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

Hindawi Complexity Volume 2018, Article ID 2036809, 9 pages https://doi.org/10.1155/2018/2036809 Research Article New Stability Tests for Discretized Fractional-Order Systems Using the Al-Alaoui and Tustin Operators Rafał Stanisławski , Marek Rydel, and Krzysztof J. Latawiec Department of Electrical, Control and Computer Engineering, Opole University of Technology, ul. Prószkowska 76, 45-758 Opole, Poland Correspondence should be addressed to Rafał Stanisławski; Received 17 May 2018; Accepted 18 September 2018; Published 7 November 2018 Academic Editor: Rongqing Zhang Copyright © 2018 Rafał Stanisławski et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper provides new results on a stable discretization of commensurate fractional-order continuous-time LTI systems using the Al-Alaoui and Tustin discretization methods. New, graphical, and analytical stability/instability conditions are given for discretetime systems obtained by means of the Al-Alaoui discretization scheme. On this basis, an analytically driven stability condition for discrete-time systems using the Tustin-based approach is presented. Finally, the stability of discrete-time systems obtained by finite-length approximation of the Al-Alaoui and Tustin operators are discussed. Simulation experiments confirm the effectiveness of the introduced stability tests. 1. Introduction Stable discretization of continuous-time fractional-order systems is an important issue in various areas of science and technology, including system science, signal processing, and control theory. In this field, we have three main discretization operators which can generate discrete-time counterparts for continuous-time fractional-order systems, in terms of the Euler, Tustin, and Al-Alaoui methods. There are two main problems to be solved during the discretization process for fractional-order systems. Firstly, the three discretization schemes lead to infinite complexity of rational, discrete-time counterparts of fractional-order derivative. Therefore, in practical applications various finite-length approximations of the discretization operators have been used, involving the most popular finite fractional difference (FFD) approximation in the Euler approach [1, 2] and finite-length implementations of the continuous fraction expansion (CFE) method in the Tustin and Al-Alaoui approaches [2–6]. Also, a number of papers have presented some other approximation/discretization methods for the fractional-order derivative [7–10]. Secondly, it is well known that the discretization process affects stability conditions for the discrete-time counterparts of continuous-time fractional-order systems [11–14]. Moreover, the stability conditions of fractional-order discretetime systems depend on a type of operator used in the discretization process. This can be easily seen when we compare stability results for discrete-time systems obtained for the “forward-shifted” Euler operator [15, 16] with those for the classical backward Euler operator [13]. The first results in the area of stability analysis for discrete-time fractionalorder systems have been developed in [11], and they concern sufficient conditions only. More complete results have been obtained in a special case of discrete-time fractional-order positive systems [17–20]. Simple, analytical, necessary, and sufficient stability results for discrete-time systems have been obtained for both “forward-shifted” and backward Euler operators [13, 15, 16]. Specific, numerical stability results for discrete-time fractional-order systems based on the Euler expansion have been presented in [21–23]. On the other hand, it is well known that finite-length implementations of the Euler and Tustin operators affect the stability conditions for the underlying discrete-time systems [2, 11, 13, 15]. 2 Complexity In this paper, we introduce simple, either analytically driven or purely analytical stability tests for discrete-time systems obtained by the use of the Al-Alaoui operator. 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]. This paper is organized as follows. Having introduced in Section 1 the stability problem for discretized commensurate fractional-order continuous-time systems, the system representations based on the Euler, Tustin, and Al-Alaoui discretizers are given in Section 2. Section 3 presents new stability results involving both graphical and analytical criteria, which are the main results of the paper. Moreover, Section 3 adopts the analytical stability criterion for the Al-Alaoui operator to the Tustin-based one. Discussion on the stability of discretization based on finite-length approximations of the Al-Alaoui operator is presented in Section 4. Simulation examples of Section 5 confirm the effectiveness of the proposed stability results. Section 5 summarizes the achievements of the paper. where A sα and B sα are the coprime polynomials in the f f variable sα and λ j , j = 1, … , n, and γ j , j = 1, … , m, are the poles and zeros of G sα , called f -poles and f -zeros, f respectively (compare [16]). Note that the f -poles λ j , j = 1, … , n, are the eigenvalues of the state matrix A f . In the discretization process, we seek for a discrete-time equivalent of the fractional-order system (2), in form of the Z-transform w z X z = A f X z + BU z , 4 Y z = CX z , where w z will be used as a discrete-time model of sα . Alternatively, for the SISO case, we can obtain a discrete-time fractional-order transfer function in form of = Gwz = Bwz Awz = bm w z m + bm−1 w z m−1 + ⋯ + b0 an w z n + an−1 w z n−1 + ⋯ + a0 bm w z − γ1 f f w z − γ2 … w z − γm f w z − λ2 … w z − λnf an w z − λ1 f f , 5 2. Preliminaries f Consider a linear continuous-time state space system of commensurate fractional order α ∈ 0, 2 described by α 0D x t = A f x t + Bu t , x0 = x 0 , y t = Cx t , 1 where 0 Dα denotes a fractional-order derivative of order α; x t ∈ ℝn , u t ∈ ℝnu , y t ∈ ℝny are the state, control, and output vectors, respectively; and A f ∈ ℝn×n , B ∈ ℝn×nu , and C ∈ ℝny ×n are the system matrices, with nu and ny being the number of inputs and outputs, respectively. Here, the fractional-order derivative will be described by three various representations, involving the Caputo, Riemann-Liouville, or Grünwald-Letnikov definitions. But regardless of the specific definition of the fractional-order derivative, assuming the zero initial conditions in (1), that is 0 Dk x 0 = 0, for any k ∈ ℝ, the Laplace transform of system (1) (...truncated)