Analysis And Control Of Several Classes Of Fractional Order Systems

As pointed out by many researchers in the last few decades,differential equations with fractional(non-integer)order differential operators,in compar-ison with classical integer order ones,have apparent advantages in modelling mechanical and electrical properties of various real materials and in some oth-er fields.Especially in terms of control,fractional order dynamic systems can better reflect the real-world situations,which may improve the freedom of the controller design and the control performance.Therefore,it is very important and meaningful to study the fractional order dynamic systems.Based on the ex-isting work,this dissertation further discusses the analysis and control synthesis problems of singular fractional order linear systems,nonlinear fractional order systems and fractional order impulsive switched systems under the definition of Caputo.The main work is as follows:In Chapter 1,the research backgrounds and motivations of fractional order systems are introduced firstly.Then,from the aspects of singular systems,sliding mode control,impulsive switched systems and unknown input observer design,the current,research status in the field of fractional order is described.In Chapter 2,the three mainstream definitions of the fractional calculus and some properties are introduced in the control field,also including the tran-scendental functions involved in the above definitions.Then,the main results concerning some exiting stability of fractional order linear and nonlinear sys-tems are given.In Chapter 3,the existing admissibility condition of singular fractional or-der linear systems is extended,which is convenient to design feedback controller with linear matrix inequality.Firstly,based on the stability of regular fractional linear systems,a new sufficient and necessary condition for the admissibility of singular fractional-order systems is presented.Then,according to the admissi-bility criterion,not only are sufficient conditions for designing pseudostate and static output feedback controllers obtained,but also sufficient and necessary conditions are presented by using different methods that guarantee the admis-sibility of the closed-loop systems.In the process of the controller design,no requirements are introduced for the original coefficient matrix of the system and the positive matrix in the constructed Lyapunov function.Moreover,the obtained stability conditions are the strict linear matrix inequalities,which are less conservative than the existing literature.In Chapter 4,the problem of output feedback sliding mode control(OF-SMC)for fractional order nonlinear systems is studied.First,combining the sliding mode and the switching surface into a singular system,a necessary and sufficient condition for the existence of a sliding surface is obtained by the ma-trix inequality and a new singular system conclusion,which is discussed in the previous chapter.Compared with the sufficient conditions given in most lit-eratures,the conservativeness of the system is reduced to some extent.Then,by introducing a free matrix,an OFSMC law is designed based on a fractional order Lyapunov method,which ensures that the resulting fractional closed-loop system is asymptotically stable and the states of the fractional closed-loop sys-tem converge to the sliding surface in finite time.The obtained condition only depends on the original coefficient matrices of the system and sliding surface,which can directly avoid the implicit limitations of system decomposition.In Chapter 5,considering the disturbances or partial inaccessible inputs,the problem of designing the observers for fractional order one-sided Lipschitz nonlinear systems with unknown inputs is investigated.By using the matrix generalized inverse technique and fractional direct Lyapunov method,the full-order and reduced-order unknown input observers are designed,which ensure that the observer error dynamic systems are asymptotically stable.Then,the design method is extended to the observer design of the singular fractional order nonlinear system.Further,the dimension of the reduced-order observer we designed is freely chosen within a certain range.Finally,compared with existing results,numerical examples and actual circuit diagrams are presented to show the effectiveness and practicability of our method.In Chapter 6,in the practical systems,impulsive effects can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching.At present,regardless of the integer order or the fractional field,most of the results consider the simultaneous occurrence of the pulse and the switching,which may leads to some limitations.In this case,we mainly discuss the stability of the fractional order switched systems with mode-dependent impulsive effects when the impulsive instant is inconsistent with the system switching.By building the relationship between the Mittag-Leffler function and the exponential function,sufficient condition for the stability of the fractional order impulsive switched systems and the impulsive switched rule are given,which are based on the model-dependent average impulsive interval and the average dwell time approaches.Moreover,the designed switching rule is closely related to the fractional order.In particular,when the fractional order is equal to 1,the results obtained can also be applied to the integer-order impulsive switched systems.Even if the impulse signal plays negative effect,the stability of the system can still be achieved.Finally,numerical examples are used to demonstrate the effectiveness of the method.In the last chapter,we summarize this dissertation and list the prospects for future work.