Gaussian–Versoria Mixed Kernel Correntropy-Based Robust Parameter and State Estimation for Bilinear State–Space Systems with Non-Gaussian Process and Measurement Noises

Abstract

This paper investigates the joint state and parameter estimation issue of the bilinear state–space system with non-Gaussian process noise and non-Gaussian measurement noise. Tackling such an issue is challenging because either of these noises may seriously degrade the estimation performance. To significantly counteract the negative effect of these non-Gaussian noises, a Gaussian–Versoria mixed kernel correntropy (GVMKC)-based cost function is introduced by integrating two different types of kernel functions into a mixed kernel. Subsequently, a GVMKC-based Kalman filtering and a GVMKC-based robust recursive least squares method are derived for estimating the system states and parameters, respectively. Thus, a robust joint parameter and state estimation method is developed by implementing the interactive computation. The effectiveness of the proposed method is confirmed by simulation examples.

1. Introduction

In practical industrial processes, the actual system model is generally unknown. Although first principles can be used for modeling according to physics and chemistry laws, this approach faces huge challenges due to the complexity of the system and incomplete understanding of the mechanism [1,2]. Under a realistic scenario, system identification provides an ideal approach to establish a system model, since it does not need in-depth understanding of the internal mechanism and only relys on externally collected measurements [3,4]. An appropriate model structure is a prerequisite for system identification. Over decades, various nonlinear models have been proposed, such as the Volterra series [5,6], block-oriented nonlinear systems [7,8,9], neural networks [10] and bilinear models [11,12]. Compared with other nonlinear models, bilinear models have good abilities to capture nonlinear characteristics and relatively simple structures since they only have one more additional term (i.e., the product term of the control variable and the system state) than the linear model [13]. Bilinear models can be regarded as a nonlinear extension of linear models and have been validated as feasible for heat exchangers, chemical processes and biological processes [14,15].

Several identification methods have been proposed for bilinear systems [16,17]. In ref. [18], the linear subspace identification method was extended to bilinear systems and the suitability of linear subspace methods versus bilinear subspace methods was investigated. Although the subspace method is widely used in industrial process modeling, it can result in large storage demands and heavy computational complexity because the size of the block Hankel matrix significantly increases as the dimensionality of the input–outputs increases [19]. To address this, the system states can be removed from the state–space system to convert the bilinear system into an equivalent input–output representation, which is then identified using conventional parameter estimation methods [20]. In addition, an alternative choice is to directly identify the bilinear system based on the state–space representation, which can achieve state estimation except for parameter estimation [21,22,23]. Recently, a Kalman filter-based data iterative estimation algorithm was developed for a bilinear system, where the output noise is assumed to be colored Gaussian noise [24]. The works referred to above focus on the identification of bilinear systems under Gaussian noise interference.

As is well-known, Gaussian noise is a strong indicator of external interference [25,26,27]. For various reasons, such as unreliable sensors and communication interferences, the external noise may be non-Gaussian heavy-tailed noise, which has a pivotal influence on parameter and state estimation [28,29,30]. As a means to gauge the resemblance between two stochastic variables X and Y defined in the reproducing kernel Hilbert space, the correntropy can capture high-order statistics of an arbitrary error distribution and adaptively switches between different norms (i.e., ${\ell }_2$-norm, ${\ell }_1$-norm and ${\ell }_0$-norm) according to the distance of X and Y; therefore, it is inherently insensitive to non-Gaussian noise [31]. Unfortunately, traditional Gaussian kernel-based correntropy is strongly affected by the bandwidth of the Gaussian kernel and thus it may be vulnerable to non-Gaussian noise [32].

To overcome this problem, many alternative kernels have been proposed, such as the exponential kernel, the Laplace kernel and the generalized Gaussian kernel [33]. These kernels can reduce sensitivity to kernel bandwidths, but their derivatives do not exist at zero and thus their applications are restricted. To address this, a Versoria kernel-based fractional-order adaptive filtering method was developed for linear systems with $\alpha$-stable impulsive noise, where the Versoria kernel correntropy was applied to reduce the steady–state error and to enhance robustness [34]. Moreover, the concept of mixture correntropy was introduced by uniting multiple Gaussian kernels with different bandwidths into a convex combination [35]. Recently, an iterated maximum mixture correntropy Kalman filter was proposed for nonlinear systems with non-Gaussian noise and was applied to tracking and navigation, where the kernel function was taken as the mixture of two Gaussian kernels [36]. Although the mixture correntropy can achieve better robustness than the single kernel correntropy, the homogeneous kernels-based mixture correntropy is essentially based on a single type of kernel; thus, it faces a risk of performance deterioration when a complex noise environment is encountered.

This paper concerns joint parameter and state estimation for a bilinear state–space system with non-Gaussian noise. Compared with homogeneous kernels-based mixture correntropy [35,36], this paper integrates the Gaussian kernel and the Versoria kernel into a Gaussian–Versoria mixed kernel and leverages different kinds of kernels to improve robustness and flexibility against non-Gaussian noise. In contrast to existing works where process noise is either not involved or is Gaussian [21,24,37], this paper considers a bilinear state–space system where both the process noise and output noise are non-Gaussian. Due to the nonlinear nature of bilinear systems and the heavy-tailed characteristic of multiple complex non-Gaussian noises, standard Kalman filtering may suffer from severe state estimation performance degradation and the conventional quadratic criterion-based parameter estimation method may be sensitive to non-Gaussian noise [38,39]. Therefore, it is essential to develop a robust parameter and state estimation method to resist the negative influences of the non-Gaussian process and measurement noise and to achieve high-precision estimation. The key contributions of this paper are as follows:

• Integration of two different types of kernels (i.e., a Gaussian kernel function and a Versoria function) into a mixed kernel to produce a Gaussian–Versoria mixed kernel correntropy-based cost function for enhancing robustness to complex non-Gaussian noise.
• Comprehensive consideration of the influence of non-Gaussian process noise and non-Gaussian measurement noise and development of a Gaussian–Versoria mixed kernel correntropy-based Kalman filtering method to obtain robust state estimation of a bilinear system.
• Presentation of a Gaussian–Versoria mixed kernel correntropy-based Kalman filtering robust recursive least squares method for jointly estimating the states and parameters of a bilinear system under non-Gaussian noise environments.

The remainder of the paper is arranged as follows. Section 2 gives the problem description. Section 3 presents a Gaussian–Versoria mixed kernel correntropy-based robust parameter and state estimation method for a bilinear system with non-Gaussian noise. Section 4 provides a simulation example to validate the proposed method. Section 5 presents the conclusions.

2. Problem Description

Consider the following bilinear state–space system described by the observability canonical form:

$$\begin{array}{ccc}\hfill {\mathit{x}}_{k+1}& =& \mathit{G}{\mathit{x}}_k+\mathit{F}{\mathit{x}}_k{u}_k+\mathit{h}{u}_k+{\mathit{w}}_k,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill y_k& =& \mathit{c}{\mathit{x}}_k+v_k,\hfill \end{array}$$

(2)

where ${\mathit{x}}_k:={[x_{1,k},x_{2,k},\cdots ,x_{n,k}]}^{\mathrm{T}}\in {\mathbb{R}}^n$, $u_k\in \mathbb{R}$ and $y_k\in \mathbb{R}$ are the system state, input and output at the instant k, ${\mathit{w}}_k:={[w_{1,k},w_{2,k},\cdots ,w_{n,k}]}^{\mathrm{T}}\in {\mathbb{R}}^n$ and $v_k\in \mathbb{R}$ are the stochastic noises with zero means and variances ${\mathit{Q}}_k\in {\mathbb{R}}^{n\times n}$ and $R_k\in \mathbb{R}$ and are irrelevant with each other, and $\mathit{G}\in {\mathbb{R}}^{n\times n}$, $\mathit{F}\in {\mathbb{R}}^{n\times n}$ and $\mathit{h}\in {\mathbb{R}}^n$ are the system parameters:

$$\begin{array}{ccc}\hfill \mathit{G}& :=& \left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 1\\ -g_n& -g_{n-1}& -g_{n-2}& \cdots & -g_1\end{array}\right]\in {\mathbb{R}}^{n\times n},\hfill \\ \hfill \mathit{F}& :=& \left[\begin{array}{c}{\mathit{f}}_1\\ {\mathit{f}}_2\\ ⋮\\ {\mathit{f}}_{n-1}\\ {\mathit{f}}_n\end{array}\right]\in {\mathbb{R}}^{n\times n},{\mathit{f}}_i\in {\mathbb{R}}^{1\times n},i=1,2,\cdots ,n,\mathit{h}:=\left[\begin{array}{c}{h}_1\\ h_2\\ ⋮\\ h_{n-1}\\ h_n\end{array}\right]\in {\mathbb{R}}^n,\hfill \\ \hfill \mathit{c}& :=& [1,0,\cdots ,0,0]\in {\mathbb{R}}^{1\times n}.\hfill \end{array}$$

From (1), we have

$$\begin{array}{ccc}\hfill x_{i,k+1}& =& x_{i+1,k}+{\mathit{f}}_i{\mathit{x}}_k{u}_k+h_i{u}_k+w_{i,k},i=1,2,\cdots ,n-1,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill x_{n,k+1}& =& -g_n{x}_{1,k}-g_{n-1}{x}_{2,k}-\cdots -g_1{x}_{n,k}+{\mathit{f}}_n{\mathit{x}}_k{u}_k+h_n{u}_k+w_{n,k}.\hfill \end{array}$$

(4)

Define

$$\begin{array}{ccc}\hfill {\mathit{\phi }}_k& :=& [-x_{n,k-n},-x_{n-1,k-n},\cdots ,-x_{1,k-n},{\mathit{x}}_{k-1}^{\mathrm{T}}{u}_{k-1},{\mathit{x}}_{k-2}^{\mathrm{T}}{u}_{k-2},\cdots ,{\mathit{x}}_{k-n}^{\mathrm{T}}{u}_{k-n},\hfill \\ & & u_{k-1},u_{k-2},\cdots ,u_{k-n}{]}^{\mathrm{T}}\in {\mathbb{R}}^{n^2+2n},\hfill \\ \hfill \mathit{\theta }& :=& {[g_1,g_2,\cdots ,g_n,{\mathit{f}}_1,{\mathit{f}}_2,\cdots ,{\mathit{f}}_n,h_1,h_2,\cdots ,h_n]}^{\mathrm{T}}\in {\mathbb{R}}^{n^2+2n}.\hfill \end{array}$$

Let $e_k:={\sum }_{i=1}^n{w}_{i,k-i}+v_k\in \mathbb{R}$. Multiplying both sides of the i-th equation in (3) and (4) by $z^{-i}(i=1,2,\cdots ,n)$ and adding all n equations together, we have

$$\begin{array}{ccc}\hfill y_k& =& x_{1,k}+v_k\hfill \\ & =& -\sum _{i=1}^n{g}_i{x}_{n-i+1,k-n}+\sum _{i=1}^n{f}_i{x}_{k-i}{u}_{k-i}+\sum _{i=1}^n{h}_i{u}_{k-i}+\sum _{i=1}^n{w}_{i,k-i}+v_k\hfill \\ & =& {\mathit{\phi }}_k^{\mathrm{T}}\mathit{\theta }+e_k.\hfill \end{array}$$

(5)

Remark 1.

