The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an improved second-order fractional differential equation operation method based on power series expansion. The algorithm's operation speed and accuracy performance are analyzed. The research found that the algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.
- second-order fractional differential equation
- fractional-order system
- recursive algorithm
- power series expansion
For a long time, we have focused on the mathematical theory of fractional calculus (FOC). It was not until the mid and late 20th century that FOC theory was introduced and applied in some fields such as electrochemistry, rheology, signal processing, and power transmission theory. Recently, the research of fractional order systems has aroused the interest of many scholars . In the field of automatic control, a new branch of fractional control has emerged.
The research of fractional order control should cover at least three aspects: (1) The establishment of a fractional-order system (FOS) model and its analysis based on describing the fractional-order object more accurately and concisely. (2) The fractional-order control strategy is selected to obtain better control performance. (3) Apply fractional arithmetic to process signals, data, etc. The above work is based on the calculation of FOC and FOS.
Based on the Tustin transformation theory and the analysis of the characteristics of the discrete generating function formula used for fractional operators, and using the Maclaurin series expansion of the binomial power function, this paper proposes an improved method based on the power series expansion (PSE) and the Tustin transformation. Fractional calculation method . We apply it to the solution of the linear fractional system and derive the corresponding recurrence formula. The performance of the algorithm's operation speed and accuracy is analyzed through specific example simulations and comprehensive comparisons with several conventional methods. The algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.
The essence of the numerical method is to perform a discrete approximation of FDEs to obtain corresponding approximate numerical solutions . The core problem is the discretization of the FOC operator. The basic idea of discretization is to use grid function Euler backward difference method
Tustin method (trapezoidal method or bilinear transformation)
Euler backward difference method
Tustin method (trapezoidal method or bilinear transformation)
Since the corresponding generating function is irrational, it is not convenient to be directly used for solving. We can approximate it rationally . Commonly used methods are PSE and continued fraction expansion (CFE). The combination of the above-mentioned different methods can get different forms of FOC operator discrete algorithm (PSE-Tustin, CFE-Tustin). Of course, no matter whether it is the PSE method or the CFE method, the solution expression obtained is still an infinite number of items. In its practical application, it can be considered to approximate its finite term, that is, the short memory method (SMP).
The research-based on traditional integer-order systems is relatively mature, and it is a basic research idea and method to approximate FOSs with integer-order systems . So the problem is transformed into using standard integer-order operators to approximate fractional-order operators. This also overcomes the difficulty of not allowing direct fractional operator operations to calculate and simulate existing software. The fractional operator
In the research, it is found that the integral transformation formula of the analytical algorithm is complicated, the calculation amount is large, and the dimension of the state space method is often too high. Models after rational approximation are often inconvenient for the solution and further analysis because of their high order and complexity. The existing analytical algorithms are all aimed at special forms of equations, but many equations often do not have analytical solutions . On the other hand, the application premise of the analytical method is that the prototype function is known, and the problem of calculus for the known data is often encountered in practical engineering applications. Such as system parameter identification, signal analysis, etc. When applying PSE or CFE to the Eqs. (2) to (4) to realize the discretization approximation, it is necessary to call the related functions in Matlab or Maple. The calculation speed is very slow, and the resulting form cannot be directly called in programming. But modern control is based on the computer as the main realization tool. The meaning of its ‘realization’ is not limited to the ‘realization’ of its analysis and design process, but more importantly, it is put into actual operation. There are obvious obstacles to the above methods.
It is well known that the mapping relationship between the
In the research based on the numerical example simulation, it is found that the Tustin method and Al-Alaoui method with the same number of expansion terms or orders are better than the Euler method in terms of accuracy no matter whether PSE or CFE is used. The Tustin method is slightly better than the Al-Alaoui method . Therefore, from the perspective of response distortion or error, Tustin transform is a better choice.
The convergence of the algorithm is one of the important issues that must be considered. Due to the possible limitations of the algorithm itself, when it is applied to the approximation of fractional-order operators, the system's output will diverge due to improper selection of the order (Figure 1). The fundamental reason is that the algorithm cannot guarantee convergence, or the convergence is not good . Theoretically, because FOC has a unique attenuation memory characteristic, it must be shown as the uniform attenuation of the weight coefficients of the power terms of
Tustin transform formula (3) is a fractional power function, and its numerator and denominator have the same structure as formula (2). Suppose the numerator and denominator are approximated as polynomials, and the two are regarded as the numerator and denominator of the pulse transfer function of a filter. In that case, this filter is a discrete FOC filter . Furthermore, we can use the Maclaurin series expansion of the binomial power function to ensure convergence, and the weight coefficient of the expansion term can be quickly and recursively obtained.
We carry out the Maclaurin series expansion of the numerator and denominator of the function
Based on the above analysis, this paper proposes an improved FOC algorithm based on PSE and Tustin transform and applies it to the solution of linear fractional systems.
We assume that the FOC of function
Step 2: The initial value of the parameter
Step 3: Take
Step 4: Let
First consider the general form of the differential equation of a special fractional-order linear system:
Step 1: Calculate the sequence
Step 2: We take the initial value:
Step 5: Use the results of the previous steps to calculate
First, to verify the effectiveness of the FOC recursive solution algorithm proposed in this paper, here is a simple simulation curve of the 0.5-order derivative, 0.5-order integral, and the first-order integral of the sine function
It can be seen that the algorithm curve in this paper is the same as the result of the conventional PSE expansion approximation of the Tustin transform operator as a whole. This is almost the same as the curve obtained by the most direct and simple G-L definition discrete method . The G-L method is almost identical to the theory obtained by applying the Cauchy definition to remove the area around
The unit step response is obtained through a specific FOS model. Based on the accuracy and speed, a comprehensive analysis and comparison between the method in this paper and the commonly used algorithms are made. The calculation and simulation tool is Matlab7. Let the FDEs of the simulation example be
Algorithm error and accuracy indicators
PSE, power series expansion
The error is larger for step
Figure 6 shows that using the recursive discrete approximation algorithm given in this article to obtain the system's step response can obtain a relatively satisfactory result. The response curve of the memory length
It can be seen from Figure 8 that the initial response of the comparison curve between the algorithm in this paper and the conventional PSE-Tustin method is not much different. After 5 s, the difference is obvious, but this phenomenon is not reflected in Figure 5. The reason may be that different FOC discrete approximation methods exhibit different accuracy due to the difference in signal behavior. For this reason, we apply the fractional-order Al-Alaoui discrete operator to this example using the conventional PSE method and compare and analyze the results to show (Figure 9 and Table 1). The response curve shape of PSE-Al-Alaoui is similar to that of PSE-Tustin. In addition, the difference between the number of expansion terms of PSE-Tustin and the obvious influence on the accuracy is that the number of expansion terms has a less obvious influence on the accuracy. This shows that the convergence of the algorithm is better than that of PSE-Tustin. Overall, the results of the algorithm in this paper are the best. The PSE-Tustin method is better than the PSE-Al-Alaoui method.
This paper proposes an improved PSE-Tustin recursive algorithm. This algorithm is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical models are unknown and the solution of linear systems. A simulation example verifies the effectiveness of the algorithm. We show the characteristics and advantages of the algorithm in this paper by comparing the accuracy with several commonly used fractional-order algorithms.
Algorithm error and accuracy indicators