Parameter Estimation of Fractional Wiener Systems with the Application of Photovoltaic Cell Models

Abstract

Fractional differential equations are used to construct mathematical models and can describe the characteristics of real systems. In this paper, the parameter estimation problem of a fractional Wiener system is studied by designing linear filters which can obtain smaller tunable parameters and maintain the stability of the parameters in any case. To improve the identification performance of the stochastic gradient algorithm, this paper derives two modified stochastic gradient algorithms for the fractional nonlinear Wiener systems with colored noise. By introducing the forgetting factor, a forgetting factor stochastic gradient algorithm is deduced to improve the convergence rate. To achieve more efficient and accurate algorithms, we propose a multi-innovation forgetting factor stochastic gradient algorithm by means of the multi-innovation theory, which expands the scalar innovation into the innovation vector. To test the developed algorithms, a fractional-order dynamic photovoltaic model is employed in the simulation, and the dynamic elements of this photovoltaic model are estimated using the modified algorithms. Concurrently, a numerical example is given, and the simulation results verify the feasibility and effectiveness of the proposed procedures.

1. Introduction

The parameter estimation of mathematical models is highly important for system analysis and controller design [1,2,3,4], and the quality of models determines the performance of control systems [5,6,7,8]. The majority of interactions in engineering practice are nonlinear, and a linear interaction is only the approximation of a nonlinear interaction under certain conditions. Therefore, the identification of nonlinear systems is still an open research topic for the purpose of improving the control performance of the controlled object. The block-oriented nonlinear model, which consists of nonlinear static blocks and linear dynamic blocks, is used to describe the complex characteristics of industrial processes and is widely recognized [9,10]. Different block combinations form different nonlinear systems, and among them, the most representative nonlinear systems are the Hammerstein system [11,12,13,14,15], the Wiener system, and the Hammerstein–Wiener system.

Wiener systems are typical block-oriented nonlinear systems composed of linear blocks and static nonlinear blocks. In the fields of engineering and science, they have been widely used to model reaction processes or control processes, such as the simulation of chemical processes [16], electrical control [17], and biological processes [18]. Consequently, an accurate model that characterizes the input–output behavior of Wiener systems would be highly useful for the control process. Li et al. developed a recursive maximum likelihood identification algorithm to interactively estimate the unknown parameters and the system states for Wiener nonlinear systems whose linear subsystems were observable state-space models [19]. The modified extended-Kalman-filter-based recursive estimation algorithms were proposed for Wiener nonlinear systems with process noise and measurement noise [20]. The scope of parameter identification research has been expanded using the nonlinear part of the Wiener systems with inverse functions. Ding et al. presented the hierarchical-gradient- and least-squares-based iterative algorithms for input nonlinear output-error systems using the key term separation [21].

The fractional-order systems like the classical integer-order system are also suitable for block-oriented systems [22,23,24,25]. They offer a better representation of the long memory behavior and infinite dimensional structure, and the nonlinear systems can be improved by introducing the fractional order into the block structure, which also increases the difficulty of identification for block-oriented nonlinear systems. Recently, relevant studies on the parameter estimation of fractional-order nonlinear systems have raised concerns. Hammar et al. studied the identification of the fractional Hammerstein systems using the Levenberg–Marquardt algorithm, which was combined with the over-parametrization principle and the key term separation principle. An output error approach was developed using the robust Levenberg–Marquardt algorithm to identify the parameters of fractional-order nonlinear systems based on the Hammerstein–Wiener models [26]. In some identification research, the order is assumed to be known and is not identified, which leads to the incompleteness of the parameter identification. Numerous experiments have shown that it is essential to use fractional differential equations instead of integer differential equations. Using fractional nonrecursive filters to construct a mathematical model, smaller tunable parameters can be obtained and remain stable at any coefficient values [27].

As the available data become increasingly larger, the effectiveness of the second-order Levenberg–Marquardt identification algorithm decreases. In contrast, simple gradient algorithms are usually more practical. The gradient identification method is to search for the parameter estimation along the negative gradient direction of the criterion function and implement it in the form of recursive or iterative algorithms. Stochastic gradient algorithms can be applied to obtain the optimal solutions of nonlinear problems, which requires less computation than least squares methods [28]. In order to obtain better parameter estimation accuracy, researchers adopt different methods to optimize the stochastic gradient algorithm. Using the hierarchical identification principle, Ji et al. decomposed a class of nonlinear systems into three subsystems with fewer variables and interactively identified each subsystem [29]. The computational load of the algorithm was reduced, which sped up its operating efficiency. Innovation is a useful tool that can improve the estimation performance. The multi-innovation identification theory extends the scalar innovation to the innovation vector or the innovation vector to the innovation matrix, which effectively improves parameter estimation accuracy [30].

The recursive methods can estimate the parameters of the systems in real time and are suitable for online identification through capturing the real-time information regarding actual production processes [31,32,33,34]. In contrast, the iterative methods update the parameters of the systems using a batch of measurement data and are suitable for off-line identification [35,36,37,38,39]. This paper studies the parameter identification of the fractional-order Wiener systems and improves the performance of the stochastic gradient algorithm. The main contributions are as follows:

• For the fractional nonlinear Wiener systems with colored noise, by applying the gradient search method and introducing the forgetting factor, a forgetting factor stochastic gradient algorithm is derived, which has a higher convergence rate than the stochastic gradient algorithm.
• Based on the multi-innovation identification theory, a multi-innovation forgetting factor stochastic gradient algorithm is proposed to estimate the parameters, and the accuracy of parameter estimation is improved.
• The ability of the proposed algorithms to estimate the parameters is verified with a numerical example, and they are applied to a fractional model of photovoltaic cells. The results are obtained through the process of parameter variation and the fitting of the model outputs.

The rest of this paper is organized as follows. Section 2 introduces the mathematical background of the fractional-order calculus and describes the identification model of the fractional-order nonlinear systems with colored noise. Section 3 presents a forgetting factor stochastic gradient identification algorithm, and a multi-innovation forgetting factor stochastic gradient algorithm is proposed in Section 4. Section 5 offers a numerical example to illustrate the effectiveness of the proposed algorithms and their applications to the dynamic photovoltaic model. Finally, Section 6 offers some concluding remarks.

2. System Description and Identification Model

Let us introduce some symbols. $A=:X$ or $X:=A$ indicates that X is defined by A. The symbol ${\mathit{I}}_n$ stands for an identity matrix of size $n\times n$. Here, ${\mathbf{1}}_n$ represents an n-dimensional column vector whose elements are all 1, that is, ${\mathbf{1}}_n:={[1,1,\cdots ,1]}^{\mathrm{T}}\in {\mathbb{R}}^n$.

With the development of fractional-order calculus theory, it is worth researching its application in system modeling and control. As a generalization of classical calculus, there are three commonly used definitions for fractional calculus: the Riemann–Liouville definition, Caputo definition, and Grünwald–Letnikov definition. According to the literature, different definitions of the different integral operators have been proposed, and the most used definition for the discrete case is the Grünwald–Letnikov definition. The Grünwald–Letnikov definition is given as follows.