To explain how the structure of the bilinear state space model was developed, take the microbial fermentation process [40] as an example. According to the mass balance, the rate of cell growth can be described by:

$$\begin{array}{ccc}\hfill {\dot{m}}_{a,k}& =& {\gamma }_{1,k}{m}_{a,k}-u_k{m}_{a,k},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\dot{m}}_{e,k}& =& {\gamma }_{2,k}{m}_{a,k}-u_k{m}_{e,k},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\dot{s}}_k& =& -{\gamma }_{3,k}{m}_{a,k}-u_k{s}_k+s_r{u}_k,\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill y_k& =& m_{a,k}+m_{e,k},\hfill \end{array}$$

(9)

where $m_{a,k}$ and $m_{e,k}$ are the active cell concentration and the nonliving cell concentration, respectively, $u_k$ is the flow rate of fresh nutrient into the growth vessel of unit volume, $s_k$ is the substrate concentration in growth vessel, $y_k$ is the total concentration of cells in the culture, $s_r$ is the constant concentration of nutrient flowing into growth vessel, ${\gamma }_{1,k}$, and ${\gamma }_{2,k}$ and ${\gamma }_{3,k}$ are the specific growth rates of cells. In general, the specific growth rate is influenced by the substrate concentration and external environment, such as the temperature and pH. Assume that the environment conditions remain unchanged and nutrient substrate is stably supplied such that ${\gamma }_{1,k}$, ${\gamma }_{2,k}$ and ${\gamma }_{3,k}$ are constants. Take $u_k$ as the system input and let $x_k:={[m_{a,k},m_{e,k},s_k]}^{\mathrm{T}}\in {\mathbb{R}}^3$ be the state vector. Define the parameter matrices/vectors G, F, h and c as

$$\begin{array}{ccc}\hfill \mathit{G}& :=& \left[\begin{array}{ccc}{\gamma }_1& 0& 0\\ {\gamma }_2& 0& 0\\ -{\gamma }_3& 0& 0\end{array}\right],\mathit{F}:=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& -1\end{array}\right],\mathit{c}:=\left[\begin{array}{ccc}1& 1& 0\end{array}\right],\mathit{h}:=\left[\begin{array}{c}0\\ 0\\ s_r\end{array}\right].\hfill \end{array}$$

Equations (6)–(9) can be written as the continuous bilinear state–space system:

$$\begin{array}{ccc}\hfill {\dot{\mathit{x}}}_k& =& \mathit{G}{\mathit{x}}_k+\mathit{F}{\mathit{x}}_k{u}_k+\mathit{h}{u}_k,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill y_k& =& \mathit{c}{\mathit{x}}_k.\hfill \end{array}$$

(11)

Discretizing the continuous system (10) and (11) and introducing the stochastic noises result in the bilinear state–space system in (1) and (2). Bilinear systems have been widely applied to biological processes and industrial processes [41].

Define the cost function

$$J_1\left(\mathit{\theta }\right):=\sum _{j=1}^k{(y_j-{\mathit{\phi }}_j^{\mathrm{T}}\mathit{\theta })}^2.$$

Let ${\widehat{\mathit{\theta }}}_k$ be the estimate of the parameter vector $\mathit{\theta }$ at the instant k, ${\widehat{x}}_k^{-}$ and ${\widehat{x}}_k^{+}$ be the prior and posterior state estimates of $x_k$, ${\tilde{x}}_k^{-}:=x_k-{\widehat{x}}_k^{-}\in {\mathbb{R}}^n$ and ${\tilde{x}}_k^{+}:=x_k-{\widehat{x}}_k^{+}\in {\mathbb{R}}^n$ be the prior and posterior state estimation errors, and ${\tilde{\mathit{P}}}_k^{-}:\mathrm{E}=\left[\left({\tilde{\mathit{x}}}_k^{-}\right){\left({\tilde{\mathit{x}}}_k^{-}\right)}^{\mathrm{T}}\right]$ and ${\tilde{\mathit{P}}}_k^{+}:\mathrm{E}=\left[\left({\tilde{\mathit{x}}}_k^{+}\right){\left({\tilde{\mathit{x}}}_k^{+}\right)}^{\mathrm{T}}\right]$ be the prior and posterior state estimation covariances, respectively. Substituting the unknown ${\mathit{x}}_k$ in ${\mathit{\phi }}_k$ with ${\widehat{x}}_k^{+}$ and minimizing $J_1\left(\mathit{\theta }\right)$ yield the least squares method:

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_k& =& {\widehat{\mathit{\theta }}}_{k-1}+{\mathit{L}}_k(y_k-{\widehat{\mathit{\phi }}}_k^{\mathrm{T}}{\widehat{\mathit{\theta }}}_{k-1}),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\mathit{L}}_k& =& {\displaystyle \frac{{\mathit{P}}_{k-1}{\widehat{\mathit{\phi }}}_k}{1+{\widehat{\mathit{\phi }}}_k^{\mathrm{T}}{\mathit{P}}_{k-1}{\widehat{\mathit{\phi }}}_k}},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\mathit{P}}_k& =& (\mathit{I}-{\mathit{L}}_k{\widehat{\mathit{\phi }}}_k^{\mathrm{T}}){\mathit{P}}_{k-1},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\phi }}}_k& =& [-{\widehat{x}}_{n,k-n}^{+},-{\widehat{x}}_{n-1,k-n}^{+},\cdots ,-{\widehat{x}}_{1,k-n}^{+},{\left({\widehat{x}}_{k-1}^{+}\right)}^{\mathrm{T}}{u}_{k-1},{\left({\widehat{x}}_{k-2}^{+}\right)}^{\mathrm{T}}{u}_{k-2},\cdots ,{\left({\widehat{x}}_{k-n}^{+}\right)}^{\mathrm{T}}{u}_{k-n},\hfill \\ & & u_{k-1},u_{k-2},\cdots ,u_{k-n}{]}^{\mathrm{T}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_k& =& {[{\widehat{g}}_{1,k},{\widehat{g}}_{2,k},\cdots ,{\widehat{g}}_{n,k},{\widehat{\mathit{f}}}_{1,k},{\widehat{\mathit{f}}}_{2,k},\cdots ,{\widehat{\mathit{f}}}_{n,k},{\widehat{h}}_{1,k},{\widehat{h}}_{2,k},\cdots ,{\widehat{h}}_{n,k}]}^{\mathrm{T}},\hfill \end{array}$$

(16)

where ${\widehat{g}}_{i,k}$, ${\widehat{\mathit{f}}}_{i,k}$ and ${\widehat{h}}_{i,k}$ represent the estimates of the true parameters $g_i$, $f_i$ and $h_i$, respectively. To obtain ${\widehat{\mathit{x}}}_k^{+}$, the following Kalman filtering can be used:

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{-}& =& ({\widehat{\mathit{G}}}_{k-1}+{\widehat{\mathit{F}}}_{k-1}{u}_{k-1}){\widehat{\mathit{x}}}_{k-1}^{+}+{\widehat{\mathit{h}}}_{k-1}{u}_{k-1},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{-}& =& ({\widehat{\mathit{G}}}_{k-1}+{\widehat{\mathit{F}}}_{k-1}{u}_{k-1}){\mathit{P}}_{k-1}^{+}{({\widehat{\mathit{G}}}_{k-1}+{\widehat{\mathit{F}}}_{k-1}{u}_{k-1})}^{\mathrm{T}}+{\mathit{Q}}_{k-1},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{+}& =& {\widehat{\mathit{x}}}_k^{-}+{\mathit{K}}_k(y_k-c{\widehat{\mathit{x}}}_k^{-}),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_k& =& {\displaystyle \frac{{\mathit{P}}_k^{-}{c}^{\mathrm{T}}}{c{\mathit{P}}_k^{-}{c}^{\mathrm{T}}+R_k}},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{+}& =& (\mathit{I}-{\mathsf{\Gamma }}_{k}c){\mathit{P}}_k^{-}{(\mathit{I}-{\mathsf{\Gamma }}_{k}c)}^{\mathrm{T}}+{\mathsf{\Gamma }}_k{R}_k{\mathsf{\Gamma }}_k^{\mathrm{T}},\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{G}}}_k& =& \left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 1\\ -{\widehat{g}}_{n,k}& -{\widehat{g}}_{n-1,k}& -{\widehat{g}}_{n-2,k}& \cdots & -{\widehat{g}}_{1,k}\end{array}\right],\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{F}}}_k& =& {[{\widehat{\mathit{f}}}_{1,k}^{\mathrm{T}},{\widehat{\mathit{f}}}_{2,k}^{\mathrm{T}},\cdots ,{\widehat{\mathit{f}}}_{n,k}^{\mathrm{T}}]}^{\mathrm{T}},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{h}}}_k& =& {[{\widehat{h}}_{1,k},{\widehat{h}}_{2,k},\cdots ,{\widehat{h}}_{n,k}]}^{\mathrm{T}}.\hfill \end{array}$$

(24)

Equations (12)–(24) form the Kalman filtering-based recursive least squares (KF-RLS) method for the bilinear state–space system in (1) and (2). When both the process noise and the measurement noise are Gaussian, the KF-RLS method can perform well. However, as is known, both the RLS method and the KF method are sensitive to non-Gaussian impulsive noise. Therefore, the performance of the KF-RLS method may be drastically deteriorated when the bilinear system encounters not only the non-Gaussian process noise ${\mathit{w}}_k$ but also the non-Gaussian measurement noise $v_k$. To rectify this limitation, the aim of this paper is to present a robust parameter and state estimation method for the bilinear state–space systems under non-Gaussian process noise and non-Gaussian measurement noise environments based on correntropy theory.

