Fractional order calculus has a history of more than 300 years, which extends the order of the classical calculus from integer number to arbitrary real number and even complex number. Compared with integral order calculus, the fractional order calculus could describe the dynamic characteristics of the actual system more accurately. Therefore, fractional order control is increasingly becoming one of the most important topics in control theory in recent years  .
The discretization of the fractional-order differentiator and integrator is the foundation of the digital realization of fractional order controller. Generally, there are two methods for the discretization of the fractional-order differentiator and integrator  , i.e., direct discretization and indirect discretization, while the former is more practical in real applications  . Several algorithms have been proposed for the direct discretization method, e.g., the power series expansion (PSE) of the Euler operator, the continued fractional expansion (CFE) of the Tustin operator, etc.  .
The discretization of the fractional-order differentiator is taken for example. The direct discretization method could be summarized as the following two steps. Firstly, some kind of generating function is used to discretize the differentiator , i.e., , where is the order of the fractional- order differentiator and is usually expressed as a function of the com- plex variable or the shift operator , and then some kind of expansion method is applied to generate the approximate digital filter of the differentiator. For example, Chen and Vinagre propose an IIR (infinite impulse response)-type digital fractional-order differentiator with weighted sum of Simpson integration rule and the trapezoidal integration rule  . Al-Alaoui operator and CFE are applied in the discretization of fractional-order operator for better discretization approximation  . Zhu and Zou propose an improved recursive algorithm for fractional-order system solution based on PSE and Tustin operator  . Mila- dinovic and his colleagues use genetic algorithm to minimize the deviation in magtitude and phase responses between the original fractional order element and the rationalized discrete time filter in IIR structure  . The discretization methods for fractional-order differentiator are compared based on Tustin ope- rator and three different expansion algorithms in  . A class of generating function called parameterized Al-Alaoui transform is presented and the simu- lation analysis indicates that the variable parameter could be adjusted to achieve certain optimal digital filter approximation for the fractional-order operator  . However, the variable parameter of the optimal digital filter is different for the fractional-order operator with different orders, this issue has not been inten- sively discussed in previous papers and will be studied in this paper. The main contributions of this paper are outlined as threefold. (1) The parameterized Al-Alaoui transform is analysed thoroughly to show the effect of the variable pa- rameter. (2) A weighted square integral index about the magtitude and phase is defined as the objective functions to obtain the optimal variable parameter for different fractional orders. (3) Some simulations are implemented to demon- strate the great differences on the optimal variable parameter for differential and integral operators with different fractional orders.
The remainder of this paper is organized as follows. The preliminary of frac- tional calculus is briefly introduced in Section II. The discretization of fractional- order operator with different orders is discussed in Section III. Finally, the con- clusions are given in the last section.
2. Preliminary of Fractional Calculus
A. Definition of Fractional Calculus
Fractional order calculus is a natural generalization of the classical integral order calculus, which extends the order of the integration and differentiation to the non-integer or fractional order  . Two of the commonly used definitions of the fractional calculus are the Grünwald-Letnikov (GL for short) definition and the Riemann-Liouville (RL for short) definition. The GL definition is defined as
where is the fractional-order calculus operator, and are the limits of the operator respectively, is the order of the operator and means the integer part. The RL definition is defined as
where, and is the Gamma function. Actually, the afore- mentioned definitions are equivalent to each other in the real physical systems and engineering applications.
The Laplace transform of fractional order calculus with zero initial conditions for order is
where denotes the operation of Laplace transform, is the Laplace transform of function and denotes the fractional-order operator  . The discretization of fractional-order operator is to design a digital filter to ap- proximate the operator , where could be positive or negative for differen- tiator and integrator, respectively.
B. Parameterized Al-Alaoui Transform
The discretization of the fractional-order differentiator and integrator is to design a digital filter for the fractional-order operator . Firstly, a generating function is used to realize the transform of the Laplace operator from the continuous complex frequency domain to the discrete complex frequency domain. According to the definition of transform, the mapping relationship between the two domains is
where is the sampling period. Furthermore, the equivalent relation can be formulated
where . Take power series expansion to the numerator and denomi- nator of Equation (5) and neglect the high-order terms, Equation (5) can be approximated as
and then the complex variable can be solved to yield
Equation (7) is defined as the transform from continuous complex frequency domain to the discrete complex frequency domain in  and  . The analyses presented in  and  prove that the so-called transform is actually equivalent to the Al-Alaoui transform   with variable parameter when parameter is set to , i.e.,
Specially, three commonly used generating functions, i.e., Tustin transform, Al-Alaoui transform and Euler transform can be formulated when is set to 0, 3/4 and 1, respectively.
