Fractional-Order Discrete-Time Laguerre Filters : A New Tool for Modeling and Stability Analysis of Fractional-Order LTI SISO Systems

This paper presents new results on modeling and analysis of dynamics of fractional-order discrete-time linear time-invariant single-input single-output (LTI SISO) systems by means of new, two-layer, “fractional-order discrete-time Laguerre filters.” It is interesting that the fractionality of the filters at the upper system dynamics layer is directly projected from the lower Laguerre-based approximation layer for the Grünwald-Letnikov difference. A new stability criterion for discrete-time fractional-order Laguerrebased LTI SISO systems is introduced and supplemented with a stability preservation analysis. Both the stability criterion and the stability preservation analysis bring up rather surprising results, which is illustrated with simulation examples.


Introduction
It is well known that orthonormal basis functions (OBF) can be a good alternative to, for example, state space or ARX models when solving the problems in modeling and identification of some classes of "regular" or integer-order dynamical systems, both linear [1][2][3] and nonlinear ones [4,5].Recently, the OBF machinery has increasingly been used in modeling and identification tasks for noninteger or fractional-order systems [6][7][8][9], a broad research field attracting a huge interest both from the academia and industrial environments [10][11][12][13][14][15][16][17].In this paper we limit ourselves to discrete-time fractionalorder systems.There are two main research directions for potential use of OBF, in particular Laguerre functions, in the fractional-order modeling context, namely, Laguerrebased approximation of fractional-order Grünwald-Letnikov (GL) derivatives [8,9] and Laguerre-based modeling of fractional-order dynamical systems [18,19].We pursue both research directions, employing discrete-time Laguerre filters to effectively model both the GL fractional-order difference (FD) [8,9] and dynamics of a discrete-time fractional-order plant to be covered here.Such a two-layer Laguerre modeling idea is a new contribution to modeling of linear discrete-time fractional-order systems.Our new Laguerre-based approximators to FD, called FLD and FFLD [8,9], follow the lines of the introduction of the "classical" (truncated or) finite FD, that is, FFD.
The FD and FFD [20] are often used in the context of modeling of discrete-time fractional-order LTI state space systems [14,[20][21][22].It is the asymptotic stability that is considered the most important aspect of an analysis of fractional-order LTI systems [23][24][25][26][27][28].As for continuous-time systems, the celebrated Matignon's stability criterion [29] has established a simple argument condition for eigenvalues of a state matrix.It is only recently that a thorough asymptotic stability analysis has been made for discrete-time fractionalorder LTI state space systems [30,31] and the long-awaited stability criterion has been introduced for such systems [31].That criterion will be reformulated here for LTI systems described by a new fractional-order model based on "fractionalized" Laguerre filters.
In this paper, a new concept of "fractional-order discrete-time Laguerre filters" is introduced for modeling of fractional-order LTI SISO systems and a new analytical stability criterion is announced.Also, the stability preservation problem is solved.
The remainder of this paper is organized as follows.Fundamentals of the regular OBF system description, in particular the Laguerre filters, are recalled in Section 2, with an important factorization of the Laguerre transfer functions presented.The first significant result of this paper is provided in a unified framework in Section 3, where a new term of "fractional-order discrete-time Laguerre filters" is coined, supported with a series of preliminary outcomes preparing for the main result of Section 4. That section provides a new general criterion for the asymptotic stability of discretetime LTI SISO fractional-order Laguerre systems, being a nice supplementation of the criterion of [31].The criterion is followed by a stability preservation analysis in Section 5.The parameter estimation problem for fractional-order Laguerre systems is outlined in Section 6 and the reorthonormalization issue is addressed in Section 7. Conclusions of Section 8 summarize the achievements of the paper.

