Iterative Parameter Identification for Fractional-Order Block-Oriented Nonlinear Systems with Application to a Battery Model


 This paper investigates parameter and order identification of a class of block-oriented nonlinear systems. By using the hierarchical identification principle, the system is divided into two subsystems, which are a linear block system and a nonlinear block system. For the purpose of solving the difficulty of estimating two sets of parameter vectors, the over-parameterization method and the key item separation technique are used, respectively. Therefore, a two-stage over-parameterization gradient-based iterative algorithm and a key term separation two-stage gradient-based iterative algorithm are derived. The simulation results indicate that the proposed algorithms are effective. Finally, the proposed method is evaluated through a battery model. The results show well agreement with the real system outputs.


Introduction
Nonlinear phenomena widely exist in various systems, e.g., photovoltaic cell models [1], lithium-ion batteries [2], and stirred-tank reactors [3]. On the identification of nonlinear systems, Ghani et al. presented a new method for photovoltaic cells, their method is based on the examination of the power voltage data from which a system of five residual equations are derived and solved via the multi-variable Newton-Raphson method [4]. For a lithium-ion battery model, Li et al. combined the bias compensation recursive least squares algorithms and the extended Kalman filter to alleviate the impact of the noises [5]. Moreover, some methods have been proposed for the nonlinear systems identification, e.g., the maximum likelihood methods [6,7,8], the least squares methods [9,10,11], and the gradient search methods [12,13,14].
The block-oriented nonlinear systems are popular for their simplicity and ability to accurately describe a wide variety of nonlinear systems [15,16,17,18]. The block structures, like Hammerstein systems, have the ability of flexible combination of various static nonlinear elements and various dynamic linear elements. Wang et al. considered switch detection and robust identification for slowly switched Hammerstein systems, and proposed a two-identifier-based switch detection scheme to improve the precision of the time-invariant parameter estimation [19]. For the multivariable Hammerstein systems, it is difficult to parameterize the system into an auto-regression form to which the standard least square method can be applied. Wang et al. maximized the logarithmic likelihood function about each parameter vector to get their estimates [20]. It is worth noting that some complex systems are difficult to be accurately described by the traditional integer-order systems. This motivates us to study the parameter identification of the fractional-order systems.
In many practical nonlinear processes, the fractional behavior cannot be ignored. Moreover, the existence of orders increase the difficulty of system identification, which makes many scholars study the fractional-order systems [21,22,23,24]. Hu et al. established a improved second-order equivalent circuit model based on the fractional calculus theory, and used the mixed-swarm-based cooperative particle swarm optimization algorithm to identify the parameters of the equivalent circuit model [25]. For fractional-order Hammerstein-Wiener systems, an output-error approach was developed by using the robust Levenberg-Marquardt algorithm, and was applied to the benchmark test experiment of the robot arm [26]. In the application of heating processes, Hammar et al. researched the parameterization of the Hammerstein system by transforming the fractional-order polynomial nonlinear state-space model [27]. Sersour et al. used the heuristic particle swarm optimization to identify unmeasurable internal variables combined with the key item separation technique [28].
Considering the nonlinearity of the channel, we use the hierarchical identification principle [29,30,31] to divide the original system into several subsystems or sub-identification models. For the purpose of solving the difficulty of estimating the two sets of parameter vectors, we use the over-parameterization method and the key term separation technique, respectively. The gradient search is basic for nonlinear optimization problems [32,33,34], a two-stage gradient-based iterative algorithm was developed for the fractional-order block-oriented nonlinear systems. The main contributions of this paper are as follows.
• By using the model decomposition technique, a fractional-order block-oriented nonlinear system is decomposed into two subsystems, which are a linear subsystem and a nonlinear subsystem.
• Based on the iterative search, we propose a two-stage over-parameterization gradient-based iterative (2S-OP-GI) algorithm, and a key term separation two-stage gradient-based iterative (KT-2S-GI) algorithm.
In addition, the comparison of computation amount between the two algorithms is given.
• Two examples are given, in which the two-stage over-parameterization gradient-based iterative algorithm is applied to a battery model by using the proposed hierarchical identification method.
This paper is organized as follows. Section 2 describes the identification model for a class of fractional-order nonlinear systems. A 2S-OP-GI algorithm is presented in Section 3. Section 4 utilizes the key term separation technique to deduce a KT-2S-GI algorithm. Section 5 analyzes the computational efficiency of the 2S-OP-GI algorithm and the KT-2S-GI algorithm. The numerical examples illustrate the performance of the proposed methods in Section 6. Finally, some conclusions are given in Section 7.

