Filtering-Based Parameter Identification Methods for Multivariable Stochastic Systems

This paper presents an adaptive filtering-based maximum likelihood multi-innovation extended stochastic gradient algorithm to identify multivariable equation-error systems with colored noises. The data filtering and model decomposition techniques are used to simplify the structure of the considered system, in which a predefined filter is utilized to filter the observed data, and the multivariable system is turned into several subsystems whose parameters appear in the vectors. By introducing the multi-innovation identification theory to the stochastic gradient method, this study produces improved performances. The simulation numerical results indicate that the proposed algorithm can generate more accurate parameter estimates than the filtering-based maximum likelihood recursive extended stochastic gradient algorithm.


Introduction
System identification is the theory and methods of establishing the mathematical models of dynamical systems [1][2][3][4][5] and some identification approaches have been proposed for scalar systems and multivariable systems [6][7][8][9][10][11]. Multivariable systems exist more widely in modern large-scale industrial processes, multivariable systems can more accurately describe the characteristics of dynamic processes, and have extensive application prospects to study the identification methods of multivariable systems [12][13][14]. The identification methods of multivariable systems can be regarded as an extension of those of scalar systems [15,16]. Therefore, how to identify the multivariable systems by extending the identification methods of scalar systems has attracted much attention. This paper focuses on the identification issues of multivariable systems with complex structures and many parameters. For decades, many parameter estimation methods have been developed for multivariable systems, such as the stochastic gradient methods [17], the iterative methods [18], the recursive least-squares methods [19,20] and the blind identification methods [21]. The maximum likelihood algorithm has good statistical properties and can deal with colored noises directly [22][23][24]. The present study aims to investigate a more efficient algorithm based on the maximum likelihood principle, the negative gradient search, the data filtering, and the multi-innovation identification theory.
The complex structures and high dimensions in the parameter matrices of the multivariable systems lead to the increase in computational complexity [25][26][27]. Inspired by the hierarchical control based on the decomposition-coordination principle for large-scale systems, the hierarchical identification can be served as the solution to reduce the computational intensity by decomposing the identification model into several subsystems with smaller dimension and fewer variables [28].
Differing from the hierarchical identification [29], the model decomposition technique, which is based on the matrix row and column multiplication expansion, is an effective method to reduce the computational burden. Recently, the model decomposition technique are used in [30,31] to reduce the computational complexity by transforming the multivariable system into several small-scale subsystems with only the parameter vectors to be determined. By changing the noise model structure of the subsystem to whiten the colored noise, an adaptive filter is designed to filter the observed data, then the subsystem identification model is further simplified and the parameter estimation accuracy is improved [32][33][34]. For ARX models with unmeasurable outputs, a modified Kalman filter was designed and a new multi-step-length formulation was derived to improve the performance of the gradient iterative algorithm [35].
The advantage of the stochastic gradient methods is that they need less computational effort compared to existing identification methods [36,37]. Due to their zigzagging behavior, the stochastic gradient methods have slow convergence rates [38,39]. The focus of this paper is to investigate a new method with computational efficiency by introducing the multi-innovation identification theory into the stochastic gradient method. The innovation is the useful information that can improve the accuracy of parameter estimation or state estimation. From the viewpoint of innovation modification, the multi-innovation identification theory improves the convergence rate and parameter estimation accuracy from the following two aspects [40,41]. Firstly, the multi-innovation method uses not only the current data but also the past data in each recursive calculation step, which is the reason to improve the convergence rate. Secondly, the multi-innovation method repeatedly utilizes the available data in the neighboring two recursions, which is the reason to improve the parameter estimation accuracy. In this aspect, multi-innovation methods have been developed in [42,43]. It is well known that an increasing innovation length leads to better parameter estimation accuracy, but the price paid is a large computational effort [44,45]. The difficulty arises as to how to choose the innovation length.
In summary, although a filtering and maximum likelihood-based recursive least-squares algorithm is available for multivariable systems with complex structures and colored noises [32], there remains a need for enhancing the parameter estimation accuracy with computational efficiency. Motivated by these considerations, this paper has the following contributions:

•
The data filtering and model decomposition techniques are used to reduce the computational complexity of the multivariable systems contaminated by uncertain disturbances.
• A filtering-based multivariable maximum likelihood multi-innovation extended stochastic gradient (F-M-ML-MIESG) algorithm is proposed for improved parameter estimation accuracy while retaining desired computational performance.
X := A: X is defined by A. k: The time variable.
Consider the following multivariable controlled autoregressive autoregressive moving average (M-CARARMA) model: where y k ∈ R m and u k ∈ R r are the output and input vectors, respectively, v k ∈ R m denotes random white noise vector with zero mean and variance σ 2 . The polynomials A(q), Q(q), C(q), and D(q) are expressed as Assume that y(k) = 0, u(k) = 0, and v(k) = 0 for k 0, the orders n a , n b , n c , and n d are known. Differing from the work in [32], the focus of this paper is to derive a new method to identify the polynomial coefficients A l , Q l , c l , and d l . Referring to the work in [32], in order to reduce the computational complexity, Equation (1) is decomposed into several subsystems. Then, the ith subsystem can be represented as Define From (2), it follows that Multiplying both sides of the above equation by C(q) gives That is, Then, the subsystem identification model can be expressed as Define an intermediate variable or From (4), it follows that
The flowchart of computing the estimatesθ i1,k ,ĉ i,k andΘ k by the F-M-ML-RESG algorithm in (9)-(30) is shown in Figure 1.
Obtain the estimateΘ k ? End

