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# State Estimation Moving Window Gradient Iterative Algorithm for Bilinear Systems Using the Continuous Mixed *p*-norm Technique

Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, China

* Corresponding Author: Weili Xiong. Email:

(This article belongs to the Special Issue: Advances on Modeling and State Estimation for Industrial Processes)

*Computer Modeling in Engineering & Sciences* **2023**, *134*(2), 873-892. https://doi.org/10.32604/cmes.2022.020565

**Received** 30 November 2021; **Accepted** 28 March 2022; **Issue published** 31 August 2022

## Abstract

This paper studies the parameter estimation problems of the nonlinear systems described by the bilinear state space models in the presence of disturbances. A bilinear state observer is designed for deriving identification algorithms to estimate the state variables using the input-output data. Based on the bilinear state observer, a novel gradient iterative algorithm is derived for estimating the parameters of the bilinear systems by means of the continuous mixed*p*-norm cost function. The gain at each iterative step adapts to the data quality so that the algorithm has good robustness to the noise disturbance. Furthermore, to improve the performance of the proposed algorithm, a dynamic moving window is designed which can update the dynamical data by removing the oldest data and adding the newest measurement data. A numerical example of identification of bilinear systems is presented to validate the theoretical analysis.

## Keywords

Although many physical dynamic behaviors are characterized as linear systems in the neighborhood of a single operating point, when they exhibit strong nonlinearities or must be described over the entire operating range, linear models may not yield appropriate results [1–3]. Owing to that, the study of system identification and parameter estimation for nonlinear systems has drawn considerable attention of academic researchers [4–6]. Various models are exploited to describe actual nonlinear systems with relatively simple structures [7–11] such as Hammerstein systems, Bilinear systems, and Wiener systems.

The parameter estimation of the system models is important for control system analysis and design. The parameters of the models can be estimated by using some identification methods [12–15] such as the hierarchical algorithms [16–19]. In this paper, we confine our discussion to bilinear systems, which are simplicity in model structures and capable of approaching arbitrary nonlinear systems with much higher accuracy than traditional linear approximations theoretically [20]. Bruni et al. designed a bilinear time-series model-based self-tuning controller for a multi-machine power system to enhance the region of stability of the system, and return the states to their stable equilibrium [21]. Yeo et al. developed the bilinear model predictive control algorithm for paper plant systems to control the grade change operations in paper production [22]. Wang et al. simulated the bridge nonlinear boundary as bilinear translational, and applied the nonlinear least squares optimization algorithm to identify the nonlinear translational and rotational boundary parameters of the bridge [23].

Many researches have been studied in the parameter estimation of bilinear state space systems. A stochastic gradient algorithm and a gradient-based iterative algorithm have been proposed for the parameter identification of bilinear systems by using the auxiliary model. The gradient-based iterative algorithm uses fixed batch data to update the parameters, so that the parameter estimation accuracy can be greatly improved compared with the stochastic gradient algorithm. Li et al. combined the maximum likelihood theory with the data filtering technique for bilinear systems with colored noises. The state observer is vital in the field of state estimation of bilinear systems [24]. Tsai considered a H-infinity fuzzy observer for bilinear systems by means of a linear matrix inequality approach [25]. Phan et al. presented a full-order bilinear state observer for the bilinear system, which optimized the observer gain by interaction matrices [26]. Zhang et al. considered the state estimation problem of bilinear systems and proposed a state filtering method for the single-input single-output bilinear systems and multiple-input multiple-output bilinear systems by minimizing the covariance matrix of the state estimation errors [27]. Recently, some state and parameter estimation methods have been proposed for linear and bilinear state space systems in the presence of exogenous noises [28–30].

However, in industrial applications, the dynamical processes often work on various noise environments, and some works used nonlinear filtering technique such as median filtering and least mean

Inspired by the above researches, we study the parameter estimation algorithms for bilinear state space models with the noise disturbances. Based on the iterative search and the state estimator, a bilinear state continues mixed

• A state observer is designed to obtain the system states variables consisting of the product terms of state variables and control variables.

• A bilinear state observer-based continues mixed

• A bilinear state observer-based moving window CMPN-GI (BSO-MW-CMPN-GI) algorithm is derived to update collected data and thus maintain high data utilization.

The outlines of this paper are organized as follows. Section 2 introduces some definitions and proposes the identification model of the bilinear state space system and introduces a bilinear state observer for the state estimation. Sections 3 and 4 derive a BSO-CMPN-GI and a MW-CMPN-GI algorithms based on the bilinear state estimator, respectively. An example to illustrate the effectiveness of the proposed algorithms in Section 5. Finally, Section 6 gives some concluding remarks.