The Grünwald–Letnikov fractional derivative formula for $\alpha$ is defined as:

$${\Delta }^{\alpha }f\left(t\right)=\frac{1}{h^{\alpha }}\sum _{j=0}^{[t/h]}{(-1)}^j\left(\genfrac{}{}{0pt}{}{\alpha }{j}\right)f(t-jh),$$

where ${\Delta }^{\alpha }$ denotes the fractional-order difference operator of order $\alpha$, $f\left(t\right)$ is a function of time t, h is the sampling interval, and the Newton’s binomial $\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{j}}\right)$ is generalized to non-integer orders using Euler’s function as:

$$\begin{array}{ccc}\hfill \left(\genfrac{}{}{0pt}{}{\alpha }{j}\right)& =& \frac{\Gamma (\alpha +1)}{\Gamma (j+1)\Gamma (\alpha -j+1)}.\hfill \end{array}$$

The Euler’s Gamma function $\Gamma (·)$ is denoted as:

$$\Gamma \left(n\right)={\int }_0^{\infty }{e}^{-t}{t}^{n-1}dt.$$

The Wiener system consists of a linear block followed by a static nonlinear part. Considering the Wiener system shown in Figure 1, the linear block is a fractional linear part. Thus, the system considered in this paper will be described using the following mathematical equations:

$$\begin{array}{cc}\hfill m\left(t\right)=& u^{*}\left(t\right)+\frac{1}{H\left(z\right)}v\left(t\right),\hfill \\ \hfill y\left(t\right)=& f\left(m\right(t\left)\right),\hfill \end{array}$$

where $u\left(t\right)\in \mathbb{R}$ is the input of the system, $y\left(t\right)\in \mathbb{R}$ is the output of the system, $v\left(t\right)\in \mathbb{R}$ is a stochastic white noise with zero mean and variance ${\sigma }^2$, and $G\left(z\right)$ and $H\left(z\right)$ are the fractional-order polynomials in the backward shift operator $z^{-{\alpha }_i}$, as follows:

$$\begin{array}{cc}\hfill G\left(z\right):=& g_1{z}^{-{\overline{\alpha }}_1}+g_2{z}^{-{\overline{\alpha }}_2}+\cdots +g_{n_g}{z}^{-{\overline{\alpha }}_{n_g}},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill H\left(z\right):=& 1+h_1{z}^{-{\beta }_1}+h_2{z}^{-{\beta }_2}+\cdots +h_{n_h}{z}^{-{\beta }_{n_h}},\hfill \end{array}$$

(2)

The variable $u^{*}\left(t\right)$ is the output of the linear block, i.e., the input of the nonlinear block, and $w\left(t\right)$ and $m\left(t\right)$ are unmeasurable intermediate variables. In order to deal with these variables, we define:

$$\begin{array}{cc}\hfill u^{*}\left(t\right):=& G\left(z\right)u\left(t\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill w\left(t\right):=& \frac{1}{H\left(z\right)}v\left(t\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill m\left(t\right):=& u^{*}\left(t\right)+w\left(t\right).\hfill \end{array}$$

(5)

Rewriting Equations (3)–(5) using the linear filter with fractional difference, we obtain:

$$\begin{array}{cc}\hfill u^{*}\left(t\right)=& \sum _{i=1}^{n_g}{g}_i{\Delta }^{{\overline{\alpha }}_i}u\left(t\right),\hfill \\ \hfill w\left(t\right)=& [1-H\left(z\right)]w\left(t\right)+v\left(t\right)=-\sum _{j=1}^{n_h}{h}_j{\Delta }^{{\beta }_j}w\left(t\right)+v\left(t\right),\hfill \\ \hfill m\left(t\right)=& G\left(z\right)u\left(t\right)+w\left(t\right)=\sum _{i=1}^{n_g}{g}_i{\Delta }^{{\overline{\alpha }}_i}u\left(t\right)-\sum _{j=1}^{n_h}{h}_j{\Delta }^{{\beta }_j}w\left(t\right)+v\left(t\right).\hfill \end{array}$$

(6)

Equation (6) is called a non-commensurate-order linear part in which the fractional orders are completely different; otherwise, when the latter are multiples of the same basis, $\alpha (0<\alpha <1)$ with $({\overline{\alpha }}_i=i\alpha$ and ${\beta }_j=j\alpha )$, the models are of commensurate order. In this paper, we take the fractional commensurate order system. The recurrence equations in (3)–(5) can be expressed as:

$$\begin{array}{cc}\hfill u^{*}\left(t\right)=& \sum _{i=1}^{n_g}{g}_i{\Delta }^{{\alpha }_i}u\left(t\right),\hfill \\ \hfill w\left(t\right)=& [1-H\left(z\right)]w\left(t\right)+v\left(t\right)=-\sum _{j=1}^{n_h}{h}_j{\Delta }^{{\alpha }_j}w\left(t\right)+v\left(t\right),\hfill \\ \hfill m\left(t\right)=& G\left(z\right)u\left(t\right)+w\left(t\right)=\sum _{i=1}^{n_g}{g}_i{\Delta }^{{\alpha }_i}u\left(t\right)-\sum _{j=1}^{n_h}{h}_j{\Delta }^{{\alpha }_j}w\left(t\right)+v\left(t\right).\hfill \end{array}$$

Using the discrete fractional-order operator $\Delta$ in the time domain and according to the properties of the unit backward shift operator $z^{-1}$ (i.e., $z^{-1}x\left(t\right)=x(t-1)$), we obtain:

$$\begin{array}{cc}\hfill u^{*}\left(t\right)=& \sum _{i=1}^{n_g}{g}_i{\Delta }^{\alpha }u(t-i),\hfill \\ \hfill w\left(t\right)=& -\sum _{i=1}^{n_h}{h}_i{\Delta }^{\alpha }w(t-i)+v\left(t\right),\hfill \\ \hfill m\left(t\right)=& \sum _{i=1}^{n_g}{g}_i{\Delta }^{\alpha }u(t-i)-\sum _{j=1}^{n_h}{h}_j{\Delta }^{\alpha }w(t-j)+v\left(t\right).\hfill \end{array}$$

(7)

Assume that the static nonlinear function $f(·)$ is considered to be invertible and the inverse $f^{-1}(·)$ can be described as a linear combination of basis functions in the following form:

$$m\left(t\right):=f^{-1}\left(y\left(t\right)\right)=\sum _{k=1}^{n_c}{c}_k{f}_k\left(y\left(t\right)\right),$$

(8)

where the basis functions $f_k(·)$ are known nonlinear functions, and the unknown parameters $c_k$$(k=1,2,\cdots ,n_c)$ are the coefficients of the nonlinear functions. Substituting (8) into (7) yields:

$$\sum _{k=1}^{n_c}{c}_k{f}_k\left(y\left(t\right)\right)=\sum _{i=1}^{n_g}{g}_i{\Delta }^{\alpha }u(t-i)-\sum _{j=1}^{n_h}{h}_j{\Delta }^{\alpha }w(t-j)+v\left(t\right),$$

or

$$f_1\left(y\left(t\right)\right)=\frac{1}{c_1}\left[\sum _{i=1}^{n_g}{g}_i{\Delta }^{\alpha }u(t-i)-\sum _{j=1}^{n_h}{h}_j{\Delta }^{\alpha }w(t-j)-\sum _{k=2}^{n_c}{c}_k{f}_k\left(y\left(t\right)\right)+v\left(t\right)\right].$$

(9)

