Research Article | Open Access
New Laguerre Filter Approximators to the Grünwald-Letnikov Fractional Difference
This paper presents a series of new results in modeling of the Grünwald-Letnikov discrete-time fractional difference by means of discrete-time Laguerre filers. The introduced Laguerre-based difference (LD) and combined fractional/Laguerre-based difference (CFLD) are shown to perfectly approximate its fractional difference original, for fractional order . This paper is culminated with the presentation of finite (combined) fractional/Laguerre-based difference (FFLD), whose excellent approximation performance is illustrated in simulation examples.
Noninteger or fractional-order dynamic models have recently attracted a considerable research interest. Their specific properties can make them more adequate in modeling of selected industrial systems [1–4]. Our interest is in discrete-time representations of fractional-order systems, so we proceed with the Grünwald-Letnikov fractional-order difference (FD) [5–9]. An infinite-memory filter incorporated in FD may lead to a computational explosion. Therefore, a number of discrete-time FD-based systems have been modeled both via transfer function or difference equation models [10–13] and state space ones [7, 9, 14].
Various approximations to the fractional difference have been pursued. 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-difference IIR filter to that of the associated FD [15–17]. These methods give digital rational approximations (IIR filters) to continuous fractional-order integrators and differentiators. The problem here is to propose a good structure of the integer-difference 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 , with a high-order optimal filter providing a good approximation accuracy, at the cost of a remarkable computational effort, however.
Another approach relies on the approximation of the FD filter with its truncated, finite-memory version [14, 19, 20]. In analogy to finite impulse response (FIR) the term finite FD, or FFD, has been coined . Additionally, a series of results in finite and infinite-memory modeling of a discrete-time FD by FFD-like models has been presented in .
An approach behind that research direction has been the employment of an approximating filter incorporating orthonormal basis functions (OBF) [21, 23]. Another attempt at the application of OBF in modeling of FD has been presented in . This paper provides a nice theoretical background for those rather intuitive approaches to the OBF-based approximation of FD, in that the so-called Laguerre-based difference (LD) is shown to be equivalent, in some sense, to FD.
The proposed approximation method is solved for the model parameters in an analytical way. The paper is culminated with the introduction of a new model of FD, being an effective combination of FFD and finite LD (or FLD), whose excellent performance results from expert a priori knowledge used when constructing the model.
Having introduced the FD modeling problem, the Grünwald-Letnikov discrete-time fractional difference is recalled, together with its FFD approximation, in Section 2. Section 3 presents the OBF modeling problem, in particular via discrete-time Laguerre filters. Laguerre-based difference (LD) is covered in Section 4, followed by a Laguerre-based approximation to FD in terms of finite LD (FLD). Finally, Section 4 provides tools for selection of optimal Laguerre pole for FLD approximation and presents a series of simulations, which show the approximation efficiency of FLD modeling. Finally, combined fractional/Laguerre-based difference (CFLD) and its finite approximation called finite (combined) FLD (or FFLD) have been introduced in Section 5. That section also presents a method for selection of optimal Laguerre pole for FFLD and includes a series of simulation examples which present a high approximation accuracy of FFLD modeling. Conclusions of Section 6 summarize the achievements of the paper.
2. Fractional Discrete-Time Difference
In our considerations, we use a simple generalization of the familiar Grünwald-Letnikov difference , that is the fractional difference (FD) in discrete time , described by the following equation [7–9]: where the fractional order , is the backward shift operator and with Note that each element in (2.1) from time back to is nonzero so that each incoming sample of the signal increases the complication of the model equation. In the limit, with , we have an infinite number of FD components leading to computational explosion.
Remark 2.1. Possible accounting for the sampling period when transferring from a continuous-time derivative to the discrete-time difference results in dividing the right-hand side of (2.1) by . Operating without as in the sequel corresponds to putting or to the substitution of for .
2.1. Finite Fractional Difference
In , 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 to assume 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.
