A Novel Zeroing Neural Network for Solving Time-Varying Quadratic Matrix Equations against Linear Noises

Abstract

The solving of quadratic matrix equations is a fundamental issue which essentially exists in the optimal control domain. However, noises exerted on the coefficients of quadratic matrix equations may affect the accuracy of the solutions. In order to solve the time-varying quadratic matrix equation problem under linear noise, a new error-processing design formula is proposed, and a resultant novel zeroing neural network model is developed. The new design formula incorporates a second-order error-processing manner, and the double-integration-enhanced zeroing neural network (DIEZNN) model is further proposed for solving time-varying quadratic matrix equations subject to linear noises. Compared with the original zeroing neural network (OZNN) model, finite-time zeroing neural network (FTZNN) model and integration-enhanced zeroing neural network (IEZNN) model, the DIEZNN model shows the superiority of its solution under linear noise; that is, when solving the problem of a time-varying quadratic matrix equation in the environment of linear noise, the residual error of the existing model will maintain a large level due to the influence of linear noise, which will eventually lead to the solution’s failure. The newly proposed DIEZNN model can guarantee a normal solution to the time-varying quadratic matrix equation task no matter how much linear noise there is. In addition, the theoretical analysis proves that the neural state of the DIEZNN model can converge to the theoretical solution even under linear noise. The computer simulation results further substantiate the superiority of the DIEZNN model in solving time-varying quadratic matrix equations under linear noise.

1. Introduction

The quadratic matrix equation (QME), as a fundamental nonlinear paradigm, arises frequently in a variety of applications. The application of the QME extensively appears in optimal control [1,2,3], the analysis of structural systems and vibration problems [4,5,6,7,8,9], block tridiagonal transition probability matrices which are used in two-dimensional Markov chains with quasi-birth-death processes [10,11], damped mass-spring systems [11], telecommunication stochastic models [12], computer performance and inventory control [13]. To solve the QME, many techniques have been proposed and investigated [11,14,15,16,17,18,19,20]. For instance, Davis [14,15] used Newton’s method to solve the QME, with theoretical analysis and implementation details provided. Benner and Byers [16] studied the use of exact line searching in Newton’s methods for solving algebraic Riccati equations, which are a special type of QME. Higham et al. [11] solved the QME with the Bernoulli iterative method and a convergence time less than that in Newton’s methods.

The aforementioned approaches were instinctively designed to solve static QMEs in a serial processing manner. In recent years, dynamic solvers with a parallel processing ability have been developed and attracted attention from research communities [21,22,23]. The neural dynamic method can be a strong alternative for matrix computation problems due to its parallelism and ease of hardware implementation. For example, the GNNs [24,25,26] that evolve along the negative gradient descent direction of the scalar-valued energy function cause the residual errors to gradually approach zero over time. However, even after an infinite amount of time, the GNN model cannot converge to the exact solution under time-varying situations; it can only arrive at an approximation of the theoretical solution of the time-varying matrix. This is because the GNNs do not use the time derivative information of the coefficient matrix or there is a lack of velocity compensation for the time-varying coefficient. As a special kind of recurrent neural network solver, zeroing neural networks (ZNNs) have recently been developed and applied in various time-varying matrix problems in an online manner [27,28,29,30,31]. Compared with the GNNs, a ZNN makes full use of the time derivative of the time-varying term for superior convergence performance. Furthermore, the ZNN model can be effectively applied to different types of redundant robot manipulators. For example, the Li function-activated zeroing neural network (LFAZNN) has been applied to the four-link planar manipulator and PA10 manipulator [32], and the integration-enhanced zeroing neural network (IEZNN) has been applied to the two-link planar robot manipulator [33].