In order to ensure the uniqueness of the system in (9), one parameter in one block has to be fixed in the mathematical model. Without a loss of generality, the coefficient $c_1$ of the nonlinear block polynomial can be normalized and set equal to 1. We define the parameter vectors as follows:

$$\begin{array}{cc}\hfill \mathit{g}:=& {[g_1,g_2,\cdots ,g_{n_g}]}^{\mathrm{T}}\in {\mathbb{R}}^{n_g},\hfill \\ \hfill \mathit{h}:=& {[h_1,h_2,\cdots ,h_{n_h}]}^{\mathrm{T}}\in {\mathbb{R}}^{n_h},\hfill \\ \hfill \mathit{c}:=& {[c_2,c_3,\cdots ,c_{n_c}]}^{\mathrm{T}}\in {\mathbb{R}}^{n_c-1},\hfill \\ \hfill \mathit{\vartheta }:=& \left[\begin{array}{c}\mathit{g}\\ \mathit{h}\\ \mathit{c}\end{array}\right]\in {\mathbb{R}}^{n_g+n_h+n_c-1},\hfill \\ \hfill \mathit{\theta }:=& \left[\begin{array}{c}\mathit{\vartheta }\\ \alpha \end{array}\right]\in {\mathbb{R}}^n,n=n_g+n_h+n_c,\hfill \end{array}$$

and the information vectors as follows:

$$\begin{array}{cc}\hfill {\mathit{\phi }}_1(t,\alpha ):=& {[{▵}^{\alpha }u(t-1),{▵}^{\alpha }u(t-2),\cdots ,{▵}^{\alpha }u(t-n_g)]}^{\mathrm{T}}\in {\mathbb{R}}^{n_g},\hfill \\ \hfill {\mathit{\phi }}_2(t,\alpha ):=& {[-{▵}^{\alpha }w(t-1),-{▵}^{\alpha }w(t-2),\cdots ,-{▵}^{\alpha }w(t-n_h)]}^{\mathrm{T}}\in {\mathbb{R}}^{n_h},\hfill \\ \hfill {\mathit{\phi }}_3(t,\alpha ):=& {[-f_2\left(y\left(t\right)\right),-f_2\left(y\left(t\right)\right),\cdots ,-f_{n_c}\left(y\left(t\right)\right)]}^{\mathrm{T}}\in {\mathbb{R}}^{n_c-1},\hfill \\ \hfill \mathit{\phi }(t,\alpha ):=& {[{\mathit{\phi }}_1^{\mathrm{T}}(t,\alpha ),{\mathit{\phi }}_2^{\mathrm{T}}(t,\alpha ),{\mathit{\phi }}_3^{\mathrm{T}}(t,\alpha )]}^{\mathrm{T}}\in {\mathbb{R}}^{n_g+n_h+n_c-1}.\hfill \end{array}$$

Equation (16) can be written as:

$$\begin{array}{cc}\hfill f_1\left(y\left(t\right)\right)=& {\mathit{\phi }}_1^{\mathrm{T}}(t,\alpha )\mathit{g}+{\mathit{\phi }}_2^{\mathrm{T}}(t,\alpha )\mathit{h}+{\mathit{\phi }}_3^{\mathrm{T}}(t,\alpha )\mathit{c}+v\left(t\right)\hfill \\ \hfill =& {\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }+v\left(t\right).\hfill \end{array}$$

(10)

The parameter estimation algorithms proposed in this paper are based on the identification model in (10). Many identification methods are derived based on the identification models of such systems [40,41,42,43,44,45], and these methods can be used to estimate the parameters of other linear systems and nonlinear systems [46,47,48,49,50,51] and can be applied to other fields [52,53,54,55,56,57], such as communication and transportation systems. The linear identification model in (10) is established for the fractional Wiener system in Figure 1, and the follow-up work is carried out in regard to this identification model. The purpose of this paper is to develop new recursive algorithms for identifying the parameters and order of fractional models using the measured input–output data. Thus, we provide a detailed derivation of the algorithms in the following sections.

3. The Forgetting Factor Stochastic Gradient Algorithm

For parameter identification, online identification can realize the parameter updated using the latest input–output data point. As a classical algorithm, the stochastic gradient (SG) algorithm has a simple process of recursive operation and low computational complexity. However, the SG algorithm has a slow convergence rate. In this section, we introduce the forgetting factor to improve the performance of the SG algorithm and derive the forgetting factor stochastic gradient (FFSG) algorithm to estimate the parameter vector $\mathit{\theta }$.

We define a criterion function as:

$$J_1\left(\mathit{\theta }\right)=J_1(\mathit{\vartheta },\alpha ):=\frac{1}{2}{[f_1\left(y\left(t\right)\right)-{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }]}^2.$$

The gradient vector of $J_1\left(\mathit{\theta }\right)$, with respect to $\mathit{\theta }$, is:

$$\mathrm{grad}\left[J_1\left(\mathit{\theta }\right)\right]:=\left[\begin{array}{c}\frac{\partial J(\mathit{\vartheta },\alpha )}{\partial \mathit{\vartheta }}\\ \frac{\partial J(\mathit{\vartheta },\alpha )}{\partial \alpha }\end{array}\right]=-\left[\begin{array}{c}\mathit{\phi }(t,\alpha )\\ \frac{\partial \left[{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }\right]}{\partial \alpha }\end{array}\right][f_1\left(y\left(t\right)\right)-{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }].$$

We define the information vector $\psi (t,\alpha )$ and the innovation $e\left(t\right)$ as:

$$\begin{array}{cc}\hfill \mathit{\psi }(t,\alpha ):=& \left[\begin{array}{c}\mathit{\phi }(t,\alpha )\\ \frac{\partial \left[{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }\right]}{\partial \alpha }\end{array}\right]\in {\mathbb{R}}^n,\hfill \\ \hfill e\left(t\right):=& f_1\left(y\left(t\right)\right)-{\mathit{\phi }}^{\mathrm{T}}(t,\alpha ){\widehat{\mathit{\vartheta }}}_{t-1}.\hfill \end{array}$$

Using the negative gradient search to minimize $J_1\left(\mathit{\theta }\right)$, we can obtain:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_1\left(t\right)}\left[\begin{array}{c}{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\\ \frac{\partial \left[{\mathit{\phi }}^{\mathrm{T}}(t,\alpha )\mathit{\vartheta }\right]}{\partial \alpha }\end{array}\right][f_1\left(y\left(t\right)\right)-{\mathit{\phi }}^{\mathrm{T}}(t,\alpha ){\widehat{\mathit{\vartheta }}}_{t-1}]\hfill \end{array}$$

$$\begin{array}{ccc}& =& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_1\left(t\right)}\mathit{\psi }(t,\alpha )e\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill r_1\left(t\right)=& r(t-1)+{\parallel \mathit{\psi }(t,\alpha )\parallel }^2,r_1\left(0\right)=1.\hfill \end{array}$$

(12)

Let $\widehat{\mathit{\vartheta }}\left(t\right)=\left[\begin{array}{c}\widehat{\mathit{g}}\left(t\right)\\ \widehat{\mathit{h}}\left(t\right)\\ \widehat{\mathit{c}}\left(t\right)\end{array}\right]$, $\widehat{\mathit{\theta }}\left(t\right)=\left[\begin{array}{c}\widehat{\mathit{\vartheta }}\left(t\right)\\ \widehat{\alpha }\left(t\right)\end{array}\right]$, and $\widehat{\alpha }\left(t\right)$ represent the estimates of $\mathit{\vartheta }$, $\mathit{\theta }$, and $\alpha$ at time t and $\widehat{\mathit{\vartheta }}\left(t\right)={\widehat{\mathit{\vartheta }}}_t$. Note that the information vector $\psi (t,\alpha )$ and the innovation $e\left(t\right)$ in the above equations contain the unknown variables $\alpha$ and $w\left(t\right)$, so that the recursive computing process cannot proceed as such. The auxiliary model identification idea is an effective method to solve those problems. The unknown parameter $\alpha$ is replaced with its estimate $\widehat{\alpha }\left(t\right)$ and the unknown terms $w(t-i)$$(i=1,2,\cdots ,n_h)$ are replaced with the estimates $\widehat{w}(t-i)$. We can obtain:

$$\begin{array}{cc}\hfill \widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))=& \left[\begin{array}{c}\widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))\\ \frac{\partial \left[{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{\vartheta }}\left(t\right)\right]}{\partial \alpha }\end{array}\right],\hfill \\ \hfill \widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))=& {[{\widehat{\mathit{\phi }}}_1^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_3^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))]}^{\mathrm{T}},\hfill \\ \hfill {\widehat{\mathit{\phi }}}_2(t,\widehat{\alpha }\left(t\right))=& {[-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-1),-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-2),\cdots ,-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-n_h)]}^{\mathrm{T}},\hfill \\ \hfill \widehat{w}\left(t\right)=& {\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{h}}\left(t\right)+\widehat{v}\left(t\right),\hfill \\ \hfill \widehat{v}\left(t\right)=& f_1\left(y\left(t\right)\right)-{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1},\hfill \\ \hfill \widehat{e}\left(t\right)=& f_1\left(y\left(t\right)\right)-{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1}.\hfill \end{array}$$

