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

This paper provides a series of new results in both steady-state accuracy and frequency-domain analyses for two Laguerre-based approximators to the Grunwald-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.


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 discretetime integer-order IIR filter to that of the associated FD [1][2][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 fractionalorder derivatives [9][10][11][12][13][14][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][17][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 Laguerrebased 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 Laguerrebased 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-Alaouibased approximator.We advocate the contribution of the FFD on the one hand since in the high frequency range it is 2 Mathematical Problems in Engineering 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 frequencydomain 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.

Grünwald-Letnikov Fractional-Order Difference
It is well known [23,24] that continuous-time fractionalorder 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  ∈ (0, 2) is the fractional order,  −1 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   () ≈ 0 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  = min(, ) 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][27][28], or a truncated/finite difference [21,23,29], or a short-memory difference [30].

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   = (),  = 1, 2, . .., can be described in the Laurent expansion form [31,32] including a series of orthonormal basis functions (OBF)   () and the weighting parameters   ,  = 1, 2, . .., 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  = √1 −  2 and  ∈ (−1, 1), consist of a first-order low-pass factor and ( − 1)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   (),  = 1, 2, . .., do not include .
Remark 5. Depending on the domain context, we will use various arguments in   (⋅), for example,   () in the domain and   () or   ( −1 ) 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).

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.
Remark 8. Again, whenever substantial we will comment on the effect of the sampling period  when using an LDbased form Δ  LD ()/  of DTD (compare Remark 1).This will also hold true for the forthcoming CFLD and its two approximators FLD and FFLD.
Since  FD () in ( 6) represents a sort of IIR and so does  LD () as in (12), the question arises as to what a relationship between  FD () and  LD () is and, moreover, if and when it is possible to obtain  LD =  FD .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  = √1 −  2 ,  ∈ (−1, 1) \ {0} 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.
Here is another fundamental equivalence result.

Finite Approximations of
Fractional-Order Differences 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.

Finite Laguerre-Based Difference.
In analogy to the presented finite fractional difference (FFD), the convergence to zero of the series   enables assuming   ≈ 0 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   ,  = 1, 2, . . ., , are calculated as in (13).

Finite
Fractional/Laguerre-Based Difference.The idea behind combining FFD and FLD comes from a priori knowledge about the natures of (1) FFD versus FD in the initial (or high-frequency) part of the model [8] and (2) FLD versus classical FIR in the remaining (or medium/low-frequency) part.In fact, FFD ≡ FD 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].

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 steadystate 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 ().

Lemma 18. The steady-state error for FFLD-based model of DTD with respect to the DTD one defined as
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.
Proof.The proof is immediate from Lemmas 17 and 18, taking into account that ∑  =1   () is always negative [8].
We are in a position now to recall two important theoretical results for LD and CFLD.
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  −1 = 1.
Lemma 25.The steady-state error for the Al-Alaoui-based discretization model (24) with respect to DTD defined as Proof.The proof comes immediately from (24), with  −1 = 1.
Remark 26.Note that the steady-state error equations ( 26), ( 29), (33), and ( 35) incorporate the factor 1/  in the same manner.Therefore, in a comparative analysis we can use, for example,  = 1.Example 27.Recall the steady-state errors as in ( 26), ( 29), (33), and ( 35) for the FLD-, FFLD-, Tustin-, and Al-Alaouibased 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  = 10 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  > 18 due to numerical problems, in particular in the Matlab environment.
Remark 28.It is worth mentioning that the steady-state accuracy issue is very important as even low steady-state approximation errors for the fractional difference may be propagated to high modeling errors for a fractional-order dynamical system [8].On the other hand, the responses of DTD and its FLD-, FFLD-, and Al-Alaoui-based approximations are hardly distinguishable, which suggests that a frequency-domain analysis could be welcome here.On the other hand, the mean square prediction errors (MSPE), shown in Table 1, indicate that the time-domain fit of the FFLD-based model of DTD is the best.
Remark 30.The "strange" behavior of the step response for the Tustin-based approximation results from the fact that one pole of the transfer function   ( −1 , )/  ( −1 , −) is highly negative (in the range of, e.g., −0.9), which itself may raise doubts on the adequacy of the Tustin-based approximation.