Various internal and external noises are not considered primitively in the design and implementation of dynamic recurrent neural networks, such as the OZNN model for the solution of a time-varying matrix square root [34,35] or the FTZNN model for the solution of a time-varying matrix square root [36]. These models can be solved effectively in a noiseless environment, and the network state solution of the neural network model can converge to the theoretical solution of the square root of the time-varying matrix. However, the neural network models could be contaminated by implementation errors when they are applied to computational issues. Eventually, accurate solutions may not be found under such unexpected exerted noises. As noises may always appear and exist in the coefficients of the neural network models after the solution process, involving the denoising loop and pretreating noises prior to the solution process can be time-consuming and unnecessary. Therefore, it is crucial to diminish the negative effects of noise during the solution. The IEZNN model shows excellent convergence performance when solving the time-varying matrix inverse with constant noise interference, but when there is linear noise interference, the IEZNN model cannot completely suppress linear noise, and the residual error of the IEZNN model cannot converge to zero [33].

In this paper, in order to solve the time-varying QME under simultaneous linear noise, a novel ZNN model with a double-integrated enhanced error-processing loop is proposed and studied. Such a model, termed as the DIEZNN, is able to make its neural state converge to the theoretical solution globally even under linear noise. The proposed DIEZNN model can be substantiated to possess improved convergence performance compared with other state-of-the-art models; that is, unbounded linear noise will greatly affect the solving ability of existing models, and the DIEZNN model proposed can suppress this unbounded linear noise well. In other words, when solving the time-varying QME under linear noise, the neural state solution of the DIEZNN model can converge to the theoretical solution of the time-varying QME, and the residual error of the DIEZNN model can be kept very small. The efficiency of the model is further verified by example computational results.

The rest of this paper is organized into the following sections. The problem formulation is presented in Section 2. The dynamic recurrent neural network method is presented in Section 3. In Section 4, the theoretical analysis and results are addressed to investigate the convergence properties of the DIEZNN model for the solution of the time-varying QME without and with linear noise. Section 5 provides three experimental examples to verify the superiority of the DIEZNN model compared with other existing models. The conclusion is given in Section 6. The main contributions of this paper are listed below:

• In order to suppress linear noise perturbation for the solution of the time-varying QME, a DIEZNN model is first proposed with a new error-processing method.
• Theoretical analysis demonstrates that the proposed DIEZNN model converges to the theoretical solution of the QME globally. More importantly, the DIEZNN model is proved to also be able to converge to the theoretical solution of the QME in the case of linear noise interference.
• The superiority of the DIEZNN model to solve the time-varying QME under linear noise, compared with other methods such as the OZNN, FTZNN and IEZNN, is further verified by three simulation examples.

2. Problem Formulation

In this section, the problem formulation of time-varying QME subject to noise is presented first. Afterward, the proposed novel DIEZNN model with other comparative models is addressed. In this work, let us consider the following time-varying QME:

$$A\left(t\right)X^2\left(t\right)+B\left(t\right)X\left(t\right)+C\left(t\right)=0.$$

(1)

where $X\left(t\right)\in R^{n\times n}$ denotes an unknown time-varying matrix to be obtained, $A\left(t\right)\in R^{n\times n},B\left(t\right)\in R^{n\times n}$ and $C\left(t\right)\in R^{n\times n}$ are known time-varying matrices and $X^{*}\left(t\right)$ denotes the theoretical solution to Equation (1). We assume that $A\left(t\right),B\left(t\right)$ and $C\left(t\right)$ are non-singular at any time $(t\in \left[0,\infty \right))$ and the $A\left(t\right),B\left(t\right)$ and $C\left(t\right)$ time derivatives are uniformly bounded.

When the noises come into the time-varying QME (Equation (1)), it further becomes

$$(A\left(t\right)+\mathsf{\Delta }A\left(t\right))X^2\left(t\right)+(B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right)=0,$$

(2)

where $\mathsf{\Delta }A\left(t\right)$, $\mathsf{\Delta }B\left(t\right)$ and $\mathsf{\Delta }C\left(t\right)$ are noise disturbances to the time-varying coefficients. The accurate solution of the time-varying QME (Equation (1)) will be perturbed by the degraded model with coefficient noise. In this work, we propose to solve such a time-varying QME under linear noise with our newly developed DIEZNN models.