Replacing $\mathit{\psi }(t,\alpha )$ and $e\left(t\right)$ in (11) and (12) with the estimates $\widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))$ and $\widehat{e}\left(t\right)$, the SG algorithm is given as:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_1\left(t\right)}\widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))\widehat{e}\left(t\right),\hfill \\ \hfill r_1\left(t\right)=& r_1(t-1)+{\parallel \widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))\parallel }^2,r_1\left(0\right)=1.\hfill \end{array}$$

By introducing a forgetting factor $\lambda$ to achieve the performance improvement of the SG algorithm, we obtain the FFSG algorithm for the fractional Wiener models:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_1\left(t\right)}\widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))\widehat{e}\left(t\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill r_1\left(t\right)=& \lambda r_1(t-1)+{\parallel \widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))\parallel }^2,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))=& \left[\begin{array}{c}\widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))\\ \frac{\partial \left[{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{\vartheta }}\left(t\right)\right]}{\partial \alpha }\end{array}\right],\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \widehat{e}\left(t\right)=& f_1\left(y\left(t\right)\right)-{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))=& {[{\widehat{\mathit{\phi }}}_1^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_3^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))]}^{\mathrm{T}},\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\widehat{\mathit{\phi }}}_1(t,\widehat{\alpha }\left(t\right))=& {[{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-1),{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-2),\cdots ,{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-n_g)]}^{\mathrm{T}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\widehat{\mathit{\phi }}}_2(t,\widehat{\alpha }\left(t\right))=& {[-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-1),-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-2),\cdots ,-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-n_h)]}^{\mathrm{T}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathit{\phi }}_3(t,\widehat{\alpha }\left(t\right))=& {[-f_2\left(y\left(t\right)\right),-f_2\left(y\left(t\right)\right),\cdots ,-f_{n_c}\left(y\left(t\right)\right)]}^{\mathrm{T}},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \widehat{w}\left(t\right)=& {\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{h}}\left(t\right)+\widehat{v}\left(t\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \widehat{v}\left(t\right)=& f_1\left(y\left(t\right)\right)-{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& {[{\widehat{\mathit{\vartheta }}}^{\mathrm{T}}\left(t\right),\widehat{\alpha }\left(t\right)]}^{\mathrm{T}},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \widehat{\mathit{\vartheta }}\left(t\right)=& {[{\widehat{\mathit{g}}}^{\mathrm{T}}\left(t\right),{\widehat{\mathit{h}}}^{\mathrm{T}}\left(t\right),{\widehat{\mathit{c}}}^{\mathrm{T}}\left(t\right)]}^{\mathrm{T}}.\hfill \end{array}$$

(24)

The model parameter estimation methods proposed in this paper can combine certain filtering and estimation techniques [58,59,60,61,62,63,64,65] to study the parameter identification problems of linear and nonlinear systems with different disturbances [66,67,68,69,70,71,72,73] and can be applied to other fields [74,75,76,77,78,79,80,81], such as signal processing and engineering application systems. The pseudo-code for implementing the FFSG algorithm in (13)–(24) for the fractional Wiener models is shown in Algorithm 1.

Algorithm 1 The pseudo-code for implementing the FFSG algorithm.

Require:  $\left\{u\right(t),y(t):t=1,2,\cdots ,L\}$, $\{f_j(·):j=2,3,\cdots ,n_c\}$, $\epsilon$, n, $n_0>max\{n_g,n_h,n_c\}$ Ensure:  $\widehat{\mathit{\theta }}\left(t\right)$  1: Initialization: $\widehat{\mathit{\theta }}\left(0\right)={\mathbf{1}}_n/p_0$, $r_1\left(0\right)=1$, $\widehat{w}\left(i\right)=1/p_0$ for $i\le$0, where ${\mathbf{1}}_n$ denotes an n-dimensional column vector whose elements are all 1 and $p_0$ is a large number, e.g., $p_0={10}^6$.  2: for $t=n_0:L$  do  3:     Form ${\widehat{\mathit{\phi }}}_1(t,\widehat{\alpha }\left(t\right))$, ${\widehat{\mathit{\phi }}}_2(t,\widehat{\alpha }\left(t\right))$ and ${\mathit{\phi }}_3(t,\widehat{\alpha }\left(t\right))$ by (18)–(20), and $\widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))$ by (17);  4:     Compute $\frac{\partial \left[{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{\vartheta }}\left(t\right)\right]}{\partial \alpha }$, form $\widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))$ by (15) and $r_1\left(t\right)$ by (14);  5:     Compute $\widehat{e}\left(t\right)$ by (16);  6:     Update the parameter estimation vector $\widehat{\mathit{\theta }}\left(t\right)$ by (13);  7:     Compute $\widehat{v}\left(t\right)$ by (22) and $\widehat{w}\left(t\right)$ by (21);  8:     if $\mathbf{then}\parallel \widehat{\mathit{\theta }}\left(t\right)-\widehat{\mathit{\theta }}(t-1)\parallel >\epsilon$  9:         $t=t+1$, go to Step 3; 10:    else 11:        Obtain the parameter estimation vector $\widehat{\mathit{\theta }}\left(t\right)$, break; 12:    end if 13: end for

4. The Multi-Innovation Forgetting Factor Stochastic Gradient Algorithm

In order to improve the convergence rate and parameter estimation accuracy of the FFSG algorithm in (13)–(24), according to the multi-innovation identification theory, we expand the innovation $e\left(t\right)$ to the innovation vector as:

$$\mathit{E}(p,t):=\left[\begin{array}{c}e\left(t\right)\\ e(t-1)\\ ⋮\\ e(t-p+1)\end{array}\right]=\left[\begin{array}{c}{f}_1\left(y\left(t\right)\right)-{\mathit{\phi }}^{\mathrm{T}}(t-1,\alpha ){\widehat{\mathit{\vartheta }}}_{t-1}\\ f_1\left(y(t-1)\right)-{\mathit{\phi }}^{\mathrm{T}}(t-2,\alpha ){\widehat{\mathit{\vartheta }}}_{t-1}\\ ⋮\\ f_1\left(y(t-p+1)\right)-{\mathit{\phi }}^{\mathrm{T}}(t-p+1,\alpha ){\widehat{\mathit{\vartheta }}}_{t-1}\end{array}\right]\in {\mathbb{R}}^p,$$

(25)

where p denotes the innovation length.

