The monograph is based on the author works on the stabilization and control design for discrete and continuous Fractional Order Systems (FOS). The first two chapters and the initial parts of the third chapter are devoted to basic concepts of FOS and sliding mode control for FOS.
In Chapter 3, existence results of the solution of Fractional Order Differential Equation (FODE) with discontinuous right-hand side are presented. Then, the sliding surface design methodology is introduced for FOS with application to FODEs. A short review on point to point control of fractional chain integrator is given. A stabilization method of uncertain fractional chain of integrator is given which is illustrated by a numerical example of the proposed control scheme.
In Chapter 4, the higher sliding mode control of FOS in terms of output and their chain of fractional derivatives is formulated. At first, some useful properties of fractional calculus are presented. Then, the concept of finite-time steering of a controllable fractional linear system from initial to final states is introduced and reviewed. The main result on finite-time stabilization of uncertain chain of integrator with numerical examples is formulated with its extension on more general uncertain FOS.
Chapter 6 deals with the problem of cooperative control of networked fractional order multiagent systems over a directed interaction graph which has many applications in military and civil services. Main contributions of this chapter are: agents are modeled in a more realistic way using fractional order integrator; for achieving the specified goals, a new fractional order control law is designed based on sliding mode theory; finite time reach ability to the sliding surface is proved at the usage of fractional order extensions of the Lyapunov stability criterion; robustness of proposed controllers to matched uncertainties and continuous.
After some preliminaries on Fractional Systems with Discrete Time (FSDT) in Chaper 7, a methodology for stabilization of the FSDT based on discrete sliding mode approach is suggested. Quai-sliding band for FSDT containing matched uncertainty obtained.
Chapter 8 presents one of the open problems – the robust controller design for discrete FOS. For achieving specific goals, the disturbance observer based on full state information is constructed. After that, the extra compensator based on the theory of disturbance observer is added to the classic state feedback. Also it is shown that – after applying the control – the closed loop system would become robust against the disturbance when the estimation error gets small.
Chapter 9 “Contraction Analysis by Integer Order and Fractional Order Infinitesimal Variations” presents a short review of fractional order Routh-Hurwitz condition, introduces the concepts of contraction theory and gives the motivation for finite time stabilization of an integrator chain with subsequent its generalization for FOS (the main result) accompanied by the illustrative numerical example. Then, the contraction analysis using fractional infinitesimal variation is presented with the extension of the same approach for checking the contraction behavior of FOSs.