3. Dynamic Recurrent Neural Network Method

3.1. OZNN Model

When the GNN model is used to solve the time-varying problem, only the approximate solution of the time-varying problem can be obtained, and it cannot converge to the exact solution of the time-varying matrix problem while the residual error cannot converge to zero [24]. Moreover, the residual error of the GNN model is larger when it is used to solve time-varying problems under linear noise conditions [33].

For comparative purposes, other models are presented to show the differences with the proposed DIEZNN model. In order to solve the time-varying QME with linear noise, instead of using a GNN model, the original zeroing neural network (OZNN) model needs to process the error as follows:

$$R\left(t\right)=(A\left(t\right)+\mathsf{\Delta }A\left(t\right))X^2\left(t\right)+(B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right).$$

(3)

In order to make $R\left(t\right)$ approach zero as time evolves, the following error processing formula is adopted [24]:

$$\dot{R}\left(t\right)=-\gamma R\left(t\right)$$

(4)

By expanding the design formula, the OZNN model for solving the time-varying QME under linear noise is derived as follows:

$$\begin{array}{cc}\hfill & (A\left(t\right)+\mathsf{\Delta }A\left(t\right))\dot{X}\left(t\right)X\left(t\right)+((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))\dot{X}\left(t\right)\hfill \\ \hfill =& -((\dot{A}\left(t\right)+\mathsf{\Delta }\dot{A}\left(t\right))X\left(t\right)+\dot{B}\left(t\right)+\mathsf{\Delta }\dot{B}\left(t\right))X\left(t\right)-\dot{C}\left(t\right)-\mathsf{\Delta }\dot{C}\left(t\right)\hfill \\ \hfill & -\gamma \left(\right(\left(A\right(t)+\mathsf{\Delta }A(t\left)\right)X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right)\left)X\right(t)+C(t)+\mathsf{\Delta }C(t\left)\right).\hfill \end{array}$$

(5)

where $\gamma$ denotes the design parameters to control the convergence rate.

3.2. FTZNN Model

To solve for the time-varying matrix QME with linear noise, the following error function is applied to the finite-time zeroing neural network (FTZNN) model:

$$R\left(t\right)=(A\left(t\right)+\mathsf{\Delta }A\left(t\right))X^2\left(t\right)+(B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right).$$

(6)

To make $R\left(t\right)$ approach zero in a limited time, the following error processing manner is applied [36]:

$$\dot{R}\left(t\right)=-\gamma (k_{1}R\left(t\right)+k_2{R}^{(p/q)}\left(t\right)).$$

(7)

As a result, the FTZNN model for solving the time-varying QME under linear noise is as shown below:

$$\begin{array}{cc}\hfill & (A\left(t\right)+\mathsf{\Delta }A\left(t\right))\dot{X}\left(t\right)X\left(t\right)+((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))\dot{X}\left(t\right)\hfill \\ \hfill =& -((\dot{A}\left(t\right)+\mathsf{\Delta }\dot{A}\left(t\right))X\left(t\right)+\dot{B}\left(t\right)+\mathsf{\Delta }\dot{B}\left(t\right))X\left(t\right)-\dot{C}\left(t\right)-\mathsf{\Delta }\dot{C}\left(t\right)\hfill \\ \hfill & -{\beta }_1(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))\hfill \\ \hfill & -{\beta }_2{(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))}^{(p/q)},\hfill \end{array}$$

(8)

where ${\beta }_1=k_1\gamma >0$ and ${\beta }_2=k_2\gamma >0$ denote the design parameters and p and q denote positive odd integers that satisfy $p>q$. In this paper, for computation illustration, we can choose $k_1=k_2=1$, $p=1$ and $q=5$.

