Research Of Some Problems About Fractional-order System

In practical engineering control field,many dynamical systems can be modeled by fractional-order systems.As a result,fractional-order systems have attracted more and more attention from many scholars,and some research results have been obtained.However,the research on the dynamic properties of fractional-order systems and the synthesis design of controllers are not perfect.This paper mainly focuses on the dynamic properties of the fractional-order posi-tive systems,fractional-order cone invariant systems and fractional degree generalized frequency domain linear systems.The main contents of are detailed as followsIn Chapter 1,we introduce the research background of this article and some important preliminaries.In Chapter 2,the control problem of fractional-order positive systems LP(p ? {l,2,?})is investigated.This paper investigates the analysis and synthesis control problems of L1 and L2 performance on finite time interval,and also gives a necessary and sufficient condition of L? performance on infinite time interval for fractional-order positive systems.Like the integer-order positive system,the L1 and L? performance characterisations can be expressed by linear programming and L2 performance characterisation can be expressed in term of linear matrix inequality.Secondly,based on these performance conditions,the design method of state feedback controller is given.In Chapter 3,the problems of stability and cone-induced performance analysis for fractional-order delay systems constrained on a proper cone are investigated.Based on the partial order relationship defined on a cone,a cone preserving condition is firstly given for fractional-order systems with bounded time-varying delays.Then by comparing the trajectory of the time-varying delay system with that of the constant delay system,the asymptotic stability conditions of cone-invariant fractional-order time-varying delay systems are given.In addition,it is shown that the cone-induced performance of a cone-preserving fractional-order time-varying delay sys-tem is independent of the magnitude of delays and can be fully determined by system matrices.Finally,a numerical example is given to show the validity of the theoretical results.In Chapter 4,the problems of controllability(observability)and stability for fractional degree generalized frequency domain variable linear system axe investigated.Firstly,some nec-essary and sufficient conditions are given to guarantee controllability and observability of such linear system.Then by discriminating the distribution of the roots of the fractional degree poly-nomials,the stability conditions of the linear system are given.The stability results are applied to the cooperative control problem of fractional-order multi-agent system,and two examples are provided to illustrate the effectiveness of the proposed methods.