This paper deals with the identification of the fractional order Hammerstein model by using proposed adaptive differential evolution with the Local search strategy (ADELS) algorithm with the steepest descent method and the overparameterization based auxiliary model recursive least squares (OAMRLS) algorithm. The parameters of the static nonlinear block and the dynamic linear block of the model are all unknown, including the fractional order. The initial value of the parameter is obtained by the proposed ADELS algorithm. The main innovation of ADELS is to adaptively generate the next generation based on the fitness function value within the population through scoring rules and introduce Chebyshev mapping into the newly generated population for local search. Based on the steepest descent method, the fractional order identification using initial values is derived. The remaining parameters are derived through the OAMRLS algorithm. With the initial value obtained by ADELS, the identification result of the algorithm is more accurate. The simulation results illustrate the significance of the proposed algorithm.
Currently, systems in industrial processes have become more and more increasingly complex, such as chemical plants, robotic arms, etc. This requires us to provide a more accurate mathematical model. Therefore, many parameter identification methods have been studied for system modelling and identification, such as bilinear systems, linear systems, and nonlinear systems [
As a widely represented mathematical model, the nonlinear model has attracted more and more attention. The Hammerstein model is a typical nonlinear model composed of static nonlinear links and dynamic linear links. It can reflect the characteristics of process characteristics and describe a series of nonlinear processes such as neutralization processes [
Due to the wide application of the Hammerstein model, how to identify accurate mathematical models has become the research direction of many researchers. At present, some methods that can effectively identify the Hammerstein model are proposed, such as the least squares algorithm [
In recent years, many scholars have researched the static nonlinear module of the Hammerstein model, hoping to obtain a mathematical model with higher accuracy and wider applicability. Some methods to describe the nonlinear part are proposed, such as radial basis functions [
At present, some identification methods of fractional order systems have been studied. The enhanced response sensitivity approach can reduce the sensitivity of the identification parameter results in measurement noise [
The intelligent optimization algorithm has attracted increasingly scholars’ attention because of its generalization, simple parameter setting, and easy programming. It has been applied to the field of parameter optimization. In [
Based on the above background, this paper will adopt a new method to parameterize the nonlinear and linear coefficients and fractional order of the fractional Hammerstein model, which has rarely been considered before. The identification of each parameter has a corresponding mathematical derivation. In the proposed method, the ADELS algorithm adds a domain search strategy and improves the setting of algorithm parameters. Parameters of the fractional order Hammerstein model including the fractional order are identified. The identification result provides a relatively accurate initial value for the subsequent algorithms and solves the problem that most algorithms rely on the initial value. Then, the fractional order identification method is proposed by using the principle of steepest gradient descent, and the auxiliary model recursive least squares algorithm is used to estimate the coefficients of the fractional order Hammerstein model. Numerical simulations prove the effectiveness of the proposed methods.
The main contributions are to propose an adaptive differential evolution with local search strategy (ADELS) algorithm for identifying the initial parameters of the fractional order Hammerstein system and develop new recursive identification methods for identifying the fractional order and coefficients of the fractional order nonlinear system by using the auxiliary model. Compared with the classic DE, the proposed ADELS algorithm has higher estimation accuracy because of the adaptive strategy and the introduction of Chebyshev sequence. The self-adaptive operator and mutation strategy make the individual take the corresponding search strategy according to the fitness function value, and the Chebyshev sequence has the characteristics of periodicity, randomness, and ergodicity, which makes up for the defect that DE is easy to fall into local optimum. Based on the auxiliary model, the recursive algorithm of fractional order Hammerstein model is given by using the overparameterized least squares and gradient descent algorithm. Both fractional orders and coefficients of the system are considered. The proposed ADELS algorithm can be applied to many fields like robust control, multi-objective optimization, and economic dispatch problems [
In this paper, the mathematical background of identifying the fractional Hammerstein model is discussed in Section
Fractional calculus is the general form of integer calculus. So far, there are many definitions of fractional calculus. In some cases, different definitions of fractional calculus are completely equivalent. The rationality of these definitions has been proven in the literature. Among them, the definitions of three types of fractional calculus are widely used: Grünwald–Letnikov (GL), Riemann–Liouville (RL), and Caputo definitions [
In this article, considering the ease of calculation, the GL definition will be used [
By using
The
Differential evolution (DE) algorithm has better stability and global search ability. The population of each generation of DE is composed of NP parameter vectors with dimension D. Each individual is represented as
In DE, random initial populations are generated by a uniform probability distribution. After initialization, the population evolves by using three steps: mutation, crossover, and selection.
For each target vector
To increase the diversity of trial vectors, crossover operations are introduced as follows:
The trial vector
Repeat the three steps above until the termination condition is met, such as the total number of iterations or the accuracy requirement of the algorithm.
