Mathematical Problems in Engineering

Volume 2015, Article ID 512104, 10 pages

http://dx.doi.org/10.1155/2015/512104

## A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

Department of Electrical, Control and Computer Engineering, Opole University of Technology, Ulica Sosnkowskiego 31, 45-272 Opole, Poland

Received 17 August 2014; Revised 30 December 2014; Accepted 31 December 2014

Academic Editor: Baocang Ding

Copyright © 2015 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.

#### Abstract

This paper provides a series of new results in both steady-state accuracy and frequency-domain analyses for two Laguerre-based approximators to the Grünwald-Letnikov difference. In a comparative study, the Laguerre-based approximators are found superior to the classical Tustin- and Al-Alaoui-based approximators, which is illustrated in simulation examples.

#### 1. Introduction

Various approximations to a discrete-time fractional difference (FD) have been pursued in order to prevent its possible computational explosion problem and provide high approximation accuracy. Since FD represents in fact (a sort of) an infinite impulse response (IIR) filter, one solution has been to least-squares (LS) fit an impulse/step response of a discrete-time integer-order IIR filter to that of the associated FD [1–3]. However, the problem is to propose a “good” structure of the integer-order filter, possibly involving a low number of parameters. On the other hand, an LS fit of the FIR filter to FD has been analyzed in the frequency domain [4], with the high-order optimal filter providing a good approximation accuracy, at the cost of a remarkable computational effort however. Similar results are reported in other FIR-based approximations to FD [5, 6]. New time-domain modeling concepts for FD have been introduced in [7, 8].

The above introductory reference review is, deliberately, far from completeness. We refer the reader to the excellent surveys of the state of the art in discretization of fractional-order derivatives [9–15], providing a broad spectrum of the discretization machinery. For space saving reasons, we refrain from repeating the discretization principles and technologies covered therein. Rather, we will recall from [16] the main mathematical results on our unique, Laguerre-based approach [16–18] to direct discretization of the Grünwald-Letnikov (GL) fractional-order derivative. The approach advocates the use of the Laguerre filters, rather than, for example, FIR ones. Indeed, the number of FIR components used in, for example, LS-based Pade, Prony, or Shanks discretization schemes [1, 19] is dramatically higher than the number of their Laguerre counterparts. Thus, our Laguerre-based approach is highly competitive in terms of computational efficiency, in addition to a very high approximation accuracy. Also, our discretization approach is computationally superior to the optimization-based competitors of [14]. It is also worth mentioning that another Laguerre-based discretization approach of [20] is related to the Tustin operator which will be shown essentially inferior to our approximation concept.

This paper extends an original concept of the employment of the Laguerre filters in approximation of the Grünwald-Letnikov fractional difference as intimated in [16]. In particular, new effective solutions are offered as a result of time- and frequency-domain analyses of various versions of Laguerre-based fractional differences. An excellent approximator to FD, which is a combination of the classical finite fractional difference (FFD) [8] and finite Laguerre-based difference (FLD) [16], is found superior to the celebrated Al-Alaoui-based approximator. We advocate the contribution of the FFD on the one hand since in the high frequency range it is identical to the original FD [8, 21, 22]. On the other hand, we advantage approximating the medium/low-frequency “tale” of the FD by means of the Laguerre filter FLD [16, 18].

The remainder of this paper is structured as follows. Section 2 outlines the fundamentals of the Grünwald-Letnikov/Riemann-Liouville fractional-order discrete-time derivative (DTD) comprising the Grünwald-Letnikov fractional-order difference (FD). Also, the finite-length approximation to FD, namely, finite fractional difference (FFD), is recalled. The basics of orthonormal basis functions, in particular Laguerre functions, are presented in Section 3, and their application in the construction of the Laguerre-based difference (LD) and combined fractional/Laguerre-based difference (CFLD) is given in Section 4. Finite approximations of LD and CFLD, called finite Laguerre-based difference (FLD) and finite (combined) fractional/Laguerre-based difference (FFLD), respectively, are shown in Section 5, also comprising, for comparison purposes, the Tustin- and Al-Alaoui-based approximations. This most important Section also includes original analyses of both steady-state errors and frequency-domain behaviors of the FLD/FFLD-based versus Tustin- and Al-Alaoui-based models of DTD. The Section is culminated with an important technical theorem enabling estimation of a sampling interval for the FD-based DTD, guaranteeing the prespecified phase accuracy requirement, which can be projected to the FLD- and FFLD-based models of DTD. Simulation examples of this section demonstrate high performances of the FLD- and, in particular, FFLD-based approximations to DTD as compared with the Tustin/Al-Alaoui-based ones. Conclusions of Section 6 summarize the contributions of this paper.

#### 2. Grünwald-Letnikov Fractional-Order Difference

It is well known [23, 24] that continuous-time fractional-order derivatives of Grünwald-Letnikov and Riemann-Liouville can be discretized at the sampling interval to obtain the (fractional-order) discrete-time derivative (DTD): where the Grünwald-Letnikov fractional-order difference (FD) in discrete time is described by equation where is the fractional order, is the backward shift operator, and with

*Remark 1. *For brevity, we will proceed with the FD instead of the more general DTD. Whenever substantial, however, we will comment on the effect of on the results to follow.

