Stability Of Some Fractional Order Linear Systems

The development of modern industry and information technology is continuously improving the need of modern science and technology, so the today’s research needs more realistic and accurate mathematical models to analyze and synthesize the system for qualitative, stability and controllability and other aspects, improving the performance of the system to a greater extent, to achieve high precision required for modern science research by improving and controlling system precision by tracking error and restraining disturbance and flexibility of parameters control and so on. To solve these problems,fractional system stability, control theory, and qualitative theory are emerging. However,due to the complexity of fractional differential definition, the research-related aspects of the fractional system is relatively slow progress.This paper mainly studies the stability of several kinds of linear fractional-order systems by seeking to a more compact form, more efficient judgement theorems for the stability of several classes of fractional order linear systems. The main contents are as follows: Firstly, we study the stability of the class of uncertain fractional order linear systems without time delay, by linear matrix inequality method, under three circumstances when the order of fractional differential, the system matrix uncertainty and even the system matrix and order fractional differential are both uncertainty. Secondly, by applying a series inequalities on matrix elements and matrix eigenvalues, the stability of a class of fractional order linear time-invariant systems are established. Specially, we find the relationship between fractional differential 0 < α < 1 and 1 < α < 2 for two kinds of linear system stabilities and broaden the existing research results applied scope.In addition, the stability of three types of linear systems are studied by using the characteristic function method, including a fractional linear systems with time delay and two types of neutrality fractional linear systems. This thesis discussed for the three systems under the constant and uncertain conditions for a given system, respectively. Characteristic function method for integer order system has been very mature, but for fractional order system is far from perfect, and even is confined to a delayed fractional order linear systems. The reason is that correct characteristic function of fractional order linear systems are transcendental equation, which increases the difficulty of the study.From the relationship of fractional order linear systems and corresponding integer-order linear systems, we find the relationship between the characteristic function of fractional order linear systems and corresponding integer-order linear system. Based on this relationship and theorems in matrix theory of eigenvalues, by minimizing conservative algebra reduction, a series of the form simple, easy to determine conditions of the stability of the three kinds of fractional order linear systems with time delay are deduced. Finally, according to the conclusions, some specific numerical examples are given, which confirm the validity of these conclusions.