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

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.


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].
Up to now, plenty of methodologies have been used to identify block-oriented models. In general, the identi cation approaches based on input-output signals for block-oriented models can be divided into two parts: the synchronous method and the separate step algorithm. e synchronous method identi es 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 identi cation method [10], the subspace algorithm [11], the overparameterization approach [12], and the hierarchical identi cation methods [13]. e 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 identi cation 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 intelligencebased 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 R vu (τ) � b 0 R u (τ) holds for an arbitrary static nonlinearity if the input signal is separable, where R vu (τ) � E(v(k)u(k − τ)) is the cross-correlation between the input signal u(t) and the output signal v(t), R u (τ) � E(u(k)u(k − τ)) is the autocorrelation of the input signal, and b 0 � E(f ′ (u(k))), 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 [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][30][31], and the correlation analysis algorithm was applied to the SISO Wiener model in [32]. However, most of the real industry processes [33][34][35] are inherently multivariable systems which can describe the relationships among different variables more accurately. us, multivariable nonlinear identification techniques are required. In addition, according to [36][37][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. erefore, motivated by our previous works, a novel identification algorithm for correlation analysis algorithm-based MISO-distributed Wiener model with noise is presented. e 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. en, 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. e rest of the paper is organized as follows. e identification problem of the MISO neurofuzzy Wiener model with noise is described in Section 2. 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.

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 where u i (t)(i � 1, 2, . . . , m) and y(t) 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, is the static nonlinearity of the i-th branch of the MISO Wiener model. y i (t)(i � 1, 2, . . . , m) which cannot be measured is the output of the nonlinear block. η(t) denotes the white noise, q − 1 is the backward shift, and n ia and n ib are integers concerned with the model order.
In this paper, the target of the proposed identification approach is to obtain a MISO Wiener model such that the following cost function E is made acceptable small, that is, where ε is a given tolerance, f(·) is the estimated nonlinear function, y(t) is the estimated output of the MISO Wiener model, v i (t) is the estimated intermediate variable of i-th 2 Complexity branch, N P is the number of the input and output data, and "∧" represents "model". As shown in Figure 2, the nonlinear function of each branch of the MISO Wiener model is represented by a fourlayer neurofuzzy system which integrates the Takagi-Sugeno fuzzy system and radial basis function-based feed-forward network into a connection structure. e first layer is the input layer in which the nodes just transmit the input signal to the next layer. e second layer is the membership layer where the membership function of the input variable is calculated. e third layer is the fuzzy rule layer, and the number of the nodes in this layer denotes the fuzzy rules. e last layer is the output layer. e output of the i-th neurofuzzy model is given by where where c il and σ il denote the center and width of the membership function, respectively, w il is the consequent parameters of the neurofuzzy model, and N i represents the fuzzy rule of the nonlinear block of the i-th branch.

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 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., And presume that the cross-correlations between different input signals are zero. en, the intermediate variable v i (t) of the MISO Wiener model can be written as follows: en, the conditional expectation of v i (t) can be deduced: Layer2 Layer3 Layer4 Fuzzifier Inferrence Defuzzifier Figure 2: e structure of the neurofuzzy model. Complexity 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, v i (t) subjects to normal distribution with zero means and variance Y i,t subjects to normal distribution with zero means and variance σ Y i � σ 2 As can be seen from (7), and thus, the joint probability density function of U i,t− τ− k and V i,t is as follows: where Jac is the Jacobian determinant, that is (see [39]), us, the conditional probability density function of U i,t− τ− k is given by where the coefficients are defined in the following equation and the constant c is the value of v i (t): As a result, the conditional expectation of From (7) and (12), we can obtain the following equation: . , m is a constant. In conclusion, according to [26], the intermediate variable v i (t) satisfies the separability of the process. (1), if the input signals are Gaussian signals with independent identical distribution and the expectations of inputs satisfy E(u i (t)) � 0 (i � 1, 2, . . . , m), the following equation holds:

