A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equation

Abstract

Abstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example confirms the usefulness of the method. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with ny approximator to the operator expressed by a discrete transfer function, e.g. CFE-based Al-Alaoui approximation. A simulation example onfirms the usefulness of the method. Many real applications, to mention model-based conrol, model-based fault detection, require to implement a oninteger-order model at a digital platform like PLC or PGA. Known discrete-time state space models of nonintegerrder systems are typically based on the Grünvald-Letnikov GL) definition. An accurate implementation of this model, in articular at the bounded resource platforms, requires a (very) ong-length approximation of the GL-based system [33].The purpose of this paper is to propose a new, discrete-time tate space model of a noninteger-order system, constructed ith the use of the continuous fraction expansion (CFE) imlemented for the Al-Alaoui operator. The use of such an aproximant enables to obtain a much more effective model in hat the memory length is quite low. This recommends its use t industrial digital platforms.The paper is organized as follows. Heaving recalled the ackground of the paper in Section 1, Section 2 outlines the undamentals of fractional-order calculus and introduces a operator is defined aswhere a and t denote time limits for calculation of the operator, α ∈ R denotes the noninteger order of the operation.Next, an idea of the Gamma (Euler) function (see for example [12]) can be given: DEFINITION 2. The Gamma function is defined asThe fractional-order, integro-differential operator (1) can be described by different definitions, given by Grünvald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). All these definitions are given below. With respect to particular additional assumptions these definitions can be considered equivalent. A new algorithm for a CFE-approximated solution of a discrete-time noninteger-order state equationAbstract. In the paper, a new method for solution of linear discrete-time fractional-order state equation is presented. The proposed method is simpler than other methods using directly discrete-time version of the Grünwald-Letnikov operator. The method is dedicated to use with any approximator to the operator expressed by a discrete tra...

A Memory–Efficient Noninteger–Order Discrete–Time State–Space Model of a Heat Transfer Process

“…Stability of the discrete system considered will be analyzed with the use of the approach presented by Stanislawski and Latawiec (2013a; 2013b), Ostalczyk (2016, pp. 202-223) and Oprzędkiewicz et al (2017b).…”

“…The accuracy of the model we deal with can be estimated using the approach presented by Oprzędkiewicz et al (2017b) with the use of the steady-state error of the model considered. This error can be estimated using Proposition 1 by Oprzędkiewicz et al (2017b):…”

“…The model considered follows directly from the semigroup model given by Oprzędkiewicz and Gawin (2016) as well as Oprzędkiewicz et al (2016a). It employs a new, continuous fraction expansion (CFE) based solution method proposed by Oprzędkiewicz et al (2017b). The approach allows us to obtain a model accurate and significantly smaller than the analogous model using power series expansion (PSE) approximation, discussed by Oprzędkiewicz et al (2017a).…”

“…A detailed analysis of various forms of CFE approximators is presented by Stanisławski et al (2015). Oprzędkiewicz et al (2017b). Its idea consists in replacing the continuous operator s α in the Laplace transform of the FO state equation 9by its discrete CFE approximant expressed by (10), with coefficients given in Table 1.…”

New Stability Tests for Discretized Fractional‐Order Systems Using the Al‐Alaoui and Tustin Operators

“…These results are then used to propose a stability test for systems based on the discretized Tustin operator, which can be considered as a special case of the Al-Alaoui approach. Also, practically oriented results for discretization using a finite-length implementation of the Tustin approach of [2] are extended to finite-length approximation using the Al-Alaoui method [24].…”