# Correlation Analysis Algorithm-Based Multiple-Input Single-Output Wiener Model with Output Noise

Authors: Li Jia; Yangyang Li; Feng Li
Journal: Complexity
 (2019)

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## Abstract

A novel identification algorithm for the correlation analysis-based multiple-input single-output (MISO) neurofuzzy Wiener model with noise is proposed. Firstly, several sets of Gaussian signals are utilized to realize the decoupling between the dynamic linear blocks and the static nonlinear blocks of a MISO Wiener system. Then, the correlation analysis is adopted to identify the parameters of the linear parts, and the problem that the output of static nonlinear block is immeasurable can be solved. As a result, it can circumvent the problem of initialization and convergence of the model parameters encountered by the existing iterative approach used for the identification of the Wiener model. Next, the least-squares algorithm is employed to estimate the consequent parameters of the nonlinear blocks which can avoid the influence of noise. Finally, examples are used to illustrate the performance of the proposed method.


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## Body

## 1. Introduction

Wiener model consists of the cascade of a dynamic linear block followed by a static nonlinear block which can approximate a lot of industry processes, such as PH neutralization [1], continuous stirred tank reactor (CSTR) [2], heat exchanger [3], and distillation columns [4]. Thus, block-oriented models have gained wide attention in the system identification and automatic control fields [5–8].Up to now, plenty of methodologies have been used to identify block-oriented models. In general, the identification approaches based on input-output signals for block-oriented models can be divided into two parts: the synchronous method and the separate step algorithm. The synchronous method identifies the block-oriented system by directly constructing the hybrid model of the dynamic linear block and the static nonlinear block [9], such as the direct identification method [10], the subspace algorithm [11], the overparameterization approach [12], and the hierarchical identification methods [13]. The separate step algorithm decomposes the block-oriented model into the dynamic linear block and the static nonlinear block, such as the correlation method [14], the iterative algorithm [15], the blind approach [16], and stochastic technique [17].Few papers concentrated on the MISO Wiener model. Guo and Papanicolaou transformed the MISO Wiener model into a pseudolinear system, and then, the adaptive recursive pseudolinear regressions and the smoothing technique were employed to estimate the intermediate variables so that the parameters of the MISO Wiener model were estimated [18]. Tiels et al. [19] formulated the dynamical linear part of the MISO Wiener–Schetzen model with a set of orthonormal basis functions, and the system poles were estimated by the best linear approximation algorithm. Feroldi, Gomez, and Roda introduced a MISO Wiener model identification method based on support vector (SV) regression which is used to estimate a nonlinear dynamical model for the air supply system of laboratory polymer electrolyte membrane fuel cells [20]. However, the abovementioned methods assume that the static nonlinear block is of the polynomial model, which has a lot of limitations. If the nonlinearity is discontinuous or not in the polynomial form, the algorithms do not converge. Moreover, high-degree polynomials have oscillatory behaviour. Zhou et al. developed a gradient-based iterative identification approach for the MISO Wiener model which can approximate the glutamate fermentation process [21]. In 2011, a prediction error identification principle for the MISO Wiener model was proposed by Silva to solve the problem in anesthesia [22]. A hybrid genetic and swarming intelligence-based hierarchical cultural algorithm was applied to identify the MISO Wiener model in [23]. But the aforementioned results considered the identification of the MISO Wiener model without output noise, which is not the case in the real industrial processes.A useful theorem which is summarized in Bussgang’s classic principle (see [24]) about Gaussian signal presents that Rvuτ=b0Ruτ holds for an arbitrary static nonlinearity if the input signal is separable, where Rvuτ=Evkuk−τ is the cross-correlation between the input signal ut and the output signal vt, Ruτ=Eukuk−τ is the autocorrelation of the input signal, and b0=Ef′uk, where f′⋅ is the first-order derivative of f⋅. Nuttall generalized this theorem to a larger kind of separable signals, such as sine signals, random binary signals, and several kinds of modulated signals [25]. Enqvist and Ljung further proposed a principle: the process is separable if and only if the condition expectation of the signal uk satisfies Euk−τuk=aτuk, where aτ=Ruτ/Ru0 [26]. Greblicki and Pawlak studied the white noise incentive signal-based identification algorithm for the Hammerstein model in [27]. Goethals et al. generalized the linear subspace method and combined correlation analysis with least squares support vector machines (LS-SVM) regression to estimate the parameters of the dynamics and nonlinearity representation of the Hammerstein–Wiener model [28]. In our previous works, Jia et al. extended Bussgang’s theorem to Hammerstein model [29–31], and the correlation analysis algorithm was applied to the SISO Wiener model in [32]. However, most of the real industry processes [33–35] are inherently multivariable systems which can describe the relationships among different variables more accurately. Thus, multivariable nonlinear identification techniques are required. In addition, according to [36–38], sometimes for reason of making the system as close as possible to the real one, MISO block-oriented models with distributed dynamic linear blocks are chosen to represent some nonlinear industry processes. Therefore, motivated by our previous works, a novel identification algorithm for correlation analysis algorithm-based MISO-distributed Wiener model with noise is presented. The nonlinearities of the MISO Wiener model are approximated by neurofuzzy models, which can avoid the inevitable restrictions on static nonlinearities encountered by using the polynomial models. Gaussian signals are adopted to separate the dynamic linear blocks and the static nonlinear blocks. Then, the correlation analysis is employed for the identification of the dynamic linear parts so as to obtain the intermediate variables indirectly. At last, least squares approach is used to identify the nonlinearity and the estimation is proved to be unbiased.The rest of the paper is organized as follows. The identification problem of the MISO neurofuzzy Wiener model with noise is described in Section2. A correlation analysis-based identification approach is presented in detail in Section 3. Section 4 gives the simulation examples, followed by the conclusion given in Section 5.