Define the stacked information matrices $\mathit{\Phi }(p,t,\alpha )$ and $\mathit{\Psi }(p,t,\alpha )$ and the stacked vector $\mathit{F}(p,t)$ as:

$$\begin{array}{cc}\hfill \mathit{\Phi }(p,t,\alpha ):=& [\mathit{\psi }(t,\alpha ),\mathit{\psi }(t-1,\alpha ),\cdots ,\mathit{\psi }(t-p+1,\alpha )]\in {\mathbb{R}}^{n\times p},\hfill \\ \hfill \mathit{\Psi }(p,t,\alpha ):=& [\mathit{\phi }(t,\alpha ),\mathit{\phi }(t-1,\alpha ),\cdots ,\mathit{\phi }(t-p+1,\alpha )]\in {\mathbb{R}}^{(n_g+n_h+n_c-1)\times p},\hfill \\ \hfill \mathit{F}(p,t):=& {[f_1\left(y\left(t\right)\right),f_1\left(y(t-1)\right),\cdots ,f_1\left(y(t-p+1)\right)]}^{\mathrm{T}}\in {\mathbb{R}}^p.\hfill \end{array}$$

Equation (25) can be expressed as:

$$\mathit{E}(p,t)=\mathit{F}(p,t)-{\mathit{\Psi }}^{\mathrm{T}}(p,t,\alpha )\mathit{\vartheta }.$$

By minimizing the criterion function:

$$J_2\left(\mathit{\theta }\right):=\frac{1}{2}{\parallel \mathit{F}(p,t)-{\mathit{\Psi }}^{\mathrm{T}}(p,t,\alpha )\mathit{\vartheta }\parallel }^2,$$

Equations (11) and (12) can be rewritten as:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_2\left(t\right)}\mathit{\Phi }(p,t,\alpha )\mathit{E}(p,t),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill r_2\left(t\right)=& r_2(t-1)+{\parallel \mathit{\Phi }(p,t,\alpha )\parallel }^2,r_2\left(0\right)=1.\hfill \end{array}$$

(27)

Similarly, this recursion formula in (26) and (27) cannot be realized for the reason that there are unknown internal variables, $\alpha$ and $w\left(t\right)$. In order to complete the parameter estimation algorithm, the unknown variables are also replaced with their estimates $\widehat{\alpha }\left(t\right)$ and $\widehat{w}\left(t\right)$ using the auxiliary model identification idea, and we can obtain the multi-innovation forgetting factor stochastic gradient (MI-FFSG) algorithm for the parameter estimation:

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& \widehat{\mathit{\theta }}(t-1)+\frac{1}{r_2\left(t\right)}\widehat{\mathit{\Phi }}(p,t,\widehat{\alpha }\left(t\right))\widehat{\mathit{E}}(p,t),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill r_2\left(t\right)=& \lambda r_2(t-1)+{\parallel \widehat{\mathit{\Phi }}(p,t,\widehat{\alpha }\left(t\right))\parallel }^2,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \widehat{\mathit{\Phi }}(p,t,\widehat{\alpha }\left(t\right))=& [\widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right)),\widehat{\mathit{\psi }}(t-1,\widehat{\alpha }\left(t\right)),\cdots ,\widehat{\mathit{\psi }}(t-p+1,\widehat{\alpha }\left(t\right))],\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \widehat{\mathit{E}}(p,t)=& \mathit{F}(p,t)-{\widehat{\mathit{\Psi }}}^{\mathrm{T}}(p,t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \widehat{\mathit{\psi }}(t,\widehat{\alpha }\left(t\right))=& \left[\begin{array}{c}\widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))\\ \frac{\partial \left[{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{\vartheta }}\left(t\right)\right]}{\partial \alpha }\end{array}\right],\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \widehat{\mathit{\Psi }}(p,t,\alpha \left(t\right))=& [\widehat{\mathit{\phi }}(t,\alpha \left(t\right)),\widehat{\mathit{\phi }}(t-1,\alpha \left(t\right)),\cdots ,\widehat{\mathit{\phi }}(t-p+1,\alpha \left(t\right))],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \mathit{F}(p,t)=& {[f_1\left(y\left(t\right)\right),f_1\left(y(t-1)\right),\cdots ,f_1\left(y(t-p+1)\right)]}^{\mathrm{T}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))=& {[{\widehat{\mathit{\phi }}}_1^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)),{\widehat{\mathit{\phi }}}_3^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))]}^{\mathrm{T}},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\widehat{\mathit{\phi }}}_1(t,\widehat{\alpha }\left(t\right))=& {[{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-1),{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-2),\cdots ,{▵}^{\widehat{\alpha }\left(t\right)}\widehat{u}(t-n_g)]}^{\mathrm{T}},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\widehat{\mathit{\phi }}}_2(t,\widehat{\alpha }\left(t\right))=& {[-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-1),-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-2),\cdots ,-{▵}^{\widehat{\alpha }\left(t\right)}\widehat{w}(t-n_h)]}^{\mathrm{T}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\mathit{\phi }}_3(t,\widehat{\alpha }\left(t\right))=& {[-f_2\left(y\left(t\right)\right),-f_2\left(y\left(t\right)\right),\cdots ,-f_{n_c}\left(y\left(t\right)\right)]}^{\mathrm{T}},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill \widehat{w}\left(t\right)=& {\widehat{\mathit{\phi }}}_2^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right))\widehat{\mathit{h}}\left(t\right)+\widehat{v}\left(t\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \widehat{v}\left(t\right)=& f_1\left(y\left(t\right)\right)-{\widehat{\mathit{\phi }}}^{\mathrm{T}}(t,\widehat{\alpha }\left(t\right)){\widehat{\mathit{\vartheta }}}_{t-1},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \widehat{\mathit{\theta }}\left(t\right)=& {[{\widehat{\mathit{\vartheta }}}^{\mathrm{T}}\left(t\right),\widehat{\alpha }\left(t\right)]}^{\mathrm{T}},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \widehat{\mathit{\vartheta }}\left(t\right)=& {[{\widehat{\mathit{g}}}^{\mathrm{T}}\left(t\right),{\widehat{\mathit{h}}}^{\mathrm{T}}\left(t\right),{\widehat{\mathit{c}}}^{\mathrm{T}}\left(t\right)]}^{\mathrm{T}}.\hfill \end{array}$$

(42)

The pseudo-code for implementing the MI-FFSG algorithm in (28)–(42) for fractional Wiener models is shown in Algorithm 2.

Algorithm 2 The pseudo-code for implementing the MI-FFSG algorithm.

Require:  $\left\{u\right(t),y(t):t=1,2,\cdots ,L\}$, $\{f_j(·):j=2,3,\cdots ,n_c\}$, $\epsilon$, n, p, $n_0\ge max\{n_g+p,n_h+p,n_c+p\}$ Ensure:  $\widehat{\mathit{\theta }}\left(t\right)$  1: Initialization: $\widehat{\mathit{\theta }}\left(0\right)={\mathbf{1}}_n/p_0$, $r_2\left(0\right)=1$, $\widehat{w}\left(i\right)=1/p_0$ for $i\le$0, where ${\mathbf{1}}_n$ denotes an n-dimensional column vector whose elements are all 1, and $p_0$ is a large number, e.g., $p_0={10}^6$.  2: for $t=n_0:L$  do  3:     Form $\mathit{F}(p,t)$ by (34);  4:     Form ${\widehat{\mathit{\phi }}}_1(t,\widehat{\alpha }\left(t\right))$, ${\widehat{\mathit{\phi }}}_2(t,\widehat{\alpha }\left(t\right))$ and ${\mathit{\phi }}_3(t,\widehat{\alpha }\left(t\right))$ by (36)–(38), $\widehat{\mathit{\phi }}(t,\widehat{\alpha }\left(t\right))$ by (35) and $\widehat{\mathit{\Psi }}(p,t,\alpha \left(t\right))$ by (33);  5:     Compute $\widehat{\mathit{E}}(p,t)$ by (31) and $r_2\left(t\right)$ by (29);  6:     Update the parameter estimation vector $\widehat{\mathit{\theta }}\left(t\right)$ by (28);  7:     Compute $\widehat{v}\left(t\right)$ by (40) and $\widehat{w}\left(t\right)$ by (39);  8:      if $\mathbf{then}\parallel \widehat{\mathit{\theta }}\left(t\right)-\widehat{\mathit{\theta }}(t-1)\parallel >\epsilon$  9:         $t=t+1$, go to Step 3; 10:    else 11:        Obtain the parameter estimation vector $\widehat{\mathit{\theta }}\left(t\right)$, break; 12:    end if 13: end for

5. Simulation Examples

This section investigates the estimation performance of the developed algorithms. In the first part, a numerical example is presented to illustrate the feasibility of the FFSG algorithm and the MI-FFSG algorithm. In the second part, the algorithms’ performance is assessed on the basis of a dynamic photovoltaic model.

Example 1.

