**1. Introduction**

In the typical control systems, the measurement is the primary component that senses the system status and provides the input information. The measurement accuracy influences the control effect directly. Especially in intelligent terminals, such as industrial robots, unmanned aerial vehicles and unmanned vehicles, various sensors are implemented to measure the motion state and working condition. The sensors are expected to be of high precision. However, the precision is limited by the manufacturing technique when the intelligent terminals are with small size and low cost relying on the micro-electromechanical sensors [1,2]. For the complicated noises, it is essential to filter the noises. Then the motion state and condition can be estimated accurately for the target tracking and control.

In the noise filtering and state estimation field, many methods have been studied, such as the wavelet filter [3], time-frequency peak filtering [4], empirical mode decomposition [5] and the Kalman filter [6]. These filtering and estimation algorithms are often based on the mathematical models and established using the iterative schemes [7–9] or recursive schemes [10,11]. Some filtering-based estimation algorithms use input-output representations [12,13], and others use state-space models [14–16]. Among these methods, Kalman filter is a state estimation algorithm based on the state-space model. It introduces the state space into the stochastic estimation theory and obtains the optimal estimation without requiring a vast amount of historical data. However, it is obvious that the Kalman filter depends highly on some assumptions that the system model is linear, the process and measurement noises are standard Gaussian, and their covariance matrixes are all known. When these assumptions are seriously limited in reality, two categories of methods have been explored. On the one hand, the adaptive Kalman filter (AKF) was proposed focusing on the parameter adjustment to approximate

the filtering process to the practical system. Then common AKF includes innovation-based adaptive estimation (IAE) [17], multiple model adaptive estimation (MMAE) [18] and adaptive fading Kalman filter (AFKF) [19]. On the other hand, some methods focus on nonlinear systems, such as extended Kalman filter (EKF) [20], unscented Kalman filter (UKF) [21], noise-robust filter [22] and other estimation methods [23,24]. The two categories of the methods try to describe and represent the system features with the variable approximation. The filters will be efficient if the alternative expression of the model and parameter is similar to the system dynamic characteristic.

The methods above mainly extract and represent the system characteristics with the existing information. While the characteristic extraction is the specialty of machine learning, the artificial neural network (ANN) has been introduced in noise filtering and state estimation. The leading solution is the distributed mode in which Kalman filter and ANN are applied separately in sequential order [25–27]. In the mode, the neural network mainly preprocesses or reprocesses the data before or after the filtering process. However, the inner relation in the Kalman filter has not been explored deeply with ANN. Then it becomes an issue on how to extract the relationship of parameters in the Kalman filter and optimize the filtering results with the limited existing information.

Because of the advantages in Kalman filter and the neural network, a new neuron-based Kalman filter is built in this paper. It mainly enhances the filtering process with the existing information. The potential numerical relation of the intermediate variables in the Kalman filter is explored with the feature extraction and nonlinear fitting ability of the neural network. In the paper, neurocomputing is integrated with the inner components of the Kalman filter. The nonlinear autoregressive model is introduced and constructed to predict and modify the critical intermediate variables in the Kalman filter. The simulation and practical experiments have verified the precision and feasibility of the proposed filter.

This paper is organized as follows: Section 2 introduces the underlying theory and related works on noise filtering. Section 3 presents the main proposed filter with the framework and network design. The simulation and experiment are designed and conducted in Section 4. The results and work are discussed in Sections 5 and 6 finally concludes the paper.

### **2. Related Work**

As the typical filtering method, the Kalman filter is selected as the basic framework in this paper. The basic theory and developments of the Kalman filter are introduced firstly. Then the related work is presented on the integration of the filter and neural networks.

### *2.1. Kalman Filter and Its Improvement*

Because of its clearness and convenience in computer calculation, the Kalman filter has been the classical method in the filtering and estimation of Gaussian stochastic systems [28,29]. It is applied widely in target tracking [30], integrated navigation [31], communication signal processing [32], etc. Kalman filter introduces the state space description in the time domain, in which the estimated signal is set as the output of the stochastic linear system in the action of white noise. Kalman filter is appropriate for the stationary process and the non-stationary Markov sequence.