## 2. MISO Wiener Neurofuzzy Model with Noise

Consider a MISO Wiener model (see [18]) with noise which consists of the dynamic linear blocks and the static nonlinear blocks as shown in Figure 1 as given by(1)vit=BiqAiquit,yit=fivit,yt=∑i=1myit+ηt,i=1,2,…,m,Aiq=1+ai1q−1+ai2q−2+⋯+ainaq−na,Biq=1+bi1q−1+bi2q−2+⋯+binbq−nb,where uiti=1,2,…,m and yt which can be measured represent the input and output processes at the t-th sampling time, respectively. i denotes the i-th branch of the MISO Wiener model, i=1,2,…,m. viti=1,2,…,m corresponds to the intermediate variable; fiviti=1,2,…,m is the static nonlinearity of the i-th branch of the MISO Wiener model. yiti=1,2,…,m which cannot be measured is the output of the nonlinear block. ηt denotes the white noise, q−1 is the backward shift, and nia and nib are integers concerned with the model order.Figure 1
The structure of the MISO Wiener model.In this paper, the target of the proposed identification approach is to obtain a MISO Wiener model such that the following cost functionE is made acceptable small, that is,(2)EA^1,A^2,…,A^m,B^1,B^2,…,B^m,f^1v^1t,f^2v^2t,…,f^mv^mt=12NP∑t=1NPy^t−yt2≤ε,(3)s.t.v^it=B^iqA^iquit,y^it=f^iv^it,i=1,2,…,m,y^t=∑i=1my^it+et,where ε is a given tolerance, f^⋅ is the estimated nonlinear function, y^t is the estimated output of the MISO Wiener model, v^it is the estimated intermediate variable of i-th branch, NP is the number of the input and output data, and “∧” represents “model”.As shown in Figure2, the nonlinear function of each branch of the MISO Wiener model is represented by a four-layer neurofuzzy system which integrates the Takagi–Sugeno fuzzy system and radial basis function-based feed-forward network into a connection structure. The first layer is the input layer in which the nodes just transmit the input signal to the next layer. The second layer is the membership layer where the membership function of the input variable is calculated. The third layer is the fuzzy rule layer, and the number of the nodes in this layer denotes the fuzzy rules. The last layer is the output layer. The output of the i-th neurofuzzy model is given by(4)y^it=∑j=1Niϕijv^itwij,i=1,2,…,m,where(5)ϕilv^it=exp−v^it−cil2/σil2∑l=1Niexp−v^it−cil2/σil2,where cil and σil denote the center and width of the membership function, respectively, wil is the consequent parameters of the neurofuzzy model, and Ni represents the fuzzy rule of the nonlinear block of the i-th branch.Figure 2
The structure of the neurofuzzy model.


## 3. Identification of Correlation Analysis-Based MISO Wiener Model

Gaussian signals with independent and identical distribution are adopted to separate the linear blocks from the nonlinear blocks by this proposed algorithm so that the immeasurable intermediate variableviti=1,2,…,m is obtained indirectly.Assume that the input and output of the model are zero initial state and the input signals are Gaussian sequences with independent and identical distribution, i.e.,uit∼0,σui2i=1,2,…,m. And presume that the cross-correlations between different input signals are zero. Then, the intermediate variable vit of the MISO Wiener model can be written as follows:(6)vi1=0,vi2=bi1ui1,vi3=−ai1vi2−ai2vi1+bi1ui2+bi2ui1,i=1,2,…,m,=−ai1bi1+bi2ui1+bi1ui2⋯vit=∑k=1t−1hi,tkuit−k,where hi,tk denotes the coefficient of uit−k of the i-th branch at t-th time.Then, the conditional expectation ofvit can be deduced:(7)Evit−τvit=∑k=1t−τ−1hi,t−τkEuit−τ−k∑ξ=1t−1hi,tξuit−ξ=∑k=1t−τ−1hi,t−τkEuit−τ−khi,tτ+kuit−τ−k+∑ξ=1t−1hi,tξuit−ξ,ξ≠τ+k=∑k=1t−τ−1hi,t−τkEUi,t−τ−kXi,t−τ−k+Yi,t,where Ui,t−τ−k=uit−τ−k, Xi,t−τ−k=hi,tτ+kuit−τ−k, and Yi,t=∑ξ=1t−1hi,tξuit−ξ,ξ≠τ+k. Assume that the input signals are Gaussian signals with independent and identical distribution, a linear combination of a finite number of Gaussian signal is necessarily also Gaussian signal; thus, vit subjects to normal distribution with zero means and variance σvi2=σui2∑k=1t−1hi,t2k, i.e., vit∼0,σvi2. Yi,t subjects to normal distribution with zero means and variance σYi=σui2∑ξ=1t−1hi,t2ξ,ξ≠τ+k, i.e., Yi,t∼0,σYi2.As can be seen from (7), Ui,t−τ−k and Xi,t−τ−k are independent of Yi,t, respectively. Moreover, we can get Ui,t−τ−k=Xi,t−τ−k/hi,tτ+k. And assume that Vi,t=vit=Xi,t−τ−k+Yi,t, and thus, the joint probability density function of Ui,t−τ−k and Vi,t is as follows:(8)gUi,t−τ−k,Vi,tui,vi=gXi,t−τ−k,Yi,thi,tτ+kui,vi−hi,tτ+kui×Jac−1=gXi,t−τ−khi,tτ+kui×gYi,tvi−hi,tτ+kui×Jac−1=12πhi,tτ+kσui×e−hi,tτ+kui2/2hi,tτ+kσui2×12πσYi×e−vi−hi,tτ+kui2/2σYi2×hi,tτ+k=12πσuiσYie−ui2σYi2+vi−hi,tτ+kui2σui2/2σui2σYi2,where Jac is the Jacobian determinant, that is (see [39]),(9)Jac=∂Ui,t−τ−k,Vi,t∂Xi,t−τ−k,Yi,t=∂Ui,t−τ−k∂Xi,t−τ−k∂Ui,t−τ−k∂Yi,t∂Vi,t∂Xi,t−τ−k∂Vi,t∂Yi,t=1hi,tτ+k.Thus, the conditional probability density function ofUi,t−τ−k is given by(10)gUi,t−τ−kVi,t=Cuivi=c=gUi,t−τ−k,Vi,tui,cgVi,tc=σvi2πσuiσYi×ec2/2σvi2−ui2σYi2+vi−hi,tτ+kui2σui2/2σui2σYi2=kce−kaui2−2kbcui/2σui2σYi2=kde−kaui−kbc/ka2/2σui2σYi2,where the coefficients are defined in the following equation and the constant c is the value of vit:(11)ka=σYi2+σui2hi,tτ+k2,kb=σu12hi,tτ+k,kc=σvi2πσuiσYieσYi2−σvi2c2/2σvi2σYi2,kd=kc⋅ekb2c2/2kaσui2σYi2.As a result, the conditional expectation ofuit−τ−k is given by(12)Euit−τ−k∑ξ=1t−1hi,tξuit−ξ=c=kbkac,From (7) and (12), we can obtain the following equation:(13)Evit−τvit=kbka⋅c⋅∑k=1t−τ−1hi,t−τk=aiτ⋅c,where aiτ=kb/ka⋅∑k=1t−τ−1hi,t−τk,i=1,2,…,m is a constant. In conclusion, according to [26], the intermediate variable vit satisfies the separability of the process.Theorem 1.
For a MISO Wiener model as expressed in (1), if the input signals are Gaussian signals with independent identical distribution and the expectations of inputs satisfy Euit=0i=1,2,…,m, the following equation holds:(14)Ryu=BRu,where Ryu=Ryu1,Ryu2,…,RyumT, B=diagb01L1,b02L2,…,b0mLm, and Ru=Ru1,Ru2,…,RumT, and Ryuiτ denotes the cross-correlation function between yt and the i-th input uit, Ruiτ represents the autocorrelation of uit, Liq is the linear part, i.e.,Liq=Biq/Aiq, and b0i=Eyitvit/Evitvit is a constant.Proof.
As can be seen from (13), when the input is Gaussian signal, the output vit of the dynamic of the i-th branch satisfies the separability of the process. Thus, for a MISO Wiener model, we can get the following equation:(15)Ryviτ=Eytvit−τ=Ey1t+y2t+⋯+ymt+ηtvit−τ=Ey1tvit−τ+Ey2tvit−τ+⋯+Eymtvit−τ+Eηtvit−τ,i=1,2,…,m=Eyitvit−τ=EyitEvit−τvit=bτEyitvti.
From (15), we know that the cross-correlation function between vit and the immeasurable variable yit can be replaced by the cross-correlation function between vit and yt:(16)Rviτ=Evitvit−τ=EvitEvit−τvit=bτEvitvit.
According to (15) and (16), the following equation holds:(17)Ryviτ=EyitvitEvitvitRviτ=b0iRviτ.
According to [40], for a stable and time-invariant linear system, its impulse response can be represented by(18)Liq=∑k=1∞likq−k.
Thus, the output of the dynamic linear block ofi-th branch can be written as(19)vit=Liquit=∑k=1∞likq−k⋅uit=∑k=1∞likuit−k.
And (15) and (16) can be rewritten as follows:(20)Ryviτ=Eytvit−τ=Eyt∑k=1∞likuit−τ−k=∑k=1∞likRyuiτ+k=∑k=1∞likqk⋅Ryuiτ,(21)Rviτ=Evitvit−τ=E∑k=1∞likuit−k∑ς=1∞liςuit−τ−ς=∑k=1∞∑ς=1∞likRuiτ+ς−kliTς=∑ς=1∞liςqξ⋅Ruiτ⋅∑k=1∞likq−k.
Then, by substituting (20) and (21) into (17), we can get the following equation:(22)∑k=1∞likqk⋅Ryuiτ=b0i∑ς=1∞liςqξ⋅Ruiτ⋅∑k=1∞likq−k,where ∑k=1∞gikqk on the left side of (22) is equivalent to ∑ς=1∞giςqξ on the right side of (22).
Thus, we have(23)Ryuiτ=b0iRuiτ⋅∑k=1∞likq−k=b0iLiq⋅Ruiτ.
Then, (14) holds. This completes the proof.Remark 1.
In this paper, the MISO-distributed block-oriented nonlinear Wiener model which is consistent with the model in [18] is mainly considered. The content before Theorem 1 has proved that vit satisfies the separability of the process when the input signals uit are Gaussian signals. Thus, Ryiviτ=b0iRviτ holds according to Bussgang’s classic theorem. However, as shown in Figure 1, v1,v2,…,vm and y1,y2,…,ym are immeasurable variables, and Ryiviτ=b0iRviτ cannot be used to identify the parameters of the proposed Wiener model directly. Therefore, Theorem 1 is proposed to solve this problem. The input and output relationship of the MISO Wiener model, that is Ryuiτ=b0iLiq⋅Ruiτ, is deduced so that the measurable variables u1,u2,…,um and y can be utilized to estimate the parameters of dynamic linear blocks.