Theorem 1. For a MISO Wiener model as expressed in
where Proof. As can be seen from (13), when the input is Gaussian signal, the output v i (t) of the dynamic of the i-th branch satisfies the separability of the process. us, for a MISO Wiener model, we can get the following equation: From (15), we know that the cross-correlation function between v i (t) and the immeasurable variable y i (t) can be replaced by the cross-correlation function between v i (t) and y(t): According to (15) and (16), the following equation holds: According to [40], for a stable and time-invariant linear system, its impulse response can be represented by us, the output of the dynamic linear block of i-th branch can be written as And (15) and (16) can be rewritten as follows: en, by substituting (20) and (21) into (17), we can get the following equation: where ∞ k�1 g i (k)q k on the left side of (22) is equivalent to ∞ ς�1 g i (ς)q ξ on the right side of (22). us, we have en, (14) holds. is 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. e content before eorem 1 has proved that v i (t) satisfies the separability of the process when the input signals u i (t) are Gaussian signals.
us, R y i v i (τ) � b 0i R v i (τ) holds according to Bussgang's classic theorem. However, as shown in Figure 1,  v 1 , v 2 , . . . , v m and y 1 , y 2 , . . . , y m are immeasurable variables, and R y i v i (τ) � b 0i R v i (τ) cannot be used to identify the parameters of the proposed Wiener model directly. erefore, eorem 1 is proposed to solve this problem. e input and output relationship of the MISO Wiener model, that is , is deduced so that the measurable Complexity 5 variables u 1 , u 2 , . . . , u m and y can be utilized to estimate the parameters of dynamic linear blocks.

Identification of the Dynamic Linear Blocks.
Correlation analysis is adopted to identify the linear blocks of the MISO Wiener model with noise. According to eorem 1 and (23), we have where b ij � b 0i b ij . en, the following equation holds: us, the parameters of dynamic linear block can be estimated by least squares algorithm: where where R yu i (τ) and R u i (τ) are as follows:

Identification of the Static Nonlinear Blocks. After the linear blocks have been identified, the intermediate variable v i (t)
can be further estimated according to the input signal u i (t) 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 w ij . From (1) and (4), we have where . L i is the clustering number of nonlinear block of the i-th branch.
As a result, the estimation of W can be obtained by using least squares approach: According to (30), substitute y(t) with Wψ(t) + η(t), and then (31) can be rewritten as As the noise is the white noise with zero means, thus we have In conclusion, W LS is the unbiased estimation of W. e identification algorithm for the correlation analysisbased MISO Wiener model is concluded as follows: Step 1. e parameters a ij and b ij of the dynamic linear blocks of the MISO Wiener model are estimated according to (26)- (29) in Section 3.1 Step 2. e antecedent parameters c il and σ il are obtained by using the clustering approach [41,42] 6 Complexity Step 3.
e consequent parameter W of each static nonlinear block is identified according to (30) and (31) in Section 3.2
In order to identify the above TISO Wiener model, the following signals as shown in Figure 3 are adopted: (1) Gaussian signal u 1 (t) with zero means and variance 4; (2) Gaussian signal u 2 (t) with zero means and variance 4; and (3) the corresponding output signal y(t).
At first, u 1 (t), u 2 (t), and y(t) 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 b 01 � E(y(t)v(t))/E(v(t)v(t)) � 0.4517 and b 02 � E(y(t)v(t))/E(v(t)v(t)) � 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. e 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.
Next, the intermediate variables v 1 (t) and v 2 (t) are estimated by using the input signals u 1 (t) and u 2 (t) and the estimated parameters of linear blocks. en, the antecedent parameters of two nonlinear blocks are identified by using the clustering algorithm according to the following design parameters [41,42]: S 01 � 0.98, ρ 1 � 1.61, λ 1 � 0.02 and S 02 � 0.988, ρ 2 � 1.05, λ 2 � 0.1, which result in two neurofuzzy models whose rules are 6 and 80, respectively. Moreover, the consequent parameters W 1 and W 2 of two nonlinear blocks are obtained according to (30) and (31). e polynomial method described in [18] is also constructed by using the identical data to identify the nonlinear blocks. e comparisons between the proposed algorithm and the method in [18] are shown in Figures 6 and 7. e 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.

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: In order to identify the above TISO Wiener model, the following signals as shown in Figure 8 are adopted: (1) Gaussian signal u 1 (t) with zero means and variance 1.5; (2) Gaussian signal u 2 (t) with zero means and variance 1.9; and (3) the corresponding output signal y(t).
Next, the design parameters are set as: S 01 � 0.9991, ρ 1 � 1.991, λ 1 � 0 and S 02 � 0.995, ρ 1 � 1.85, λ 1 � 0, resulting in two neurofuzzy models whose rules are 113 and 55, respectively. en, the weights W 1 and W 2 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. e 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 neurofuzzybased 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 neurofuzzybased TISO Wiener model can achieve better approximation of nonlinearities than the polynomial-based TISO Wiener model.

Conclusion
is 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. e 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.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.