Full-order observers for linear fractional multi-order difference systems

Abstract

Abstract. The paper is devoted to the construction of observers for linear fractional multi-order difference systems with Riemann-Liouvilleand 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.Key words: difference operators, fractional order difference system, observer. Full-order observers for linear fractional multi-order difference systemsM. WYRWAS * Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A St., 15-351 Białystok, Poland 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 RiemannLiouville-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. [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 physically build and the experimental results were compared with numerical implementations. PreliminariesThe paper is organized as follows. At the beginning, in Sect. 2, the few types of fractional variable order derivatives are recalled, together with their discrete approximations and matrix forms. In Sect. 3 the solution of linear control system in state-space form for time-variant and time-invariant noncommensurate fractional variable order system is presented. An analog model of particular type of fractional variable order state-space system is introduced in Sect. 4. The experimental and numerical results are collected in Sect. 5. Finally, Sect. 6 summarizes the main results. Fractional variable order operatorsBelow, we recall the already known different types of fractional constant and variable order derivatives and differences. Definitions of variable order operatorsThe following fractiona...