3. Orthonormal Basis Functions
It is well known that an open-loop stable linear discrete-time IIR system governed by the transfer function: with the impulse response , , can be described in the Laurent expansion form [28, 29]: including a series of orthonormal basis functions (OBF) and the weighting parameters , , characterizing the model dynamics.
Various OBFs can be used in (3.2). 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 3.1. 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 .
The coefficients , , can be calculated form the scalar product of and  as follows: 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 with , , , and is the Kronecker delta.
4. Laguerre-Based Difference
In analogy to FD, let us firstly define a “sort of” a difference to be referred to as the Laguerre-based difference.
Now, a new fundamental result in this respect is announced as follows.
Theorem 4.2. Let the FD be defined as in (2.1) through (2.3) or, equivalently, as in (2.5), and let the LD be defined as in Definition 4.1. Then LD is identical with FD, that is, , if and only if with being the dominant Laguerre pole and
Proof. See Appendix A.
Remark 4.3. Note that, rather surprisingly, an actual value of is meaningless for the validity of Theorem 4.2. This intriguing fact has been confirmed in a plethora of our simulations, both in time and frequency domains. Well, on the other hand, the infinite expansion as in (3.2) can also perfectly model any rational transfer function irrespectively of an actual value of .
4.1. Finite Approximation of LD
Like for FD, we have an infinite number of LD components leading to computational explosion. In analogy to the presented finite fractional difference (FFD), the convergence to zero of the series enables to assume 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 4.5. Let the Laguerre-based discrete-time difference (LD) be defined as in Definition 4.1. Then, the finite Laguerre-based difference (FLD) is defined as where is the number of the Laguerre filters used to calculate the difference FLD, and , , are calculated as in (4.2).
Of course, an introduction of the bound in FLD will lead to generation of an approximation error as compared to the original FD/LD. Define this error in the time domain as The energy of the sequence, , , is given by where is the scalar product. For the considered FLD, the value of can be easily computed as (compare ) follows: with as in (2.2) and (2.3), and as in (4.2). The value of depends on three parameters: the limit , the fractional order , and the dominant Laguerre pole . Accounting for the fact that increasing the limit enhances the complexity of the FLD model, “costless” optimization of the FLD model with respect to can only be realized by selection of a Laguerre pole . So, in contrast to LD, selection of an optimal Laguerre pole in the FLD model is important from the accuracy point of view.
4.2. Selection of Dominant Laguerre Pole
The optimal Laguerre poles for various values of and are presented in Figure 1. Since FLD is not quite effective for , which will be commented in the sequel, we refrain from showing analogous results for that range of .
On the basis of a plethora of simulations, in Appendix G we propose an approximation of an optimal Laguerre pole for FLD as a (heuristic) function of and .
Example 4.6. Consider the fractional difference FD and its FLD model, with selected as in (4.9). Figure 2 presents Bode plots for the FLD models versus FD = LD, with and various values of . Table 1 presents the approximation errors of the FLD model for various values of and . It can be seen from Table 1 that unsurprisingly, increasing the value of increases an approximation accuracy of the FLD model, generally, for the same values of the approximation accuracy of FLD modeling is higher for greater (excluding the area where is close to ).
Qualitatively, the above results are quite similar to those for the FFD model [22, 27]. However, higher values of lead to reduction of the approximation error for the FFD model much faster as compared to the FLD one. So, FLD is effective (and, in fact, more effective than FFD) for . This is illustrated in Table 2 which shows the values of in the FFD model, providing an equivalent approximation accuracy to the FLD model with specified , for various values of and . For instance, for , the FLD model with is equivalent, in terms of the approximation accuracy, to the FFD model with , but for is equivalent to the FFD one with .
Taking into account that the FLD model needs a priori knowledge about the optimal Laguerre pole and is more complex than FFD from the computational point of view, the FLD model can be recommended for only.
Let us finally show some interesting feature related with the FLD model.
