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

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1 Complexity, Article ID , 9 pages Research Article New Stability Tests or Discretized Fractional-Order Systems Using the Al-Alaoui and Tustin Operators Raał Stanisławski, Marek Rydel, and Krzyszto J. Latawiec Department o Electrical, Control and Computer Engineering, Opole University o Technology, ul. Prószkowska 76, Opole, Poland Correspondence should be addressed to Raał Stanisławski; r.stanislawski@po.opole.pl Received 7 May 08; Accepted 8 September 08; Published 7 November 08 Academic Editor: Rongqing Zhang Copyright 08 Raał 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 o commensurate ractional-order continuous-time LTI systems using the Al-Alaoui and Tustin discretization methods. New, graphical, and analytical stability/instability conditions are given or discretetime systems obtained by means o the Al-Alaoui discretization scheme. On this basis, an analytically driven stability condition or discrete-time systems using the Tustin-based approach is presented. Finally, the stability o discrete-time systems obtained by inite-length approximation o the Al-Alaoui and Tustin operators are discussed. Simulation experiments conirm the eectiveness o the introduced stability tests.. Introduction discretization o continuous-time ractional-order systems is an important issue in various areas o science and technology, including system science, signal processing, and control theory. In this ield, we have three main discretization operators which can generate discrete-time counterparts or continuous-time ractional-order systems, in terms o the Euler, Tustin, and Al-Alaoui methods. There are two main problems to be solved during the discretization process or ractional-order systems. Firstly, the three discretization schemes lead to ininite complexity o rational, discrete-time counterparts o ractional-order derivative. Thereore, in practical applications various inite-length approximations o the discretization operators have been used, involving the most popular inite ractional dierence (FFD) approximation in the Euler approach [, ] and inite-length implementations o the continuous raction expansion (CFE) method in the Tustin and Al-Alaoui approaches [ 6]. Also, a number o papers have presented some other approximation/discretization methods or the ractional-order derivative [7 0]. Secondly, it is well known that the discretization process aects stability conditions or the discrete-time counterparts o continuous-time ractional-order systems [ 4]. Moreover, the stability conditions o ractional-order discretetime systems depend on a type o operator used in the discretization process. This can be easily seen when we compare stability results or discrete-time systems obtained or the orward-shited Euler operator [5, 6] with those or the classical backward Euler operator [3]. The irst results in the area o stability analysis or discrete-time ractionalorder systems have been developed in [], and they concern suicient conditions only. More complete results have been obtained in a special case o discrete-time ractional-order positive systems [7 0]. Simple, analytical, necessary, and suicient stability results or discrete-time systems have been obtained or both orward-shited and backward Euler operators [3, 5, 6]. Speciic, numerical stability results or discrete-time ractional-order systems based on the Euler expansion have been presented in [ 3]. On the other hand, it is well known that inite-length implementations o the Euler and Tustin operators aect the stability conditions or the underlying discrete-time systems [,, 3, 5].

2 Complexity In this paper, we introduce simple, either analytically driven or purely analytical stability tests or discrete-time systems obtained by the use o the Al-Alaoui operator. These results are then used to propose a stability test or systems based on the discretized Tustin operator, which can be considered as a special case o the Al-Alaoui approach. Also, practically oriented results or discretization using a inite-length implementation o the Tustin approach o [] are extended to inite-length approximation using the Al-Alaoui method [4]. This paper is organized as ollows. Having introduced in Section the stability problem or discretized commensurate ractional-order continuous-time systems, the system representations based on the Euler, Tustin, and Al-Alaoui discretizers are