3.3. IEZNN Model

In order to solve the time-varying QME in a linear noise environment, the integration-enhanced zeroing neural network (IEZNN) model uses the following error function to process the error:

$$R\left(t\right)=(A\left(t\right)+\mathsf{\Delta }A\left(t\right))X^2\left(t\right)+(B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right).$$

(9)

In order to suppress noise interference in solving the time-varying QME, an error-processing design formula with a single integral loop is used [33]:

$$\dot{R}\left(t\right)=-\gamma R\left(t\right)-\lambda {\int }_0^{t}R\left(t\right)dt.$$

(10)

According to the IEZNN design formula, the IEZNN model for solving the time-varying QME under linear noise is as follows:

$$\begin{array}{cc}\hfill & (A\left(t\right)+\mathsf{\Delta }A\left(t\right))\dot{X}\left(t\right)X\left(t\right)+((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))\dot{X}\left(t\right)\hfill \\ \hfill =& -((\dot{A}\left(t\right)+\mathsf{\Delta }\dot{A}\left(t\right))X\left(t\right)+\dot{B}\left(t\right)+\mathsf{\Delta }\dot{B}\left(t\right))X\left(t\right)-\dot{C}\left(t\right)-\mathsf{\Delta }\dot{C}\left(t\right)\hfill \\ \hfill & -\gamma \left(\right(\left(A\right(t)+\mathsf{\Delta }A(t\left)\right)X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right)\left)X\right(t)+C(t)+\mathsf{\Delta }C(t\left)\right)\hfill \\ \hfill & -\lambda {\int }_0^t(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))dt,\hfill \end{array}$$

(11)

where $\gamma$ and $\lambda$ denote the design parameters. We chose $\gamma =\lambda$ for this paper.

3.4. DIEZNN Model

In order to monitor the time-varying QME solution process, we define the following error function in matrix form:

$$R\left(t\right)=(A\left(t\right)+\mathsf{\Delta }A\left(t\right))X^2\left(t\right)+(B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right).$$

(12)

To eliminate or reduce the linear noise, in this work, we propose the following DIEZNN design formula:

$$\dot{R}\left(t\right)=-b_{1}R\left(t\right)-b_2{\int }_0^{t}R\left(t\right)dt-b_3{\int }_0^t{\int }_0^{t}R\left(t\right)dt.$$

(13)

The design formula possesses proportion parts and integral parts in which $b_1>0$, $b_2>0$ and $b_3>0$, where $b_1$, $b_2$ and $b_3$ denote the convergence scaling parameters. Throughout this paper, we chose $b_1=3\gamma$, $b_2=3{\gamma }^2$ and $b_3={\gamma }^3$. With the combination of Equations (12) and (13), we can obtain the following DIEZNN model:

$$\begin{array}{cc}\hfill & (A\left(t\right)+\mathsf{\Delta }A\left(t\right))\dot{X}\left(t\right)X\left(t\right)+((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))\dot{X}\left(t\right)\hfill \\ \hfill =& -((\dot{A}\left(t\right)+\mathsf{\Delta }\dot{A}\left(t\right))X\left(t\right)+\dot{B}\left(t\right)+\mathsf{\Delta }\dot{B}\left(t\right))X\left(t\right)-\dot{C}\left(t\right)-\mathsf{\Delta }\dot{C}\left(t\right)\hfill \\ \hfill & -b_1(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))\hfill \\ \hfill & -b_2{\int }_0^t(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))dt\hfill \\ \hfill & -b_3{\int }_0^t{\int }_0^t(((A\left(t\right)+\mathsf{\Delta }A\left(t\right))X\left(t\right)+B\left(t\right)+\mathsf{\Delta }B\left(t\right))X\left(t\right)+C\left(t\right)+\mathsf{\Delta }C\left(t\right))dt,\hfill \end{array}$$

(14)

where $X\left(t\right)$, starting from a random initial state $X\left(0\right)$, is the state matrix corresponding to the theoretical solution to Equation (1).

Remark 1.

