A new IIR-type digital fractional order differentiator

Abstract

A new infinite impulse response (IIR)-type digital fractional order differentiator (DFOD) is proposed by using a new family of first-order digital differentiators expressed in the second-order IIR filter form. The integer first-order digital differentiators are obtained by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule. The distinguishing point of the proposed DFOD lies in an additional tuning knob to compromise the high-frequency approximation accuracy.

Introduction

Fractional calculus is a 300-years-old topic. The theory of fractional-order derivative was developed mainly in the 19th century. Recent books [11], [12], [21], [24] provide a good source of references on fractional calculus. However, applying fractional-order calculus to dynamic systems control is just a recent focus of interest [7], [17], [18], [22], [23]. For pioneering works, we cite [4], [9], [10], [13].

In theory, the control systems can include both the fractional order dynamic system or plant to be controlled and the fractional-order controller (FOC). However, in control practice, it is more common to consider the fractional-order controller. This is due to the fact that the plant model may have already been obtained as an integer-order model in the classical sense. In most cases, our objective is to apply fractional-order control to enhance the system control performance. For example, as in the CRONE¹ control [14], [17], [18], fractal robustness is pursued. The desired frequency template leads to fractional transmittance [16], [19] on which the CRONE controller synthesis is based. In the CRONE controller, the major ingredient is the fractional-order derivative s^r, where r is a real number and s is the Laplacian operator. Another example is the PI^λD^μ controller [20], [22] which is actually an extension of the classical PID controller. In general form, the transfer function of PI^λD^μ is given by K_p+T_is^−λ+T_ds^μ, where λ and μ are positive real numbers; K_p is the proportional gain, T_i the integration constant and T_d the differentiation constant. Clearly, taking λ=1 and μ=1, we obtain a classical PID controller. If λ=0 (T_i=0) we obtain a PD^μ controller, etc. All these types of controllers are particular cases of the PI^λD^μ controller. It can be expected that the PI^λD^μ controller may enhance the systems control performance due to more tuning knobs introduced. Actually, in theory, PI^λD^μ itself is an infinite dimensional linear filter due to the fractional order in the differentiator or integrator. It should be pointed out that a band-limit implementation of FOC is important in practice, i.e., the finite dimensional approximation of the FOC should be done in a proper range of frequencies of practical interest [15], [16]. Moreover, the fractional order can be a complex number as discussed in [15]. In this paper, we focus on the case where the fractional order is a real number.

The key step in digital implementation of a FOC is the numerical evaluation or discretization of the fractional-order differentiator s^r. In general, there are two discretization methods: direct discretization and indirect discretization. In indirect discretization methods [15], two steps are required, i.e., frequency domain fitting in continuous time domain first and then discretizing the fit s-transfer function. Other frequency-domain fitting methods can also be used but without guaranteeing the stable minimum-phase discretization. Existing direct discretization methods include the application of the direct power series expansion (PSE) of the Euler operator [8], [27], [28], [29], continuous fractional expansion (CFE) of the Tustin operator [5], [27], [28], [29], and numerical integration-based method [8], [5]. However, as pointed out in [1], [2], [3], the Tustin operator-based discretization scheme exhibits large errors in high-frequency range. A new mixed scheme of Euler and Tustin operators is proposed in [5] which applies the Al-Alaoui operator [1].

The above discretization methods for s^r lead naturally to the DFODs usually in IIR form. Recently, there are some methods to directly obtain the DFODs in finite impulse response (FIR) form [25], [26]. However, using an FIR filter to approximate s^r may be less efficient due to the very high order of the FIR filter. In this paper, a new IIR (infinite impulse response)-type digital fractional order differentiator (DFOD) is proposed by using a new family of first-order digital differentiators expressed in the second-order IIR filter form. The integer first-order digital differentiators are obtained by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule [2]. The distinguishing point of the proposed DFOD lies in an additional tuning knob to compromise the high-frequency approximation accuracy.

This paper is organized as follows: in Section 2, after a brief introduction of a new family of first-order digital differentiators expressed in the second-order IIR filter form by the stable inversion of the weighted sum of Simpson integration rule and the trapezoidal integration rule [2], [3], the corresponding fractional-order digital differentiator via CFE truncation is presented. Section 3 presents some illustrative examples. Section 4 concludes this paper.