Example 4.7. Consider the fractional difference FD and its FLD model as in Example 4.6. The approximation errors for the FLD model with various values of and consecutive values of are presented in Table 3.
It can be seen from that table that the adjacent values of provide the same approximation accuracy for the FLD model. It is interesting to note that for we obtain the same approximation errors for the pairs , but for the same errors are obtained for the pairs . Accounting for the computational aspect, we, thus, recommend to use odd values of for and even values of for .
The above examples demonstrate that FLD is effective for . For , the FLD is not as effective as FFD in approximation of FD. The motivation of the work presented in the next section is searching for a “good” FLD-like model also for .
5. Combined Fractional/Laguerre-Based Difference
To cope with the problem, we introduce a new difference, which is a combination of the “classical” FFD and our FLD.
Definition 5.1. Define the combined fractional/Laguerre-based difference (CFLD) as where and the first component at the right-hand side of (5.2) constituting the FFD share in the CFLD, the second one being the (-delayed) LD share, with , , as in (2.2) and (2.3), and , , as in (3.3) and (3.4), respectively.
Now, we have another fundamental result in perfect modeling of FD via CFLD.
Theorem 5.2. Let the Grünwald-Letnikov fractional difference (FD) be defined as in (2.1) through (2.3), Laguerre-based difference (LD) is as in Definition 4.1 and combined fractional/Laguerre-based difference (CFLD). Then CFLD is equivalent to FD in that (or ) if and only if with , being the dominant Laguerre pole and
Proof. See Appendix D.
Remark 5.3. Note that regardless of an actual value of we have , in the sense that , .
An interesting CFLD orthonormality result can now be obtained.
Proof. See Appendix F.
Remark 5.5. Like in the FD, possible accounting for the sampling period in LD and CFLD models when transferring from a continuous-time derivative to the discrete-time difference results in dividing the right-hand side of (4.1) and (5.1) by , respectively.
Like in the FD/LD, the infinite length expansion incorporated in CFLD leads to a computational explosion. Therefore, in analogy to FLD, we introduce a finite-length approximation to CFLD called finite (combined) fractional/Laguerre-based difference (FFLD).
5.1. Finite Approximation of CFLD
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  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, the number of Laguerre filters is essentially lower than a number of FIR components (and FD is a “sort of” IIR, 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 5.6. Let the combined fractional/Laguerre-based difference (CFLD) be defined as in Definition 5.1. Then the finite (combined) fractional/Laguerre-based difference (FFLD) is defined as where is the number of Laguerre filters used in the model.
Again, the bound in FFLD leads to an approximation error in FFLD modeling. Immediately, based on Theorem 5.4, an approximation error for the FFLD model can be calculated like for the FLD one (compare (4.8)): with as in (5.6).
Remark 5.7. It is essential that, like for FLD, the approximation error for FFLD can be made arbitrarily small by selection of sufficiently high , which is the well-known feature of OBF. However, the power of FFLD is that, owing to the FFD contribution, the value of can be much lower than that for FLD.
5.2. Selection of Dominant Laguerre Pole
Here, an optimal Laguerre pole is selected by minimization of the approximation error (5.9) in a similar way as in (4.9). Figures 3 and 4 present the optimal Laguerre pole as a function of the order for and various values of , and for and various values of , respectively.
As in the case of the FLD model, on the basis of a number of simulations, an (heuristic) approximation of an optimal Laguerre pole in the FFLD model, as a function , and , is presented in Appendix B.
Example 5.8. Consider the fractional difference FD and its FFLD model with . Table 4 presents the approximation error for the FFLD model with and various values of and . Table 5 shows values of in the FFD model that are accuracy-equivalent to the FFLD with specified and .
It can be seen from Tables 4 and 5 that the FFLD model is much more effective than FFD in modeling of FD in that FFD needs a huge number of to provide equivalent approximation accuracy to FFLD. Figure 5 presents Bode plots for the FFLD model versus FD = LD = CFLD, with , and various values of .