Because the DIEZNN model (Equation (14)), OZNN model (Equation (5)), FTZNN model (Equation (8)) and IEZNN model (Equation (11)) are shown in terms of implicit dynamics, we can use MATLAB for the simulation experiments. Specifically, we can use the routine “ode45” with the mass matrix property (i.e, $M(t,x)\dot{x}=z(t,x)$). In our program, $M(t,x)=X^T\otimes (A+\mathsf{\Delta }A)+I\otimes ((A+\mathsf{\Delta }A)X+B+\mathsf{\Delta }B)$, $z(t,x)$ is the vectorization form of the right part of the above model equation (i.e., Equations (5), (8), (11) and (14)), the identity matrix is represented by I, the matrix transpose is represented by a superscript T, and the Kronecker-product is represented by ⊗. The above model was transformed into an initial-value ODE problem with a mass matrix such that it could be simulated and computed more easily and effectively.

In addition, the comparison of the above models is shown in

Table 1. Comparison of OZNN model, FTZNN model, IEZNN model and DIEZNN model.

Model	OZNN model	FTZNN model	IEZNN model	DIEZNN model
Problem	QME	QME	QME	QME
Design Formula	$\dot{R}\left(t\right)=-\gamma R\left(t\right)$	$\dot{R}\left(t\right)=-{\beta }_{1}R\left(t\right)-{\beta }_{2}R{\left(t\right)}^{(p/q)}$	$\dot{R}\left(t\right)=-\gamma R\left(t\right)-\lambda {\int }_0^{t}R\left(t\right)dt$	$\dot{R}\left(t\right)=-b_{1}R\left(t\right)-b_2{\int }_0^{t}R\left(t\right)dt-b_3{\int }_0^t{\int }_0^{t}R\left(t\right)dt$
Noise	zero noise	zero noise	linear noise	linear noise
Residual Error	infinity	infinity	constant	zero
Robustness	rare	rare	weak	strong

. In Table 1, we can see the different design formulas and noise tolerance levels of the above models.

4. Theoretical Analysis and Results

For the problem of solving a time-varying QME, in this section, we provide two theorems to substantiate the convergence ability of the DIEZNN model:

Theorem 1.

In a noise-free environment, the neural state matrix of the DIEZNN model converges globally to the theoretical solution of the time-varying QME.

Proof of Theorem 1.

Let us take the second derivative of $\dot{R}\left(t\right)=-b_{1}R\left(t\right)-b_2{\int }_0^{t}R\left(t\right)dt-b_3{\int }_0^t{\int }_0^{t}R\left(t\right)dt$ with respect to both sides: $\stackrel{⃛}{R}\left(t\right)=-b_1\ddot{R}\left(t\right)-b_2\dot{R}\left(t\right)-b_{3}R\left(t\right)$. Take the Laplace transform of this, and we can obtain

$$\begin{array}{cc}\hfill S^{3}R{\left(t\right)}_S-S^{2}R{\left(t\right)}_0-S\dot{R}{\left(t\right)}_0-\ddot{R}{\left(t\right)}_0=& -b_1(S^{2}R{\left(t\right)}_S-SR{\left(t\right)}_0-\dot{R}{\left(t\right)}_0)\hfill \\ \hfill & -b_2(SR{\left(t\right)}_S-R{\left(t\right)}_0)\hfill \\ \hfill & -b_{3}R{\left(t\right)}_S\hfill \end{array}$$

(15)

We further obtain

$$R{\left(t\right)}_S=\frac{S^{2}R{\left(t\right)}_0+S\dot{R}{\left(t\right)}_0+\ddot{R}{\left(t\right)}_0+b_{1}SR{\left(t\right)}_0+b_1\dot{R}{\left(t\right)}_0+b_{2}R{\left(t\right)}_0}{S^3+b_1{S}^2+b_{2}S+b_3}$$

(16)

By setting $e=S^{2}R{\left(t\right)}_0+S\dot{R}{\left(t\right)}_0+\ddot{R}{\left(t\right)}_0+b_{1}SR{\left(t\right)}_0+b_1\dot{R}{\left(t\right)}_0+b_{2}R{\left(t\right)}_0$ and substituting $b_1=3S_0,b_2=3S_0^2$ and $b_3=S_0^3$, we can obtain

$$R{\left(t\right)}_S=\frac{e}{S^3+3S_0{S}^2+3S_0^{2}S+S_0^3}$$

(17)

In other words, we have

$$R{\left(t\right)}_S=\frac{e}{{(S+S_0)}^3}$$

(18)

Because the real part of $S_0$ is in the left half-plane, $R{\left(t\right)}_S$ is stable, and we can use the final value theorem:

$$\underset{t\to \infty }{lim}R\left(t\right)=\underset{S\to 0}{lim}SR{\left(t\right)}_S=\underset{S\to 0}{lim}\frac{Se}{{\left(S+S_0\right)}^3}=0$$

(19)

□

Theorem 2.