Consider the following fractional Wiener model:

$$\begin{array}{cc}\hfill u^{*}\left(t\right)=& -0.43{▵}^{-0.5}u(t-1)-0.23{▵}^{-0.5}u(t-2),\hfill \\ \hfill w\left(t\right)=& 0.26{▵}^{-0.5}w(t-1)+0.23{▵}^{-0.5}w(t-2)+v\left(t\right),\hfill \\ \hfill m\left(t\right)=& y\left(t\right)-0.8y^2\left(t\right).\hfill \end{array}$$

The parameter vector given by:

$$\begin{array}{ccc}\hfill \mathit{\theta }& =& {[g_1,g_2,h_1,h_2,c_2,\alpha ]}^{\mathrm{T}}={[-0.43,-0.23,0.26,0.23,-0.8,0.5]}^{\mathrm{T}}.\hfill \end{array}$$

In the simulation, the input $\left\{u\right(t\left)\right\}$ is taken as persistent signal sequences with zero mean and unit variance, $\left\{v\right(t\left)\right\}$ is taken as a white noise sequence with zero mean and variance ${\sigma }^2=0.{10}^2$, and the simulation model parameters and the input signal are used to generate the output signal $\left\{y\right(t\left)\right\}$.

Applying the SG, FFSG, and MI-FFSG algorithms to estimate the parameters of this model, the parameter estimates and their errors are presented in

Table 1. The parameter estimates and their estimation errors.

	t	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\alpha }$	$\mathit{\delta }\mathbf{(}\mathbf{\%}\mathbf{)}$
SG	5	−0.79197	−0.20372	0.24820	0.22446	−0.45337	0.58542	45.57930
	10	−0.66692	−0.27068	0.25422	0.23299	−0.59555	0.55514	28.67874
	50	−0.57926	−0.25043	0.25813	0.23730	−0.67221	0.49868	17.69470
	100	−0.54557	−0.25195	0.25839	0.23756	−0.69244	0.50039	14.28345
	400	−0.52051	−0.24888	0.25846	0.23763	−0.70704	0.50413	11.76147
FFSG $\lambda$ = 0.9	5	−0.78798	−0.20820	0.24882	0.22513	−0.45999	0.58565	44.90559
	10	−0.66462	−0.27438	0.25492	0.23384	−0.60189	0.55542	28.21456
	50	−0.57089	−0.25360	0.25910	0.23843	−0.68478	0.48936	16.47133
	100	−0.53013	−0.25472	0.25944	0.23878	−0.70966	0.48852	12.33969
	400	−0.48467	−0.24700	0.25955	0.23889	−0.73630	0.49825	7.70894
FFSG $\lambda$ = 0.8	5	−0.78462	−0.21196	0.24935	0.22570	−0.46555	0.58586	44.34192
	10	−0.66272	−0.27746	0.25551	0.23456	−0.60723	0.55567	27.83422
	50	−0.56379	−0.25627	0.25993	0.23940	−0.69568	0.47999	15.49219
	100	−0.51643	−0.25689	0.26035	0.23983	−0.72515	0.47508	10.78248
	400	−0.44504	−0.24189	0.26052	0.24000	−0.76782	0.48792	3.63496
MI-FFSG $\lambda$ = 0.8 p = 3	5	−0.78327	−0.21347	0.24956	0.22593	−0.46779	0.58595	44.11592
	10	−0.66196	−0.27870	0.25575	0.23484	−0.60937	0.55577	27.68385
	50	−0.44163	−0.24069	0.26104	0.24059	−0.77687	0.50698	2.75356
	100	−0.45951	−0.23909	0.26103	0.24059	−0.76709	0.47331	4.78808
	400	−0.43065	−0.23750	0.26104	0.24059	−0.78265	0.50327	1.96427
True values	0	−0.43000	−0.23000	0.26000	0.23000	−0.80000	0.50000

, and the parameter estimation errors $\delta :=\parallel \widehat{\mathit{\theta }}\left(t\right)-\mathit{\theta }\parallel /\parallel \mathit{\theta }\parallel$ versus t are shown in Figure 2. The estimates for each parameter are shown in Figure 3. Figure 4 displays the estimated outputs, the true process outputs, and their errors.

From Table 1 and Figure 2, Figure 3 and Figure 4, we can draw the following conclusions:

• Figure 2 and Table 1 show that the parameter estimation errors of these three methods decay when t is increased. The parameter estimation accuracy of the FFSG algorithm increases as $\lambda$ decreases, and the MI-FFSG algorithm has a faster convergence rate and can perform more accurate parameter estimation.
• From Figure 3, it can be seen that the parameter estimates of the MI-FFSG algorithm gradually approach their true values as t increases.
• As shown in Figure 4, the predicted outputs of the MI-FFSG algorithm are close to the actual outputs, which shows that the estimates model can reflect the output characteristics of the system.

Example 2.

An accurate equivalent model can achieve a proper photovoltaic (PV) system design by precisely identifying the electrical characteristics of the PV equipment. Several static PV models have been proposed, such as the single-diode model, the double-diode model, and the three-diode model. However, dynamic models are necessary for most theoretical research or simulation purposes. In this subsection, a fractional-order dynamic PV model (FOM) is selected for testing the method’s performance.

The single-diode model is the most commonly utilized static PV model due to its moderate complexity. Based on its equivalent circuit shown in Figure 5a, the output current is calculated as follows [82]:

$$I=I_{ph}-I_0\left[exp\left(\frac{q(V+R_{s}I)}{akT}\right)-1\right]-\frac{V+R_{s}I}{R_{sh}}.$$

In order to omit verbose descriptions, the abbreviations of some terms are given in

Table 2. Abbreviations of some important nomenclatures that appear in Example 2.

Nomenclature
I	Current(A)	T	Cell absolute temperature in Kelvin(K)
V	Voltage(V)	$I_L$	Load current
$I_{ph}$	Photo-generated current(A)	$V_o$	Voltage(V)
$I_0$	Diode saturation current(A)	$C_{\alpha }$	Capacitance in fractional-order dynamic PV model
$R_s$	Series resistance($\mathsf{\Omega }$)	$L_{\beta }$	Inductance in fractional-order dynamic PV model
$R_{sh}$	Shunt resistance($\mathsf{\Omega }$)	$R_c$	Conductance
a	Ideality factor	$R_L$	Load
q	Electronic charge ($1.6\times {10}^{19}$C)	$\alpha$	Derivative order
k	Boltzmann’s constant ($1.38\times {10}^{23}$ J/K)	$\beta$	Derivative order

. For this static model, five parameters ($I_{ph}$, $I_0$, $R_s$, $R_{sh}$, a) need to be identified to model an accurate simulation model of the PV system.

Figure 5. A static PV model and a dynamic PV model. (a) Static single-diode PV model. (b) Fractional-order dynamic PV model.

Figure 5. A static PV model and a dynamic PV model. (a) Static single-diode PV model. (b) Fractional-order dynamic PV model.