Regular OBF System Description
An output of a "regular" (or integer-order) discrete-time OBF-modeled dynamical system or shortly OBF system can be described as where  −1 is the backward shift operator, () and () are the system input and output, respectively, at discrete-time  = 0, 1, . .., and   ( −1 ) and   ,  = 1, . . ., , are orthonormal transfer functions and weighting parameters, respectively.For discrete-time Laguerre filters there is with  = √ 1 −  2 and  being a dominant pole.We proceed with the practically justified case of 1 >  > 0. The unknown parameters   ,  = 1, . . ., , can be easily estimated via, for example, Recursive Least Squares (RLS) or Least Mean Squares (LMS) algorithms using the linear regression formalism.We discriminate between the parameters of the Laguerre model of a dynamical system (, , ,   ,  = 1, . . ., ) and those for the Laguerre-based fractional-order difference (, , ,   ,  = 1, . . ., ) as in [8,9].Remark 1.Note that we do not need to account for the sampling period  in  as it will be indirectly included in the estimated parameters   ,  = 1, . . ., .
Depending on the time-domain or -domain contexts, we will interchangeably use the notations   ( −1 ) or   ( −1 ), respectively.The same manner will be applied to other functions of  −1 or  −1 .
Then FD/FFD/LD/FLD/CFLD/FFLD-based fractionalorder LTI SISO Laguerre system can be described as where () and () are the system input and output, respectively,   ,  = 1, . . ., , are the unknown parameters, and the "fractional-order discrete-time Laguerre filters" are as follows: with  being the dominant Laguerre pole and with  = √ 1 −  2 and pertaining to specifications (i) to (vi) for the particular differences.
An outstanding value of Theorem 3 is at least threefold.Firstly, a new concept of "fractional-order discrete-time Laguerre filters" is introduced.Secondly, the factorization of the fractional-order Laguerre filters holds true in the same way as for the regular (nonfractional) ones, with (10) being still valid.This means that the block diagram for the fractional Laguerre system is identical with that for the regular one, with    and    substituted for   and   , respectively.This provides a nice technical tool for modeling in, for example, Matlab.Thirdly, the whole calculation process for the model output of the fractional Laguerre system is quite similar to that for the regular Laguerre system, with an additional fractional component ( −1 ) involved in the former case (see (11)).
Remark 4. Note that FFLD as in ( 7) and ( 8) can be considered as one general unified model from which all the other considered models can be obtained according to specifications (i) through (v), with FFLD itself covering specification (vi).
Remark 5. Note that the whole "fractionality" of the otherwise (two-layer) fractional-order Laguerre system is contained in the (lower-layer) transfer function ( −1 ).Also note that for the FD-based fractional Laguerre system we have the familiar GL transfer function ( −1 ) = (1 −  −1 )  (compare [30]).Of course, for  = 1 we arrive at the classical Laguerre filters   and   .Remark 6.Let us emphasize that the transfer function ( −1 ) is related to the function Ψ() (or Ψ()) introduced in [30] in the asymptotic stability analysis of fractional-order discretetime state space systems.In fact, we have with  being the imaginary unit and  = arg , 0 ≤  ≤ 2.
Remark 8.In addition to computational simplicity, the Laguerre-based difference is attractive also in that it can model both well-damped ( ∈ (0, 1)) and oscillatory behaviors ( ∈ (1, 2)), without referring to, for example, the Kautz filters in the latter case.
An exemplary block diagram of the output calculation process for the FLD-based fractional Laguerre system is presented in Figure 1, with the "internal" (or lower-layer) FLD being additionally zoomed.

Stability Results for Fractional-Order
Laguerre Systems 4.1.Stability of FD/FFD/LD/FLD/CFLD/FFLD-Based Laguerre Systems: A Unified Framework.We adopt the stability results of [30,31] to the Laguerre-based fractional systems to obtain an original outcome which is the main result of this paper.
It is rather surprising that the Laguerre-based FFD system can be asymptotically stable even for  > 1 (and  < 1).This stability result yielded from ( 22) is a nice theoretical confirmation of our earlier surprising simulation outcome of [35].In fact, with ∑  =1   () > −1 under  ∈ (0, 1) and finite , the pole  can be admitted higher than unity.But on the other hand, for  ∈ (1, 2) the term ∑  =1   () can be lower than −1 so that even for  < 1 the FFD system may be unstable, this being yet another nice stability result confirming our earlier surprising simulation observations.Out of a plethora of simulations runs, we firstly present two selected examples.
Example 10.Consider the FFD-based fractional Laguerre system with  = 20 and two different values of  = 0.5 and  = 1.5.It can be concluded from Theorem 9 that for  = 0.5 the system is asymptotically stable for  < 1.12537.On the other hand, for  = 1.5 the system is unstable for  > 0.99678.
In a similar way, the FLD/FFLD-based fractional Laguerre systems can be stable even for  > 1, which is illustrated in the following.22) is satisfied and the two systems are asymptotically stable.
Remark 12.Note that the "closedness" of Ψ(0) to zero is an excellent indicator of the quality of the specific approximator FFD/FLD/FFLD to FD/LD/CFLD.
Remark 13.In order to account for the sampling period  when transferring from the continuous-time derivative to FD/LD/CFLD/FFD/FLD/FFLD, in the stability analysis for the fractional Laguerre system we replace (22) with with Ψ(0) as in Theorem 9.
Remark 14.It is interesting that the stability condition for fractional-order Laguerre systems based on the FD difference (and equivalently, LD and CFLD ones [8,9]), that is,  < 1, is identical with that for the regular integer-order Laguerre system.Also note that in this case (Ψ(0) = 0) the stability condition is, rather surprisingly, independent of both the sampling period  and the order .In contrast, criterion (23) for the FFD/FLD/FFLD-based fractional Laguerre systems is always dependent on both  and , and this is because Ψ(0) ̸ = 0 in that case.