The state matrix of the DIEZNN model converges to the theoretical solution of the time-varying QME under linear noise.

Proof of Theorem 2.

Let us take the second derivative of $\dot{R}\left(t\right)=-b_{1}R\left(t\right)-b_2{\int }_0^{t}R\left(t\right)dt-b_3{\int }_0^t{\int }_0^{t}R\left(t\right)dt+N\left(t\right)$ with respect to both sides: $\stackrel{⃛}{R}\left(t\right)=-b_1\ddot{R}\left(t\right)-b_2\dot{R}\left(t\right)-b_{3}R\left(t\right)dt+\ddot{N}\left(t\right)$. Because of linear noise $N\left(t\right)=a_0+a_{1}t$, therefore, $\ddot{N}\left(t\right)=0$, and we can obtain $\stackrel{⃛}{R}\left(t\right)=-b_1\ddot{R}\left(t\right)-b_2\dot{R}\left(t\right)-b_{3}R\left(t\right)$. By taking the Laplace transform of this, we can obtain

$$\begin{array}{cc}\hfill S^{3}R{\left(t\right)}_S-S^{2}R{\left(t\right)}_0-S\dot{R}{\left(t\right)}_0-\ddot{R}{\left(t\right)}_0=& -b_1(S^{2}R{\left(t\right)}_S-SR{\left(t\right)}_0-\dot{R}{\left(t\right)}_0)\hfill \\ \hfill & -b_2(SR{\left(t\right)}_S-R{\left(t\right)}_0)\hfill \\ \hfill & -b_{3}R{\left(t\right)}_S\hfill \end{array}$$

(20)

We further obtain

$$R{\left(t\right)}_S=\frac{S^{2}R{\left(t\right)}_0+S\dot{R}{\left(t\right)}_0+\ddot{R}{\left(t\right)}_0+b_{1}SR{\left(t\right)}_0+b_1\dot{R}{\left(t\right)}_0+b_{2}R{\left(t\right)}_0}{S^3+b_1{S}^2+b_{2}S+b_3}$$

(21)

Set $e=S^{2}R{\left(t\right)}_0+S\dot{R}{\left(t\right)}_0+\ddot{R}{\left(t\right)}_0+b_{1}SR{\left(t\right)}_0+b_1\dot{R}{\left(t\right)}_0+b_{2}R{\left(t\right)}_0$ and substitute $b_1=3S_0,b_2=3S_0^2$ and $b_3=S_0^3$. Then, we obtain

$$R{\left(t\right)}_S=\frac{e}{S^3+3S_0{S}^2+3S_0^{2}S+S_0^3}$$

(22)

That is, we have

$$R{\left(t\right)}_S=\frac{e}{{(S+S_0)}^3}$$

(23)

Because the real part of $S_0$ is in the left half-plane, $R{\left(t\right)}_S$ is stable, and we can use the final value theorem:

$$\underset{t\to \infty }{lim}R\left(t\right)=\underset{S\to 0}{lim}SR{\left(t\right)}_S=\underset{S\to 0}{lim}\frac{Se}{{\left(S+S_0\right)}^3}=0$$

(24)

□

5. Illustrative Verification

In this section, we verify the convergence ability and solution performance of the DIEZNN model (Equation (14)) in solving the time-varying QME under a linear noise environment. Other models such as the OZNN model (Equation (5)), FTZNN model (Equation (8)) and IEZNN model (Equation (11)) are also utilized for solving such QMEs under linear noise for comparisons.

Example 1.