In [8, 25], truncated or finite fractional difference (FFD) has (in analogy to FIR) been considered for practical, feasibility reasons, with the convergence to zero of the series enabling assuming for some , where is the number of backward signal samples used to calculate the fractional difference. We will further proceed with FFD, to be formally defined below.

*Definition 2 (see [8]). *Let the fractional difference (FD) be defined as in (2) to (4). Then the finite fractional difference (FFD) is defined as
where and is the upper bound for when .

The FFD has been analyzed in some papers under the heading of a practical implementation of FD [26–28], or a truncated/finite difference [21, 23, 29], or a short-memory difference [30].

*Remark 3. *It is well known [8] that, equivalent to (2), FD can be rewritten as the limiting FFD (for ) in the form
with for all .

#### 3. Orthonormal Basis Functions

It is well known that an open-loop stable linear discrete-time IIR system governed by the transfer function where the impulse response , , can be described in the Laurent expansion form [31, 32] including a series of orthonormal basis functions (OBF) and the weighting parameters , , characterizing the model dynamics.

Various OBF can be used in (8). Two commonly used sets of OBF are simple Laguerre and Kautz functions. These functions are characterized by the “dominant” dynamics of a system, which is given by a single real pole or a pair of complex ones , respectively. In case of discrete-time Laguerre filters to be exploited hereinafter, the orthonormal functions with and , consist of a first-order low-pass factor and th-order all-pass filters.

*Remark 4. *It is important that the factor need not include the sampling interval (which can be set to unity) and this is because the FD components , , do not include .

*Remark 5. *Depending on the domain context, we will use various arguments in , for example, in the -domain and or in the time-domain. The same concerns the arguments in .

*Remark 6. *Our interest in the Laguerre filters also results from the fact that their well-damped behavior fit the nonoscillatory dynamics of DTD (in addition to a low number of Laguerre model parameters involved).

The coefficients , , can be calculated from the scalar product of and [31]: where is the complex conjugate of and is the unit circle. Note that and , , must be analytic in . It is also possible to calculate the scalar product in the time-domain where the impulse response of the system , , , , and is the Kronecker delta.

#### 4. Laguerre-Based Fractional-Order Differences

##### 4.1. Laguerre-Based Difference

Let us firstly define a “sort of” a difference to be referred to as the Laguerre-based difference.

*Definition 7 (see [16]). *Let and , , be described as in (8) through (10). Then the Laguerre-based difference (LD) is defined as
with for all .

*Remark 8. *Again, whenever substantial we will comment on the effect of the sampling period when using an LD-based form of DTD (compare Remark 1). This will also hold true for the forthcoming CFLD and its two approximators FLD and FFLD.

Since in (6) represents a sort of IIR and so does as in (12), the question arises as to what a relationship between and is and, moreover, if and when it is possible to obtain .

Now, a fundamental equivalence result in this respect is recalled.

Theorem 9 (see [16]). *Let the FD be defined as in (2) through (4) or, equivalently, as in (6) and let the LD be defined as in Definition 7. Then LD is identical to FD, that is, , if and only if
**
with , being the dominant Laguerre pole and
*

*4.2. Combined Fractional/Laguerre-Based Difference*

*Let us finally define a combined fractional/Laguerre-based difference, which is a combination of the “classical” FD and our LD.*

*Definition 10 (see [16]). *Let the FD and LD be defined as in (2) and (12), respectively. Then the combined fractional/Laguerre-based difference (CFLD) is defined as
where
with the first component at the right-hand side of (16) constitutes the FFD share in the CFLD and the second one is the (-delayed) LD share, with , , as in (3) and (4), and and , , as in (9) and (10), respectively.

*Here is another fundamental equivalence result.*

*Theorem 11 (see [16]). Let the Grünwald-Letnikov fractional difference (FD) be defined as in (2) through (4), the Laguerre-based difference (LD) is as in Definition 7 and the combined fractional/Laguerre-based difference (CFLD) is as in Definition 10. Then CFLD is equivalent to FD in that if and only if
with and being the dominant Laguerre pole and
*

*Remark 12. *Note that regardless of an actual value of we have , in the sense that , .

*Note that the above-presented fractional-order differences FD, LD, and CFLD may lead to computational explosion. So, in the next section, finite approximations of the above will be considered.*

*5. Finite Approximations of Fractional-Order Differences*

*5. Finite Approximations of Fractional-Order Differences*

*5.1. Finite Fractional Difference*

*5.1. Finite Fractional Difference*

*In Section 2, the “classical” finite fractional difference (FFD) has been redefined. In a similar way, we define two finite fractional approximators to LD and CFLD.*

*5.2. Finite Laguerre-Based Difference*

*5.2. Finite Laguerre-Based Difference*

*In analogy to the presented finite fractional difference (FFD), the convergence to zero of the series enables assuming for some , where is the number of the Laguerre filters used to calculate the finite LD. We will further proceed with the finite Laguerre-based difference (FLD), to be formally defined below.*

*Definition 13 (see [16]). *Let the Laguerre-based discrete-time difference (LD) be defined as in Definition 7. Then the finite Laguerre-based difference (FLD) is defined as
where is the number of the Laguerre filters used do calculate the difference FLD and , , are calculated as in (13).