### 3.1. Identification of the Dynamic Linear Blocks

Correlation analysis is adopted to identify the linear blocks of the MISO Wiener model with noise.According to Theorem 1 and (23), we have(24)Ryuiτ=−∑n=1naiainRyuiτ−n+b0i∑j=1nbibijRuiτ−j,i=1,2,…,m,=−∑n=1naiainRyuiτ−n+∑j=1nbib˜ijRuiτ−j=Φiθi,where b˜ij=b0ibij.Then, the following equation holds:(25)Ryu=Φθ.Thus, the parameters of dynamic linear block can be estimated by least squares algorithm:(26)θ=ΦTΦ−1ΦRyu,where Ryu=Ryu1,Ryu2,…,RyumT, Ryui=Ryui1,Ryui2,…,RyuiPT∈RP×1, θ=θ1,θ2,…,θmT, θ^i=a^i1,a^i2,…,a^inai,b˜^i1,b˜^i1,,…,b˜^inbiT∈Rnai+nbi×1, τ=1,2,…,PP≥nai+nbi, Φ=diagΦ1,Φ2,…,Φm, and(27)Φi=−Ryui0−Ryui1−Ryui2⋯−RyuiP−10−Ryui0−Ryui1⋯−RyuiP−2⋮⋮⋮⋱⋮000⋯−RyuiP−naRui0Rui1Rui2⋯RuiP−10Rui0Rui1⋯RuiP−2⋮⋮⋮⋱⋮000⋯RuiP−nbT∈RP×nai+nbi,where Ryuiτ and Ruiτ are as follows:(28)R^yuiτ=1N∑t=1Nytuit−τ,(29)R^uiτ=1N∑t=1Nuituit−τ.


### 3.2. Identification of the Static Nonlinear Blocks

After the linear blocks have been identified, the intermediate variablev^it can be further estimated according to the input signal uit and the estimated parameter θ^i. Next, the clustering approach used in [41, 42] is employed to identify the antecedent parameters of static nonlinear blocks. And the proposed algorithm focuses on the consequent parameters wij. From (1) and (4), we have(30)yt=Wψt+ηt,where W=W1,W2,…,Wm∈R1×L,L=∑i=1mLi, Wi=wi1,wi2,…,wiLi∈R1×Lii=1,2,…,m and ψt=ψ1t,ψ2t,…,ψmtT∈RL×1, ψit=ϕi1v^it,ϕi2v^it,…,ϕiLiv^itT∈RLi×1. Li is the clustering number of nonlinear block of the i-th branch.As a result, the estimation ofW can be obtained by using least squares approach:(31)WLS=∑t=1NytψTt∑t=1NψtψTt−1.According to (30), substitute yt with Wψt+ηt, and then (31) can be rewritten as(32)WLS=∑t=1NWψt+ηtψTt∑t=1NψtψTt−1=W∑t=1NψtψTt∑t=1NψtψTt−1+∑t=1NηtψTt∑t=1NψtψTt−1=W+∑t=1NηtψTt∑t=1NψtψTt−1.As the noise is the white noise with zero means, thus we have(33)EWLS=EW+E∑t=1NetψTt∑t=1NψtψTt−1=W+∑t=1NEηtψTt∑t=1NψtψTt−1=W.In conclusion,WLS is the unbiased estimation of W.The identification algorithm for the correlation analysis-based MISO Wiener model is concluded as follows:Step 1. The parametersaij and bij of the dynamic linear blocks of the MISO Wiener model are estimated according to (26)–(29) in Section 3.1Step 2. The antecedent parameterscilandσil are obtained by using the clustering approach [41, 42]Step 3. The consequent parameterW of each static nonlinear block is identified according to (30) and (31) in Section 3.2