In this example, for solving a time-varying QME under linear noise, we give the following time-varying matrix A(t):

$$A\left(t\right)=\left[\begin{array}{cc}1& cos\left(t\right)\\ sin\left(t\right)& 3\end{array}\right]\in {\mathbb{R}}^{2\times 2}.$$

(25)

The time-varying matrices $B\left(t\right)$ and $C\left(t\right)$ are as follows:

$$B\left(t\right)=\left[\begin{array}{cc}1& cos\left(t\right)\\ sin\left(t\right)& 5\end{array}\right]\in {\mathbb{R}}^{2\times 2}.$$

(26)

$$C\left(t\right)=\left[\begin{array}{cc}-(7c^2+sc+6)& -(7s+20c+s{c}^2)\\ -(23c+6s+s^{2}c)& -(7s^2+3sc+68)\end{array}\right]\in {\mathbb{R}}^{2\times 2}.$$

(27)

where s and c denote $sin\left(t\right)$ and $cos\left(t\right)$, respectively. For the purpose of comparison with the neural network solution, according to Equation (1), we directly give the theoretical solution of the time-varying QME as follows:

$$X^{*}\left(t\right)=\left[\begin{array}{cc}2& sin\left(t\right)\\ cos\left(t\right)& 4\end{array}\right]\in {\mathbb{R}}^{2\times 2}.$$

(28)

In this example, the four models have the same linear noise and design parameters when solved. The computer simulation results are shown in Figure 1. In Figure 1, with linear noise $N\left(t\right)=10+10t$ and design parameters $\gamma =5$, starting from an randomly generated initial state $X_0\in R^{2\times 2}$, the network solution to the FTZNN model (Equation (8)) could not fit the theoretical solution, and the error was large. The network solution to the OZNN model (Equation (5)) also could not fit the theoretical solution. The network state solution to the IEZNN model (Equation (11)) was close to the theoretical solution but still did not track it. In contrast, the network state solution to the DIEZNN model (Equation (14)) could quickly fit the theoretical solution. The error function norm ${\parallel R\left(t\right)\parallel }_{\mathrm{F}}$ is shown in Figure 2. As shown in Figure 2a, with design parameters $\gamma =5$ and linear noise $N\left(t\right)=10+10t$, the error function norm of the OZNN model (Equation (5)) and the FTZNN model (Equation (8)) were kept at a relatively high level. The error function norm of the IEZNN model (Equation (11)) was relatively small but could not converge to zero. By contrast, the error function norm of the DIEZNN model (Equation (14)) converged to zero within 2 s. In Figure 2b, we increased the design parameter $\gamma$ to 10, while the linear noise $N\left(t\right)=10+10t$ remained unchanged. The error function norm of the OZNN model (Equation (5)) and the FTZNN model (Equation (8)) still showed an upward trend. The error function norm of the IEZNN model (Equation (11)) still could not converge to zero. In contrast, the error function norm of the DIEZNN model (Equation (14)) could converge to zero, and the convergence time was reduced to about 1 s. Through the above simulation results, the DIEZNN model (Equation (14)) showed its superiority in solving time-varying QMEs under linear noise.

Example 2.

We use the following time-varying matrix to further verify the solving ability of the DIEZNN model (Equation (14)):

$$A\left(t\right)=B\left(t\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1+t\end{array}\right]\in {\mathbb{R}}^{3\times 3}.$$

(29)

$$C\left(t\right)=\left[\begin{array}{ccc}-2& 0& 0\\ 0& -2& 0\\ 0& 0& -(1+t)(2+sin\left(t\right)+{sin}^2\left(t\right))\end{array}\right]\in {\mathbb{R}}^{3\times 3}.$$

(30)

According to Equation (1), the theoretical solution to the time-varying QME is given directly as follows:

$$X^{*}\left(t\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1+sin\left(t\right)\end{array}\right]\in {\mathbb{R}}^{2\times 3}.$$

(31)