System description and identification model
Let us introduce some symbols. A =: X or X := A represents that X is defined by A. The symbol I n stands for an identity matrix of size n × n. 1 n represents an n-dimensional column vector whose elements all are 1, that is, 1 n := [1, 1, · · · , 1] T ∈ R n . The superscript T denotes the matrix/vector transpose. Consider the following fractional-order block-oriented nonlinear system, where y(l) is the output of the system, u(l) is the input of the system, e(l) : is an autoregressive noise and v(l) is the random white noise with zero mean and variance σ 2 , f (·) is the nonlinear element of the system, A(z), B(z) and C(z) are three fractional-order polynomials in the unit backward shift operator [z −1 y(l) = y(l − 1), zy(l) = y(l + 1)], define as where α na , α n b and α nc are fractional orders, and the nonlinear relation is given by (2) In this paper, the commensurate orders are the multiple of the same value α, such as α j = jα. e(l) can be obtained According Equations (1)-(3), we get Define the parameter vectors a, b and c of the linear part and γ of the nonlinear part as Define the information vectors/matrix: Then Equation (4) can be rewritten as Equation (5) is the identification model of the fractional-order block-oriented nonlinear system in (1). The objective of this paper is to propose the iterative identification algorithms to estimate the unknown parameters of the fractional-order block-oriented nonlinear systems by using the decomposition technique.

The two-stage over-parameterization gradient-based iterative algorithm
The section aims to apply the gradient search to derive an iterative estimation algorithm for the system. Let F i (l) ∈ R 1×n b be the ith row of the information matrix F (l). Redefine the parameter vectors The corresponding information vector is defined as The (5) can be written as Introduce two transition variables: Thus, Equation (6) can be decomposed into two submodels: Consider the input-output data {u(l), y(l) : l = 1, 2, · · · , L} and define the stacked output vectors and the stacked information matrices: For the subsystems in (7) and (8), we define two criterion functions: Minimizing J 1 (ϑ) and J 2 (θ), we can obtain the estimates of the parameter vectors ϑ, θ and the order α, , andθ k be the kth iterative estimate of the parameter vector θ,α k be the kth iterative estimate of the parameter α, and µ 1,k , µ 2,k and µ 3,k be the kth iterative step sizes. Furthermore, we can obtain the gradient-based iterative relations: However, it is not difficult to find that the above iterative relations is incapable of calculating the parameter estimates. First, the information vector Φ(l) contains the unknown intermediate variable △ α e(l). Second, the order of the information vectors Φ(l) and Φ F (l) are non-integer, and the order has to be identified. Replacing the unknown terms △ α e(l) and α with their iterative estimates △α k−1ê k−1 (l) andα k−1 , combining (9)- (14), we can obtain the two-stage over-parameterized gradient-based iterative (2S-OP-GI) algorithm for estimating the parameter vectors ϑ, θ and the order α: After the estimateθ k of θ is obtained by using the 2S-OP-GI algorithm, the parameter estimation vectorθ k contains the product of γ and b.
The iterative estimateb 1,k can be computed bŷ Then, we can compute the estimateγ j,k of γ j , and the estimateb i, Equations (15)-(33) make up the 2S-OP-GI algorithm. The steps of computing the parameter estimatesθ k ,θ k nadα k are listed as follows.
2. Collect the input and output data u(l) and y(l) and set the data length L.
increase k by 1 and go to Step 3; otherwise, obtain the parameter estimation vectorsθ k andθ k , and the orderα k and terminate the process.

The key term separation two-stage gradient-based iterative algorithm
In this section, a key term separation identification model is established based on the key term separation technique. The goal is to derive a key term separation two-stage gradient-based iterative algorithm for estimating the parameter vectors a, b, c, γ and α from available observation data.

The key term separation identification model
To obtain the unique parameter estimates, one has to let b 1 = 1. Choose x(k) as the key term to parameterize the fractional-order block-oriented nonlinear system. Then the Equation (5) output can be expressed as where the corresponding information and parameter vectors are defined as . . . . . . . . .
Based on the model in (34), the decomposed two submodels are given by Therefore, we can get the identification model and derive the key term separation two-stage gradient-based iterative algorithm.