## 3.1. Identification of the Dynamic Linear Blocks

Correlation analysis is adopted to identify the linear blocks of the MISO Wiener model with noise.According to Theorem 1 and (23), we have(24)Ryuiτ=−∑n=1naiainRyuiτ−n+b0i∑j=1nbibijRuiτ−j,i=1,2,…,m,=−∑n=1naiainRyuiτ−n+∑j=1nbib˜ijRuiτ−j=Φiθi,where b˜ij=b0ibij.Then, the following equation holds:(25)Ryu=Φθ.Thus, the parameters of dynamic linear block can be estimated by least squares algorithm:(26)θ=ΦTΦ−1ΦRyu,where Ryu=Ryu1,Ryu2,…,RyumT, Ryui=Ryui1,Ryui2,…,RyuiPT∈RP×1, θ=θ1,θ2,…,θmT, θ^i=a^i1,a^i2,…,a^inai,b˜^i1,b˜^i1,,…,b˜^inbiT∈Rnai+nbi×1, τ=1,2,…,PP≥nai+nbi, Φ=diagΦ1,Φ2,…,Φm, and(27)Φi=−Ryui0−Ryui1−Ryui2⋯−RyuiP−10−Ryui0−Ryui1⋯−RyuiP−2⋮⋮⋮⋱⋮000⋯−RyuiP−naRui0Rui1Rui2⋯RuiP−10Rui0Rui1⋯RuiP−2⋮⋮⋮⋱⋮000⋯RuiP−nbT∈RP×nai+nbi,where Ryuiτ and Ruiτ are as follows:(28)R^yuiτ=1N∑t=1Nytuit−τ,(29)R^uiτ=1N∑t=1Nuituit−τ.


## 3.2. Identification of the Static Nonlinear Blocks

After the linear blocks have been identified, the intermediate variablev^it can be further estimated according to the input signal uit and the estimated parameter θ^i. Next, the clustering approach used in [41, 42] is employed to identify the antecedent parameters of static nonlinear blocks. And the proposed algorithm focuses on the consequent parameters wij. From (1) and (4), we have(30)yt=Wψt+ηt,where W=W1,W2,…,Wm∈R1×L,L=∑i=1mLi, Wi=wi1,wi2,…,wiLi∈R1×Lii=1,2,…,m and ψt=ψ1t,ψ2t,…,ψmtT∈RL×1, ψit=ϕi1v^it,ϕi2v^it,…,ϕiLiv^itT∈RLi×1. Li is the clustering number of nonlinear block of the i-th branch.As a result, the estimation ofW can be obtained by using least squares approach:(31)WLS=∑t=1NytψTt∑t=1NψtψTt−1.According to (30), substitute yt with Wψt+ηt, and then (31) can be rewritten as(32)WLS=∑t=1NWψt+ηtψTt∑t=1NψtψTt−1=W∑t=1NψtψTt∑t=1NψtψTt−1+∑t=1NηtψTt∑t=1NψtψTt−1=W+∑t=1NηtψTt∑t=1NψtψTt−1.As the noise is the white noise with zero means, thus we have(33)EWLS=EW+E∑t=1NetψTt∑t=1NψtψTt−1=W+∑t=1NEηtψTt∑t=1NψtψTt−1=W.In conclusion,WLS is the unbiased estimation of W.The identification algorithm for the correlation analysis-based MISO Wiener model is concluded as follows:Step 1. The parametersaij and bij of the dynamic linear blocks of the MISO Wiener model are estimated according to (26)–(29) in Section 3.1Step 2. The antecedent parameterscilandσil are obtained by using the clustering approach [41, 42]Step 3. The consequent parameterW of each static nonlinear block is identified according to (30) and (31) in Section 3.2