3. Gaussian–Versoria Mixed Kernel Correntropy-Based Robust Parameter and State Estimation Method

3.1. Gaussian–Versoria Mixed Kernel Correntropy

The local similarity of two random variables X and Y can be measured by the correntropy, which is defined as  

$$\begin{array}{c}\hfill J(X,Y):=\mathrm{E}[m(X,Y)]=\int m(x,y)\mathrm{d}F(x,y),\end{array}$$

where $\mathrm{E}(·)$ is the expectation operator, $m(·,·)$ is a Mercer kernel function, and $F(x,y)$ is the joint distribution function of X and Y. Due to the excellent characteristics such as positive definiteness and shift-invariance, the Gaussian kernel $G_{\sigma }(x-y):=exp\left\{-{\displaystyle \frac{{(x-y)}^2}{2{\sigma }^2}}\right\}$ is a default choice. However, the Gaussian kernel is sensitive to the kernel bandwidth $\sigma$ and is incapable of dealing with complex bimodal non-Gaussian noise. To avoid such a dilemma, the Versoria kernel function can also be selected and is denoted by

$$V_{\gamma }(x-y):={\displaystyle \frac{8{\gamma }^{\prime 3}}{4{\gamma }^{\prime 2}+{(x-y)}^2}}={\displaystyle \frac{2{\gamma }^{\prime }}{1+{\left(\frac{x-y}{2{\gamma }^{\prime }}\right)}^2}}={\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{(x-y)}^2}},$$

where ${\gamma }^{\prime }$ is the radius of the circle generated by the Versoria function and $\gamma :=2{\gamma }^{\prime }$. The advantage of the Versoria function lies in that it has an even thicker tail than the density function of the Student’s t distribution; thus, it is suitable for handling strong outliers [42]. In addition, alternative robust kernels are proposed, such as the exponential kernel $E_{\sigma }(x-y)=exp\left({\displaystyle \frac{|x-y|}{2{\sigma }^2}}\right)$, the Laplace kernel $L_{\sigma }(x-y):=exp\left({\displaystyle \frac{|x-y|}{\sigma }}\right)$ and the generalized Gaussian kernel $G_{\alpha ,\beta }(x-y)={\displaystyle \frac{\alpha }{2\beta \Gamma (1/\alpha )}}exp\left\{-{\left|{\displaystyle \frac{x-y}{\beta }}\right|}^{\alpha }\right\}$ [33]. However, a single kernel may be inadequate to cope with complex non-Gaussian noise. Thus, multiple kernels can be combined into a mixed kernel [35]:

$$\kappa (x-y):=\sum _{i=1}^m{\omega }_i{\kappa }_i(x-y),$$

where $m⩾2$ is the number of the kernel functions, ${\omega }_i⩾0(i=1,2,\cdots ,m)$ are the mixing coefficients with ${\sum }_{i=1}^m{\omega }_i=1$, and ${\kappa }_i(x-y)$ is the i-th kernel function. If all the kernel functions ${\kappa }_i(x-y)(i=1,2,\cdots ,m)$ are of the same type, the mixed kernel $\kappa (x-y)$ is essentially restricted to a single type and its performance may be deteriorated when confronted with complex noises. Therefore, heterogeneous kernel functions can be combined to enhance robustness. Considering that the derivatives of the exponential kernel, the Laplace kernel and the generalized kernel do not exist at zero, the Gaussian kernel and the Versoria kernel are combined herein and the Gaussian–Versoria mixed kernel is constructed:

$$\begin{array}{ccc}\hfill m(x-y)& =& \omega G_{\sigma }(x-y)+(1-\omega )V_{\gamma }(x-y)\hfill \\ & =& \omega exp\left\{-{\displaystyle \frac{{(x-y)}^2}{2{\sigma }^2}}\right\}+(1-\omega ){\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{(x-y)}^2}},\hfill \end{array}$$

where $\omega \in [0,1]$ denotes the mixing coefficient. Compared with the single kernel correntropy, the Gaussian–Versoria mixed kernel correntropy (GVMKC) leverages the strengths of different kinds of kernels, providing a more flexible and robust scheme to handle non-Gaussian noise. For the multi-dimensional random variables $\mathit{X}\in {\mathbb{R}}^n$ and $\mathit{Y}\in {\mathbb{R}}^n$, the Gaussian–Versoria mixed kernel can be defined as

$$m(\mathit{X}-\mathit{Y}):=\omega exp\left\{-{\displaystyle \frac{{(\mathit{X}-\mathit{Y})}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}(\mathit{X}-\mathit{Y})}{2{\sigma }^2}}\right\}+(1-\omega ){\displaystyle \frac{\gamma }{1+{\frac{1}{{\gamma }^2}(\mathit{X}-\mathit{Y})}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}(\mathit{X}-\mathit{Y})}},$$

where $\mathsf{\Sigma }\in {\mathbb{R}}^{n\times n}$ is the covariance matrix of the error vector $\mathit{e}:=\mathit{X}-\mathit{Y}\in {\mathbb{R}}^n$. Thus, the Gaussian–Versoria mixed kernel correntropy can be written as  

$$\begin{array}{ccc}\hfill J(\mathit{X},\mathit{Y})& =& \mathrm{E}\left[m\left(\mathit{e}\right)\right]=\mathrm{E}\left[\omega exp\left(-{\displaystyle \frac{{\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}}{2{\sigma }^2}}\right)+(1-\omega ){\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}}}\right]\hfill \\ & =& \mathrm{E}\left[\omega G_{\sigma }\left({\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}\right)+(1-\omega )V_{\gamma }\left({\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}\right)\right]\hfill \\ & =& \mathrm{E}\left[\omega G_{\sigma }{(\parallel \mathit{e}\parallel }_{{\mathsf{\Sigma }}^{-1}}^2)+(1-\omega )V_{\gamma }{(\parallel \mathit{e}\parallel }_{{\mathsf{\Sigma }}^{-1}}^2)\right],\hfill \end{array}$$

where ${\parallel \mathit{e}\parallel }_{{\mathsf{\Sigma }}^{-1}}^2:={\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}\in \mathbb{R}$. When $\omega =0$ or $\omega =1$, the correntropy $\mathrm{E}\left[G_{\sigma }\left(\mathit{e}\right)\right]$ or $\mathrm{E}\left[V_{\gamma }\left(\mathit{e}\right)\right]$ reduces to the Versoria or Gaussian kernel correntropy. Maximizing $\mathrm{E}\left[G_{\sigma }\left(\mathit{e}\right)\right]$ or $\mathrm{E}\left[V_{\gamma }\left(\mathit{e}\right)\right]$ is equivalent to minimizing the weighted quadratic cost function ${\parallel \mathit{e}\parallel }_{{\mathsf{\Sigma }}^{-1}}^2={\mathit{e}}^{\mathrm{T}}{\mathsf{\Sigma }}^{-1}\mathit{e}$. By integrating the advantages of the Gaussian kernel and the Versoria function, the GVMKC may be more superior than the single-kernel correntropy and homogeneous multi-kernel correntropy in complex non-Gaussian noise environments.

3.2. Gaussian–Versoria Mixed Kernel Correntropy-Based Kalman Filtering

Note that the bilinear system in (1) and (2) contains a nonlinear term (i.e., the product of the state and the input) and the system is corrupted by the non-Gaussian noise $w_k$ and $v_k$. To suppress the effect of the non-Gaussian noise and to obtain an effective and robust state estimation, the following derives the Gaussian–Versoria mixed kernel correntropy-based Kalman filtering using the GVMKC.

Let ${\overline{\mathit{G}}}_k:=\mathit{G}+\mathit{F}{u}_k\in {\mathbb{R}}^{n\times n}$. The bilinear system in (1) and (2) can be written as the following time-varying system:

$$\begin{array}{ccc}\hfill {\mathit{x}}_{k+1}& =& {\overline{\mathit{G}}}_k{\mathit{x}}_k+\mathit{h}{u}_k+{\mathit{w}}_k,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill y_k& =& \mathit{c}{\mathit{x}}_k+v_k.\hfill \end{array}$$

(26)

According to the GVMKC, define the cost function:

$$\begin{array}{ccc}& & J_2\left({\mathit{x}}_k\right):=m\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)+m\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)\hfill \\ & & =\omega exp\left(-{\displaystyle \frac{{({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})}^{\mathrm{T}}{\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})}{2{\sigma }^2}}\right)+(1-\omega ){\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})}^{\mathrm{T}}{\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})}}\hfill \\ & & +\omega exp\left(-{\displaystyle \frac{{(y_k-\mathit{c}{\mathit{x}}_k)}^{\mathrm{T}}{R}_k^{-1}(y_k-\mathit{c}{\mathit{x}}_k)}{2{\sigma }^2}}\right)+(1-\omega ){\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{(y_k-\mathit{c}{\mathit{x}}_k)}^{\mathrm{T}}{R}_k^{-1}(y_k-\mathit{c}{\mathit{x}}_k)}}.\hfill \end{array}$$

The cost function $J_2\left({\mathit{x}}_k\right)$ consists of the weighted errors $m\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)$ and $m\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)$; thus, it considers the effects of the process noise ${\mathit{w}}_k$ and the measurement noise $v_k$. The derivative of $J_2\left({\mathit{x}}_k\right)$ with respect to ${\mathit{x}}_k$ can be computed by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial J_2\left({\mathit{x}}_k\right)}{\partial {\mathit{x}}_k}}& =& -{\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right){\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})\hfill \\ & & -{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right){\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})\hfill \\ & & +{\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right){\mathit{c}}^{\mathrm{T}}{R}_k^{-1}(y_k-\mathit{c}{\mathit{x}}_k)\hfill \\ & & +{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right){\mathit{c}}^{\mathrm{T}}{R}_k^{-1}(y_k-\mathit{c}{\mathit{x}}_k).\hfill \end{array}$$

Maximizing $J_2\left({\mathit{x}}_k\right)$ and letting its gradient ${\displaystyle \frac{\partial J_2\left({\mathit{x}}_k\right)}{\partial {\mathit{x}}_k}}$ be a zero vector yield

$$\begin{array}{ccc}& & \left[{\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)\right]{\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})\hfill \\ & & ={\mathit{c}}^{\mathrm{T}}\left[{\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)\right]R_k^{-1}(y_k-\mathit{c}{\mathit{x}}_k),\hfill \end{array}$$

i.e.,

$$\begin{array}{ccc}\hfill & & W_{P_k}{\left({\mathit{P}}_k^{-}\right)}^{-1}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})={\mathit{c}}^{\mathrm{T}}{W}_{R_k}{R}_k^{-1}[y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}-\mathit{c}({\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-})],\hfill \end{array}$$

(27)

where

$$\begin{array}{ccc}\hfill W_{P_k}& :=& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}{\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2\right)\in \mathbb{R},\hfill \\ \hfill W_{R_k}& :=& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel y_k-\mathit{c}{\mathit{x}}_k{\parallel }_{R_k^{-1}}^2\right)\in \mathbb{R}.\hfill \end{array}$$

Define ${\overline{\mathit{P}}}_k^{-}:=W_{{\mathit{P}}_k}^{-1}{\mathit{P}}_k^{-}\in {\mathbb{R}}^{n\times n}$ and ${\overline{R}}_k:=W_{R_k}^{-1}{R}_k\in \mathbb{R}$. From (27), we have

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{+}& =& {\widehat{\mathit{x}}}_k^{-}+{\left[W_{{\mathit{P}}_k}{\left({\mathit{P}}_k^{-}\right)}^{-1}+{\mathit{c}}^{\mathrm{T}}{W}_{R_k}{R}_k^{-1}\mathit{c}\right]}^{-1}{\mathit{c}}^{\mathrm{T}}{W}_{R_k}{R}_k^{-1}(y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-})\hfill \\ & =& {\widehat{\mathit{x}}}_k^{-}+{\left[{\left({\overline{\mathit{P}}}_k^{-}\right)}^{-1}+{\mathit{c}}^{\mathrm{T}}{\overline{R}}_k^{-1}\mathit{c}\right]}^{-1}{\mathit{c}}^{\mathrm{T}}{\overline{R}}_k^{-1}(y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-})\hfill \\ & =& {\widehat{\mathit{x}}}_k^{-}+{\mathsf{\Gamma }}_k(y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}),\hfill \end{array}$$