2 System Description and Identification Model

Consider the following single-input single-output bilinear state space modelblue:

where

where

Assumption 1: The dimension n of the system state vector is known,

Assumption 2: The bilinear system in Eqs. (1) and (2) is observable and controllable.

Eq. (1) can be written as

Adding both sides of the above equations has

Substituting Eq. (5) into Eq. (2) obtains the regression form of the bilinear state model

Define the information vector

Redefine the noise term as

Eq. (6) can be rewritten as

The proposed parameter estimation algorithms in this paper are based on this identification model in (16). Many identification methods are derived based on the identification models of the systems [34–43] and these methods can be used to estimate the parameters of other linear systems and nonlinear systems [44–50] and can be applied to other fields such as chemical process control systems. The purpose of this paper is to obtain the estimates of the unknown parameters in the bilinear state space model by means of the measurement data

where

Remark 1: Differently from the identification of bilinear-in-parameter systems which involve the product terms of the parameter vectors and the information matrices, this paper studies the identification problem of bilinear state-space systems with the product terms of state variables and control variables, which makes it difficult for the parameter and state estimation.

3 Bilinear State Observer Based Continuous Mixed

Set the data length be L, based on the input and output data

To suppress the effect of noise interference, define the continuous mixed

where

Using the negative gradient search method to minimize the cost function

where

Therefore, Eq. (19) can be expressed as

where

Let

where

Eqs. (21) and (22) cannot figure out the parameter estimation vector

where

Replace the information matrix

To obtain a closed form formula for

The flowchart of the BSO-CMPN-GI algorithm in Eqs. (24)–(30) is shown in Fig. 1.

Remark 2: The parameter p in BSO-CMPN-GI is adapted by continuous

Remark 3: The BSO-CMPN-GI algorithm makes full use of the measurement data in each iteration of the calculation process, but it requires a batch of data to be collected in advance, and thus is implemented offline. Therefore, an on-line identification algorithm derived from the BSO-CMPN-GI algorithm will be introduced by exploiting the past and current measurement data to estimate the unknown parameters in Section 4.

4 Bilinear State Observe Moving Window Continuous Mixed

In this section, we introduce the moving window method to derive an on-line identification algorithm and enhance the performance of the BSO-CMPN-GI algorithm. The length of the moving window is set as a fixed value m. Define the stacked output vector

Consider the measurements from

where

Taking the gradient of

where

Therefore, Eq. (4) can be expressed as

where

Let

where

As pointed earlier in Section 3, the unknown states

where

Based on the above derivation, a bilinear state observe-based moving window continuous mixed

The flowchart of the BSO-MW-CMPN-GI algorithm in Eqs. (39)–(45) is shown in Fig. 2.

Remark 4: The parameter estimate given by the BSO-MW-CMPN-GI algorithm depends only on the iterative counter s, but also time t. As the sampling time t increases, the BSO-MW-CMPN-GI algorithm can utilize a batch of data to calculate the parameter estimate simultaneously.

Case 1: About parameter estimation

Consider a second-order bilinear state space system:

The parameter vector to be estimated is

In simulation, the input

• The parameter estimation errors given by the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm become smaller as the iteration s increases. It thus to say the proposed algorithms are effective for bilinear systems.

• The state estimates are close to their true values with t increasing.

• Under the same data length, a lower noise variance leads to higher parameter estimation accuracy by the BSO-GI algorithm, the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm.

• The BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm possess higher parameter estimation accuracy at the same noise variance compared with the BSO-GI algorithm.

• When comparing the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm, the parameter estimation errors of the BSO-MW-CMPN-GI algorithm become smaller with m increasing, and approach to zero if m is large enough.

Case 2: About model validation

Applying the BSO-CMPN-GI algorithm and the BSO-MW-CMPN-GI algorithm to construct the estimated model for the model validation, respectively. Take the data from

where

From Fig. 9, we can see that the predicted outputs of the BSO-CMPN-GI and the BSO-MW-CMPN-GI are very close to the true outputs, and the RMSEs of the two algorithms are very close to the noise standard deviation

This paper studies the parameter identification problems of the nonlinear systems described by the bilinear state space models with the noise disturbances. A bilinear state observe-based continuous mixed

Funding Statement: This research was funded by the National Natural Science Foundation of China (No. 61773182) and the 111 Project (B12018).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

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*p*-norm technique.

*Computer Modeling in Engineering & Sciences*,

*134*

*(2)*, 873-892. https://doi.org/10.32604/cmes.2022.020565

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*p*-norm Technique,"

*Comput. Model. Eng. Sci.*, vol. 134, no. 2, pp. 873-892. 2023. https://doi.org/10.32604/cmes.2022.020565

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