Stability Preservation Analysis
The stability preservation issue for fractional-order systems under direct discretization methods has been considered in [36].The problem reduces to the provision of a stable discretized system approximating its stable continuous-time original.Here we present our stability preservation results for the FFD/FLD/FFLD-based Laguerre systems.
Some remarks are due, now.Firstly, the results for variants 1 and 3 of Theorem 15 are quite surprising.Secondly, we have verified all the variants of Theorem 15 in a plethora of simulation runs, the selected examples of which are tabulated.Table 1 collects the results, with the particular variants of Theorem 15 superindexed in the Comment column.The results of Table 1 are self-explanatory.

Parameter Estimation for Fractional Laguerre Filter
When output (9) (or (20)) of the fractional Laguerre filter is corrupted with noise and the order  is known, the model output ŷ() can be described in a linear regression fashion ŷ() =   ()Θ, with   () = [  1 (),   2 (), . . .,    ()] and the unknown parameters Θ  = [ 1 ,  2 , . . .,   ] estimated analytically, for example, (R)LS method.When  is unknown the parameter vector Θ  = [ 1 ,  2 , . . .,   , ] can be estimated numerically using, for example, LS or, preferably, GA procedures.The  parameter is selected (heuristically) to be equal to 16 and   ,  = 1, . . ., , parameters are estimated by the RLS method.The input is a discrete-time zero mean unityvariance Gaussian random signal.The MSPEs for the outputs of the three Laguerre-based models with respect to that for the FD-based state space system are presented in Table 2.
Table 2 shows that the best performance is obtained for the FFLD-based Laguerre model.In this case, the FFLDbased fractional Laguerre model with 45 parameters gives essentially lower MSPE than the FFD-based one with 500 parameters.This shows that the FFLD is a very good approximation to FD.Also, a simple FLD-based fractional Laguerre model, with 35 parameters only, gives satisfactory results.
The frequency responses of the actual system and (hardly distinguishable) FFD/FLD/FFLD-based fractional Laguerre models are presented in Figure 2. Table 3 presents standard deviations of the magnitude and phase of the frequency responses for the FFD/FLD/FFLD-based fractional Laguerre models with respect to the FD-based system.closer to unity than the original   = 0.85.Also, "strong" oscillatory fractional behaviors may be Laguerre modeled with  ∈ (1, 2); compare Remark 8.

The Reorthonormalization Issue
A side effect of factorizations (3) or ( 10) is that the orthonormality property is lost for fractional Laguerre systems; that is, the functions  The simulation experiments show that both models give quite the same MSPE errors equal to some 4.88.So, the reorthonormalization process may not improve the model accuracy.However, it is well known that the reorthonormalization process improves conditioning of the covariance matrix in the RLS estimation process.This is shown in Figure 3, which presents the condition numbers of the covariance matrix () as a function of time  for both orthonormalized and nonorthonormalized fractional Laguerre filters.

Conclusion
This paper has presented a series of new results in modeling and analysis of discrete-time fractional-order LTI SISO systems using Laguerre filters.In the "lower" modeling layer described in [8,9], the Grünwald-Letnikov fractionalorder difference has been effectively approximated by the Laguerre filters.This paper has produced new results related to modeling and analysis of the "upper" system dynamics 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Orthonormalized fractional Laguerre filters Nonorthonormalized fractional Laguerre filters  Grünwald-Letnikov fractional-order difference FFD: Finite fractional difference FFLD: Finite (combined) fractional/Laguerre-based difference FLD: Finite Laguerre-based difference GL: Grünwald-Letnikov LD: Laguerre-based difference OBF: Orthonormal basis functions.

Figure 1 :
Figure 1: Block diagram of output calculation process for the FLD-based Laguerre model.

Figure 3 :
Figure 3: Conditioning of the covariance matrix versus time; Example 18.

Table 2 :
Mean square prediction errors for the analyzed models; Example 16.

Table 3 :
Standard deviations of frequency responses for the analyzed models; Example 16.