Research paper
Fractional derivative truncation approximation for real-time applications

https://doi.org/10.1016/j.cnsns.2023.107096Get rights and content

Highlights

  • Short Memory Principle.

  • Fractional Relaxation.

  • System identification.

Abstract

Fractional derivatives are non local operators as they are well-suited for modeling long-memory phenomena such as heat transfers or fluid dynamics. However, such non-locality implies a constant knowledge of the full past of the functions to differentiate. In the context of real-time (online) system identification, such a global property may limit the implementation as calculations can become slower as time grows. Also, instead of using the full data length to compute fractional order derivatives, truncated fractional derivatives are used. However, such a truncation windowing will bring inaccuracy on the computation of fractional derivatives. This study deals with the relationship between the signal frequency content, the fractional derivative truncated approximation as well as the fractional system relaxation in order to get a good approximation of the truncated fractional derivative and consequently, to help providing real-time exploitable system identification algorithms that uses such truncated fractional derivatives.

Introduction

Fractional calculus has been a mere mathematical concept for many centuries. However, during the current and past centuries its properties have proven to be useful for diffusive system modeling [1], as well as robust control design [2], [3]. Diffusive phenomena have proven to be analytically well expressed and well modeled through fractional derivatives. Thermal impedances at middle and high frequencies has also been proven to behave like a half-order integrator [4], [5]. Its properties have proven to be useful in medical scenarios, as is the case of lung impedance modeling [6], [7], [8], [9], cardiac tissue [10] and muscle relaxation [11], [12]. Therefore, there is an increasing interest regarding fractional order models and their properties.

Even though fractional differentiation definition is not unique [13], [14], [15], all of the proposed definitions share the same long-memory property to model diffusive phenomena. The past history of a given function f(t) may still have a notorious influence on its fractional derivative after a long period of time. Fractional order system dynamics can be interpreted as an infinite distribution of time constants [1], [16]. Unfortunately, this implies the existence of infinitely slow time constants. A main drawback regarding this property is the non local nature of fractional differentiation: the whole knowledge of the function past is required in order to well compute its fractional derivative at any given time. As time increases, a signal history becomes progressively longer, thus leading to slower computation times for computing a fractional derivative. Furthermore, computation time may be critical for real-time (online) applications and a continuous increase in the computation time can lead to infeasible algorithms. As a consequence, an appropriate moving window in time is sought so that the (truncated) approximation of the fractional operator may be accurate enough to still make online implementable algorithms.

The fractional derivative may be elegantly truncated in a really simple way through Podlubny’s short memory principle [17]. The principle allows to use a truncated time window of a function past and still get an accurate estimation of the fractional derivative. Even though Podlubny’s principle provides an accuracy guarantee, it gives a pessimistic limit that could still be too slow for implementable real-time scenarios. In system identification, experiments usually imply an input signal exciting a defined bandwidth (see [18]), in order to be sufficiently persistent (rich in frequency). Therefore, the spectrum of the input signal may influence the required time window of the past signal and its associated truncation error. One of the aims of this paper is to analyze a signal spectrum and the differentiation order influence on the truncation error in order the design a correct time window so that the fractional derivative approximation on a time window gets sufficiently precise enough. This will be performed through a simple scenario.

Besides the fractional differentiation long-memory property, another issue may arise when dealing with real-time fractional order system identification: output estimation at a given time. The fractional order model non-locality and the presence of noise in the recorded data may lead to prefer an estimation of the output based upon an auxiliary model such as the instrumental variable method. The use of a truncated auxiliary model can lead to an initial condition problem for simulation. It has been proven that fractional order systems can be interpreted as an infinite state representation [19], [20]. Formal methods to estimate the response of a fractional order system with initial conditions are available. However, their solutions are too complex to be easily computable for real-time scenarios. Therefore, an analysis based on the transient response will be presented for a simple fractional order model of the first kind.

The earliest proposed techniques for fractional order system identification come from the 90s [21], [22], [23]. These first methods only estimated the coefficients of fractional order transfer functions. However, they were also limited to the Grünwald–Letnikov discrete-time definition of the fractional derivative. By considering continuous-time system identification, methods relying on state variable filters, least squares and instrumental variables have been introduced [24], [25], [26] as well as validation through experimental applications [4], [25], [27]. Recently, experiment design for optimal identification has also been studied for fractional order systems [28].

