Fractional‐order iterative learning control for fractional‐order linear systems

Abstract

The paper is devoted to the construction of observers for linear fractional multi-order difference systems with Riemann-Liouville-and Grünwald-Letnikov-type operators. Basing on the Z-transform method the sufficient condition for the existence of the presented observers is established. The behaviour of the constructed observer is demonstrated in numerical examples. h = 1 has been studied in [24, 25]. The aim of the present paper is to study the construction of the full-order observers for linear fractional muti-order discrete-time systems with the Riemann-Liouville-and Grünwald-Letnikov-type difference operators with the step h > 0. We restrict the design of the observers for the systems whose fractional orders are from the interval (0, 1], because the systems with fractional orders higher than one can be always transform to systems with orders less than or equal to one, see for instance [26]. The paper is organized in the following way. Section 2 gathers preliminary notations, facts and definitions needed in the sequel. In Section 3 the initial value problems for fractional multi-order systems are presented. The main results of the paper, namely the construction of the fractional observer, that estimates the unknown state vector, is presented in Section 5. Since the fractional order system corresponding to the error vector should be asymptotically stable in order to guarantee the estimation of the unknown state of the system by the observer , the condition for asymptotic stability of fractional order systems is given in Section 4. Additionally, two examples that illustrate our results are presented. Finally, the conclusions are drawn. 2. Preliminaries introduce a matrix approach for approximate solving non-commensurate fractional variable order linear he approach is based on switching schemes that realize variable order derivatives. The obtained numerical and analog model results calculus, differential equations, analog modeling aditional integer order inte-on-integer order operators. 1695 by Leibniz and de ury, Liouville and Riemann fractional derivative. How-entury, the idea drew atten-kground of fractional calcu-Fractional calculus has been behavior of many materials volving diffusion processes. be modeled more efficiently emonstrated in [4, 5]. is time-varying, begun to be l variable order behavior can mistry when system's prop-cal reactions. Experimental mple of physical fractional presented in [6]. The vari-sed to describe time evolu-Numerical implementations ators and differentiators can actional variable order cal-be variable order fractional order interpretation of the ders integrators, realized as red. Applications of variable e also in control [12, 13, 14]. of variable order derivative native definitions of variable in [17, 18]. Numerical and onal variable order differen-ectively in [19, 20] and [21]. ing a numerical solution of ystem in a state-space form iant as well as time-variant lts are also valid for system rent types of variable orders ch the fractional variable order state-space system was p...

Lebesgue‐p NORM Convergence OF Fractional‐Order PID‐Type Iterative Learning Control for Linear Systems

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Fractional differential repetitive processes with Riemann–Liouville and Caputo derivatives

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ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems

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