Elsevier

Signal Processing

Volume 91, Issue 3, March 2011, Pages 386-426
Signal Processing

Studies on fractional order differentiators and integrators: A survey

https://doi.org/10.1016/j.sigpro.2010.06.022Get rights and content

Abstract

Studies on analysis, design and applications of analog and digital differentiators and integrators of fractional order is the main objective of this paper. Time and frequency domain analysis, different ways of realization of fractance device is presented. Active and passive realization of fractance device of order 12 using continued fraction expansion is carried out. Later, time and frequency domain analysis of fractance based circuits is considered. The variations of rise time, peak time, settling time, time constant, percent overshoot with respect to fractional order α is presented.

Digital differentiators and integrators of fractional order can be obtained by using direct and indirect discretization techniques. The s to z transforms used for this purpose are revisited. In this paper by using indirect discretization technique fractional order differentiators and integrators of order 12 and 14 are designed. These digital differentiators and integrators are implemented in real time using TMS320C6713 DSP processor and tested using National instruments education laboratory virtual instrumentation system (NIELVIS). The designed fractional order differentiators have been used for the detection of QRS sequences as well as the occurrence of Sino Atrial Rhythms in an ECG signal and also for the detection of edges in an image. The obtained results are in comparison with the conventional techniques.

Introduction

This paper deals with the analysis, design and applications of analog and digital fractional order differentiators and integrators. Fractional order differentiators and integrators are examples of fractional order systems. Fractional order systems are described by fractional order differential equations [52], [54]. A method for modeling and simulation of fractional systems using state-space representation is presented in [33]. Fractional order differentiators and integrators are used to compute the fractional order time derivative and integral of the given signal [1], [2], [3], [4]. Geometrical and physical interpretations of fractional order integrators and differentiators is discussed in [6], [8], [9], [13], [37]. In [35] Machado presented a probabilistic interpretation of fractional order derivative based on Grunwald–Letnikov definition. In analog domain such an operation can be called as fractance device. The expression for impedance function of a fractance device is given by, Z(s)=k0/sα where k0 is a constant and α is a fractional order. Depending upon the values of α the behavior of the element changes from Inductor to Capacitor [24], [44]. Terms fractance, fractional order differ integral, fractional order capacitor can be used synonymously.

In order to solve the fractional order differential equations, which characterize such systems, a combination of fractional calculus and Laplace transform techniques can be used. The solution of fractional order differential equations contain Mittag–Leffler functions [7], [10], [12], [36], [43], [56]. In this paper expressions for time-domain response of a fractance device of order α,12 for different excitations are derived.

With the advantages of fractance device in various fields its realization has gained importance. The fractance device can be realized by using fractal structure. Nakagawa and Sorimachi proposed a tree type circuit using resistors and capacitors [24]. Oldham has proposed a chain type circuit for the realization of fractance device [1]. Recently a net grid type circuit was proposed by Pu [38], [42]. But the fractal based realization suffer from the problem of occupying high space and high cost.

The crucial point in the realization of fractance device is finding a rational approximation of its impedance function. There are so many procedures that can be used to calculate the rational function approximation of fractance device. Oustaloup method, Newton's method, Matsudas method, etc. were some of them [17], [22], [32], [47], [58]. It has been proposed that using continued fraction expansion fractance device can be realized. The rational approximation thus obtained is synthesized as a ladder network. The results compare well with the previous techniques [58].

The fractance can be used in circuits along with the three passive elements resistor, inductor and capacitor either as series or as a shunt element [18], [21], [26], [34], [60], [62]. As part of the paper time and frequency domain analysis of inverted-L type fractance based circuits has been performed. The effect of fractional order of the circuit on frequency response is also studied. It can be observed that the performance of the higher integer order circuit could be obtained from the circuit with lesser fractional order. This also reduces the cost and space. Later, the expressions for peak overshoot, rise time, time constant, settling time, etc. were obtained for the fractional order circuit that is considered [48].

The second part of this article concentrates on the design, application and real time implementations of fractional order digital differentiators and integrators. The design of digital differentiators and integrators involves the discretization of the fractional-order operator, sα [77], [78]. Direct discretization and indirect discretization were the commonly used discretization techniques. A lot of literature is available for direct discretization technique. In this paper indirect discretization technique is followed. An s to z transform has to be used to perform the discretization [61], [62], [63]. As the s to z transform maps the left hand plane of s-domain into the unit circle in z domain it has to preserve the stability properties. Some of the common s to z transforms are Bilinear and Backward transform. Every integration rule can produce a new s to z transform. Al-Alaoui has proposed a method for the calculation of different s to z transforms from the integration rules [64], [65], [66], [67], [68], [69], [70], [71], [72]. The first order s to z transform, called Al-Alaoui transform has shown to be much more efficient than the previous transforms. Fractional order differentiators and integrators can also be designed by using least squares method [80], [85], power series expansion [87], adaptive technique [81], etc.