The fractional order Hammerstein model is shown in Figure
The fractional order Hammerstein model.
The system inputs and outputs are
The fractional order transfer function of the linear part is defined as follows:
In summary, the fractional Hammerstein system discussed in this article can be expressed as follows:
The DE algorithm is an intelligent search algorithm worked by cooperation and competition among individuals in the population. It still remains the population-based global search strategy in the evolutionary algorithm. As an efficient parallel search algorithm, the DE algorithm has strong global convergence and robustness, which is worthy of theoretical and applied research.
The adaptive differential evolution with the local search strategy (ADELS) algorithm proposed in this paper has been innovatively improved in three places. Mutation and crossover operators in the DE algorithm control the enlargement ratio of the deviation vector and the probability that the trial vector comes from the mutation vector, which plays a pivotal role. Therefore, the operators in the proposed algorithm adopt the strategy of adaptive change according to the individual fitness value, which improves the optimization efficiency of the algorithm. Different mutation strategies also affect the accuracy of parameter identification. In this paper, different mutation strategies are selected according to the fitness value, which further improves the efficiency of the algorithm. Then, the local search ability of the algorithm is improved by using the Chebyshev chaotic sequence, which makes up for the shortcoming that the DE algorithm is easy to fall into the local optimum.
Mutation operator and crossover operator affect the optimization ability and the convergence speed of the algorithm. Generally, these operators are fixed constants. In this paper, the individual score is obtained based on the fitness value of each individual in each generation by introducing a nonlinear scoring method, and the score ranges from 0 to 1. The operator will be determined based on the score of each individual, which makes the operator more adaptable. The scoring rules based on the minimized objective function are as follows:
It can be seen from the above formulas that the value of the operator changes with the individual’s fitness value score. When the individual’s fitness value is low, the score will be high according to equation (
In each iteration, individuals are sorted according to the fitness value, and the population will be divided into two subpopulations with different individual fitness function. Two subpopulations adopt different mutation strategies as follows:
To avoid the algorithm falling into local optimum, a Chebyshev chaotic sequence is introduced to perform a local search for the optimal individual. Chaotic systems have the characteristics of periodicity, randomness, and ergodicity, which can increase the diversity of individual populations. The family of Chebyshev polynomial can be expressed as
Then, the Chebyshev map can be expressed as
Then, a dual search strategy is introduced to search for the best gene for each dimension of the optimal individual. Due to the randomness and ergodicity of chaos, chaotic mapping has a better performance in improving the diversity of the population compared to a uniform probability distribution. The local search strategy is as follows:
Because of adaptive operators, different mutation strategies, and local search based on the Chebyshev chaotic map, the algorithm’s optimization ability and convergence speed have been significantly improved according to the following numerical simulations. The effectiveness of the ADELS algorithm will be proved in the numerical simulation in
Define the objective function Initialize parameters of the Chebyshev map: Initialize individuals Evaluate all the individuals in the population by the objective function Initialize the number of iteration While ( For each individual Update operators adaptively (equations ( The mutation vector If the generated mutation vector exceeds the boundary, a new mutation vector is generated randomly, until it is within the boundary; The trial vectors The best individuals Find the current best If a new optimal individual End End while; Postprocess results and visualization.
In the fractional order Hammerstein model, the polynomial coefficients of the static nonlinear block, the coefficients of the dynamic linear block, and the fractional order are all needed to be determined, which is rarely concerned before. In this paper, a set of input and output data will be first used to obtain the initial values of coefficients and the fractional order through the proposed ADELS algorithm. Then, the initial value will be used to get the final parameter identification result of the fractional order Hammerstein model through an over-parameterized based auxiliary model recursive least squares (OAMRLS) algorithm and the steepest descent method.
The OAMRLS algorithm is required to get the initial values of the parameters, which have been calculated in the ADELS algorithm. According to equations (
The input-output relations can be written in the regression form as
The estimated vector
It should be noted that the vector
The fractional order Hammerstein based on the auxiliary model.
According to Figure
Then, the output of the auxiliary model
Define the criterion function as
Then, the estimated
Provided the inverse
Then, a recursive version is given as the following formula:
For the problem of overparameterization, to ensure the uniqueness of parameter identification, the value of
Based on the identification result of the ADELS algorithm, the OAMRLS algorithm cooperates with the following order identification method to complete the identification of the system parameters.
The initial value of the fractional order has been given by the ADELS algorithm; then, the iterative optimization of the fractional order is performed according to the steepest descent method. Define the criterion function as
The criterion function can be minimized by the steepest descent method:
According to formula (
Then,
Consequently,
Combined with ADELS and OAMRLS algorithms, the identification process of the fractional Hammerstein model is shown as follows.
