Stability Analysis Of A Class Of Singular Fractional-order System

In essence, the human society and the natural world are actually fractional-order systems. The so-called’Fractional-order’is more accurate to say that the’non-integer order’. Also refers to the order of their calculus is not the traditional first, second, third and so on but arbitrary. In practice, if we can use the fractional-order model to build the dynamic response of the systems, then,there is no doubt we can improve the accuracy of description, and our ability to research, design, and control systems. As for singular fractional-order systems, since the singular form not only contains the static information of the system but also the dynamic constraints, therefore singular description is more accurate than the traditional description. For a practical system, because of the various factors that can not be avoided, such as errors generated when the system modeling, changes in the working environment and operating conditions, the aging, wear, etc of executive component and control element, such that there often exists uncertainty in the mathematical model of the describing object. It is also true in the singular fractional-order systems. For this reasons, in the practical engineering applications, considering these uncertain factors are very necessary during the modeling, analysis and design of control object.First, for a class of deterministic singular fractional-order system, when the state of the system is not directly detect or even some state variables can not be detected, put for-ward the asymptotically stable conditions which based on the method of state observer and output feedback when the fractional-order is taken (0.1) and (1,2). Problems can effectively deal with when the state variables can not be directly detected by the state observer based approach. For this method, we give a rigorous theoretical proof.Secondly, for a class of norm-bounded uncertain singular fractional-order system, designing an output feedback controller when the fractional-order is (1,2). The con-troller consists of two parts, one part can be seen as a regular controller, through its design to make the uncertain singular fractional-order system can be regular, and then through another part of the design makes the regular uncertain fractional order system to be robust asymptotically stable. Finally, two numerical examples prove the correctness and rationality of the proposed method by a feasible solution of solving LMI.Finally, for a class of determine singular fractional-order system, we study the asymptotically stable conditions of singular fractional order linear time-invariant sys-tems when the fractional-order is (1,2) and the BIBO stable conditions when the fractional-order is (0,1). In both cases, we give the sufficient and necessary conditions when the system does not input and the conditions when the system under the state feedback. In spite of, there have been some relevant conclusions about this when the fractional-order is (1,2), but different with some existing conclusions, we based on the method which to make the stability fractional-order system is equivalent to the stability of an integer-order system. Consideingr a more general form of singular fractional-order system, the stability of the system is equivalent to the stability of a general integer order system and obtain the stability condition. Numerical simulation results show the correctness and rationality of conclusions.