Frequency-Domain Analysis.
Let us start with an instructive simulation example.
Example 31.Consider a fractional-order derivative represented by   and its discrete-time approximations as in Example 29. Figure 3 presents Bode plots for the fractional-order derivative and its FLD/FFLD/Tustin-based approximations at the sampling period  = 1.In Figure 3, the gray-marked area shows the ±3 dB error from the actual value of ()  in the magnitude spectrum and its equivalent in the phase spectrum  ∈ [(/2)(1/ √ 2), (/2)(2 − 1/ √ 2)], respectively, with the error bound fulfilled within the frequency spectrum  ∈ ( min mod ,  max mod ) and a specific type of the model subindexed as "mod." Magnitude plots for the FFLD-and FLD-based models are within the error bound over a remarkably wider area of the frequency spectra  ∈ (9.5 − 5, ) and  ∈ (7.3 − 4, ), respectively, as compared to the Tustin-based approximation, with  ∈ (1 − 2, 2.1).So, in this regard the FFLD and FLD approximations are much more effective than the Tustin-based approach.For lucidity, we refrain from plotting the magnitude characteristic for the Al-Alaoui-based approximator as it is very close to the FLD-based one.In case of phase plots, the FFLD-and FLD-based models are within the error bound over a remarkably wider area of the frequency spectra  ∈ (3.9 − 4, 9.8 − 1) and  ∈ (3.0 − 3, 9.8 − 1), respectively, as compared to the Tustin-based approximation, with  ∈ (3.8 − 2, 3.1).As for the Al-Alaoui-based approximation, the phase plot is slightly better than the FLD-based one, with  ∈ (2.0 − 3, 1.1), but remarkably worse than the FFLD-one.(Note: for lucidity, we refrain from marking the  max Al value as it is very close to  max FLD and  max FFLD .)Note that the FLD/FFLD/Al-Alaoui-based approximators generate a high phase error for high frequencies (close to the sampling frequency).This phase error is a result of the backward difference based discretization scheme for the continuous-time derivative.
Finally, on the basis of both magnitude and phase plots we obtain adequacy ranges for the FFLD-based model:  ∈ (3.9 − 4, 9.8 − 1), Al-Alaoui-based model:  ∈ (2.0 − 3, 1.1), FLD-based model:  ∈ (3.0 − 3, 9.8 − 1), and Tustin-based model:  ∈ (3.8 − 2, 2).So, the Tustin-based approximation has two times higher upper frequency limit as compared to the FFLD/FLD/Al-Alaoui-based approximations.However, taking into account that the FFLD/FLD/Al-Alaoui-based models include the sampling period in their denominators only, we can easily left/right-shift the frequency spectrum by changing the sampling period .For our FFLD-and FLDbased models we obtain the adequacy ranges  ∈ ((3.9 − 4)/, (9.8 − 1)/) and  ∈ ((3.0 − 3)/, (9.8 − 1)/), respectively, whereas for the Al-Alaoui model we have  ∈ ((2.0 − 3)/, 1.1/).Therefore, using the sampling period  = 0.5 for the FFLD/FLD/Al-Alaoui-based models we obtain the adequacy ranges  ∈ (7.8 − 4, 1.96) for FFLD,  ∈ (6.0 − 3, 1.96) for FLD, and  ∈ (4.0 − 3, 2.2) for the Al-Alaoui model, with the upper frequency bounds being similar to the Tustin-based approximation for  = 1.(Note: for lucidity, we refrain from showing both magnitude and phase plots for the FD-based DTD model, with the former one being identical to that for ()  and the latter one being very close to that for ()  at the low and medium frequency ranges and very close to that for the FLD/FFLDbased approximation for high frequencies.)Example 31 illustrates that possible high argument errors for the FLD/FFLD-based models in the high frequency range are related to the backward difference argument error which depends on the sampling period .Now, the argument error analysis deserves a special attention.Here we present a new method to calculate this error.