The differentiators and integrators obtained using direct and indirect discretizations are compared. The proposed approach is tested for differentiators and integrators of order 14 and 12. The results obtained compare favorably with the ideal characteristics. Fractional order digital differentiators and integrators designed are implemented in real time and the practical behavior is compared with the theoretical behavior. The digital fractional order differentiators and integrators are implemented in real time using TMS320C6713 DSP kit and tested using NIELVIS. For the real time implementation Cascaded Direct Form-II structure is chosen. The theoretical and practical results compare well within the reasonable limits. The error can be reduced by increasing the gain.

QRS detection is an important topic in the area of Biomedical Engineering. The electrocardiogram (ECG) is a graphical representation of the electrical activity of the heart and is obtained by connecting specially designed electrodes to the surface of the body. Variety of methods use digital differentiators for the QRS detection [94], [95], [96]. The template matching technique using digital differentiator is one of the traditional technique used by the research community [92]. In this paper, fractional order digital differentiator has been replaced with the traditional differentiator. The results are comparable with the previous techniques. Edge detection refers to the identification of changes in brightness of an image. Applying fractional order differentiators to detect edges of an image is also performed in this paper. The performance of the fractional order differentiators is comparable to the conventional differentiators.

The paper is organized as follows. Section 2 deals with the basic definition, time-domain response calculations of fractance device. Realization of fractance device using different approximations is also presented in this section. Fractance based circuits, their time and frequency domain response calculations, time-domain parameter calculations are presented in Section 3. An s to z transform is to be used for the discretization of continuous time systems. Different types of s to z transforms (digital differentiators) and their comparisons are presented in Section 4. Section 5 deals with the indirect discretization technique used for the design of fractional order digital differentiators and integrators. Design and real-time implementation of the digital differentiators of fractional order are also discussed in this section. Some of the applications of fractional order differentiators such as detection of QRS sequences in an ECG signal, edge detection are discussed in Section 6. Finally, Section 7 deals with results and conclusions.

Section snippets

Fractance device

Of late, many researchers are paying attention to the fractance device. The origin of this device is from the working principle of well known passive element capacitor [20], [23]. According to Curie's Law when the initial stored energy is zero, in a capacitor and if DC Voltage V has been applied, the current flowing through the device will bei(t)=Vhtαfort>0where h and V are real. Taking Laplace transform of Eq. (1),I(s)=Γ(1α)Vhs1αWhen the applied voltage signal is DC,Z(s)=hsαΓ(1α)=k0sαwhere

Fractance based circuits

The six possible inverted-L type circuits using fractance device as series or as shunt element were shown in Fig. 23 [21], [32], [59]. It has been observed that the transfer function H(s) can be expressed in two ways as,

  • For R–F, L–F and C–F circuits, H(s)=η/(η+sβ).

  • For F–R, F–L and F–C circuits H(s)=sβ/(η+sβ) where β=αforRF&FRα+1forLF&FLα1forCF&FCand η=k0RforRF&FRk0LforLF&FLCk0forCF&FC

IIR type digital differentiators

It is well-known that a digital differentiator can be used for the purpose of discretization [64], [65], [66], [67], [68], [69], [70], [71], [72]. Digital differentiators are used to find the time-derivative of the incoming signal. A differentiator is defined asG(jω)=jωwhere j=1. In 1992, Al-Alaoui has proposed a procedure for the design of IIR type digital differentiators which are obtained by the inversion and magnitude stabilization of digital integrators [64].

Some of the commonly available

Fractional order digital differentiators and integrators

An ideal fractional order digital differentiator is defined as [74], [75], [76], [77], [78]Hd(jω)=(jω)αwhere α is fractional order. Similarly an ideal fractional order integrator is defined asHI(jω)=1(jω)α

In general, there are two discretization methods, namely direct discretization and indirect discretization [50], [77], [78], [79], [82], [83], [84]. Chen and others proposed an IIR type fractional order digital differentiator based on direct discretization method. The simplest and straight

Applications of fractional order digital differentiators

Fractional order differentiators and integrators are gaining importance in many fields. The application of fractional calculus in the area of control systems, robotics, instrumentation is illustrated in [15], [16], [19], [39], [41], [53], [55], [89]. Debnath [11] has summarized the application of fractional calculus in various fields of science and engineering. In [40] Malti et al. have discussed about the application of fractional order differentiation for system identification.

Results and conclusions

In this paper, numerical calculations have been performed for the response characteristics of a typical fractance device. The parameters used in the calculation are, R=1KΩ, C=1 nF and f=10 kHz. The response to complicated input functions for a fractance device can be computed easily by employing fractional calculus. With fractional calculus approach the equations are simple and are easily amenable for manipulation. A new method of realization of fractance device of order 12 using continued

Acknowledgements

The author wishes to express his gratitude to Prof. Yang Quan Chen from Utah State University, USA, Prof. M.A. Al-Alaoui from American university of Beirut, Lebanon, Dr. Virginia Kiryakova, Editor, FCAA Journal and Dr. Manuel D. Ortigueira for their encouragement and suggestions. The author would also like to thank the anonymous reviewers for their useful comments. The author express the deep sense of gratitude to the Board of Management, GITAM University, Visakhapatnam for their encouragement.

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