(28)

where

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_k& :=& {\left[{\left({\overline{\mathit{P}}}_k^{-}\right)}^{-1}+{\mathit{c}}^{\mathrm{T}}{\overline{R}}_k^{-1}\mathit{c}\right]}^{-1}{\mathit{c}}^{\mathrm{T}}{\overline{R}}_k^{-1}\in {\mathbb{R}}^n.\hfill \end{array}$$

(29)

Applying the matrix inversion lemma

$${(\mathit{A}+\mathit{B}\mathit{C}\mathit{D})}^{-1}={\mathit{A}}^{-1}-{\mathit{A}}^{-1}\mathit{B}{({\mathit{C}}^{-1}+\mathit{D}{\mathit{A}}^{-1}\mathit{B})}^{-1}\mathit{D}{\mathit{A}}^{-1}$$

to (29), we have

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_k& =& \left[{\overline{\mathit{P}}}_k^{-}-{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}{\left({\overline{R}}_k+\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right)}^{-1}\mathit{c}{\overline{\mathit{P}}}_k^{-}\right]{\mathit{c}}^{\mathrm{T}}{\overline{R}}_k^{-1}\hfill \\ & =& {\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\left[\mathit{I}-{\left({\overline{R}}_k+\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right)}^{-1}\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right]{\overline{R}}_k^{-1}\hfill \\ & =& {\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}{\left(\mathit{I}+{\overline{R}}_k^{-1}\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right)}^{-1}{\overline{R}}_k^{-1}\hfill \\ & =& {\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}{\left({\overline{R}}_k+\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right)}^{-1}.\hfill \end{array}$$

(30)

Note that both ${\overline{\mathit{P}}}_k^{-}$ (or $W_{{\mathit{P}}_k}$) and ${\overline{R}}_k$ (or $W_{R_k}$) contain the unknown state ${\mathit{x}}_k$. To implement the algorithm, the unknown ${\mathit{x}}_k$ in $W_{{\mathit{P}}_k}$ and $W_{R_k}$ can be substituted by its prior estimates ${\widehat{\mathit{x}}}_{k-1}^{+}$ and ${\widehat{\mathit{x}}}_k^{-}$, respectively. Moreover, the weighted ${\ell }_2$-norm ${\parallel ·\parallel }_{{\left({\mathit{P}}_k^{-}\right)}^{-1}}^2$ in $W_{P_k}$ is replaced by the ordinary ${\ell }_2$-norm ${\parallel ·\parallel }^2$ to avoid the inverse of ${\mathit{P}}_k^{-}$. Thus, $W_{P_k}$ and $W_{R_k}$ can be computed by

$$\begin{array}{ccc}\hfill {\widehat{W}}_{P_k}& =& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel {\widehat{\mathit{x}}}_{k-1}^{-}-{\widehat{\mathit{x}}}_k^{-}{\parallel }^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel {\widehat{\mathit{x}}}_{k-1}^{-}-{\widehat{\mathit{x}}}_k^{-}{\parallel }^2\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\widehat{W}}_{R_k}& =& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}{\parallel }_{R_k^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}{\parallel }_{R_k^{-1}}^2\right).\hfill \end{array}$$

(32)

To update the covariance matrices ${\mathit{P}}_k^{+}$ and ${\mathit{P}}_k^{-}$, using (2) and (28) gives

$$\begin{array}{ccc}\hfill {\tilde{\mathit{x}}}_k^{+}& =& {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{+}\hfill \\ & =& {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}-{\mathsf{\Gamma }}_k(\mathit{c}{\mathit{x}}_k+v_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-})\hfill \\ & =& (\mathit{I}-{\mathsf{\Gamma }}_k\mathit{c}){\tilde{\mathit{x}}}_k^{-}-{\mathsf{\Gamma }}_k{v}_k.\hfill \end{array}$$

(33)

Note that $v_k$ is a zero-mean stochastic noise with variance $R_k$ and is independent of prior estimation error ${\tilde{\mathit{x}}}_k^{-}$. From (33), we have

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{+}& =& \mathrm{E}\left[{\tilde{\mathit{x}}}_k^{+}{\left({\tilde{\mathit{x}}}_k^{+}\right)}^{\mathrm{T}}\right]\hfill \\ & =& (\mathit{I}-{\mathsf{\Gamma }}_k\mathit{c}){\mathit{P}}_k^{-}{(\mathit{I}-{\mathsf{\Gamma }}_k\mathit{c})}^{\mathrm{T}}+{\mathsf{\Gamma }}_k{R}_k{\mathsf{\Gamma }}_k^{\mathrm{T}},\hfill \end{array}$$

(34)

From (25), the prior estimate ${\widehat{\mathit{x}}}_k^{-}$ can be computed by

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{-}& =& {\overline{\mathit{G}}}_{k-1}{\widehat{\mathit{x}}}_{k-1}^{+}+\mathit{h}{u}_{k-1}.\hfill \end{array}$$

(35)

Using (25) and (35), we have

$$\begin{array}{ccc}\hfill {\tilde{\mathit{x}}}_k^{-}& =& {\mathit{x}}_k-{\widehat{\mathit{x}}}_k^{-}\hfill \\ & =& [{\overline{\mathit{G}}}_{k-1}{\mathit{x}}_{k-1}+\mathit{h}{u}_{k-1}+{\mathit{w}}_{k-1}]-[{\overline{\mathit{G}}}_{k-1}{\widehat{\mathit{x}}}_{k-1}^{+}+\mathit{h}{u}_{k-1}]\hfill \\ & =& {\overline{\mathit{G}}}_{k-1}{\widehat{\mathit{x}}}_{k-1}^{+}-{\mathit{w}}_{k-1}.\hfill \end{array}$$

(36)

Noting that ${\mathit{w}}_k$ is a zero-mean stochastic noise with variance ${\mathit{Q}}_k$, using (36) gives

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{-}& =& \mathrm{E}\left[{\tilde{\mathit{x}}}_k^{-}{\left({\tilde{\mathit{x}}}_k^{-}\right)}^{\mathrm{T}}\right]\hfill \\ & =& {\overline{\mathit{G}}}_{k-1}{\mathit{P}}_{k-1}^{+}{\overline{\mathit{G}}}_{k-1}^{\mathrm{T}}+{\mathit{Q}}_{k-1}.\hfill \end{array}$$

(37)

Equations (28) and (30)–(37) construct the Gaussian–Versoria mixed kernel correntropy-based Kalman filtering (GVMKC-KF) for the bilinear state space system in (1) and (2).

Remark 2.

When the non-Gaussian measurement noise $v_k$ is encountered, $y_k$ in (2) grows sharply and the weighted norm $\parallel y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}{\parallel }_{R_k^{-1}}^2$ increases; thus, ${\widehat{W}}_{R_k}$ in (31) decreases and ${\overline{R}}_k$ increases, which results in the gain ${\mathsf{\Gamma }}_k$ decreasing and the posterior state estimate ${\widehat{\mathit{x}}}_k^{+}$ remains unchanged with the prior estimate ${\widehat{\mathit{x}}}_k^{-}$. Similarly, when the bilinear system suffers from the non-Gaussian process noise ${\mathit{w}}_k$, the gain ${\mathsf{\Gamma }}_k$ also remains unchanged. Therefore, the GVMKC-KF method can provide the robust state estimation of both the non-Gaussian measurement noise $v_k$ and the non-Gaussian process noise ${\mathit{w}}_k$.

3.3. Gaussian–Versoria Mixed Kernel Correntropy-Based Robust Recursive Least Squares Method

To realize the robust parameter and state estimation, the following derives the Gaussian–Versoria mixed kernel correntropy-based robust recursive least squares method. Note that the joint distribution function $F(x,y)$ is typically unknown in practice. Thus, when only limited measurements $(x_j,y_j),j=1,2,\cdots ,L$ are available, the correntropy $J(X,Y)$ can be estimated by

$$\begin{array}{c}\hfill \widehat{V}(x,y):={\displaystyle \frac{1}{L}}\sum _{j=1}^{L}m(x_j-y_j).\end{array}$$

Based on the identification model (5) of the bilinear state–space system in (1) and (2), define the following GVMKC-based cost function:  

$$\begin{array}{ccc}\hfill J_3\left(\mathit{\theta }\right)& =& \sum _{j=1}^k\left[\omega exp\left\{-{\displaystyle \frac{{(y_j-{\mathit{\phi }}_j^{\mathrm{T}}\mathit{\theta })}^2}{2{\sigma }^2}}\right\}+(1-\omega ){\displaystyle \frac{\gamma }{1+\frac{1}{{\gamma }^2}{(y_j-{\mathit{\phi }}_j^{\mathrm{T}}\mathit{\theta })}^2}}\right]\hfill \\ & =& \sum _{j=1}^k\left[\omega G_{\sigma }\left(e_j\right)+(1-\omega )V_{\gamma }\left(e_j\right)\right],\hfill \end{array}$$

where $e_j:=y_j-{\mathit{\phi }}_j^{\mathrm{T}}\mathit{\theta }\in \mathbb{R}$ represents the error. The gradient of $J_3\left(\mathit{\theta }\right)$ with respect to $\mathit{\theta }$ can be computed by

$$\begin{array}{ccc}\hfill \mathrm{grad}\left[J_3\left(\mathit{\theta }\right)\right]& =& \sum _{j=1}^k\left[{\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(e_j\right){\mathit{\phi }}_j{e}_j+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(e_j\right){\mathit{\phi }}_j{e}_j\right]\hfill \\ & =& \sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j(y_j-{\mathit{\phi }}_j^{\mathrm{T}}\mathit{\theta }),\hfill \end{array}$$

where

$$\eta \left(e_j\right):={\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(e_j\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(e_j\right).$$

Letting the gradient of $J_3\left(\mathit{\theta }\right)$ be zero gives

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_t& =& {\left[\sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{\mathit{\phi }}_j^{\mathrm{T}}\right]}^{-1}\left[\sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{y}_j\right],\hfill \end{array}$$

(38)

Define

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_k& :=& {\left[\sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{\mathit{\phi }}_j^{\mathrm{T}}\right]}^{-1}\in {\mathbb{R}}^{(n^2+2n)\times (n^2+2n)},\hfill \\ \hfill {\mathit{\xi }}_k& :=& \sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{y}_j\in {\mathbb{R}}^{n^2+2n}.\hfill \end{array}$$

