalized system solution, as well as that of its impulsive-ness is discussed, some conclusions on regularity and non-impulsiveness areproposed for the fractional generalized system.2. Studies are made on the stability and the admissibility of the frac-tional generalized system. Through the analysis of the pole distribution ofthe fractional generalized system in the complex plane, a stability criterionis obtained in the form of argument. Then a new su?cient and neces-sary LMI conditions are proposed for generalized systems, from which twoforms of su?cient and necessary condition are derived to determine theadmissibility of the fractional generalized system, i.e. strict LMI conditionand non-strict LMI one with equality constraint.3. With the help of LMI method, the problem of robust admissibilitycontrol for the fractional generalized system with parametric uncertain-ties is addressed. For two types of uncertainty, i.e. the norm-bounded uncertainty and the polytopic uncertainty, a few su?cient conditions areprovided to guarantee the robust admissibility of uncertain fractional gen-eralized system. From these conditions and with the use of variable sub-stitution, the LMI-based state feedback controller and dynamic outputfeedback controller design methods are developed. Finally the simulationresults of several examples demonstrate the validity and the feasibility ofour methods.4. The stability of the fractional-order system with constant delayis dealt with. The definition of the stability, as well as independent-of-delay stability of fractional delayed systems is first given. By adoptingcomplex Lyapunov matrix function and matrix measure methods, somedelay-independent criteria are advanced, and then delay-dependent criteriaare also presented by taking into account of the size of time-delay.