## 4. Examples

### 4.1. Example 1

Consider the following TISO (two-input single-output) Wiener model which has two branches (see [18] which is a classical MISO-distributed Wiener model):(34)v1t=0.04308q−1+0.0315q−21−1.3139q−1+0.3886q−2u1t,y1t=2tanhv1t−2exp0.1v1t+2,v2t=0.0305q−1+0.0254q−21−1.5218q−1+0.5778q−2u2t,y2t=v2t,v2t≤1.5,1.5expv2t−1.5,v2t>1.5,yt=y1t+y2t+ηt,where the white noise with zero means ηt is dependent on u1t and u2t, respectively.Define the noise-to-signal ratios asδns=varet/varyt−et×100% and the estimation error δ=θ^−θ/θ of the linear block.In order to identify the above TISO Wiener model, the following signals as shown in Figure3 are adopted: (1) Gaussian signal u1t with zero means and variance 4; (2) Gaussian signal u2t with zero means and variance 4; and (3) the corresponding output signal yt.Figure 3
Part of the training data used in Example 1.At first,u1t, u2t, and yt are employed to estimate the parameters of linear blocks of two branches according to (25) and (29) in Section 3.1. And the modifying coefficients are b01=Eytvt/Evtvt=0.4517 and b02=Eytvt/Evtvt=0.3643, respectively. Tables 1 and 2 give the results of identifying the linear parts of two branches by the proposed algorithm and the method in [18] under different noise-to-signal ratios. The estimation errors of two linear blocks are illustrated in Figures 4 and 5, respectively. It is obvious that the proposed algorithm results in a better approximation of the linear blocks than the method in [18] according to Tables 1 and 2 as well as Figures 4 and 5.Table 1
Identification results of the linear block in the first branch under differentδns.
t
The proposed method
The method in [18]
a
^
11
a
^
12
b
^
11
/
b
01
b
^
12
/
b
01
δ
1
a
^
11
a
^
12
b
^
11
/
b
01
b
^
12
/
b
01
δ
1
δ
n
s
=
5.24
%
1000
1.2829
−0.3333
0.04046
0.0312
0.0463
1.2206
−0.2576
0.03881
0.0294
0.1204
1500
1.2937
−0.3465
0.04243
0.0319
0.0341
1.2310
−0.2660
0.04077
0.0307
0.1080
2000
1.3044
−0.3668
0.04270
0.0318
0.0174
1.2508
−0.2892
0.03945
0.0292
0.0859
2500
1.3108
−0.37960.04317
0.0321
0.0069
1.2497
−0.3004
0.03994
0.0306
0.0795
3000
1.3051
−0.3837
0.04537
0.0329
0.0076
1.2452
−0.2948
0.04043
0.0309
0.0848
3500
1.3106
−0.3886
0.04387
0.0326
0.0026
1.2519
−0.3050
0.03951
0.0296
0.0760
4000
1.3088
−0.3874
0.04428
0.0325
0.0040
1.2653
−0.3211
0.03799
0.0290
0.0608
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0
δ
n
s
=
16.25
%
1000
1.3571
−0.4624
0.04853
0.0332
0.0625
1.2116
−0.2357
0.04010
0.0355
0.1372
1500
1.3879
−0.4842
0.04813
0.0291
0.0564
1.2484
−0.3041
0.04230
0.0356
0.0781
2000
1.3657
−0.4458
0.04578
0.0284
0.0564
1.2484
−0.3041
0.04230
0.0356
0.0781
2500
1.3526
−0.4311
0.04367
0.0292
0.0419
1.2645
−0.3127
0.04389
0.0348
0.0661
3000
1.3376
−0.4124
0.04375
0.0299
0.0245
1.2572
−0.2923
0.04404
0.0336
0.0815
3500
1.3205
−0.3900
0.04372
0.0307
0.0049
1.2643
−0.3020
0.04445
0.0339
0.0728
4000
1.3215
−0.3931
0.04462
0.0319
0.0066
1.2598
−0.2947
0.04617
0.0342
0.0791
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0
δ
n
s
=
35.68
%
1000
1.3992
−0.5185
0.04740
0.0281
0.1135
1.1806
−0.2297
0.04310
0.0360
0.1503
1500
1.3538
−0.4715
0.04400
0.0315
0.0671
1.2033
−0.2523
0.04599
0.0379
0.1208
2000
1.3469
−0.4702
0.04385
0.0320
0.0642
1.2176
−0.2774
0.04675
0.0379
0.1208
2500
1.3384
−0.4488
0.04311
0.0330
0.0474
1.2041
−0.2608
0.04882
0.0376
0.1230
3000
1.3001
−0.3976
0.04310
0.0339
0.0121
1.2190
−0.2733
0.05037
0.0374
0.1091
3500
1.3077
−0.4069
0.04492
0.0336
0.0142
1.2457
−0.3086
0.04919
0.0358
0.0769
4000
1.3081
−0.3999
0.04468
0.0336
0.0095
1.2304
−0.2865
0.04844
0.0368
0.0963
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0Table 2
Identification results of the linear block in the second branch under differentδns.
t
The proposed method
The method in [18]
a
^
21
a
^
22
b
^
21
/
b
02
b
^
22
/
b
02
δ
2
a
^
21
a
^
22
b
^
21
/
b
02
b
^
22
/
b
02
δ
2
δ
ns
=
5.24
%
1000
1.5968
−0.6984
0.0269
0.0230
0.0872
1.6151
−0.7697
0.0399
0.0236
0.1312
1500
1.4904
−0.4911
0.0322
0.0247
0.0566
1.6166
−0.7531
0.0384
0.0242
0.1225
2000
1.5532
−0.6055
0.0384
0.0235
0.0262
1.6164
−0.7662
0.0428
0.0204
0.1374
2500
1.5523
−0.6090
0.0359
0.0220
0.0271
1.6270
−0.7668
0.0453
0.0219
0.1332
3000
1.5546
−0.6151
0.0364
0.0228
0.0308
1.6070
−0.7248
0.0424
0.0227
0.1046
3500
1.5092
−0.5639
0.0370
0.0248
0.0122
1.6120
−0.7092
0.0381
0.0226
0.0980
4000
1.5136
−0.5715
0.0352
0.0245
0.0070
1.5787
−0.6453
0.0359
0.0240
0.0543
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0
δ
ns
=
16.25
%
1000
1.4821
−0.5452
0.0313
0.0296
0.0316
1.5969
−0.7754
0.0406
0.0271
0.1400
1500
1.5003
−0.5707
0.0298
0.0274
0.0140
1.5781
−0.7581
0.0408
0.0289
0.1162
2000
1.4947
−0.5452
0.0282
0.0257
0.0261
1.5701
−0.7379
0.0397
0.0292
0.1029
2500
1.5278
−0.5978
0.0291
0.0259
0.0128
1.5695
−0.7046
0.0399
0.0296
0.0853
3000
1.5312
−0.6073
0.0300
0.0265
0.0190
1.5422
−0.6779
0.0410
0.0263
0.0631
3500
1.5381
−0.6077
0.0270
0.0262
0.0210
1.5503
−0.7009
0.0421
0.0253
0.0779
4000
1.5444
−0.6136
0.0259
0.0244
0.0262
1.5477
−0.7011
0.0414
0.0261
0.0777
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0
δ
ns
=
35.68
%
1000
1.4290
−0.4060
0.0328
0.0267
0.1199
1.6486
−0.7773
0.0259
0.0202
0.1520
1500
1.4438
−0.4203
0.0304
0.0273
0.1080
1.6435
−0.7560
0.0261
0.0210
0.1327
2000
1.4701
−0.4830
0.0306
0.0282
0.0663
1.6275
−0.7539
0.0279
0.0222
0.1262
2500
1.4441
−0.4681
0.0275
0.0272
0.0826
1.6128
−0.7248
0.0269
0.0223
0.1062
3000
1.4605
−0.4875
0.0291
0.0279
0.0671
1.6134
−0.7176
0.0229
0.0217
0.1028
3500
1.4787
−0.5369
0.0324
0.0274
0.0365
1.6137
−0.7232
0.0236
0.0219
0.1057
4000
1.4846
−0.5424
0.0318
0.0273
0.0316
1.6224
−0.7396
0.0261
0.0212
0.1171
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0Figure 4
The parameter estimation errors of the linear block in the first branch under differentδns: (a) δns=7.33%; (b) δns=14.13%; (c) δns=22.64%; (d) δns=31.47%.
(a)
(b)
(c)
(d)Figure 5
The parameter estimation errors of the linear block in the second branch under differentδns: (a) δns=7.33%; (b) δns=14.13%; (c) δns=22.64%; (d) δns=31.47%.
(a)
(b)
(c)
(d)Next, the intermediate variablesv^1t and v^2t are estimated by using the input signals u1t and u2t and the estimated parameters of linear blocks. Then, the antecedent parameters of two nonlinear blocks are identified by using the clustering algorithm according to the following design parameters [41, 42]: S01=0.98, ρ1=1.61, λ1=0.02 and S02=0.988, ρ2=1.05, λ2=0.1, which result in two neurofuzzy models whose rules are 6 and 80, respectively. Moreover, the consequent parameters W1 and W2 of two nonlinear blocks are obtained according to (30) and (31). The polynomial method described in [18] is also constructed by using the identical data to identify the nonlinear blocks. The comparisons between the proposed algorithm and the method in [18] are shown in Figures 6 and 7. The MSEs of two neurofuzzy models are 3.5×10−3 and 2.0×10−3, respectively, while the MSEs of two polynomial functions are 2.1×10−2 and 4.1×10−2, respectively. As illustrated in Figures 6 and 7, the proposed algorithm has better identification accuracy for the nonlinearity of the actual process.Figure 6
The static nonlinear block in the first branch.Figure 7
The static nonlinear block in the second branch.