The F-M-ML-MIESG Algorithm
In order to further enhance the parameter estimation accuracy of the F-M-ML-RESG method, by introducing the multi-innovation identification theory, an F-M-ML-MIESG method is investigated. Define the information matrixΓ i1 (p, k), the filtered information matrixΓ i1f (p, k), and the stacked output vectorŶ i1 (p, k) asΓ where p is the innovation length. Define the stacked output vectorŴ i (p, k), the information matrix Γ ic (p, k), and the information matrixΓ id (p, k) aŝ Γ ic (p, k) := [φ ic,k ,φ ic,k−1 , · · · ,φ ic,k−p+1 ] ∈ R n c ×p ,

The F-M-ML-MIESG Algorithm
In order to further enhance the parameter estimation accuracy of the F-M-ML-RESG method, by introducing the multi-innovation identification theory, an F-M-ML-MIESG method is investigated. Define the information matrixΓ i1 (p, k), the filtered information matrixΓ i1f (p, k), and the stacked output vectorŶ i1 (p, k) asΓ where p is the innovation length. Define the stacked output vectorŴ i (p, k), the information matrix Γ ic (p, k), and the information matrixΓ id (p, k) aŝ Γ ic (p, k) := [φ ic,k ,φ ic,k−1 , · · · ,φ ic,k−p+1 ] ∈ R n c ×p , Referring to the work in [18,42,43], Equation (10) becomes the following equation: Equation (22) can be reformulated into the following equation: Based on the F-M-ML-RESG method in (9)-(30), the F-M-ML-MIESG method can be obtained as follows:θ The F-M-ML-RESG method is a special case of the F-M-ML-MIESG method because, when p = 1, the F-M-ML-MIESG method degenerates into the F-M-ML-RESG method. The proposed approaches in the paper can combine other estimation algorithms [46][47][48][49][50] to study the parameter identification problems of linear and nonlinear systems with different disturbances [51][52][53][54][55], and can be applied to other fields [56][57][58][59][60] such as signal processing and process control systems. The F-M-ML-MIESG method consists of the following steps for computingθ i1,k ,ĉ i,k andΘ k : The flowchart of computing the estimatesθ i1,k ,ĉ i,k andΘ k by the F-M-ML-MIESG algorithm in (31)-(58) is shown in Figure 2.
The model decomposition technique is applied to solve the coupling relationship between the input and output variables of the multivariable system. Thus, the complexity of system identification algorithms is reduced. The data filtering technique is used to filter the observed data. Hence, the subsystem identification model is simplified. The proposed method is based on the data filtering technique, the coupling identification concept, the multi-innovation identification theory, and the negative gradient search for improved parameter estimation and computational performance. The maximum likelihood principle is utilized to estimate the parameters of the noise model directly. Start Collect u k and y k , computeŷ 1,k andû 1,k , Obtain the estimateΘ k ? End

Examples
Example 1. Consider the following M-CARARMA model  Table 1, the parameter estimatesθ 1,k andθ 2,k versus k are shown in Figure 3. When σ 2 1 = σ 2 2 = 0.10 2 , the F-M-ML-MIESG parameter estimation errors versus k with different innovation lengths p are shown in Figure 4. When p = 9, the F-M-ML-MIESG parameter estimation errors versus k with different noise variances are shown in Figure 5.   Example 2. Consider the following another 3-input and 3-output system:  The simulation conditions of this example are similar to those in Example 1. Applying the F-M-ML-MIESG algorithm to estimate the parameters of this example system, the simulation results are shown in Table 2, Figures 6 and 7.

Conclusions
This paper considers the parameter estimation of the linear M-CARARMA system with an ARMA noise. By means of an adaptive linear filter, the subsystem identification model is simplified, then an F-M-ML-MIESG method is discussed by introducing the multi-innovation identification theory to the stochastic gradient method. The purpose of an adaptive filter is to improve the parameter estimation accuracy by filtering the observed data without changing the relationship between input and output data. Both the model decomposition technique and the data filtering technique are used to reduce the system complexity, and the identification model is simplified. The simulation validation demonstrates that the F-M-ML-MIESG method provides a higher parameter estimation accuracy than the F-M-ML-RESG method when p ≥ 2. The proposed filtering-based parameter identification methods for multivariable stochastic systems in this paper can be extended to study the identification issues of other scalar and multivariable stochastic systems with colored noises [61][62][63][64][65][66] and can be applied to some engineering application systems [67][68][69][70][71][72][73] such as filtering, estimation, prediction [74][75][76][77][78][79][80][81], and so on.