Let ${\widehat{e}}_k:=y_k-{\mathit{\phi }}_k^{\mathrm{T}}{\widehat{\mathit{\theta }}}_{k-1}\in \mathbb{R}$ be the estimate of $e_k$ at the instant k. According to the definitions of ${\mathsf{\Omega }}_k$ and ${\mathit{\xi }}_k$, we have

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_k^{-1}& =& \sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{\mathit{\phi }}_j^{\mathrm{T}}={\mathsf{\Omega }}_{k-1}^{-1}+\eta \left({\widehat{e}}_k\right){\mathit{\phi }}_k{\mathit{\phi }}_k^{\mathrm{T}},\hfill \\ \hfill {\mathit{\xi }}_k& =& \sum _{j=1}^k\eta \left(e_j\right){\mathit{\phi }}_j{y}_j={\mathit{\xi }}_{k-1}+\eta \left(e_k\right){\mathit{\phi }}_k{y}_k.\hfill \end{array}$$

(39)

Applying the matrix inversion lemma to (39) yields

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_k& =& {\mathsf{\Omega }}_{k-1}-{\displaystyle \frac{\eta \left(e_k\right){\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k{\mathit{\phi }}_k^{\mathrm{T}}{\mathsf{\Omega }}_{k-1}}{1+\eta \left(e_k\right){\mathit{\phi }}_k^{\mathrm{T}}{\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k}}\hfill \\ & =& \left[\mathit{I}-{\displaystyle \frac{\eta \left(e_k\right){\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k{\mathit{\phi }}_k^{\mathrm{T}}}{1+\eta \left(e_k\right){\mathit{\phi }}_k^{\mathrm{T}}{\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k}}\right]{\mathsf{\Omega }}_{k-1}\hfill \\ & =& (\mathit{I}-{\mathit{L}}_k{\mathit{\phi }}_k^{\mathrm{T}}){\mathsf{\Omega }}_{k-1},\hfill \end{array}$$

(40)

where ${\mathit{L}}_k$ is the gain vector

$$\begin{array}{ccc}\hfill L_k& :=& {\displaystyle \frac{\eta \left(e_k\right){\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k}{1+\eta \left(e_k\right){\mathit{\phi }}_k^{\mathrm{T}}{\mathsf{\Omega }}_{k-1}{\mathit{\phi }}_k}}\hfill \\ & =& \eta \left(e_k\right){\mathsf{\Omega }}_k{\mathit{\phi }}_k.\hfill \end{array}$$

(41)

Using (38) and (41) gives

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_k& =& {\mathsf{\Omega }}_k{\mathit{\xi }}_k={\mathsf{\Omega }}_k[{\mathit{\xi }}_{k-1}+\eta \left(e_k\right){\mathit{\phi }}_k{y}_k]\hfill \\ & =& {\mathsf{\Omega }}_k[{\mathsf{\Omega }}_{k-1}^{-1}{\widehat{\mathit{\theta }}}_{k-1}+\eta \left(e_k\right){\mathit{\phi }}_k{y}_k]\hfill \\ & =& {\mathsf{\Omega }}_k[{\mathsf{\Omega }}_k^{-1}-\eta \left(e_k\right){\mathit{\phi }}_k{\mathit{\phi }}_k^{\mathrm{T}}]{\widehat{\mathit{\theta }}}_{k-1}+\eta \left(e_k\right){\mathsf{\Omega }}_k{\mathit{\phi }}_k{y}_k\hfill \\ & =& {\widehat{\mathit{\theta }}}_{k-1}+\eta \left(e_k\right){\mathsf{\Omega }}_k{\mathit{\phi }}_k(y_k-{\mathit{\phi }}_k^{\mathrm{T}}{\widehat{\mathit{\theta }}}_{k-1})\hfill \\ & =& {\widehat{\mathit{\theta }}}_{k-1}+{\mathit{L}}_k(y_k-{\mathit{\phi }}_k^{\mathrm{T}}{\widehat{\mathit{\theta }}}_{k-1}).\hfill \end{array}$$

(42)

Equations (40)–(42) form the Gaussian–Versoria mixed kernel correntropy-based robust recursive least squares (GVMKC-RRLS) method for estimating the parameters of the bilinear system.

To obtain the joint parameter and state estimation, the unknown state ${\mathit{x}}_k$ in ${\mathit{\phi }}_k$ can be replaced by ${\widehat{\mathit{x}}}_k^{+}$ obtained by the GVMKC-KF method, and the unknown parameters G, F and h can be substituted by ${\widehat{\mathit{G}}}_k$, ${\widehat{\mathit{F}}}_k$ and ${\widehat{\mathit{h}}}_k$ given by the GVMKC-RRLS method. By implementing the interactive estimation, the Gaussian–Versoria mixed kernel correntropy-based Kalman filtering robust recursive least squares (GVMKC-KF-RRLS) method can be summarized as follows:

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_k& =& {\widehat{\mathit{\theta }}}_{k-1}+L_k{e}_k,\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill {\mathit{L}}_k& =& {\displaystyle \frac{\eta \left(e_k\right){\mathsf{\Omega }}_{k-1}{\widehat{\mathit{\phi }}}_k}{1+\eta \left(e_k\right){\widehat{\mathit{\phi }}}_k^{\mathrm{T}}{\mathsf{\Omega }}_{k-1}{\widehat{\mathit{\phi }}}_k}},\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill {\mathsf{\Omega }}_k& =& (\mathit{I}-{\mathit{L}}_k{\widehat{\mathit{\phi }}}_k^{\mathrm{T}}){\mathsf{\Omega }}_{k-1},\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill e_k& =& y_k-{\widehat{\mathit{\phi }}}_k^{\mathrm{T}}{\widehat{\mathit{\theta }}}_{k-1},\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill \eta \left(e_k\right)& =& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(e_k\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(e_k\right),\hfill \end{array}$$

(47)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\phi }}}_k& =& [-{\widehat{\mathit{x}}}_{n,k-n}^{+},-{\widehat{\mathit{x}}}_{n-1,k-n}^{+},\cdots ,-{\widehat{\mathit{x}}}_{1,k-n}^{+},{\left({\widehat{\mathit{x}}}_{k-1}^{+}\right)}^{\mathrm{T}}{u}_{k-1},\hfill \\ & & {\left({\widehat{\mathit{x}}}_{k-2}^{+}\right)}^{\mathrm{T}}{u}_{k-2},\cdots ,{\left({\widehat{\mathit{x}}}_{k-n}^{+}\right)}^{\mathrm{T}}{u}_{k-n},u_{k-1},u_{k-2},\cdots ,u_{k-n}{]}^{\mathrm{T}},\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{\theta }}}_k& =& {[{\widehat{g}}_{1,k},{\widehat{g}}_{2,k},\cdots ,{\widehat{g}}_{n,k},{\widehat{\mathit{f}}}_{1,k},{\widehat{\mathit{f}}}_{2,k},\cdots ,{\widehat{\mathit{f}}}_{n,k},{\widehat{h}}_{1,k},{\widehat{h}}_{2,k},\cdots ,{\widehat{h}}_{n,k}]}^{\mathrm{T}},\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{G}}}_k& =& \left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & 1\\ -{\widehat{g}}_{n,k}& -{\widehat{g}}_{n-1,k}& -{\widehat{g}}_{n-2,k}& \cdots & -{\widehat{g}}_{1,k}\end{array}\right],\hfill \end{array}$$

(50)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{F}}}_k& =& {[{\widehat{\mathit{f}}}_{1,k}^{\mathrm{T}},{\widehat{\mathit{f}}}_{2,k}^{\mathrm{T}},\cdots ,{\widehat{\mathit{f}}}_{n,k}^{\mathrm{T}}]}^{\mathrm{T}},\hfill \end{array}$$

(51)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{h}}}_k& =& {[{\widehat{h}}_{1,k},{\widehat{h}}_{2,k},\cdots ,{\widehat{h}}_{n,k}]}^{\mathrm{T}},\hfill \end{array}$$

(52)

$$\begin{array}{ccc}\hfill {\widehat{\overline{\mathit{G}}}}_{k-1}& =& {\widehat{\mathit{G}}}_{k-1}+{\widehat{\mathit{F}}}_{k-1}{u}_{k-1},\hfill \end{array}$$

(53)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{-}& =& {\widehat{\overline{\mathit{G}}}}_{k-1}{\widehat{\mathit{x}}}_{k-1}^{+}+{\widehat{\mathit{h}}}_{k-1}{u}_{k-1},\hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{-}& =& {\widehat{\overline{\mathit{G}}}}_{k-1}{\mathit{P}}_{k-1}^{+}{\widehat{\overline{\mathit{G}}}}_{k-1}^{\mathrm{T}}+{\mathit{Q}}_{k-1},\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill {\widehat{\mathit{x}}}_k^{+}& =& {\widehat{\mathit{x}}}_k^{-}+{\mathsf{\Gamma }}_k(y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}),\hfill \end{array}$$

(56)

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_k& =& {\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}{\left({\overline{R}}_k+\mathit{c}{\overline{\mathit{P}}}_k^{-}{\mathit{c}}^{\mathrm{T}}\right)}^{-1},\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill {\mathit{P}}_k^{+}& =& (\mathit{I}-{\mathsf{\Gamma }}_k\mathit{c}){\mathit{P}}_k^{-}{(\mathit{I}-{\mathsf{\Gamma }}_k\mathit{c})}^{\mathrm{T}}+{\mathsf{\Gamma }}_k{R}_k{\mathsf{\Gamma }}_k^{\mathrm{T}},\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill {\overline{\mathit{P}}}_k^{-}& =& {\displaystyle \frac{1}{{\widehat{W}}_{P_k}}}{\mathit{P}}_k^{-},\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill {\overline{R}}_k& =& {\displaystyle \frac{1}{{\widehat{W}}_{R_k}}}{R}_k,\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill {\widehat{W}}_{P_k}& =& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel {\widehat{\mathit{x}}}_{k-1}^{+}-{\widehat{\mathit{x}}}_k^{-}{\parallel }^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel {\widehat{\mathit{x}}}_{k-1}^{+}-{\widehat{\mathit{x}}}_k^{-}{\parallel }^2\right),\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill {\widehat{W}}_{R_k}& =& {\displaystyle \frac{\omega }{{\sigma }^2}}{G}_{\sigma }\left(\parallel y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}{\parallel }_{R_k^{-1}}^2\right)+{\displaystyle \frac{2(1-\omega )}{{\gamma }^3}}{V}_{\gamma }^2\left(\parallel y_k-\mathit{c}{\widehat{\mathit{x}}}_k^{-}{\parallel }_{R_k^{-1}}^2\right).\hfill \end{array}$$

(62)

The pseudo-code of implementing the GVMKC-KF-RRLS method is shown in Algorithm 1.