### 4.2. Example 2

To further verify the performance of the proposed algorithm, this example considers a TISO Wiener model whose nonlinearities are two more complicated discontinuous segment nonlinear functions:(35)v1t=0.4q−11+0.8q−1u1t,y1t=2−cos3v1t−exp−v1t,v1t≤1,−1,v1t>1,v2t=0.6q−11+0.3q−1u2t,y2t=xt=tanh2v2t,v2t≤1.5,−expv2t−1expv2t+1,v2t>1.5,yt=y1t+y2t+ηt.In order to identify the above TISO Wiener model, the following signals as shown in Figure8 are adopted: (1) Gaussian signal u1t with zero means and variance 1.5; (2) Gaussian signal u2t with zero means and variance 1.9; and (3) the corresponding output signal yt.Figure 8
Part of the training data used in Example 2.Firstly, correlation analysis theorem in Section3.1 is employed to identify the linear blocks, so the estimated linear blocks are v^1t=−0.8052v^1t−1+0.3840u1t−1 and v^2t=−0.2859v^2t−1+0.5980u2t−1, and the modifying coefficients are b01=Eytvt/Evtvt=0.9536 and b02=Eytvt/Evtvt=0.5680, respectively. The estimation errors of two linear blocks are 3.66×10−2 and 2.18×10−2, respectively.Next, the design parameters are set as:S01=0.9991, ρ1=1.991, λ1=0 and S02=0.995, ρ1=1.85, λ1=0, resulting in two neurofuzzy models whose rules are 113 and 55, respectively. Then, the weights W1 and W2 of two neurofuzzy models are obtained by using (30) and (31). For comparison purpose, the polynomial-based TISO Wiener model is constructed by using the identical training data. Table 3 shows the MSEs for different polynomials based models in two branches; the models with orders of 19 and 32 are chosen to be the best models of two subsystems, respectively. The MSEs of the proposed TISO Wiener model based on different fuzzy rules are summarized in Table 4, and the two neurofuzzy models with fuzzy rules of 113 and 55 are proved to be the best models. Table 5 compares the MSEs for both training and validation data obtained by the best neurofuzzy-based model and the best polynomial-based model, respectively. Figures 9 and 10 show the approximation results of the nonlinearities of two branches. As described in Tables 3–5 as well as Figures 9 and 10, the neurofuzzy-based TISO Wiener model can achieve better approximation of nonlinearities than the polynomial-based TISO Wiener model.Table 3
MSEs of various polynomial-based TISO Wiener models.
The nonlinear block in the first branch
The nonlinear block in the second branch
Order
MSE (training)
MSE (validation)
Order
MSE (training)
MSE (validation)
7
6.01 × 10−2
7.12 × 10−2
8
6.79 × 10−1
7.56 × 10−1
12
4.31 × 10−2
6.78 × 10−2
17
5.98 × 10−1
6.23 × 10−1
16
3.31 × 10−2
5.34 × 10−2
24
4.03 × 10−1
6.02 × 10−1
19
1.24 × 10−2
3.80 × 10−2
32
2.10 × 10−1
4.15 × 10−1Table 4
MSEs of the proposed TISO Wiener model with different fuzzy rules.
The nonlinear block in the first branch
The nonlinear block in the second branch
The number of fuzzy rules
MSE (training)
MSE (validation)
The number of fuzzy rules
MSE (training)
MSE (validation)
45
5.80 × 10−3
8.20 × 10−3
20
2.96 × 10−3
7.30 × 10−3
90
4.03 × 10−3
3.80 × 10−2
33
3.70 × 10−3
8.60 × 10−2
113
3.79 × 10−3
4.90 × 10−3
45
1.80 × 10−3
4.30 × 10−3
122
5.40 × 10−3
6.30 × 10−3
55
1.09 × 10−3
2.70 × 10−3Table 5
MSEs of the best polynomial-based and the proposed TISO Wiener model.
The nonlinear block in the first branch
The nonlinear block in the second branch
Model
MSE (training)
MSE (validation)
Model
MSE (training)
MSE (validation)
Neurofuzzy
3.79 × 10−3
4.90 × 10−3
Neurofuzzy
1.09 × 10−3
2.70 × 10−3
Polynomial
1.24 × 10−2
3.80 × 10−2
Polynomial
2.10 × 10−1
4.15 × 10−1Figure 9
The static nonlinear block in the first branch.Figure 10
The static nonlinear block in the second branch.