The key term separation two-stage gradient-based iterative algorithm
According to the key term separation identification model in (35) and (36), construct the information matrices Φ(L) and Φ (L) and the system outputs Y (L), Y 1 (L) and Y 2 (L) as Consider the input-output data {u(l), y(l) : l = 1, 2, · · · , L}, define two criterion functions: Let k = 1, 2, 3, · · · be an iterative variable,θ k ∈ R na+nc+m ,b ′ k ∈ R n b −1 andα k be the estimates of the parameter vectors ϑ and b ′ and order α at iteration k, and ρ 1,k , ρ 2,k and ρ 3,k be the iterative step sizes. Using the negative gradient search and minimizing J 3 (ϑ, α) and J 4 (γ, α) lead to the following gradient-based iterative relations for computingθ k ,b ′ k andα k : Replacing Φ(L) and Φ (L) in (37)-(39) with their estimatesΦ k (L) andΦ (L) yields the following gradientbased iterative algorithm for estimating ϑ, b ′ and α: Equations (40)-(55) make up the KT-2S-GI algorithm. The steps of computing the parameter estimatesθ k , b ′ k andα k are listed as follows: 1. For k 0, all the variables are set to zero. Let k = 1, and set the initial values:θ 0 = 1 na+nc+m /p 0 , b ′ 0 = 1 n b −1 /p 0 , p 0 = 10 6 andα 0 to be a random number, the parameter estimation accuracy ϵ.
2. Collect the input and output data u(l) and y(l), set the data length L. 5. Update the parameter estimation vectorsθ k andb ′ k using (40) and (42), update the order estimateα k using (44).

If ∥θ
increase k by 1 and go to Step 3; otherwise, obtain the parameter estimation vectorsθ k andb ′ k , and the orderα k and terminate the process.

The comparison of the computational efficiency
The following discusses the computational efficiency of the 2S-OP-GI algorithm and the KT-2S-GI algorithm. The flop (floating point operation) counting is a simple approach to the measuring of program efficiency since it ignores subscripting, memory traffic, and the countless other overheads associated with program execution, the flop counting is just a quick accounting method that captures only one of the several dimensions of the efficiency issue although multiplication (division) and addition (subtraction) with different lengths are different. The computational efficiency of the 2S-OP-GI algorithm and the KT-2S-GI algorithm is shown in Tables 1 and 2, where n 1 := n a + n c + n b m and n 2 := n a + n c + m + n b − 1. By comparing the calculation amount of the two algorithms, we have Total flops N 1 := (4k + 2)n 1 L + 2L(n b m + k) + n 1 k + k Total flops We can clearly see that the calculation amount of the KT-2S-GI algorithm is less than that of the 2S-OP-GI algorithm. For example, when n a = 10, n b = 10, n c = 10, m = 10, L = 1000 and k = 30, we have N 1 − N 2 > 1.0 × 10 7 .

A fractional-order block-oriented nonlinear system
Consider the following nonlinear system, y(l) Figure 1: The fractional-order block-oriented nonlinear system In simulation, the input {u(l)} is taken as a persistent excitation signal sequence with zero mean and unit variance, and {v(l)} is taken as a white noise sequence with zero mean and variance σ 2 . Taking the data length L = 1000, we collect the data u(l) and y(l), and use the 2S-OP-GI algorithm to estimate the parameters of the example system, the parameter estimates and their errors with variances σ 2 = 0.50 2 and σ 2 = 1.00 2 are shown in Tables 3-4. The relative parameter estimation errors δ := ∥θ k − ϑ∥/∥ϑ∥ are shown in Figures 2-3 with different variances.    • Both algorithms can produce mire accurate parameter estimates under lower noise variances.
• With the iterative variable k increasing, the parameter estimation errors given by the 2S-OP-GI and KT-2S-GI algorithms become smaller as shown in Tables 3-6 and Figures 2 [35], as follows: Obviously, the battery model has strong nonlinearity. The corresponding information vector and parameter vector of this example are given by The parameters in the battery model may be influenced by many factors, such as temperature, battery state of charge (SOC), capacity degradation, and others. In the process of the experiment, we added a white noise sequence with zero mean and variance σ 2 . Figure 7 shows the current and voltage, and use the 2S-OP-GI algorithm to estimate parameters of the example system. From Table 7 and Figures 8-11, we can draw the following conclusions.
• Figure 8 shows a better the voltage fitting effect, and the voltage fitting error is shown in Figure 9.
• It can be seen from Figure 10 that the power of the battery has a better fitting effect and the corresponding error is small.
• The parameter estimates and their errors with variance σ 2 = 0.10 2 are shown in Table 7 and Figure 11. With the iterative variable k increasing, the parameter estimation errors become smaller.

Conclusions
This paper studies two algorithms, namely a 2S-OP-GI algorithm and a KT-2S-GI algorithm, for identifying the fractional-order block-oriented nonlinear systems based on the hierarchical identification principle. The results of example 1 show that both algorithms are effective. In terms of the computational analysis, the dimension of the matrix involved in the KT-2S-GI algorithm is smaller than that of the 2S-OP-GI algorithm, so the computational efficiency of the KT-2S-GI algorithm is higher. In example 2, a good model fitting is obtained by applying the battery, which further verified the effectiveness of the proposed method. Further research will focus on exploring new identification methods for fractional-order nonlinear systems in combination with other techniques and strategies.

Data Availability Statement
All data generated or analyzed during this study are included in this article.