Real-time system identification is based upon recursive estimation methods, such as recursive least-squares or recursive prediction error method [18]. The analysis of recursive identification methods has also been extended to least squares with state variable filters and instrumental variable techniques [29], [30]. All such methods have also been adapted and analyzed for continuous-time systems [31]. Their extensions to fractional order systems have been proposed in [32] where the Long-Memory Recursive Prediction Error Method (LMRPEM) algorithm has proven to be a useful tool for accurately estimating all parameters (coefficients and differentiation orders) of fractional order models . An effective combination of a well-chosen truncation method and the LMRPEM-2 algorithm could lead to an entirely real-time implementable identification technique for fractional order models. Such an algorithm works successfully if the sampling time is not too small so that the computation time at each iteration is far smaller than the sampling time, and also when the experiment does not last too long. Consequently, if the sampling time gets smaller and if the experiments last very long such is the case for open-heart surgeries that can last for several hours, the computation of fractional derivatives should be shortened: therefore, the contribution of this paper is to propose to use a correctly designed time window to get an accurate enough approximation of the truncated fractional derivatives.

The paper is organized as follows: Section 2 introduces the basics of fractional calculus and fractional order transfer functions, Section 3 presents the Short Memory Principle and the frequency study on the truncation error, Section 4 analyzes the fractional order transient response for a simple scenario. Section 5 presents the LMRPEM algorithm and an application of a truncated estimation. Conclusions and perspectives are presented in Section 6.

Section snippets

Fractional derivatives

As previously stated, many different definitions for fractional order differentiation exist.

Definition 2.1 Riemann and Liouville Fractional Derivative

The most well-known fractional derivative definition is the one of Riemann and Liouville [15] defined as: 0RLDνf(t)=1Γ(nν)dndtn0t(tτ)nν1f(τ)dτ,where n1ν<n with n being an integer and Euler’s gamma function is defined as: Γ(x)=0ettx1dtfor xRZ.

If the initial conditions are set to zero, Riemann–Liouville’s definition is proven to be equal [17, Chap. 7] to the series definition given

Short memory principle

The Short Memory Principle is based upon Riemann–Liouville fractional derivative Definition 2.1. The principle consists in only taking the past of the function in an interval [tL,t], where L is the time window of the memory.

By expressing the fractional derivative as the sum of a long and a short memory, one gets: 0RLDνf(t)=1Γ(nν)dndtn0tL(tτ)nν1f(τ)dτ+tLt(tτ)nν1f(τ)dτ.

By applying the short memory principle, one gets: 0RLDνf(t)1Γ(nν)dndtntLt(tτ)nν1f(τ)dτ.

For a signal f(t) with

Output estimation in system identification: the initial condition problem

System identification algorithms require estimating the system output yˆ. From the LMRPEM-2 method (details are provided in [40] for the coefficient estimation and in [32] for both coefficient and differentiation order estimation and briefly recalled in Section 5.1), the estimated output is obtained from the following convolution: yˆ(t)=G(p)u(t).

Such an online algorithm has proven to provide estimates without bias and with very low variance towards high noise level. Fractional order operators

Long-memory recursive prediction error method

In this section, a recursive algorithm for fractional order system identification will be used in order to estimate its parameters. The long-memory recursive prediction error method (LMRPEM) [43] has proven to provide consistent estimates with low variance and without bias when considering the whole data set from t=0 to the current instant time kTs. A brief recall of this method is presented below.

Classic prediction error method minimizes a quadratic error criterion: J=12k=1tϵ(k)2 where the

Conclusions and perspectives

The non-locality property or long-memory property of fractional derivatives constitutes both a main advantage and drawback in mathematical modeling. It allows to capture long-memory behavior, but may impose implementation limits in real-time scenarios. Podlubny’s Short Memory Principle provides a mathematical proof of a truncation length L to be used in order to get an approximated fractional derivative without the whole past of the function. However, even though it effectively provides a

CRediT authorship contribution statement

Jean-François Duhé: Writing – original draft, Formal analysis, Software. Stéphane Victor: Writing – review & editing, Formal analysis, Methodology, Supervision. Pierre Melchior: Supervision. Youssef Abdelmounen: Supervision. François Roubertie: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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