## 4.1. Example 1

Consider the following TISO (two-input single-output) Wiener model which has two branches (see [18] which is a classical MISO-distributed Wiener model):(34)v1t=0.04308q−1+0.0315q−21−1.3139q−1+0.3886q−2u1t,y1t=2tanhv1t−2exp0.1v1t+2,v2t=0.0305q−1+0.0254q−21−1.5218q−1+0.5778q−2u2t,y2t=v2t,v2t≤1.5,1.5expv2t−1.5,v2t>1.5,yt=y1t+y2t+ηt,where the white noise with zero means ηt is dependent on u1t and u2t, respectively.Define the noise-to-signal ratios asδns=varet/varyt−et×100% and the estimation error δ=θ^−θ/θ of the linear block.In order to identify the above TISO Wiener model, the following signals as shown in Figure3 are adopted: (1) Gaussian signal u1t with zero means and variance 4; (2) Gaussian signal u2t with zero means and variance 4; and (3) the corresponding output signal yt.Figure 3
Part of the training data used in Example 1.At first,u1t, u2t, and yt are employed to estimate the parameters of linear blocks of two branches according to (25) and (29) in Section 3.1. And the modifying coefficients are b01=Eytvt/Evtvt=0.4517 and b02=Eytvt/Evtvt=0.3643, respectively. Tables 1 and 2 give the results of identifying the linear parts of two branches by the proposed algorithm and the method in [18] under different noise-to-signal ratios. The estimation errors of two linear blocks are illustrated in Figures 4 and 5, respectively. It is obvious that the proposed algorithm results in a better approximation of the linear blocks than the method in [18] according to Tables 1 and 2 as well as Figures 4 and 5.Table 1
Identification results of the linear block in the first branch under differentδns.
t
The proposed method
The method in [18]
a
^
11
a
^
12
b
^
11
/
b
01
b
^
12
/
b
01
δ
1
a
^
11
a
^
12
b
^
11
/
b
01
b
^
12
/
b
01
δ
1
δ
n
s
=
5.24
%
1000
1.2829
−0.3333
0.04046
0.0312
0.0463
1.2206
−0.2576
0.03881
0.0294
0.1204
1500
1.2937
−0.3465
0.04243
0.0319
0.0341
1.2310
−0.2660
0.04077
0.0307
0.1080
2000
1.3044
−0.3668
0.04270
0.0318
0.0174
1.2508
−0.2892
0.03945
0.0292
0.0859
2500
1.3108
−0.37960.04317
0.0321
0.0069
1.2497
−0.3004
0.03994
0.0306
0.0795
3000
1.3051
−0.3837
0.04537
0.0329
0.0076
1.2452
−0.2948
0.04043
0.0309
0.0848
3500
1.3106
−0.3886
0.04387
0.0326
0.0026
1.2519
−0.3050
0.03951
0.0296
0.0760
4000
1.3088
−0.3874
0.04428
0.0325
0.0040
1.2653
−0.3211
0.03799
0.0290
0.0608
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0
δ
n
s
=
16.25
%
1000
1.3571
−0.4624
0.04853
0.0332
0.0625
1.2116
−0.2357
0.04010
0.0355
0.1372
1500
1.3879
−0.4842
0.04813
0.0291
0.0564
1.2484
−0.3041
0.04230
0.0356
0.0781
2000
1.3657
−0.4458
0.04578
0.0284
0.0564
1.2484
−0.3041
0.04230
0.0356
0.0781
2500
1.3526
−0.4311
0.04367
0.0292
0.0419
1.2645
−0.3127
0.04389
0.0348
0.0661
3000
1.3376
−0.4124
0.04375
0.0299
0.0245
1.2572
−0.2923
0.04404
0.0336
0.0815
3500
1.3205
−0.3900
0.04372
0.0307
0.0049
1.2643
−0.3020
0.04445
0.0339
0.0728
4000
1.3215
−0.3931
0.04462
0.0319
0.0066
1.2598
−0.2947
0.04617
0.0342
0.0791
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0
δ
n
s
=
35.68
%
1000
1.3992
−0.5185
0.04740
0.0281
0.1135
1.1806
−0.2297
0.04310
0.0360
0.1503
1500
1.3538
−0.4715
0.04400
0.0315
0.0671
1.2033
−0.2523
0.04599
0.0379
0.1208
2000
1.3469
−0.4702
0.04385
0.0320
0.0642
1.2176
−0.2774
0.04675
0.0379
0.1208
2500
1.3384
−0.4488
0.04311
0.0330
0.0474
1.2041
−0.2608
0.04882
0.0376
0.1230
3000
1.3001
−0.3976
0.04310
0.0339
0.0121
1.2190
−0.2733
0.05037
0.0374
0.1091
3500
1.3077
−0.4069
0.04492
0.0336
0.0142
1.2457
−0.3086
0.04919
0.0358
0.0769
4000
1.3081
−0.3999
0.04468
0.0336
0.0095
1.2304
−0.2865
0.04844
0.0368
0.0963
True
1.3139
−0.3886
0.04308
0.0315
0
1.3139
−0.3886
0.04308
0.0315
0Table 2
Identification results of the linear block in the second branch under differentδns.
t
The proposed method
The method in [18]
a
^
21
a
^
22
b
^
21
/
b
02
b
^
22
/
b
02
δ
2
a
^
21
a
^
22
b
^
21
/
b
02
b
^
22
/
b
02
δ
2
δ
ns
=
5.24
%
1000
1.5968
−0.6984
0.0269
0.0230
0.0872
1.6151
−0.7697
0.0399
0.0236
0.1312
1500
1.4904
−0.4911
0.0322
0.0247
0.0566
1.6166
−0.7531
0.0384
0.0242
0.1225
2000
1.5532
−0.6055
0.0384
0.0235
0.0262
1.6164
−0.7662
0.0428
0.0204
0.1374
2500
1.5523
−0.6090
0.0359
0.0220
0.0271
1.6270
−0.7668
0.0453
0.0219
0.1332
3000
1.5546
−0.6151
0.0364
0.0228
0.0308
1.6070
−0.7248
0.0424
0.0227
0.1046
3500
1.5092
−0.5639
0.0370
0.0248
0.0122
1.6120
−0.7092
0.0381
0.0226
0.0980
4000
1.5136
−0.5715
0.0352
0.0245
0.0070
1.5787
−0.6453
0.0359
0.0240
0.0543
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0
δ
ns
=
16.25
%
1000
1.4821
−0.5452
0.0313
0.0296
0.0316
1.5969
−0.7754
0.0406
0.0271
0.1400
1500
1.5003
−0.5707
0.0298
0.0274
0.0140
1.5781
−0.7581
0.0408
0.0289
0.1162
2000
1.4947
−0.5452
0.0282
0.0257
0.0261
1.5701
−0.7379
0.0397
0.0292
0.1029
2500
1.5278
−0.5978
0.0291
0.0259
0.0128
1.5695
−0.7046
0.0399
0.0296
0.0853
3000
1.5312
−0.6073
0.0300
0.0265
0.0190
1.5422
−0.6779
0.0410
0.0263
0.0631
3500
1.5381
−0.6077
0.0270
0.0262
0.0210
1.5503
−0.7009
0.0421
0.0253
0.0779
4000
1.5444
−0.6136
0.0259
0.0244
0.0262
1.5477
−0.7011
0.0414
0.0261
0.0777
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0
δ
ns
=
35.68
%
1000
1.4290
−0.4060
0.0328
0.0267
0.1199
1.6486
−0.7773
0.0259
0.0202
0.1520
1500
1.4438
−0.4203
0.0304
0.0273
0.1080
1.6435
−0.7560
0.0261
0.0210
0.1327
2000
1.4701
−0.4830
0.0306
0.0282
0.0663
1.6275
−0.7539
0.0279
0.0222
0.1262
2500
1.4441
−0.4681
0.0275
0.0272
0.0826
1.6128
−0.7248
0.0269
0.0223
0.1062
3000
1.4605
−0.4875
0.0291
0.0279
0.0671
1.6134
−0.7176
0.0229
0.0217
0.1028
3500
1.4787
−0.5369
0.0324
0.0274
0.0365
1.6137
−0.7232
0.0236
0.0219
0.1057
4000
1.4846
−0.5424
0.0318
0.0273
0.0316
1.6224
−0.7396
0.0261
0.0212
0.1171
True
1.5218
−0.5778
0.0305
0.0254
0
1.5218
−0.5778
0.0305
0.0254
0Figure 4
The parameter estimation errors of the linear block in the first branch under differentδns: (a) δns=7.33%; (b) δns=14.13%; (c) δns=22.64%; (d) δns=31.47%.
(a)
(b)
(c)
(d)Figure 5
The parameter estimation errors of the linear block in the second branch under differentδns: (a) δns=7.33%; (b) δns=14.13%; (c) δns=22.64%; (d) δns=31.47%.
(a)
(b)
(c)
(d)Next, the intermediate variablesv^1t and v^2t are estimated by using the input signals u1t and u2t and the estimated parameters of linear blocks. Then, the antecedent parameters of two nonlinear blocks are identified by using the clustering algorithm according to the following design parameters [41, 42]: S01=0.98, ρ1=1.61, λ1=0.02 and S02=0.988, ρ2=1.05, λ2=0.1, which result in two neurofuzzy models whose rules are 6 and 80, respectively. Moreover, the consequent parameters W1 and W2 of two nonlinear blocks are obtained according to (30) and (31). The polynomial method described in [18] is also constructed by using the identical data to identify the nonlinear blocks. The comparisons between the proposed algorithm and the method in [18] are shown in Figures 6 and 7. The MSEs of two neurofuzzy models are 3.5×10−3 and 2.0×10−3, respectively, while the MSEs of two polynomial functions are 2.1×10−2 and 4.1×10−2, respectively. As illustrated in Figures 6 and 7, the proposed algorithm has better identification accuracy for the nonlinearity of the actual process.Figure 6
The static nonlinear block in the first branch.Figure 7
The static nonlinear block in the second branch.