The computer simulation results are shown in Figure 3. As illustrated in Figure 3, with linear noise $N\left(t\right)=10+10t$ and design parameters $\gamma =5$, the red dash-dotted line represents the theoretical solution to the time-varying QME. The network solution to the DIEZNN model (Equation (14)), represented by blue solid lines, quickly fit the theoretical solution. In comparison, the network solution to the OZNN model (Equation (5)), represented by black dashed lines, and the FTZNN model (Equation (8)), represented by orange dash-dotted lines, could not fit the theoretical solution. The network solution to the IEZNN model (Equation (11)), represented by green dotted lines, was close to the theoretical solution but still could not track the theoretical solution. The error function norm ${\parallel R\left(t\right)\parallel }_{\mathrm{F}}$ is shown in Figure 4. We can observe from Figure 4 that the error function of the DIEZNN model (Equation (14)) converged to zero in 1 s. The error function norm of the OZNN model (Equation (5)) and the FTZNN model (Equation (8)) showed an upward trend. The error function norm of the IEZNN model (Equation (11)) remained stable but could not converge to zero. In summary, the simulation results further demonstrate the superiority of the DIEZNN model (Equation (14)).

Example 3.

In this example, we consider matrices with higher dimensions:

$$A_1\left(t\right)=\left[\begin{array}{ccc}A\left(t\right)& 0& 0\\ 0& A\left(t\right)& 0\\ 0& 0& A\left(t\right)\end{array}\right]\in {\mathbb{R}}^{6\times 6},$$

(32)

$$B_1\left(t\right)=\left[\begin{array}{ccc}B\left(t\right)& 0& 0\\ 0& B\left(t\right)& 0\\ 0& 0& B\left(t\right)\end{array}\right]\in {\mathbb{R}}^{6\times 6},$$

(33)

$$C_1\left(t\right)=\left[\begin{array}{ccc}C\left(t\right)& 0& 0\\ 0& C\left(t\right)& 0\\ 0& 0& C\left(t\right)\end{array}\right]\in {\mathbb{R}}^{6\times 6}.$$

(34)

In this example, we use a time-varying matrix with a higher dimension than the above example, and the linear noise and design parameters are the same as those in the first example. In this example, we only showed the error function norm ${\parallel R\left(t\right)\parallel }_{\mathrm{F}}$ for the reason of conciseness. The simulation results are shown in Figure 5. We can see from Figure 5a that when $N\left(t\right)=10+10t$ and $\gamma =5$, starting from a randomly generated initial state $X_0\in R^{6\times 6}$, the error function norm of the DIEZNN model (Equation (14)) converged to zero at about 2 s. It can be seen from Figure 5b that when the design parameters $\gamma$ were increased to 10, the linear noise $N\left(t\right)=10+10t$ remained unchanged, and the convergence time was reduced by about half.

In summary, through the above example, we can draw the conclusion that the convergence time of the DIEZNN model (Equation (14)) for solving time-varying QMEs will not increase with the increase in the matrix dimension. In other words, the convergence ability of the DIEZNN model (Equation (14)) will not be affected by the increase in the matrix dimension.

6. Conclusions

In order to solve the problem of time-varying QMEs and simultaneously suppress linear noise, this paper presented a new DIEZNN design formula, which resulted in a novel solution model. In addition, theoretical analysis shows that the proposed DIEZNN model has global convergence and robustness. For comparison, the DIEZNN model (Equation (14)), OZNN model (Equation (5)), FTZNN model (Equation (8)) and IEZNN model (Equation (11)) solved the same problem. The proposed DIEZNN model (Equation (14)) has been demonstrated to have superior performance against linear noise; that is, when solving the time-varying QME in the presence of linear noise, the existing models would fail to solve it due to the interference of linear noise. When the new DIEZNN model (Equation (14)) was used to solve the time-varying QME in a linear noise environment, the residual error could be kept sufficiently small, and finally the neural state solution of the DIEZNN model (Equation (14)) converged to the theoretical solution of the time-varying QME. For future works, the proposed DIEZNN model can be extended for contributing to chaotic systems [37] and induction motors [38] against noise.