Algorithm 1 The GVMKC-KF-RRLS method

Input:  $\{u_k,y_k,k=1,2,\cdots ,L\}$, $\sigma$, $\gamma$, $\omega$, ${\mathit{Q}}_k$, $R_k$. Output:  ${\widehat{\mathit{\theta }}}_k$, ${\widehat{\mathit{x}}}_k^{+}$.   1: Initialization: ${\widehat{\mathit{\theta }}}_0={\mathbf{1}}_n/p_0$, ${\widehat{\mathit{x}}}_i^{+}={\mathbf{1}}_n/p_0$, $u_i=0$$(i⩽0)$, ${\mathsf{\Omega }}_0=p_0{\mathit{I}}_{n^2+2n}$, ${\mathit{P}}_0^{+}={\mathit{I}}_n/p_0$, $p_0={10}^6$.   2: for $k=1:L$ do   3:       Read ${\widehat{g}}_{i,k-1}$, ${\widehat{\mathit{f}}}_{i,k-1}$ and ${\widehat{\mathit{h}}}_{i,k-1}$ by (49). Form ${\widehat{\mathit{G}}}_{k-1}$, ${\widehat{\mathit{F}}}_{k-1}$ and ${\widehat{\mathit{h}}}_{k-1}$ by (50)–(52).   4:       Compute ${\widehat{\overline{\mathit{G}}}}_{k-1}$ by (53), ${\widehat{\mathit{x}}}_k^{-}$ by (54) and ${\mathit{P}}_k^{-}$ by (55).   5:       Compute ${\widehat{W}}_{P_k}$ and ${\widehat{W}}_{R_k}$ by (61) and (62). Compute ${\overline{\mathit{P}}}_k^{-}$ by (59) and ${\overline{R}}_k$ by (60).   6:       Compute ${\mathsf{\Gamma }}_k$ by (57) and ${\mathit{P}}_k^{+}$ by (58).   7:       Compute the posterior state estimate ${\widehat{\mathit{x}}}_k^{+}$ by (56).   8:       Form ${\widehat{\mathit{\phi }}}_k$ by (48). Compute $e_k$ by (46).   9:       Compute $\eta \left(e_k\right)$ by (47), $L_k$ by (44) and ${\mathsf{\Omega }}_k$ by (45). 10:       Update the parameter estimate ${\widehat{\mathit{\theta }}}_k$ by (43). 11:       if $k\le L$ then 12:             $k=k+1$; 13:       else obtain the parameter estimate ${\widehat{\mathit{\theta }}}_k$, break; 14:       end if 15: end for

To improve understanding of the proposed method, a flowchart of the GVMKC-KF-RRLS method is shown in Figure 1.

Remark 3.

When the non-Gaussian noise $v_k$ is encountered, the error $|e_k|$ in (46) suddenly increases, which causes $\eta \left(e_k\right)$ in (47) to decrease. Thus, the gain ${\mathit{L}}_k$ in (44) dramatically decreases such that ${\widehat{\mathit{\theta }}}_k\approx {\widehat{\mathit{\theta }}}_{k-1}$ and the negative effect of the non-Gaussian noise to the parameter estimation can be counteracted. Therefore, the GVMKC-KF-RRLS method can not only provide robust state estimation, but also generate the robust parameter estimation in non-Gaussian noise environments. However, when most or even all of the outputs are corrupted by large outliers, the GVMKC-KF-RRLS method may yield low-precision estimation since the current parameter estimate maintains almost the same value as that at a previous instant whenever a large outlier is encountered.

Remark 4.

The Gaussian–Versoria mixed kernel correntropy plays an important role in the joint parameter and state estimation resisting the non-Gaussian noise since it generates the weights $\eta \left(e_k\right)$ in (47), ${\widehat{W}}_{P_k}$ in (61) and ${\widehat{W}}_{R_k}$ in (62). Compared with the single kernel, the heterogeneous mixed kernel introduces more parameters and can utilize different kinds of kernel functions to realize better robustness when dealing with complex non-Gaussian noise.

Remark 5.

To outline how to select the key hyperparameters, one can first determine an appropriate range for each hyperparameter, i.e., the bandwidth $\sigma \in [{\sigma }_{\min },{\sigma }_{\max }]$, the mixing coefficient $\omega \in [0,1]$ and the Versoria parameter $\gamma \in [{\gamma }_{\min },{\gamma }_{\max }]$, where ${\sigma }_{\min }>0$ and ${\sigma }_{\max }>0$ represent the minimum and maximum values of σ, respectively (${\gamma }_{\min }>0$). Then, divide the feasible set $\left\{(\sigma ,\omega ,\gamma )\right|\sigma \in (0,{\sigma }_{\max }],\omega \in [0,1],\gamma \in (0,{\gamma }_{\max }]\}$ into small cuboids by inserting inner nodes for each hyperparameter interval. Finally, calculate the parameter estimation error at each node $({\sigma }_i,{\omega }_j,{\gamma }_l)$ by using the ergodic method. When the parameter estimation error is smallest, the values of σ, ω and γ can be regarded as the optimal hyperparameter values.

4. Simulation Examples

Example 1.

Consider the following bilinear state–space system:

$$\begin{array}{ccc}& & \left[\begin{array}{c}{x}_{1,k+1}\\ x_{2,k+1}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ -g_2& -g_1\end{array}\right]\left[\begin{array}{c}{x}_{1,k}\\ x_{2,k}\end{array}\right]+\left[\begin{array}{cc}{f}_{11}& f_{12}\\ f_{21}& f_{22}\end{array}\right]\left[\begin{array}{c}{x}_{1,k}\\ x_{2,k}\end{array}\right]u_k+\left[\begin{array}{c}{h}_1\\ h_2\end{array}\right]u_k+\left[\begin{array}{c}{w}_{1,k}\\ w_{2,k}\end{array}\right],\hfill \\ & & =\left[\begin{array}{cc}0& 1\\ -0.20& -0.10\end{array}\right]\left[\begin{array}{c}{x}_{1,k}\\ x_{2,k}\end{array}\right]+\left[\begin{array}{cc}-0.08& 0.13\\ -0.46& 0.14\end{array}\right]\left[\begin{array}{c}{x}_{1,k}\\ x_{2,k}\end{array}\right]u_k+\left[\begin{array}{c}-1.00\\ -1.14\end{array}\right]u_k+\left[\begin{array}{c}{w}_{1,k}\\ w_{2,k}\end{array}\right],\hfill \\ & & y_k=[1,0]\left[\begin{array}{c}{x}_{1,k}\\ x_{2,k}\end{array}\right]+v_k.\hfill \end{array}$$

The parameter vector to be estimated is

$$\begin{array}{ccc}\hfill \mathit{\theta }& :=& {[g_1,g_2,f_{11},f_{12},f_{21},f_{22},h_1,h_2]}^{\mathrm{T}}\hfill \\ & =& {[0.10,0.20,-0.08,0.13,-0.46,0.14,-1.00,-1.14]}^{\mathrm{T}}.\hfill \end{array}$$

In simulation, the input $u_k$ is a persistent excitation signal with zero mean and unit variance. To demonstrate the effectiveness of the GVMKC-KF-RRLS method in non-Gaussian noise environments, the process noises $w_{1,k}$ and $w_{2,k}$ and the measurement noise $v_k$ are modeled by the approximately mixed Gaussian distribution, i.e.,

$$\begin{array}{ccc}& & w_{i,k}\sim (1-{\epsilon }_i)\mathcal{N}(0,{\sigma }_{i1}^2)+{\epsilon }_i\mathcal{N}(0,{\sigma }_{i2}^2),i=1,2,\hfill \\ & & v_k\sim (1-{\epsilon }_3)\mathcal{N}(0,{\sigma }_{31}^2)+{\epsilon }_3\mathcal{N}(0,{\sigma }_{32}^2),\hfill \end{array}$$

where $\mathcal{N}(0,{\sigma }_{ij}^2)(j=1,2)$ represents the normal distribution with zero mean and variance ${\sigma }_{ij}^2$ (${\sigma }_{i2}^2\gg {\sigma }_{i1}^2$), respectively, and ${\epsilon }_i\in [0,1]$ is the contamination degree. The normal distribution with the larger variance ${\sigma }_{i2}^2$ produces outliers.

Take the data length $L=3200$, the Gaussian kernel bandwidth $\sigma =1.00$, the Versoria kernel parameter $\gamma =5.00$ and the mixed coefficient $\omega =0.40$. To facilitate observation of the influence of noises on parameter estimation, the noise variances and the contamination degree are taken as the same values, i.e., ${\epsilon }_i=0.15$, ${\sigma }_{i1}^2=0.{20}^2$ and ${\sigma }_{i2}^2=10.{00}^2$ ($i=1,2,3$). Based on the first 3000 samples,

Table 1. The parameter estimation errors using the GVMKC-KF-RRLS method and the KF-RLS method.

Methods	k	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{11}}$	${\mathit{f}}_{\mathbf{12}}$	${\mathit{f}}_{\mathbf{21}}$	${\mathit{f}}_{\mathbf{22}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\tau }$ (%)
HMCK-KF-RRLS	100	0.06777	0.21072	−0.04823	0.03123	−0.33294	0.00783	−0.76646	−0.83803	27.10081
	200	0.09363	0.26618	−0.09548	0.10265	−0.41861	0.12581	−0.88498	−0.98901	12.90231
	500	0.10244	0.22480	−0.08773	0.14387	−0.44554	0.15205	−0.94929	−1.07631	5.49213
	1000	0.10608	0.21143	−0.07401	0.14162	−0.45942	0.13681	−0.97418	−1.10683	2.85052
	2000	0.10113	0.21034	−0.07580	0.14314	−0.47226	0.13296	−0.98781	−1.12109	1.96369
	3000	0.09991	0.20624	−0.08001	0.13365	−0.47109	0.13724	−0.99150	−1.12771	1.24894
KF-RLS	100	0.01860	0.23386	−0.00771	0.09147	−0.36004	0.01241	−0.72558	−0.74796	32.18531
	200	−0.15103	0.42480	−0.22993	0.17817	−0.36582	0.33386	−0.82050	−0.70372	39.55145
	500	−0.04035	0.17004	−0.11716	−0.23939	−0.16535	0.21346	−1.14042	−1.03440	32.87922
	1000	0.10480	0.15849	−0.09430	−0.17452	−0.21359	0.05613	−0.92341	−1.07969	25.69518
	2000	0.04609	0.32114	−0.11738	0.08879	−0.13974	0.02426	−1.02994	−1.14131	22.98173
	3000	0.03568	0.26981	−0.07977	0.15041	−0.29377	0.10274	−1.08493	−1.08666	13.64798
True values		0.10000	0.20000	−0.08000	0.13000	−0.46000	0.14000	−1.00000	−1.14000

and Figure 2 give the GVMKC-KF-RRLS parameter estimate ${\widehat{\mathit{\theta }}}_k$ and the estimation error $\tau :=\parallel {\widehat{\mathit{\theta }}}_k-\mathit{\theta }\parallel /\parallel \mathit{\theta }\parallel \times 100\%$. Meanwhile, to show the advantages of the GVMKC-KF-RRLS method, under the same noise environment, Table 1 and Figure 2 compare the GVMKC-KF-RRLS method with the KF-RLS method. As is illustrated, although the parameter estimation error of the KF-RLS method is overall decreasing over the instant k, there are some significant fluctuations on the parameter estimation curve. This is because the quadratic criterion function of the KF-RLS method amplifies the error and is sensitive to non-Gaussian noise. In contrast, the GVMKC-KF-RRLS parameter estimation error curve smoothly decreases as k increases. This indicates that the GVMKC-KF-RRLS method has the ability to resist non-Gaussian noise.