Example 5.9. Consider the fractional difference FD with and its FFD versus FFLD models, with and , respectively, and and for FFLD. Table 6 presents the same approximation errors for both models under various values of and . It can be seen from Table 6 that increasing by in the FFLD model is equivalent to increasing by some in the FFD model. However, increasing by in both FFD and FFLD models results in roughly the same increase in the computational burden. So, in FFLD modeling we have some times better computational efficiency.
It is worth emphasizing that the approximation error is so low for FFLD that the normalization factor incorporated into FFD  may be not necessary for FFLD. Now, FFLD can be competitive to another powerful adaptive (normalized) finite fractional difference (AFFD) [22, 32], an intriguing issue to be a subject of a comparative research study.
This paper has offered a series of original results in modeling of Grünwald-Letnikov discrete-time fractional-difference (FD) using Laguerre filters. Firstly, a new quality has been presented, namely, the Laguerre-based difference (LD), which has been proven to be equivalent, under specified conditions, to the FD. For implementation reasons, a finite LD (FLD) approximator has been introduced as an analogue to the “classical” finite FD (FFD), and the two have been shown to perform in a similar way.
Another new quality, is that combined fractional/Laguerre-based difference (CFLD) has also been shown equivalent, under specified conditions, to the FD. Interestingly, a finite-length approximator to CFLD, called finite (combined) FLD, or FFLD, has been demonstrated in simulations to constitute an excellent model of FD, both in terms of the accuracy and computational efficiency. This is due to the fact that FFLD constitutes an expert combination of the high-frequency FFD component and medium/low-frequency FLD part, both efficiently balanced using the bounds and , respectively. Additionally, simple approximate derivations for optimal Laguerre poles are supplemented. Summing up, FFLD is recommended as a high-performance approximator to FD. Future research in the area will concentrate on a comparison of FFLD and AFFD models of FD and their application in fractional-order predictive control.
A. Proof of Theorem 4.2
Note that using the generalized Newton binomial and accounting for the fact that is the binomial expansion for and , and , we can write (A.2) as
It has been shown in Section 3 that (A.2) can be described by (3.2), where , , are the Laguerre filters presented in (3.3). The coefficients , , can be obtained using formula (3.4) : where is the complex conjugate of . Note that and , , are analytic in . Using the Cauchy integral formula for , we have where .
For , we have where . Now, expanding the element via the binomial theorem, we arrive at where , , are the binomial coefficients. Finally, on the basis of Cauchy integral formula, again, we obtain (4.2).
B. Exemplary Coefficients in (4.2)
Exemplary coefficients , as in (4.2) are as follows:
C. Approximated Laguerre Pole for FLD
An approximation of the optimal Laguerre pole for the FLD model is given by the following heuristic function: where Note that the approximation function can be used for and .
D. Proof of Theorem 5.2
The GL fractional difference defined in (2.1) to (2.3) can be presented in the following IIR model: where is the filter is transfer function of the form: Using the generalized Newton binomial (compare Proof of Theorem 4.2), (D.2) can be presented as follows: Getting back to Section 3, again, can be presented in the form of (3.2) so that where , , is as in (3.3). The coefficients , , are obtained from the scalar product: Following the proof of Theorem 4.2, (A.6) is substituted by with and (substituted for ) as in (5.4).
E. Exemplary Coefficients in (5.2)
F. Proof of Theorem 5.4
The functions , , are orthonormal to each other if and only if where denotes the positive integers. Observe that , , is just FIR, that is, the special case of the Laguerre filters as in (3.3) with the dominant Laguerre pole . Since the Laguerre filters are orthonormal basis functions, we have for each (independently of a value of ) and , , for () or (). In the last step, we prove that for each and . In this case, we have where and . Accounting that for each for each , and , we obtain , which completes the proof.
G. Approximated Laguerre Pole for FFLD
An approximation of the optimal Laguerre pole for the FFLD model is given by the following heuristic function: where with values of the parameters presented in Table 7. The function (G.1) can be used for .