given in Section. Section 3 presents new stability results involving both graphical and analytical criteria, which are the main results o the paper. Moreover, Section 3 adopts the analytical stability criterion or the Al-Alaoui operator to the Tustin-based one. Discussion on the stability o discretization based on inite-length approximations o the Al-Alaoui operator is presented in Section 4. Simulation examples o Section 5 conirm the eectiveness o the proposed stability results. Section 5 summarizes the achievements o the paper.. Preliminaries Consider a linear continuous-time state space system o commensurate ractional order 0, described by 0D xt = A xt + Bu t, x 0 = x 0, yt = Cx t, where 0 D denotes a ractional-order derivative o order ; x t R n, u t R n u, y t R n y are the state, control, and output vectors, respectively; and A R n n, B R n n u, and C R n y n are the system matrices, with n u and n y being the number o inputs and outputs, respectively. Here, the ractional-order derivative will be described by three various representations, involving the Caputo, Riemann-Liouville, or Grünwald-Letnikov deinitions. But regardless o the speciic deinition o the ractional-order derivative, assuming the zero initial conditions in (), that is 0 D k x 0 =0, or any k R, the Laplace transorm o system () is as ollows: s Xs = A Xs +BU s, Ys= CX s In the speciic SISO case, when n u = n y =, the system o () can be described by the transer unction where A s and B s are the coprime polynomials in the variable s and λ j, j =,, n, and γ j, j =,, m, are the poles and zeros o G s, called -poles and -zeros, respectively (compare [6]). Note that the -poles λ j, j =,, n, are the eigenvalues o the state matrix A. In the discretization process, we seek or a discrete-time equivalent o the ractional-order system (), in orm o the Z-transorm wzxz = A Xz +BU z, Yz = CX z, where w z will be used as a discrete-time model o s. Alternatively, or the SISO case, we can obtain a discrete-time ractional-order transer unction in orm o Gwz = Bwz Awz = b m wz m + b m wz m + + b 0 a n wz n + a n wz n + + a 0 b m wz γ wz γ wz γm = a n wz λ wz λ wz λ n with λ j, j =,, n, and γ j, j =,, m, being the -poles and -zeros, respectively, o the system deined in (3). Three main discretizers are used here as discretization unctions w z, namely, the Euler, Tustin, and Al-Alaoui operators. The Euler operator is used in two versions, that is, the backward Euler operator s w Eu z = h z, 6 or the orward-shited Euler one s w Eu z = h z z, 7 with h being the sampling period. The Tustin operator has the orm s w Tus z = z h +z The Al-Alaoui operator [5, 6] is obtained through an assumption that the integration rule is realized as a weighted sum o the Tustin and backward Euler rules. This leads to the Al-Alaoui operator [5, 6], Gs = Bs As = b m s m + b m s m + + b 0 a n s n + a n s n + + a 0 b m s γ s γ s γm = a n s λ s λ s λ n, 3 s wz = +a h z +az, 9 where a 0, is the weighting coeicient. It has been presented in several papers that selection o the discretization operator in the discretization process

3 Complexity 3 aects the stability conditions or discrete-time systems [, 3, 6]. For instance, the two Euler methods represented by (6) and (7) lead to quite dierent stability results or the discrete-time systems. A detailed stability analysis or discrete-time systems obtained by the use o the two Euler operators has been presented in [3, 5, 6]. In this paper, a new stability analysis or systems obtained by the discretization process using the Al-Alaoui operator will be presented. 3. Main Results Firstly, we introduce a graphical stability approach, based on the Al-Alaoui operator. Im δ 0 0 e iφ t D re iφ Theorem. The discrete-time ractional-order system (4) or (5), with w z as in (9), is asymptotically stable i and only i all -poles λ j o the system, j =,, n, are outside the instability area S = r +a a h e iφ, r 0,, φ 0, π, where 0, is the ractional-order o system (4) or (5), a 0, is the weighting coeicient, and φ is the argument o the unction w e iφ = w z z=e iφ. Proo. Consider the discrete-time ractional-order system (4) or (5) with w z as in (9). Taking into account that the Al-Alaoui operator o (9) is meromorphic, the stability area with respect to -poles depends on the contour o unction w z z=e iψ, ψ = 0, π [3, 5]. Thereore, the bound contour o the stability/instability regions is as ollows: we iψ = +a h = +a h e iψ +ae iψ ρψ + iζ ψ 0, where ρ ψ = a cos