Abstract.A new algorithm is proposed for linear dynamic systems of the identification singleinput-single-output (SISO) discrete fractional order with fractional errors-in-variables. The estimates are proved to be convergent to the true values with a probability one. The results of a simulated example indicate that the proposed algorithm provides good estimates.
IntroductionThe fractional calculus based on integro-differential operators of the generalized fractional order is a very old topic in mathematics [1]. At the end of the 19th century, Liouville and Riemann introduced the first definition of the fractional derivative.As far as the applications of fractional calculus are concerned, there is a large volume of research on viscoelasticity/damping, [2,3] and chaos/fractals [4], dielectric materials [5], electrochemical processes and flexible robots [6], traffic in information networks [7].Upon automatic control, control algorithms both in frequency [8,9] and time [10] domains based on the concepts of fractional calculus have been proposed. More examples, physical interpretations and areas of applications of fractional calculus are to be found in [11][12][13].Due to their long memory behavior, the identification of fractional order models is more difficult compared with those of the integer order models. Several algorithms, based on the frequency domain, were proposed to solve this problem [6]. In [14][15][16], time domain identification techniques of the discrete time models of the fractional order, based on the least nonrecursive squares, are presented. For an overview of different identification methods based on fractional models, let us refer to Malti et al. [17].In [18], a generalization of the Kalman filter for linear and nonlinear fractional order discrete statespace systems is presented.Identification of fractional systems is much more difficult when there is a color noise. Therefore, the choice of instrumental variables is difficult because of the long memory of the fractional systems. The paper proposes a method to identify fractional systems for one of the colored noise classes: a fractional noise.A new algorithm generalizes the results [19] in case of fractional noises. The paper is organized as follows. In the next section, we present the problem statement. In section 3, the criteria for identifying systems are defined. In section 4, the iterative algorithm isdetermined. The simulation results are presented in section 5. Finally section 6 concludes this paper.
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