Considering the load variation and switching operation of the inverter and DC/DC converter stages, the integral-order dynamic PV model (IOM) was introduced, and then the FOM was employed to reinforce the efficiency as well as the flexibility of the IOM. The considered FOM is a second-order model that takes into account the junction capacitance, conductance, and inductive effects, and the static part of the PV is reduced to a voltage $V_o$ and a series resistance $R_s$, as indicated in Figure 5b. The load current–voltage relationship in FOM can be shown in the s-domain as follows [83]:

$$\begin{array}{cc}\hfill I_L\left(s\right)=& \frac{V_o}{s}\frac{a_{21}(s^{\alpha }-b_1)+b_2(s^{\alpha }-a_{11})}{(s^{\beta }-a_{22})(s^{\alpha }-a_{11})-a_{12}{a}_{21}},\hfill \\ \hfill \left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]=& \left[\begin{array}{cc}\frac{-1}{C_{\alpha }(R_c+R_s)}& \frac{-R_s}{C_{\alpha }(R_c+R_s)}\\ \frac{R_s}{L_{\beta }(R_c+R_s)}& \frac{-[R_L{R}_c+R_s{R}_c+R_L{R}_s]}{L_{\beta }(R_c+R_s)}\end{array}\right],\left[\begin{array}{c}{b}_1\\ b_2\end{array}\right]=\left[\begin{array}{c}\frac{1}{C_{\alpha }(R_s+R_c)}\\ \frac{R_c}{L_{\beta }(R_c+R_s)}\end{array}\right].\hfill \end{array}$$

In this FOM, the unknown parameters that should be estimated are $R_c$, $C_{\alpha }$, $L_{\beta }$, $\alpha$, and $\beta$. Let $\tau =-a_{21}{b}_2$; this function can be rewritten in the time domain as

$$\begin{array}{cc}\hfill U_o\left(t\right)=& \frac{1}{\tau }{▵}^{\alpha +\beta +1}{I}_L\left(t\right)+\frac{a_{11}{a}_{22}-a_{12}{a}_{21}}{\tau }{I}_L\left(t\right)-\frac{a_{11}}{\tau }{▵}^{\beta +1}{I}_L\left(t\right)\hfill \\ & -\frac{a_{22}}{\tau }{▵}^{\alpha +1}{I}_L\left(t\right)-\frac{a_{21}+b_2}{\tau }{▵}^{\alpha }{U}_o\left(t\right)-\frac{a_{21}{b}_1}{\tau }{U}_o\left(t\right),\hfill \\ \hfill =& {\mathit{\phi }}^T\left(t\right)\mathit{\theta }.\hfill \end{array}$$

For the FOM, the upper and lower boundaries of the parameters for the FOM are set as shown in

Table 3. Lower and upper boundaries of each model parameter [83].

Parameters	${\mathit{R}}_{\mathit{c}}\left(\mathbf{\Omega }\right)$	${\mathit{C}}_{\mathit{\alpha }}\left(\mathit{F}\right)$	${\mathit{L}}_{\mathit{\beta }}\left(\mathit{H}\right)$	$\mathit{\alpha }$	$\mathit{\beta }$
Lower bound	0	20 × ${10}^{-9}$	5 × 10${}^{-6}$	0.8	0.8
Upper bound	20	600 × ${10}^{-7}$	100 × 10${}^{-6}$	1.1	1.1

. The information vector and parameter vectors are defined as:

$$\begin{array}{cc}\hfill \mathit{\theta }:=& {[k_1,k_2,k_3,k_4,k_5,k_6]}^{\mathrm{T}},\hfill \\ \hfill k_1:=& \frac{1}{\tau },k_2:=\frac{a_{11}{a}_{22}-a_{12}{a}_{21}}{\tau },k_3:=\frac{a_{11}}{\tau },k_4:=\frac{a_{22}}{\tau },k_5:=\frac{a_{21}+b_2}{\tau },k_6:=\frac{a_{21}{b}_1}{\tau },\hfill \\ \hfill \mathit{\phi }:=& {[{▵}^{\alpha +\beta +1}{I}_L\left(t\right),I_L\left(t\right),-{▵}^{\beta +1}{I}_L\left(t\right),-{▵}^{\alpha +1}{I}_L\left(t\right),-{▵}^{\alpha }{U}_o\left(t\right),-U_o\left(t\right)]}^{\mathrm{T}}.\hfill \end{array}$$

According to the correlativity between $\alpha$ and $\beta$, two cases are then considered.

At $\beta =\alpha$, the information vector is given by:

$$\begin{array}{ccc}\hfill \mathit{\phi }& =& {[{▵}^{2\alpha +1}{I}_L\left(t\right),I_L\left(t\right),-{▵}^{\alpha +1}{I}_L\left(t\right),-{▵}^{\alpha +1}{I}_L\left(t\right),-{▵}^{\alpha }{U}_{oc}\left(t\right),-U_{oc}\left(t\right)]}^{\mathrm{T}}.\hfill \end{array}$$

In this case of $\beta =\alpha$, let $R_s=100$$\mathsf{\Omega }$ and $R_L=23$$\mathsf{\Omega }$, where the parameters are given as $R_c=15$$\mathsf{\Omega }$, $C_{\alpha }=7\times {10}^{-6}$F, $L_{\beta }=5\times {10}^{-5}$H, and $\alpha =0.8$. Applying the algorithms derived in this paper to estimate the parameter vector $\mathit{\theta }$ of the FOM, the parameter estimates and their errors are shown in

Table 4. Parameter estimates of $\mathit{\theta }$ and their estimation errors ($\beta =\alpha$).

	t	${\mathit{k}}_1$	${\mathit{k}}_2$	${\mathit{k}}_3$	${\mathit{k}}_4$	${\mathit{k}}_5$	${\mathit{k}}_6$	$\mathit{\delta }(\%)$
FFSG $\lambda$ = 0.98	1000	27.73007	103.44403	−38.33561	−21.47973	59.84995	73.51337	8.48226
	5000	28.12977	103.08827	−38.49831	−21.64242	60.08411	73.46766	8.17223
	10,000	28.59150	102.70654	−38.69387	−21.83799	60.32436	73.39486	7.83989
	40,000	30.61691	100.95427	−39.53888	−22.68300	61.45685	73.12375	6.61330
MI-FFSG $\lambda$ = 0.98 p = 2	1000	27.95533	103.19742	−38.38189	−21.52600	60.03447	73.54458	8.30109
	5000	28.61277	102.64491	−38.65274	−21.79685	60.38623	73.45144	7.82317
	10,000	29.30285	102.08130	−38.93634	−22.08046	60.73911	73.34645	7.36869
	40,000	31.56301	100.17532	−39.83806	−22.98218	61.95609	73.05467	6.26271
MI-FFSG $\lambda$ = 0.98 p = 5	1000	28.31188	102.85354	−38.52594	−21.67005	60.26919	73.51818	8.02035
	5000	29.77645	101.63543	−39.12073	−22.26484	61.04057	73.31181	7.06763
	10,000	30.90760	100.72173	−39.56546	−22.70957	61.61034	73.15148	6.51056
	40,000	32.45926	99.54640	−40.05098	−23.19510	62.32073	72.99922	6.08998
True values	0	30.85833	94.30000	−38.33333	−22.24483	61.71667	66.66667

, and the parameter estimation errors $\delta :=\parallel \widehat{\mathit{\theta }}\left(t\right)-\mathit{\theta }\parallel /\parallel \mathit{\theta }\parallel$ versus t are depicted in Figure 6. By using the parameter estimates in $\widehat{\mathit{\theta }}\left(t\right)$, the estimates of $R_c$, $C_{\alpha }$, and $L_{\beta }$ and their errors are obtained, as shown in Figure 7 and

Table 5. Parameter estimates of $R_c$, $C_{\alpha }$, and $L_{\beta }$, and their estimation errors ($\beta =\alpha$).