ψ / +a +a cos ψ, ζ ψ = +a sin ψ / +a +a cos ψ, and ψ 0, π. Now, substituting ρ ψ and ζ ψ o () into the circle equation with the center in the point / a,0, we obtain ρϕ a + ζ ϕ = = a cos ϕ +a +a cos ϕ a +a sin ϕ + +a +a cos ϕ a 3 So, or a 0,, the inner contour ρ ϕ + iζ ϕ ulills the circle equation with the center in / a,0 and radius Figure : Simple closed positively oriented contour D. r =/ a. Thereore, we can write the same inner contour as a unction o another parameter φ, that is, ρφ + iζ φ = a eiφ, φ 0, π, 4 and inally we can describe the stability/instability contour or the Al-Alaoui operator o () as a unction o φ as we iφ = +a a h e iφ, φ 0, π 5 The next steps o the proo ollow the lines o Theorem o [3]. Note that the characteristic equation o system (4) or (5) wz λ wz λ wz λ n =0 6 is asymptotically stable i and only i all elements w z λ j, j =,, n, do not generate unstable poles in the z- domain. Accounting that w z is a meromorphic unction (w z is homomorphic in C \ 0 ) and the contour w e iφ presented in (6) is a simply closed contour in the complex plane, we can prove the stability o the elements w z λ j, j =,, n, on the basis on Cauchy s argument principle and Rouché s theorem [7]. Now, consider the simple closed contour D as in Figure whose interior domain is C = C \ re iφ,0 r <,0 φ π such that w D 0 and w D <. Then, the winding number W w D, λ j o w D about λ j, j =,, n, that counts the number o times the curve w D winds around the point λ j is Re W w D, λ j = w z dz = n πi D wz λ z n p, 7 j

4 Im w 4 Complexity Im a = 0.7 a = 0 a = 0.3 a = 0.5 δ 0 0 Re Figure : Contour o w D. w(d) + a ( a)h ( e iφ ) a = Re w Figure 3: Stability/instability areas or h =and various a. where w z = dw z /dz and n z and n p denote the numbers o zeros and poles o w z λ j inside the contour D, respectively. Since the transormation w z has a single pole in 0 (see (9)), the coeicient n p is 0. Thereore, the winding number W w D, λ j will describe a number o unstable elements n z o w z λ j. The contour w D is based on (6) and is presented in Figure. Now: ( ) Assume that the system is asymptotically stable. Then, all poles or all elements w z λ j, j =,, n, are outside the interior domain C o the contour D. Thereore, W w D, λ j =0 j =,, n and all λ j, j =,, n, are outside the area w D bounded by +a / a h e iφ, which can be described by (0). ( ) Assume that all eigenvalues λ j, j =,, n, are outside the instability area (0). Then, all eigenvalues are outside the contour w D and the winding number W w D, λ j = 0 j =,, n. Thereore, the elements w z λ j, j =,, n, do not generate poles in interior domain C o the contour D. So the system is asymptotically stable. This completes the proo. A graphical presentation o the stability/instability areas is shown in Figure 3. It can be seen rom Figure 3 that the stability area depends on the weighting parameter a o the Al-Alaoui equation. For a =0, we obtain the instability area as or the backward Euler operator (compare [3]) and increasing a leads to enlargement o the instability area. On the basis o Theorem, we can present a new analytical result as ollows. Theorem. The discrete-time ractional-order system (4) or (5) with w z described by the Al-Alaoui operator (9), with 0,, is not asymptotically stable i and only i there exists any φ j π, π λ j +a a h cos φ j, j =,, n, where λ j and φ j are the modulus and argument, respectively, o the j th -pole o the system and a 0, being the Al-Alaoui weighting coeicient. Proo. Taking into account that the stability/instability bounding contour is given by (5), which can be presented in orm o we iφ = +a a h = +a a h = +a a h sin φ cos φ + sin φ cos φ e iπ φ/, i arctan e 8 i arctan sin φ/ cos φ e tan π φ/ and introducing the resultant argument o w e iφ, that is, φ = π φ /, we obtain w e iφ = +a a h sin π φ = +a a h cos φ 9 e iφ 0 e iφ Note that () presents the modulus o w e iφ as a unction o its argument φ. Also note that or φ π, π, the unction w e iφ is redeined in φ π/, π/. For the -poles o the system with arg λ j π/, π/, j =,, n, the -poles are outside the instability area (see the stability/instability areas in Theorem ) and the system is asymptotically stable. ( ) Assume that the system is not asymptotically stable. Then, at least one λ j, j =,, n, lies inside or in the bound