Theorem 32 (main result).Consider an -order continuoustime derivative and its Laplace transform   ,  ∈ (0, 2).Discretize the derivative using the backward difference scheme to obtain the FD-based DTD as in ( 1) and ( 2).The frequencydomain argument error  ℎ () for DTD with respect to ()  defined as Proof.It is well known that DTD can be described in the domain as Accounting for  in the frequency-domain form  =   we obtain with the latter manipulation using the fact that  (/2) is the conjugate to  −(/2) .Finally, the argument of Now, recalling the definition in (36) and taking into account that arg(()  ) = /2 we arrive at (37).It is worth mentioning that the error | ph ()| can be quite high for high frequencies; for example, in the highest frequency range ( = ) we obtain | ph ()| = /2.On the other hand, in Example 31 it has been presented that, for high frequency ranges, the results generated by the FLD/FFLD/Al-Alaoui-based approximations are very similar to DTD.Therefore, we can write that for high frequencies we have | arg(Δ  FLD/FFLD/Al (  )/  ) − arg(Δ   (  ))| ≪  ph () and we can use (37) and (41) to estimate the sampling period  for the FLD/FFLD/Al-Alaoui-based approximations.Remark 33.It is time now to recall the "noncasual compensator" method for elimination of the phase error in the Al-Alaoui-based approximator [12], which could also be used for the FLD/FFLD-based ones.However, with such an "artificial" phase rectification, we dismiss the original timedomain interpretation (and applications) of the fractionalorder derivative/difference.Therefore we claim that our "frequency shifting" method, based on selection of the sampling period , is more practically oriented.Al ()/  ).It can be seen from Figure 4 that the Tustinbased approach shows a better performance for high frequencies, except for very high frequencies when  tends to .In medium/low frequency ranges, FLD-, Al-Alaoui-, and, particularly, FFLD-based models provide much better results than the Tustin-based approach.This can be illustrated by the aggregate relative error defined as ∑  =1 |  (   ) − (  )  |/   , which, for a set of  = 100 selected values of   , Frequency (rad) Remark 37. Clearly, the FFLD-based approximation outperforms the three other ones, also in that it covers the fractionalorder range  ∈ (0, 2), in contrast to the Tustin-and Al-Alaoui-based models.

Conclusion
This paper has presented a bunch of original results on modeling of fractional-order discrete-time derivative (DTD) by means of its two Laguerre-based approximators.The FLDand, in particular, FFLD-based approximators have been shown to substantially outperform another popular approximator, namely, the Tustin-based one but also, in case of FFLD, the Al-Alaoui-based one.New results on steady-state accuracy and, in particular, frequency-domain phase analyses, supported with simulations examples, confirm the usefulness of the considered Laguerre-based approximators, in particular the FFLD one.

Example 29 .
Consider the FD-based DTD of order  = 0.5.The FLD-(with  = 25) and FFLD-( = 20 and  = 10) based models are analyzed versus the Tustinbased approximation of order 25 (which is usually considered very high) and the Al-Alaoui-based approximation of order 17. Figure 2 presents step responses for the DTD and its FLD/FFLD/Tustin/Al-Alaoui-based approximations at the sampling period  = 1.The "strange" behavior of the response for the Tustin-based model of DTD is surprising.

Example 34 .
Consider the fractional-order derivative and its discrete-time approximations as in Example 31.
| ph ()| and the upper bound for frequency range  max we can immediately select such a sampling period  which can guarantee that | ph ()| <  max ph for  ∈ (0,  max );