Firstly, 15 benchmark functions will be used to verify the effectiveness of the ADELS algorithm, and the results will be compared with other intelligent optimization algorithms proposed in recent years. These functions include unimodal and multimodal functions. The function expression and search range are shown in Table
Benchmark functions and search range.
Function | Range |
---|---|
F1 | [−100,100] |
F2 | [−10,10] |
F3 | [−100,100] |
F4 | [−500,500] |
F5 | [−10,10] |
F6 | [−100,100] |
F7 | [−100,100] |
F8 | [−100,100] |
F9 | [−100,100] |
F10 | [−100,100] |
F11 | [−32,32] |
F12 | [−100,100] |
F13 | [−100,100] |
F14 | [−100,100] |
F15 | [−5.12,5.12] |
In this paper, the ADELS algorithm will be compared with three algorithms including JADE [
Parameter settings.
Algorithm | Parameter settings |
---|---|
JADE | CR = 0.5, |
GWO | Size = 50 |
WOA | Size = 50 |
ADELS | Size = 50, |
In this section, the population size of the four algorithms is set to 50, the number of iterations is 500, and the simulation dimensions of the benchmark function are 10 and 30. The simulation results are shown in Tables
Comparison of optimization results obtained for the 30-dimensional benchmark functions.
JADE | GWO | WOA | ADELS | |||||
---|---|---|---|---|---|---|---|---|
Ave | Std | Ave | Std | Ave | Std | Ave | Std | |
2.55 | 2.43 | 5.46 | 4.78 | 3.35 | 2.52 | 0 | 0 | |
7.65 | 5.09 | 7.26 | 6.93 | 2.13 | 5.88 | 1.80 | 0 | |
2.01 | 1.98 | 0.4587 | 0.20 | 0.086 | 6.7 | 0 | 0 | |
−11684.98 | 142.47 | −6281.5 | 861.64 | −11535.27 | 1685.05 | −12391.83 | 76.85 | |
1.96 | 4.00 | 3.46 | 7.43 | 3.12 | 2.68 | 0 | 0 | |
1.01 | 9.11 | 1.14 | 2.34 | 3.15 | 7.32 | 0 | 0 | |
17.69 | 2.06 | 26.67 | 0.87 | 27.54 | 0.33 | 0.2391 | 19.20 | |
2.3 | 0.10 | 0.94 | 0.22 | 0.49 | 0.28 | 1.50 | 5.62 | |
1.78 | 6.95 | 3.52 | 1.63 | 1.79 | 8.55 | 0 | 0 | |
5.25 | 5.43 | 7.65 | 4.43 | 2.44 | 1.12 | 0 | 0 | |
3.05 | 4.89 | 4.42 | 4.53 | 3.73 | 2.55 | 1.15 | 3.63 | |
−675.49 | 10.04 | −476.51 | 66.40 | −870.00 | 1.02 | −870.00 | 3.61 | |
9.60 | 4.28 | 1.4 | 7.8 | 5.1 | 0.0078 | 0 | 0 | |
9.11 | 6.81 | 3.68 | 5.30 | 0 | 0 | 0 | 0 | |
26.51 | 3.41 | 2.69 | 3.38 | 0 | 0 | 0 | 0 |
Comparison of optimization results obtained for the 10-dimensional benchmark functions.
JADE | GWO | WOA | ADELS | |||||
---|---|---|---|---|---|---|---|---|
Ave | Std | Ave | Std | Ave | Std | Ave | Std | |
9.16 | 3.47 | 8.14 | 3.06 | 8.55 | 2.07 | 0 | 0 | |
6.74 | 2.22 | 2.88 | 2.65 | 2.43 | 9.34 | 5.77 | 0 | |
1.46 | 1.78 | 2.56 | 7.58 | 1.99 | 2.2 | 0 | 0 | |
−3542.12 | 672.55 | −2806.47 | 322.09 | −3439.87 | 567.21 | −4189.83 | 60.45 | |
2.74 | 1.22 | 1.01 | 4.31 | 1.34 | 4.04 | 0 | 0 | |
1.42 | 6.29 | 1.94 | 4.90 | 1.79 | 5.36 | 0 | 0 | |
6.67 | 6.38 | 6.53 | 5.31 | 6.60 | 4.79 | 1.99 | 2.61 | |
6.81 | 1.01 | 1.02 | 7.3 | 4.77 | 8.06 | 1.50 | 5.62 | |
5.18 | 2.15 | 2.08 | 4.60 | 1.58 | 3.50 | 0 | 0 | |
6.55 | 2.32 | 7.91 | 1.83 | 6.55 | 2.91 | 0 | 0 | |
3.02 | 2.13 | 6.75 | 1.74 | 3.73 | 2.47 | 4.80 | 1.09 | |
−90.00 | 0 | −82.63 | 7.16 | −90.00 | 0 | −90.00 | 0 | |
6.80 | 1.02 | 1.8 | 2.7 | 5.59 | 8.67 | 3.11 | 2.67 | |
0 | 0 | 2.16 | 1.85 | 0 | 0 | 0 | 0 | |
0 | 0 | 5.99 | 1.85 | 0 | 0 | 0 | 0 |
The convergence curves of ADELS, GWO, WOA, and JADE are as shown in Figures
Comparison of convergence curves of WOA and literature algorithms obtained in some of the 30-dimensional benchmark problems.