5 Complexity 5 Imaginary M = M = 5 = 0.5, a = Real Figure 4: Stability/instability areas or h =and various M. o the instability area described in Theorem. Since the bound unction w e iφ is redeined in φ π/, π/, the argument o unstable eigenvalue λ j has to be φ j π/, π/. For arg λ j π/, π/, j =,, n, the -poles are inside the instability area when the modulus o the jth -pole is lower (or equal) than the modulus o the stability/instability bounding contour w e iφ presented in () or the same angle φ = arg λ j. Thereore, there is at least one -pole passing condition (8). ( ) Assume that an -pole λ j, j =,, n, o the system passes condition (8). Then, taking into account the stability/instability bound equation (), the -pole is inside (or in the bound) o the instability area. Taking into account Theorem, the system is not asymptotically stable. This completes the proo. On the basis o Theorem, we can immediately present the ollowing. Theorem 3. The discrete-time ractional-order system (4) or (5) with w z described by the Al-Alaoui operator (9), with 0,, is asymptotically stable i and only i Theorem 4. The discrete-time ractional-order system (4) or (5) with w z described by the Tustin operator (8), with 0,, is asymptotically stable i and only i φ j π, π, j =,, n, 3 where φ j is the argument o the j th -pole o the system. Proo. Consider the stability Theorem 3. Accounting that the Tustin-based approach is a special case o the Al- Alaoui approach with a, we arrive at the ollowing modulus condition: λ j +a > lim a a h cos φ j Taking into account that +a lim a a h cos φ j, j =,, n, 4 =+, 5 then we have λ j <+, j =,, n, and we arrive at condition (3). Remark. Note that the results o Theorem 4 are the same as those or continuous-time systems [8, 9]. Thus, using the Tustin-based discretization scheme does not aect the stability conditions or discrete-time systems. However, the Tustin-based discretization is inerior with respect to approximation accuracy [0]. φ j π, π λ j > +a a h cos φ j, j =,, n, Remark. The results o Theorems and 3 show that discretization o a stable continuous-time system using the Al-Alaoui-based approach guarantees the asymptotic stability o a discrete-time system. Moreover, or the unstable continuous-time system, we can select such a sampling period h and weighting coeicient a that lead to a stable discrete-time system. where λ j and φ j are the modulus and argument, respectively, o the j th -pole o the system and a 0, being the Al-Alaoui weighting coeicient. Proo. Immediate rom Theorem. The results o Theorems and 3 or the zero weighting coeicient o the Al-Alaoui operator (a =0) specialize to those or the backward Euler-based discretization scheme presented in [3]. Also, on the basis o the results, we can easily present the stability criterion or the Tustin-based discretization scheme. 4. Finite-Length Implementation o Discretization Operators It is important that the ideal Al-Alaoui and Tustin operators are not applicable in practice due to ininite-length implementation o the discretization equations. Thereore, in practical applications, we approximate the two above operators by use o rational inite-length discrete-time transer unctions. Usually, the approximations are obtained through the continuous raction expansion (CFE) method, but in the Tustin-approach, the Muir recursion can also be applied [0]. Regardless o the approximation method, the

6 Im 6 Complexity Table : Stability properties o discretized ractional-order system; Example. j φ j = arg λ j λ j a +a a h cos φ j Comment π π π π π rational discrete-time transer unction approximator ŵ z to w z is obtained as ollows: ŵz = M m= w mz m M m= v mz m 6 In this case, the contour o the approximation ŵ z in the complex plane is (see [, 4]) ŵe iφ = M m= w me imφ M m= v me imφ 7 Now, i curve (7) constitutes a simple closed curve in the complex plane, then the stability/instability areas with respect to λ j, j =,, M, are separated rom each other by the contour. Exemplary stability/instability areas or =05, h =, a =07, and various implementation lengths M are presented in Figure Simulation Examples Example. Consider a commensurate continuous-time ractional-order state space system as in () with A = B = 0 0 T, C = 0 0,, λ π 4 π 4 a = 0.8 a = a = 0.5 and =05. The system has three -poles, i.e., λ = and λ,3 =5±i4. Note that since arg λ,3 π/4, π/4, the system is unstable. The system is discretized by the use o the Al-Alaoui approach with the sampling period h =005 and our dierent values o the weighting coeicient a = 0 5, 0.6, 0.7, and 0.8. The stability properties o the discretized system designed by use o Theorem 3 are presented in Table. Moreover, Table shows the stability results or the Tustin method as a special case o the Al-Alaoui approach with a =. The -poles with the stability/instability areas are depicted in Figure 5. It can be seen rom Table that even though the continuous-time system is unstable, the Al-Alaoui method with the weighting coeicients a =05 and a =06 can lead to the stable discrete-time system, but or the weighting coeicients a =07 and a =08, the discrete-time system λ, Figure 5: Stability/instability areas or various a; Example. Re