Figure 3 and Figure 4 show the parameter estimates ${\widehat{g}}_{1,k}$, ${\widehat{g}}_{2,k}$, ${\widehat{f}}_{11,k}$, ${\widehat{f}}_{12,k}$, ${\widehat{f}}_{21,k}$, ${\widehat{f}}_{22,k}$, ${\widehat{h}}_{1,k}$ and ${\widehat{h}}_{2,k}$ versus the instant k. It can be found that as k increases, all the parameter estimates gradually approach and reach their true values.

To visually verify the effectiveness of the GVMKC-KF-RRLS method for the state estimation, Figure 5 and Figure 6 show the true state $x_{i,k}^0$, the noisy state $x_{i,k}$ and the estimated state ${\widehat{x}}_{i,k}^{+}$ ($i=1,2$) of the GVMKC-KF-RRLS method from 2800 to 3000, where the true state $x_{i,k}^0$ is computed by

$$\begin{array}{ccc}\hfill x_{1,k+1}^0& =& x_{2,k}^0+f_{11}{x}_{1,k}^0{u}_k+f_{12}{x}_{2,k}^0{u}_k+h_1{u}_k,\hfill \\ \hfill x_{2,k+1}^0& =& -g_2{x}_{1,k}^0-g_1{x}_{2,k}^0+f_{21}{x}_{1,k}^0{u}_k+f_{22}{x}_{2,k}^0{u}_k+h_2{u}_k.\hfill \end{array}$$

As can be seen from Figure 5 and Figure 6, the points representing the estimated states are very near the line representing the true states, which indicates that the GVMKC-KF-RRLS method can precisely estimate all the system states under the non-Gaussian noise environment.

To illustrate the influence of non-Gaussian noise on the parameter estimation,

Table 2. The parameter estimation error using the GVMKC-KF-RRLS method with different ${\varsigma }_{i2}^2$.

${\mathit{\varsigma }}_{\mathit{i}\mathbf{2}}^{\mathbf{2}}$	k	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{11}}$	${\mathit{f}}_{\mathbf{12}}$	${\mathit{f}}_{\mathbf{21}}$	${\mathit{f}}_{\mathbf{22}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\tau }$ (%)
${60.00}^2$	100	−0.01964	0.21970	−0.00081	−0.02868	−0.33531	0.12144	−0.74423	−0.82168	29.64132
	200	0.08197	0.19753	−0.01762	0.10705	−0.44611	0.23169	−0.83643	−0.94408	17.36114
	500	0.08935	0.19843	−0.01774	0.10942	−0.45003	0.20689	−0.94856	−1.06340	8.19833
	1000	0.08578	0.19969	−0.02463	0.14382	−0.45467	0.19675	−0.97305	−1.09516	6.02228
	2000	0.09822	0.19729	−0.03162	0.11818	−0.46273	0.14959	−0.98705	−1.12096	3.46150
	3000	0.09506	0.19706	−0.04113	0.11783	−0.46221	0.15047	−0.99381	−1.12816	2.76101
${20.00}^2$	100	0.01354	0.21783	−0.07134	0.01831	−0.37965	0.08900	−0.75474	−0.84329	26.11439
	200	0.09192	0.25320	−0.13028	0.17622	−0.40265	0.10861	−0.86428	−0.98117	14.59630
	500	0.09975	0.22830	−0.12440	0.16815	−0.42283	0.13374	−0.93999	−1.07477	7.20105
	1000	0.10033	0.21731	−0.11635	0.15892	−0.41996	0.10690	−0.97322	−1.11057	5.08736
	2000	0.10501	0.21682	−0.09449	0.14501	−0.45404	0.11962	−0.98885	−1.12007	2.56656
	3000	0.10085	0.21839	−0.10777	0.13155	−0.45770	0.13402	−0.99140	−1.12916	2.27257
${0.20}^2$	100	0.10820	0.18539	−0.03825	0.01925	−0.18817	−0.04694	−0.76262	−0.80486	33.47432
	200	0.11616	0.27045	−0.08163	0.08575	−0.36802	0.08646	−0.87184	−0.97253	15.55185
	500	0.11013	0.22949	−0.07679	0.13808	−0.44479	0.13892	−0.94519	−1.07740	5.61255
	1000	0.10396	0.22012	−0.07730	0.13261	−0.46404	0.13341	−0.97008	−1.11117	2.91988
	2000	0.10361	0.21174	−0.07634	0.13155	−0.46106	0.13467	−0.98498	−1.12887	1.44751
	3000	0.10229	0.20709	−0.07994	0.12983	−0.46328	0.13496	−0.99151	−1.13751	0.80762
True values		0.10000	0.20000	−0.08000	0.13000	−0.46000	0.14000	−1.00000	−1.14000

and

Table 3. The parameter estimation error using the GVMKC-KF-RRLS method with different ${\epsilon }_i$.

${\mathit{\epsilon }}_{\mathit{i}}$	k	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{11}}$	${\mathit{f}}_{\mathbf{12}}$	${\mathit{f}}_{\mathbf{21}}$	${\mathit{f}}_{\mathbf{22}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\tau }$ (%)
$0.40$	100	0.03139	0.22351	−0.06538	0.02126	−0.36181	0.05722	−0.75477	−0.84288	26.45218
	200	0.08886	0.26573	−0.12960	0.14943	−0.40277	0.10736	−0.86890	−0.98183	14.37696
	500	0.09870	0.23339	−0.11891	0.15798	−0.43124	0.13850	−0.94261	−1.07384	6.76388
	1000	0.09696	0.22511	−0.10296	0.15268	−0.44199	0.12328	−0.97599	−1.10689	3.89832
	2000	0.09643	0.22352	−0.09092	0.14739	−0.46870	0.12913	−0.98937	−1.11871	2.59053
	3000	0.09371	0.22242	−0.10773	0.14361	−0.46908	0.14297	−0.99088	−1.12796	2.64068
$0.25$	100	0.05143	0.21989	−0.05920	0.02554	−0.34834	0.02886	−0.75828	−0.84032	26.80869
	200	0.08841	0.27427	−0.11545	0.11823	−0.42308	0.12481	−0.87518	−0.98173	13.75472
	500	0.09918	0.23361	−0.10268	0.14744	−0.44826	0.15178	−0.94512	−1.07324	6.10171
	1000	0.10133	0.22027	−0.08575	0.14306	−0.45479	0.13259	−0.97523	−1.10536	3.10590
	2000	0.09578	0.22056	−0.08449	0.14570	−0.47800	0.13563	−0.98853	−1.11813	2.52469
	3000	0.09346	0.21802	−0.09588	0.13871	−0.47731	0.14070	−0.99070	−1.12699	2.19236
$0.00$	100	0.10815	0.18559	−0.03881	0.01823	−0.19023	−0.04778	−0.76266	−0.80574	33.40349
	200	0.11589	0.27195	−0.08293	0.08789	−0.36970	0.08835	−0.87166	−0.97317	15.47164
	500	0.10983	0.22953	−0.07716	0.14092	−0.44501	0.14031	−0.94503	−1.07881	5.57253
	1000	0.10382	0.22047	−0.07769	0.13359	−0.46500	0.13330	−0.96972	−1.11233	2.90940
	2000	0.10383	0.21175	−0.07595	0.13179	−0.46084	0.13410	−0.98492	−1.13017	1.42908
	3000	0.10245	0.20677	−0.08009	0.13005	−0.46322	0.13461	−0.99171	−1.13928	0.78491
True values		0.10000	0.20000	−0.08000	0.13000	−0.46000	0.14000	−1.00000	−1.14000

, and Figure 7 and Figure 8 show the GVMKC-KF-RRLS parameter estimates and errors under different noise variance ${\sigma }_{i2}^2$ and different contamination degree ${\epsilon }_i$, respectively. As can be observed from Figure 7, when the noise variance ${\sigma }_{i2}^2$ is fixed, the parameter estimation errors quickly decrease as k grows. When the instant k is fixed, the estimation errors show a minor change with increase in the noise variance ${\sigma }_{i2}^2$. It can also be seen from Figure 8 that similar cases occur. For example, as the contamination degree ${\epsilon }_i$ alters, the GVMKC-KF-RRLS estimation errors show little change under the same data length. These phenomena show that the GVMKC-KF-RRLS method is robust to variation in a non-Gaussian noise environment. In addition, it can be inferred from Figure 7 and Figure 8 that the GVMKC-KF-RRLS method maintains high estimation accuracy when ${\sigma }_{i2}^2=0.{20}^2$ and ${\epsilon }_i=0.00$, which indicates that the GVMKC-KF-RRLS method is effective in both non-Gaussian and Gaussian noise environments.

To demonstrate the robustness of the GVMKC-KF-RRLS method to the choice of key kernel hyperparameters, a sensitivity analysis is carried out by comparing the GVMKC-KF-RRLS parameter estimation errors under different kernel hyperparameters (the bandwidth $\sigma$, mixing coefficient $\omega$ and Versoria parameter $\gamma$). As can be seen from

Table 4. The GVMKC-KF-RRLS parameter estimation errors under different kernel hyperparameters.