## 4.2. Example 2

To further verify the performance of the proposed algorithm, this example considers a TISO Wiener model whose nonlinearities are two more complicated discontinuous segment nonlinear functions:(35)v1t=0.4q−11+0.8q−1u1t,y1t=2−cos3v1t−exp−v1t,v1t≤1,−1,v1t>1,v2t=0.6q−11+0.3q−1u2t,y2t=xt=tanh2v2t,v2t≤1.5,−expv2t−1expv2t+1,v2t>1.5,yt=y1t+y2t+ηt.In order to identify the above TISO Wiener model, the following signals as shown in Figure8 are adopted: (1) Gaussian signal u1t with zero means and variance 1.5; (2) Gaussian signal u2t with zero means and variance 1.9; and (3) the corresponding output signal yt.Figure 8
Part of the training data used in Example 2.Firstly, correlation analysis theorem in Section3.1 is employed to identify the linear blocks, so the estimated linear blocks are v^1t=−0.8052v^1t−1+0.3840u1t−1 and v^2t=−0.2859v^2t−1+0.5980u2t−1, and the modifying coefficients are b01=Eytvt/Evtvt=0.9536 and b02=Eytvt/Evtvt=0.5680, respectively. The estimation errors of two linear blocks are 3.66×10−2 and 2.18×10−2, respectively.Next, the design parameters are set as:S01=0.9991, ρ1=1.991, λ1=0 and S02=0.995, ρ1=1.85, λ1=0, resulting in two neurofuzzy models whose rules are 113 and 55, respectively. Then, the weights W1 and W2 of two neurofuzzy models are obtained by using (30) and (31). For comparison purpose, the polynomial-based TISO Wiener model is constructed by using the identical training data. Table 3 shows the MSEs for different polynomials based models in two branches; the models with orders of 19 and 32 are chosen to be the best models of two subsystems, respectively. The MSEs of the proposed TISO Wiener model based on different fuzzy rules are summarized in Table 4, and the two neurofuzzy models with fuzzy rules of 113 and 55 are proved to be the best models. Table 5 compares the MSEs for both training and validation data obtained by the best neurofuzzy-based model and the best polynomial-based model, respectively. Figures 9 and 10 show the approximation results of the nonlinearities of two branches. As described in Tables 3–5 as well as Figures 9 and 10, the neurofuzzy-based TISO Wiener model can achieve better approximation of nonlinearities than the polynomial-based TISO Wiener model.Table 3
MSEs of various polynomial-based TISO Wiener models.
The nonlinear block in the first branch
The nonlinear block in the second branch
Order
MSE (training)
MSE (validation)
Order
MSE (training)
MSE (validation)
7
6.01 × 10−2
7.12 × 10−2
8
6.79 × 10−1
7.56 × 10−1
12
4.31 × 10−2
6.78 × 10−2
17
5.98 × 10−1
6.23 × 10−1
16
3.31 × 10−2
5.34 × 10−2
24
4.03 × 10−1
6.02 × 10−1
19
1.24 × 10−2
3.80 × 10−2
32
2.10 × 10−1
4.15 × 10−1Table 4
MSEs of the proposed TISO Wiener model with different fuzzy rules.
The nonlinear block in the first branch
The nonlinear block in the second branch
The number of fuzzy rules
MSE (training)
MSE (validation)
The number of fuzzy rules
MSE (training)
MSE (validation)
45
5.80 × 10−3
8.20 × 10−3
20
2.96 × 10−3
7.30 × 10−3
90
4.03 × 10−3
3.80 × 10−2
33
3.70 × 10−3
8.60 × 10−2
113
3.79 × 10−3
4.90 × 10−3
45
1.80 × 10−3
4.30 × 10−3
122
5.40 × 10−3
6.30 × 10−3
55
1.09 × 10−3
2.70 × 10−3Table 5
MSEs of the best polynomial-based and the proposed TISO Wiener model.
The nonlinear block in the first branch
The nonlinear block in the second branch
Model
MSE (training)
MSE (validation)
Model
MSE (training)
MSE (validation)
Neurofuzzy
3.79 × 10−3
4.90 × 10−3
Neurofuzzy
1.09 × 10−3
2.70 × 10−3
Polynomial
1.24 × 10−2
3.80 × 10−2
Polynomial
2.10 × 10−1
4.15 × 10−1Figure 9
The static nonlinear block in the first branch.Figure 10
The static nonlinear block in the second branch.


## 5. Conclusion

This paper extends Bussgang’s classic theorem to a MISO Wiener system and proposes a novel identification method for the correlation analysis algorithm-based MISO Wiener model with noise. At first, several sets of Gaussian signals are adopted to identify the MISO Wiener model, resulting in the identification problem of the dynamic linear blocks separated from that of the static nonlinear blocks. Next, correlation analysis is employed to estimate the parameters of the linear blocks. Finally, least squares algorithm-based identification method of static nonlinear block is presented for the MISO Wiener model. The proposed approach can not only avoid the influence caused by white noise but also solve the problem of initialization and convergence of the model parameters discussed in the existing iterative method used for the identification of the MISO Wiener model. Examples demonstrate the effectiveness of the proposed algorithm.



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*Source: 9650254-2019-11-12.xml*