7 Im Complexity 7 j φ j = arg λ j λ j Table : Stability properties o discretized ractional-order system; Example. a +a a h cos φ j Comment still remains unstable. It is worth mentioning that the stability/instability o the discrete-time system is related to the unstable -poles λ and λ3. Since the -pole λ is stable, the discrete-time counterpart will still be stable with respect to λ. The outcomes o Table or the Tustin-based discretization scheme conirm the result o Theorem 4. Since the -poles λ and λ are outside the instability areas or a = 0 5 and a =06 and inside the instability areas or a =07 and a =08, the results o Figure 5 and Theorem ully conirm the stability results speciied in Table. Example. Consider a commensurate continuous-time ractional-order state space system as in () with =5 and A = B = 0 T, C = 0, 0 whose eigenvalues are λ, = 0 39 ± 0 4i. Note that the system is unstable due to arg λ, 3π/4, 3π/4. The system is discretized by the use o the Al-Alaoui operator with the sampling period h = and two dierent values o the weighting coeicient a =05 and 0.8. Stability results or the discretized system are presented in Table. It can be seen rom Table that the discrete-time system is stable or a =05 but unstable or a =08. Example 3. Consider ractional-order continuous-time system Gs =, 9 s 0 9 6s s , 30 with =03 and three -poles, i.e., λ = and λ,3 =3±i Notethatarg λ,,3 3π/0, 3π/0 and the system are stable. The system is discretized with the sampling period h = 005 and the approximation o the Al-Alaoui operator as in (6), with a = /3 and dierent implementation lengths M =3and 5 ŵ 3 z = 6779z3 3 35z z +07 z 3 0 8z, z λ ŵ 5 z =6779z z z z z /z z z z z The stability areas and -poles o the system are presented in Figure 6. It can be seen rom Figure 6 that although the original system is stable, the discretized one by use o inite implementation o the Al-Alaoui operator is unstable or M =3 and stable or M =5. Thereore, in contrast to the ininite-length Al-Alaoui operator, in some speciic cases, the inite-length implementation o the Al-Alaoui discretizer or a stable continuous time system can lead to an unstable discrete-time system. All the results presented in the above examples have been conirmed by a variety o BIBO stability experiments. Remark 3. Matlab-scripted iles or the above examples are available rom the web: doi:0.58/zenodo Conclusion 3π 0 3π 0 M = 3 M = Figure 6: Stability/instability areas or dierent implementation lengths M; Example 3. This paper has oered new, simple, graphical, and analytical stability/instability conditions or continuous-time commensurate ractional-order systems discretized by the use o the Al-Alaoui and Tustin operators. Firstly, theoretical stability condition or the Al-Alaoui operator has been given in a graphical way, which is then used in simple, analytical stability tests or both Al-Alaoui and Tustin approaches. Finally, λ λ 3 Re

8 8 Complexity the stability o discrete-time systems by use o inite-length approximations o the Al-Alaoui and Tustin discretizers has been discussed. Simulation examples ully conirm the original stability results obtained. It is important that the ractional-order stability analysis presented in the paper is based on the eigenvalues o the state matrix, exactly as or the integer-order systems. This method is very useul; however, in case o high dimensions o the state matrices, calculation o eigenvalues may lead to numerical problems. Conlicts o Interest The authors declare that they have no conlicts o interest. Reerences [] I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Nonlinear Physical Science, Springer, Berlin, Heidelberg, 0. [] M. Siami, M. S. Tavazoei, and M. Haeri, Stability preservation analysis in direct discretization o ractional order transer unctions, Signal Processing, vol. 9, no. 3, pp , 0. [3] Y. Q. Chen and K. L. 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