${\mathit{\sigma }}^{\mathbf{2}}$	$\mathit{\omega }$	$\mathit{\gamma }$	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{11}}$	${\mathit{f}}_{\mathbf{12}}$	${\mathit{f}}_{\mathbf{21}}$	${\mathit{f}}_{\mathbf{22}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\tau }$ (%)
$0.{10}^2$			0.11067	0.19461	−0.06940	0.13484	−0.46912	0.14908	−0.99194	−1.11727	1.98511
$0.{50}^2$			0.09965	0.20818	−0.07827	0.13440	−0.47275	0.13928	−0.98868	−1.12539	1.51034
$1.{00}^2$	$0.40$	$5.00$	0.09991	0.20624	−0.08001	0.13365	−0.47109	0.13724	−0.99150	−1.12771	1.24894
$10.{00}^2$			0.10013	0.20771	−0.07873	0.13396	−0.47113	0.13792	−0.98822	−1.12481	1.48533
$100.{00}^2$			0.10012	0.20767	−0.07867	0.13392	−0.47114	0.13790	−0.98822	−1.12481	1.48477
$1.{00}^2$	$0.00$	$5.00$	0.09904	0.20658	−0.07972	0.13496	−0.47547	0.14040	−0.99357	−1.13049	1.29991
	$0.20$		0.09943	0.20644	−0.07984	0.13439	−0.47356	0.13903	−0.99264	−1.12926	1.26310
	$0.40$		0.09991	0.20624	−0.08001	0.13365	−0.47109	0.13724	−0.99150	−1.12771	1.24894
	$0.60$		0.10051	0.20594	−0.08029	0.13261	−0.46775	0.13483	−0.99004	−1.12570	1.28938
	$0.80$		0.10132	0.20543	−0.08083	0.13099	−0.46300	0.13152	−0.98811	−1.12297	1.44698
	$1.00$		0.10239	0.20414	−0.08251	0.12883	−0.45528	0.12701	−0.98514	−1.11893	1.84468
		$0.50$	0.10443	0.20374	−0.07824	0.12726	−0.44671	0.12205	−0.97879	−1.11051	2.67438
		$1.00$	0.10340	0.20359	−0.07990	0.12872	−0.45146	0.12501	−0.98319	−1.11579	2.13975
$1.{00}^2$	$0.40$	$5.00$	0.09991	0.20624	−0.08001	0.13365	−0.47109	0.13724	−0.99150	−1.12771	1.24894
		$10.00$	0.09678	0.21079	−0.07765	0.13781	−0.48286	0.14741	−0.99358	−1.13185	1.83659
		$50.00$	0.09103	0.22242	−0.07501	0.14476	−0.49472	0.15992	−0.99320	−1.13213	3.12102
True values			0.10000	0.20000	−0.08000	0.13000	−0.46000	0.14000	−1.00000	−1.14000

and Figure 9, when the bandwidth $\sigma$ varies from 0.1 to 100 ($\omega =0.40$ and $\gamma =5.0$ are fixed), the GVMKC-KF-RRLS parameter estimation errors change very little and always remain below $2\%$. The cases are similar when the mixing coefficient $\omega \in [0,1]$ and Versoria parameter $\gamma \in [0.5,50]$ vary. This indicates that the GVMKC-KF-RRLS method is not sensitive to variations in the kernel hyperparameters $\sigma$, $\omega$ and $\gamma$, and that introduction of the Gaussian–Versoria heterogeneous kernel correntropy increases the robustness of the parameter estimation. Moreover, it can be seen from Figure 9 that the parameter estimation errors with $\omega =0.20,0.40$ and 0.60 are lower than those with $\omega =0.00$ and 1.00, which shows that introduction of the Gaussian–Versoria heterogeneous kernel correntropy in the GVMKC-KF-RRLS method can improve parameter estimation accuracy.

To verify the effectiveness of the GVMKC-KF-RRLS method under different kinds of non-Gaussian noises, in addition to the approximately mixed Gaussian distribution noise, the following non-Gaussian noise distributions are selected:

(1)

The Laplace distribution: $w_{i,k}\sim \mathrm{Laplace}(0.10,0.{35}^2)(i=1,2)$, $v_k\sim \mathrm{Laplace}(0.10,0.{35}^2)$;

(2)

The Cauchy distribution: $w_{i,k}\sim C(0,1)(i=1,2)$, $v_k\sim C(0,1)$;

(3)

The mixed Gaussian distribution: $w_{i,k}\sim 0.85\mathcal{N}(0,0.{20}^2)+0.15\mathcal{N}(0,10.{00}^2)(i=1,2)$, $v_k\sim 0.85\mathcal{N}(0,0.{20}^2)+0.15\mathcal{N}(0,10.{00}^2)$.

The GVMKC-KF-RRLS method is applied to identify the bilinear system under these three types of non-Gaussian noise environments, respectively. For fairness, the kernel hyperparameters of correntropy are taken to be the same in the three noise environments, i.e., the bandwidth ${\sigma }^2=1.{00}^2$, mixing coefficient $\omega =0.40$ and Versoria parameter $\gamma =5.00$.

Table 5. The GVMKC-KF-RRLS estimation errors with different kinds of non-Gaussian noises.

Noises	k	${\mathit{g}}_{\mathbf{1}}$	${\mathit{g}}_{\mathbf{2}}$	${\mathit{f}}_{\mathbf{11}}$	${\mathit{f}}_{\mathbf{12}}$	${\mathit{f}}_{\mathbf{21}}$	${\mathit{f}}_{\mathbf{22}}$	${\mathit{h}}_{\mathbf{1}}$	${\mathit{h}}_{\mathbf{2}}$	$\mathit{\tau }$ (%)
Cauchy noise	100	0.18014	−0.10092	−0.10995	0.08347	−0.47285	−0.02605	−0.62682	−0.60766	45.97503
	200	0.19772	0.03412	−0.11618	0.09050	−0.40935	0.03534	−1.05269	−1.14033	14.69448
	500	0.14952	0.15420	−0.08429	0.10980	−0.44768	0.11447	−1.06952	−1.22371	8.22579
	1000	0.14892	0.17978	−0.08107	0.11338	−0.44784	0.12044	−1.08355	−1.26395	9.98350
	2000	0.12899	0.19008	−0.07537	0.11803	−0.45362	0.12283	−1.03237	−1.19407	4.55835
	3000	0.12347	0.19237	−0.07424	0.12219	−0.45641	0.13275	−1.02098	−1.17124	2.89661
Laplace noise	100	0.16710	0.25154	−0.28676	−0.02328	−0.46359	0.06956	−0.78732	−0.69092	35.34258
	200	0.15102	0.26373	−0.09966	0.13613	−0.42173	0.20662	−0.84152	−0.83131	22.63319
	500	0.19689	0.15794	−0.19903	0.21706	−0.41123	0.16446	−0.96766	−1.03146	13.67631
	1000	0.09939	0.18023	−0.07486	0.15656	−0.42700	0.13009	−0.96488	−1.05079	6.64625
	2000	0.09737	0.19479	−0.06733	0.17793	−0.42513	0.11934	−1.00411	−1.12240	4.13900
	3000	0.11994	0.17869	−0.08971	0.14410	−0.46261	0.12163	−1.00700	−1.14119	2.43129
True values		0.10000	0.20000	−0.08000	0.13000	−0.46000	0.14000	−1.00000	−1.14000

and Figure 10 show the parameter estimation errors under three different kinds of non-Gaussian noise interferences (when the non-Gaussian noise is a mixed Gaussian one, the GVMKC-KF-RRLS parameter estimation errors are the same as those in Table 1). As can be observed, all the parameter estimation curves decrease as k increases, and the parameter estimation error is below $2.9\%$ when k reaches 3000. This indicates that the GVMKC-KF-RRLS method can effectively suppress outliers and maintain high estimation precision under the aforementioned three types of noises.

Example 2.

In the following, the proposed GVMKC-KF-RRLS method is tested on a thyristor-driven DC-motor. According to the laws of physics, the DC-motor system can be expressed by the following differential equations [43]:

$$\begin{array}{ccc}\hfill J{\dot{\omega }}_k& =& k_m{i}_{a,k}{i}_{f,k}-{\tau }_{l,k}-B{\omega }_k,\hfill \\ \hfill u_{a,k}& =& R_a{i}_{a,k}+L_a{\dot{i}}_{a,k}+k_m{\omega }_k{i}_{f,k},\hfill \end{array}$$

where ${\omega }_k$ is the motor angular speed (rad/sec), $i_{a,k}$ is the armature current (A), and $i_{f,k}$ is the field motor current (A). The other variables involved and their values at steady-state are illustrated in

Table 6. The steady-state values of the DC-motor model parameters.

Parameter	Value	Unit
Armature inductance, $L_a$	0.314	H
Armature resistance, $R_a$	12.345	$\Omega$
Motor constant, $k_m$	0.2534	Nm/A2
Total system moment of inertia, J	0.00441	Nm sec2
Total system viscous friction constant, B	0.00732	Nm sec
Load torque, ${\tau }_l$	1.47	Nm
Armature terminal voltage source, $u_a$	60	V

. The input field current $i_{f,k}$ is chosen as the input $u_k$, the angular speed ${\omega }_k$ as the output $y_k$, and the armature current $i_{a,k}$ and the motor angular speed ${\omega }_k$ as the states, i.e., ${\mathit{x}}_k:={[{\omega }_k,i_{a,k}]}^{\mathrm{T}}$. Assuming that the load torque ${\tau }_{l,k}$ and the armature terminal voltage source $u_{a,k}$ are constants, the DC-motor model can be described by a continuous-time bilinear system:

$$\begin{array}{ccc}\hfill {\dot{\mathit{x}}}_k& =& \left[\begin{array}{cc}-{\displaystyle \frac{B}{J}}& 0\\ 0& -{\displaystyle \frac{R_a}{L_a}}\end{array}\right]{\mathit{x}}_k+\left[\begin{array}{cc}{\displaystyle \frac{k_m}{J}}& 0\\ 0& -{\displaystyle \frac{k_m}{L_a}}\end{array}\right]{\mathit{x}}_k{u}_k+\left[\begin{array}{c}-{\displaystyle \frac{{\tau }_l}{J}}\\ {\displaystyle \frac{u_a}{L_a}}\end{array}\right],\hfill \\ \hfill y_k& =& [1,0]{\mathit{x}}_k.\hfill \end{array}$$

The equivalent circuit representation of the DC-motor and load is shown in Figure 11, where $B_l$ and $B_m$ are the load and motor viscous friction constant, $J_l$ and $J_m$ are the load and motor moment of inertia, and ${\tau }_m$ is the electromagnetic torque of the motor.

The sample interval is taken to be 0.01s and the forward Euler method is used to discretize the DC-motor system. The approximately mixed Gaussian noises $w_{1,k}$, $w_{2,k}$ and $v_k$ are added into the state equation and the output equation, where $w_{i,k},v_k\sim 0.85\mathcal{N}(0,0.{20}^2)+0.15\mathcal{N}(0,10.{00}^2)$. The control input $u\left(k\right)$ is taken as a random signal uniformly distributed on $[0.4,0.8]$ and 3000 data are collected. It is assumed that the system model is in its observability canonical form. Since the state–space model is equivalent with its canonical form, the sampled data are the same. The GVMKC-KF-RLS method is applied to identify this example. Based on the obtained parameter estimates, another 200 data are collected and used for model validation. Figure 12 shows the simulation results. It can be observed that the estimated outputs closely match the true outputs, which indicates that the GVMKC-KF-RLS method can capture the practical system dynamics.

5. Conclusions

This paper designs a robust parameter and state estimation method for the bilinear state–space system with non-Gaussian noise. Due to the introduction of Gaussian–Versoria mixed kernel correntropy, the time-varying gains can be generated such that the update of the parameter and state estimation can almost be halted when non-Gaussian noises are encountered but the estimation can normally be performed when Gaussian noises are encountered; thus, the proposed method can be adaptive to complex data quality. The numerical example demonstrates that the proposed method can give higher estimation accuracy than the Kalman filtering-based recursive least squares method, and the usage of the Gaussian–Versoria mixed kernel improves robust and stable estimation performance in contrast to the single kernel correntropy-based estimation methods. Moreover, the practical utility of the proposed method is verified through a thyristor-driven DC-motor experiment.