A digital integrator with adjustable parameters, which is used in the discreti- zation of fractional-order operator  , is presented in  . The expression of the generating function is
where and are gain adjusting parameter and phase adjusting parameter, respectively. The fractional-order operator can be adjusted more accurately with the adjusting parameter according to different real applications. Specially, Equation (9) is equivalent to Equation (7) when parameter is set to 1. Additionally, if the parameter in Equation (7) is set to , the generating function is formulated
which is used for the discretization of fractional-order operator in  . Table 1 shows the relationship of the common generating functions and the variable parameters.
From above analyses, the generating functions in Equations (7), (8), (9) and (10) are actually equivalent to each other when certain relations of the variable parameters are satisfied as shown in Figure 1. We call them parameterized Al-Alaoui transform in  , a certain expression could be used in certain specific issues.
C. Power Series Expansion of Generating Function
Two main methods for the expansion of generating function are continued fractional expansion and power series expansion. Generally, CFE method could
Figure 1. Relations of parameterized Al-Alaoui transform.
Table 1. Relationship of common generating functions and variable parameters.
obtain IIR-type digital filter which is easy for the digital filter design, while PSE method could obtain FIR (finite impulse response)-type digital filter and re- quires less computation cost in comparison with CFE under similar accuracy criterion  .
In this paper, Equation (10) is used as the generating function, and the power series expansion algorithm presented in  is applied to obtain an IIR-type digital filter. The general form of the discretization of fractional-order operator is
where, is the digital filter of the discrete fractional-order operator, indicates the power series expansion, and are the numerator and denominator polynomials of the digital filter, and and are their orders respectively. Without loss of generality, the approximate orders of the numerator and denominator polynomials are set to (see  for more details).
3. Discretization of Fractional-Order Operator with Different Orders
A. Objective Function
Al-Alaoui transform has better properties in the discretization of fractional- order operator in comparison with other common generating functions, which have been reported in several papers (see e.g.    ). In the parameterized Al-Alaoui transform, the parameter could be adjusted to achieve certain optimal digital filter approximation of the fractional-order operator. An objective function is defined in  , which is to obtain the optimal IIR-type digital filter realization by minimizing the weighted sum of the discrepancies between the responses of the continuous time fractional order filter and its approximate digital filter realization.
However, the papers mentioned above usually consider certain specific fractional order (e.g. fractional order 0.5) and are not concerned with the discretization of fractional-order operator with different orders, which will influence the design of the digital filter. In this paper, we define the following objective functions
where and are the magtitude responses of the fractional- order differ-integrator and its discrete counterpart; and are the corresponding phase responses; is the bandwidth betweeen the lower and upper limits and (i.e. ) within a chosen frequency band, e.g. with Nyquist frequency . The normalized and depend on not only the parameter , but also the different orders . The following simulation analyses are to find the optimal parameter for different fractional orders , which could achieve the minimal objective function with specific weight .
B. Simulation Results
In the simulation, the weight is taken as 0.75 without loss of generality. The orders of the power series expansion are taken as 5 for simplicity, the sampling period is taken as 0.001, and the orders of the fractional-order operator are typically taken as 0.1, 0.5, 0.9 and −0.1, −0.5, −0.9 for differential and integral operators respectively.
Figures 2-4 are the objective functions for differential operator with fractional order 0.1, 0.5 and 0.9, where the horizontal coordinate indicates the different
Figure 2. Objective functions for differential operator with fractional order 0.1.
Figure 3. Objective functions for differential operator with fractional order 0.5.
Figure 4. Objective functions for differential operator with fractional order 0.9.
variable parameter and the longitudinal coordinates indicate the variation of , and respectively. Figures 5-7 are the counterparts of the integral operator. All the figures demonstrate that the variation trends of the objective functions with variable parameter seem consistent with each other, while the optimal variable parameters are totally different for fractional differential and integral operator with different fractional orders under the selected objective functions. Table 2 shows the optimal variable parameters for the specific orders of the differential and integral operators.
Figure 5. Objective functions for differential operator with fractional order −0.1.
Figure 6. Objective functions for differential operator with fractional order −0.5.
Figure 7. Objective functions for differential operator with fractional order −0.9.
Table 2. Optimal variable parameter .
This paper is concerned with the discretization of the fractional-order differen- tiator and integrator with different fractional orders, which is seldom considered in previous literatures. The parameterized Al-Alaoui transform with one variable parameter is presented as a general generating function, and the objective functions are defined to achieve the optimal variable parameter for the discreti- zation. The simulation results demonstrate that there are great differences on the optimal variable parameters for the discretization of differentiator and integrator with different fractional orders.
However, the weight in the simulation is set as 0.75 without loss of generality, and it is undoubtedly arbitrary to select the proper weight for specific discreti- zation purpose, e.g. select smaller weight for more accurate phase approximation. In the future, we will consider the optimal variable parameter into the practical digital fractional order controller design to acquire the optimal control perfor- mances.
This work is supported by a Project of Shandong Province Higher Educational Science and Technology Program under Grant J14LN34.