		t	${\mathit{R}}_{\mathit{c}}$	${\mathit{C}}_{\mathit{\alpha }}$	${\mathit{L}}_{\mathit{\beta }}$	$\mathit{\delta }(\%)$
FFSG	$\lambda =0.98$	1000	13.80726	6.30246 × ${10}^{-6}$	4.65655 × ${10}^{-5}$	7.95159
		5000	13.97132	6.32917 × ${10}^{-6}$	4.73829 × ${10}^{-5}$	6.85789
		10,000	14.14278	6.35613 × ${10}^{-6}$	4.82438 × ${10}^{-5}$	5.71482
		40,000	14.66287	6.45046 × ${10}^{-6}$	5.09442 × ${10}^{-5}$	2.24752
MI-FFSG		1000	13.74761	6.29235 × ${10}^{-6}$	4.63327 × ${10}^{-5}$	8.34929
	$\lambda =0.98$	5000	13.84449	6.31015 × ${10}^{-6}$	4.68173 × ${10}^{-5}$	7.70342
	$p=2$	10,000	13.95887	6.32894 × ${10}^{-6}$	4.73963 × ${10}^{-5}$	6.94089
		40,000	14.41633	6.42001 × ${10}^{-6}$	4.98186 × ${10}^{-5}$	3.89113
MI-FFSG		1000	13.88648	6.32057 × ${10}^{-6}$	4.69757 × ${10}^{-5}$	7.42345
	$\lambda =0.98$	5000	14.24583	6.37917 × ${10}^{-6}$	4.87814 × ${10}^{-5}$	5.02778
	$p=5$	10,000	14.52039	6.42192 × ${10}^{-6}$	5.01662 × ${10}^{-5}$	3.19740
		40,000	14.94844	6.46112 × ${10}^{-6}$	5.20842 × ${10}^{-5}$	0.34370
True values		0	15.00000	7.00000 × ${10}^{-6}$	5.00000 × ${10}^{-5}$

. The prediction outputs of the MI-FFSG algorithm, the true output, and their errors are illustrated in Figure 8.

At $\beta \ne \alpha$, let $R_s=100$$\mathsf{\Omega }$, $R_L=23$$\mathsf{\Omega }$, $\alpha =0.83$, and $\beta =0.95$, where the parameters are given as $R_c=15$$\mathsf{\Omega }$, $C_{\alpha }=7\times {10}^{-6}$F, $L_{\beta }=5\times {10}^{-5}$H. By using the FFSG algorithm and the MI-FFSG algorithm, the parameter estimates and their errors are obtained as shown in

Table 6. Parameter estimates of $\mathit{\theta }$ and their estimation errors ($\beta \ne \alpha$).

	t	${\mathit{k}}_1$	${\mathit{k}}_2$	${\mathit{k}}_3$	${\mathit{k}}_4$	${\mathit{k}}_5$	${\mathit{k}}_6$	$\mathit{\delta }\mathbf{(}\mathbf{\%}\mathbf{)}$
	100	6.24364	21.83487	−18.09445	−3.59900	13.19202	29.42921	8.35927
FFSG	500	6.32221	21.56446	−18.21295	−3.68999	13.28928	29.41486	7.91340
$\lambda$ = 0.9	1000	6.44177	21.28386	−18.37800	−3.82370	13.36712	29.36396	7.43774
	2000	6.68282	20.78817	−18.70398	−4.09114	13.48436	29.24313	6.78030
MI-FFSG $\lambda$ = 0.9 p = 2	100	6.31471	21.58154	−18.19775	−3.67454	13.28329	29.38524	7.88031
	500	6.42190	21.17513	−18.36667	−3.80086	13.43134	29.36882	7.32723
	1000	6.56893	20.78416	−18.57960	−3.96828	13.54424	29.30801	6.86218
	2000	6.85940	20.16695	−18.98327	−4.29371	13.68240	29.15133	6.48975
MI-FFSG $\lambda$ = 0.9 p = 5	100	6.21775	21.64010	−17.67841	−3.54995	13.31073	28.80093	7.04659
	500	6.35952	21.07856	−17.90567	−3.71818	13.51761	28.78292	6.07952
	1000	6.55932	20.55344	−18.19623	−3.94602	13.66457	28.69559	5.30013
	2000	6.89917	19.80877	−18.67848	−4.32971	13.82636	28.50413	4.93783
True values	0	7.05833	20.16667	−18.33333	−3.65750	14.11667	26.66667

and

Table 7. Parameter estimates of $R_c$, $C_{\alpha }$, and $L_{\beta }$, and their estimation errors ($\beta \ne \alpha$).

		t	${\mathit{R}}_{\mathit{c}}$	${\mathit{C}}_{\mathit{\alpha }}$	${\mathit{L}}_{\mathit{\beta }}$	$\mathit{\delta }(\%)$
FFSG	$\lambda =0.9$	100	14.14436	6.16779 × ${10}^{-6}$	4.75388 × ${10}^{-5}$	5.70428
		500	14.07770	6.25542 × ${10}^{-6}$	4.78128 × ${10}^{-5}$	6.14863
		1000	14.12997	6.30889 × ${10}^{-6}$	4.84998 × ${10}^{-5}$	5.80021
		2000	14.35451	6.35466 × ${10}^{-6}$	5.01330 × ${10}^{-5}$	4.30329
MI-FFSG		100	14.16867	6.11078 × ${10}^{-6}$	4.73289 × ${10}^{-5}$	5.54219
	$\lambda =0.9$	500	14.14247	6.16479 × ${10}^{-6}$	4.75738 × ${10}^{-5}$	5.71684
	$p=2$	1000	14.21684	6.19462 × ${10}^{-6}$	4.81911 × ${10}^{-5}$	5.22105
		2000	14.41947	6.22814 × ${10}^{-6}$	4.95597 × ${10}^{-5}$	3.87022
MI-FFSG		100	14.36509	6.25482 × ${10}^{-6}$	4.67123 × ${10}^{-5}$	4.23276
	$\lambda =0.9$	500	14.25527	6.38177 × ${10}^{-6}$	4.70461 × ${10}^{-5}$	4.96488
	$p=5$	1000	14.33336	6.45219 × ${10}^{-6}$	4.80024 × ${10}^{-5}$	4.44425
		2000	14.58106	6.51293 × ${10}^{-6}$	4.98986 × ${10}^{-5}$	2.79293
True values		0	15.00000	7.00000 × ${10}^{-6}$	5.00000 × ${10}^{-5}$

. The parameter estimation errors versus t are shown in Figure 9 and Figure 10, and the parameter estimates versus t are plotted in Figure 11. The prediction outputs of the MI-FFSG algorithm and the true output are illustrated in Figure 12 to verify the output imitative effect.

From Table 4, Table 5, Table 6 and Table 7 and Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, the following conclusions can be obtained:

• As Figure 6, Figure 7, Figure 9, and Figure 10 and Table 4, Table 5, Table 6 and Table 7 show, the estimated values will converge to the true parameters with the increase in t, which illustrates the effectiveness of the FFSG algorithm and the MI-FFSG algorithm for the parameter estimation of the FOM.
• The MI-FFSG algorithm has a higher parameter estimation accuracy than the FFSG algorithm for the FOM.
• Figure 6, Figure 7, Figure 9, and Figure 10 demonstrate that the estimation errors of the MI-FFSG algorithm decrease as p increases.
• The model estimated by the MI-FFSG algorithm has a better match to the true one, as plotted in Figure 8 and Figure 12.
• The parameter estimates of the MI-FFSG algorithm converge to the true values with different convergence rates; this can be seen in Figure 11.

6. Conclusions

In this paper, two identification algorithms are proposed for the parameter estimation of fractional nonlinear Wiener systems. One algorithm is the FFSG algorithm, which introduces the forgetting factor into the SG algorithm. The FFSG algorithm can generate more accurate parameter estimates and a higher convergence rate than the SG algorithm. The other proposed algorithm is the MI-FFSG algorithm, which expands the single innovation to multi-innovation. The simulation results indicate that the proposed algorithms are effective for estimating parameters of fractional nonlinear Wiener systems and have a higher parameter estimation accuracy. The second example uses the FOM to verify the parameter estimation performance of the algorithms and shows that the MI-FFSG algorithm has higher precision in parameter estimation. Furthermore, the parameter estimation methods for fractional Wiener systems in this paper can combine certain estimation algorithms to study new methods for linear and nonlinear stochastic systems with colored noises [84,85,86,87,88,89] and can be applied to other applications [55,90,91,92,93,94], such as control and schedule systems [95,96,97,98,99,100], information processing and communication systems, and so on.