For the detailed analysis in the paper, the main algorithm of the Kalman filter is presented here. The discrete model can be expressed as:

$$\mathbf{x}(k+1) = A(k)\mathbf{x}(k) + w(k) \tag{1}$$

$$z(k) = \mathcal{C}(k)x(k) + v(k)\tag{2}$$

where *x*(*k*) is the to-be-estimated variable or state variable, *z*(*k*) is the measurement value from sensors, *A* is the state transition matrix or process matrix, *C* is the measurement matrix, *w*(*k*) is the process noise, *v*(*k*) is the measurement noise. The concrete Kalman filter algorithm is shown as follows:

(1) State estimation updating:

$$\pounds(k|k) = \pounds(k|k-1) + K(k)[z(k) - \mathcal{C}(k)\pounds(k|k-1)]\tag{3}$$

(2) One step forward prediction:

$$
\mathfrak{X}(k|k-1) = A(k-1)\mathfrak{X}(k-1|k-1) \tag{4}
$$

(3) Filtering gain calculation:

$$K(k) = P(k|k-1)\mathbb{C}^T(k)[\mathbb{C}^T(k)P(k|k-1)\mathbb{C}(k) + R(k)]\tag{5}$$

(4) The variance of the state estimation calculation:

*P*(*k <sup>k</sup>* <sup>−</sup> <sup>1</sup>) = *<sup>A</sup>*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*P*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup> *<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*AT*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) + *<sup>Q</sup>*(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) (6)

$$P(k|k) = [I - K(k)C(k)]P(k|k-1)\tag{7}$$

where *x*ˆ(*k k*) is the posterior estimation, *<sup>x</sup>*ˆ(*<sup>k</sup> <sup>k</sup>* <sup>−</sup> <sup>1</sup>) is the prior estimation which is also called the prediction, *K* is the filtering gain, *P* is the variance of the state estimation, *Q* is the variance of the process noise, *R* is the variance of the measurement noise.

There are assumptions in Kalman filter, namely that the process and measurement noises are standard Gaussian noises, and their covariance matrixes are all known. The assumptions deviate from the real systems. Then many studies have been carried out to improve Kalman filter from different solutions.

Some improvements were proposed for the nonlinear system, and the typical methods include EKF [20] and UKF [21]. In EKF, the Taylor expansion of the nonlinear function is truncated with the first-order linearization, and other higher-order terms are ignored. Then the nonlinear problem can be transformed into the linearity, which is suitable for the Kalman filter. In UKF, the prediction measurement values are represented with the sampling points, and the unscented transformation is used to deal with the nonlinear transfer of mean and covariance. EKF and UKF have been improved, as well as the integration with other methods [33–35].

Aside from nonlinear system methods, AKF methods have been studied to solve problems where mainly the settled and experiential parameters are given. The representative IAE [17], MMAE [18], and AFKF [19] are proposed based on the thought that the model parameter and noise statistics are modified with the observation and judgment during the filtering process. From a literature search, it was seen that some improvements in AKF [36–38] were presented recently. In the latest work [38], the colored noise is analyzed with the adaptive parameter. The second-order adaptive statistical model and Yule-Walker algorithm are used to recognize and filter the noises. The work is one of the latest representative improvements of AKF, and it can be set as a contrast in the experimental research.

The two categories of methods above, nonlinear and adaptive filters, mainly improve the filtering performance from the approximate system modeling and parameter adjustment. They are conducted based on inherent mathematics and statistic derivation. They provide an effective solution to promote the Kalman filter in the system mechanism analysis idea. The filtering and prediction are based on the mathematical models by assuming that the model parameters are known or estimated using some parameter identification methods, including the iterative algorithms [39–41], the particle-based algorithms [42–44] and the recursive algorithms [45–48].

### *2.2. Filter with Neural Network*

The methods in Section 2.1 improve the Kalman filter by modifying the system model and parameters based on the mathematic mechanism. The idea can be carried out with another data-driven solution. For the filtering parameter adjustment, the core task is to find and express the relation between parameters and process data, which meets the ability of neural networks. ANN has caused great concerns again with the trends of deep learning and artificial intelligence. ANN can fit the nonlinear model with excellent performance. It can solve the nonlinear and time-varying problems without a concrete internal mechanism model. For the problematic modeling of process and noise in the Kalman filter, ANN can be considered as a helpful tool to reconstitute the unknown elements in the filter. Scholars have made some efforts to explore the integrations of ANN and Kalman filter. The related research can be divided into two categories, including the distributed and crossed integration.

### 2.2.1. Distributed Integration of Kalman filter and ANN

For the distributed integration, Kalman filter and ANN are applied separately in sequential order. Liu et al. [25] smoothed the measurement value with the Kalman filter, and the filtered results were set as the input of the backpropagation neural network (BPNN). Hu et al. [49] estimated the target location with Kalman filter and the estimation was imported into BPNN to classify the targets. Liu et al. [50] utilized Kalman filter and fuzzy neural network (FNN) in a multi-source data fusion framework of an adaptive control system, in which data was processed firstly with Kalman filter, and the filtered results were set as the input of FNN. Others [26,27,51] used a Kalman filter and ANN in reverse order, in which ANN is constructed before the Kalman filter. Leandro et al. [26,27] built up BPNN to predict a variable, which is an important state variable in the Kalman filter. Cui et al. [51] proposed a radial basis function neural network (RBF) to train the GPS signals, and the RBF output is the input of adaptive Kalman filter, aiming at improving the processing precision.