*5.3. Finite Fractional/Laguerre-Based Difference*

*5.3. Finite Fractional/Laguerre-Based Difference*

*The idea behind combining FFD and FLD comes from a priori knowledge about the natures of FFD versus FD in the initial (or high-frequency) part of the model [8] and FLD versus classical FIR in the remaining (or medium/low-frequency) part. In fact, for so the “only” problem is to find a “good” and, on the other hand, a “good” number of the Laguerre filters, which is essentially lower than a number of FIR components, in particular in the medium/low frequency part.*

*Step by step, we arrive at the most practically important model of FD, being the truncated or finite CFLD.*

*Definition 14 (see [16]). *Let the combined fractional/Laguerre-based difference (CFLD) be defined as in Definition 10. Then the finite (combined) fractional/Laguerre-based difference (FFLD) is defined as
where is a number of the Laguerre filters used in the model.

*Remark 15. *An important problem of selection of the Laguerre pole for FLD and FFLD has been effectively solved in [16, 17].

*Remark 16. *It is essential that FLD and, in particular, FFLD have been shown to be computationally very effective, in that surprisingly low numbers of and are sufficient to provide very high modeling accuracies [16, 17].

*5.4. Tustin- and Al-Alaoui-Based Approximations*

*5.4. Tustin- and Al-Alaoui-Based Approximations*

*There are three most popular discretization schemes for fractional-order derivatives, resulting in two Tustin-based and one Al-Alaoui-based approximators [12, 33–35]. Let us recall the so-called Tustin-Muir approximator, mainly in order to rectify some error frequently repeated in the Muir recursion. The Tustin-Muir approximator is
where , , and are the polynomials in of orders , whose coefficients can be computed in a recursive way:
with
and .*

*For the Al-Alaoui-based approximator there is
where , , and are the CFE-related polynomials in of, generally, different orders [12], but we will assume equal orders here.*

*5.5. Steady-State Error*

*5.5. Steady-State Error*

*An important problem encountered in various approximations to FD is an incorrect steady-state gain of the model. This may lead to remarkable steady-state errors in modeling of DTD, the issue being sometimes disregarded, in particular in, for example, the Tustin-based discretization model. Steady-state errors for all the considered models of DTD are characterized below.*

*Lemma 17. Let the steady-state error for the FLD-based model of DTD with respect to the DTD one be defined as
Then
where is the steady-state value of .*

*Proof. *The steady-state value of the outputs from the Laguerre filters , , is given by
Accounting for (1), Remark 8, and Definition 13 and for the fact that , we immediately arrive at (26).

*Lemma 18. The steady-state error for FFLD-based model of DTD with respect to the DTD one defined as
is
*

*Proof. *It is similar to proof of Lemma 17, with Definition 14 being involved.

*Here we have a nice steady-state accuracy result for the FFLD-based model of DTD.*

*Corollary 19. Let the steady-state errors and , , be defined as in (25) and (28). Then .*

*Proof. *The proof is immediate from Lemmas 17 and 18, taking into account that is always negative [8].

*We are in a position now to recall two important theoretical results for LD and CFLD.*

*Theorem 20 (see [17, 18]). Consider LD as in Definition 7, with , , as in (13). Then
*

*Remark 21. *It is interesting that, with all the coefficients , , depending on , their infinite sum as in (30) is, rather surprisingly, independent of . Of course, the finite sum of those coefficients as in (26) remains dependent on .

*Theorem 22 (see [17, 18]). Consider CFLD as in Definition 14, with , , as in (17). Then
*

*Remark 23. *For FLD and FFLD, (30) and (31), respectively, are satisfied only approximately due to the finite summations. However, the quality of the FLD and FFLD approximations in the steady state can be assessed from the “closedness” of the right- and left-hand sides of (30) and (31), respectively.

*Let us now state a simple steady-state accuracy result for the Tustin-based model of DTD.*

*Lemma 24. The steady-state error for the Tustin-based discretization model (21) with respect to DTD defined as
is
*

*Proof. *It is immediate from (21), with .

*Lemma 25. The steady-state error for the Al-Alaoui-based discretization model (24) with respect to DTD defined as
is
*

*Proof. *The proof comes immediately from (24), with .

*Remark 26. *Note that the steady-state error equations (26), (29), (33), and (35) incorporate the factor in the same manner. Therefore, in a comparative analysis we can use, for example, .

*Example 27. *Recall the steady-state errors as in (26), (29), (33), and (35) for the FLD-, FFLD-, Tustin-, and Al-Alaoui-based approximations to DTD, respectively. The error plots presented in Figure 1 are self-explanatory. The FFLD-based model clearly outperforms the three remaining ones, of which the Tustin-based model is definitely inferior, even for very high approximation orders. Also note how low is, which when increased to, for example, 15 or 20 can contribute to further drop of the error. Also note that the Al-Alaoui approximator cannot be used for the order due to numerical problems, in particular in the Matlab environment.