Comparison of convergence curves of WOA and literature algorithms obtained in some of the 10-dimensional benchmark problems.
The unimodal function can be used to evaluate the exploitation capability of the optimization algorithm, whereas the multimodal function is very effective to evaluate the exploration capability of the algorithm. The results reported in Tables
In this section, several different algorithms will be compared to prove the effectiveness of the proposed algorithm. The fractional Hammerstein model is presented as follows:
The parameter vector to be identified is
The input
The ADELS algorithm is used to estimate initial parameter values of the nonlinear and linear parts of the system. The population size of the algorithm is set to 40, the minimum and maximum values of the mutation operator are 0.1 and 0.9, the minimum and maximum values of the crossover operator are 0.1 and 0.9, and the number of iterations of the algorithm is set to 200. The parameters estimated by the optimization algorithm converge to the true value, and the relative quadratic error is as follows:
Initial values obtained by three different heuristic algorithms.
Approach | Parameter | Rqe | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | GWO | 1.0863 | 1.7769 | 1.8239 | 1.8615 | 0.5159 | 0.2947 | 0.0932 | 0.2841 | 0.1459 |
2 | JADE | 0.3828 | 1.9053 | 1.6346 | 1.5920 | 0.5399 | 0.2728 | 0.0817 | 0.2603 | 0.3875 |
3 | ADELS | 1.1730 | 1.5398 | 1.8516 | 1.7289 | 0.5065 | 0.2882 | 0.0950 | 0.2970 | 0.0539 |
Simulation results obtained by different algorithms are given in Table
Simulation results obtained by different algorithms.
Approach | Parameter | Rqe | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | GWO & OAMRLS | 1.3233 | 1.4821 | 1.8995 | 1.7173 | 0.5039 | 0.2912 | 0.0953 | 0.3022 | 0.0133 |
2 | JADE & OAMRLS | 1.3699 | 1.4582 | 1.9044 | 1.7339 | 0.5065 | 0.2914 | 0.0946 | 0.3063 | 0.0332 |
3 | ADELS & KTSG | 1.1932 | 1.5674 | 1.8867 | 1.7289 | 0.5065 | 0.2882 | 0.0950 | 0.2969 | 0.0486 |
4 | ADELS & OAMRLS | 1.3060 | 1.4899 | 1.8977 | 1.7100 | 0.5029 | 0.2912 | 0.0956 | 0.3007 | 0.0069 |
The performance of the intelligent optimization algorithm to search for the initial value can be measured by Root Mean Square Error (RMSE) and Mean Square Error (MSE), which are defined as follows:
The output estimation errors of different intelligent optimization algorithms.
Using the initial estimation value, OAMRLS and the steepest descent method are combined to estimate the accurate parameter vector. Figure
The actual outputs and the identified outputs regarding Algorithm
Collect the input/output data { Obtain the initial of unknown parameters by using Algorithm While ( Estimate the value of system coefficient according to the equation ( Update the fractional order according to the equation ( If the criterion function value Break; End; End while; Postprocess results and visualization.
The error between the estimated and the actual outputs regarding Algorithm
Step response of real system and estimation model regarding Algorithm
The estimated fractional order regarding Algorithm
This paper outlines the identification methods for fractional order Hammerstein models. To improve the accuracy of identification, the heuristic algorithm is considered to search the initial value. Simulation results show that the heuristic algorithm is easy to fall into local optimum. To solve this problem, ADELS algorithm is proposed. In this algorithm, combined with Chebyshev mapping, genes are adaptively searched in each generation, which leads to faster convergence and more accurate results. The initial values of the coefficients of the linear part and the nonlinear part are obtained by the ADELS. Furthermore, the fractional order of the transfer function obtains the initial value through the ADELS. Then, the method of fractional order Hammerstein model coefficient estimation is proposed and compared with other algorithms. When the initial value is obtained, the parameters can be accurately identified by the corresponding OAMRLS algorithm. The fractional order is iterated by the steepest descent algorithm through derivation. Through comparison with other algorithms, the simulation results show the effectiveness of the proposed algorithm. The proposed methods in this paper can be applied to other literatures [
The code used in this paper was written in MATLAB 2018a; the data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (61673004) and the Fundamental Research Funds of the Central Universities of China (XK1802-4).