1 Introduction

Generally, how to resolve the complex time-variant Sylvester equations (TVSEs) \(A(t)X(t)-X(t)B(t)+ C(t)= 0\), which is a basic mathematical question involved in many control systems and machine vision quests, with better disturbance rejection characters and faster convergence speed, has aroused more and more concern [1]. To resolve the corresponding design problems for dynamic nonlinear systems [2], switched nonlinear systems [3], time-delay systems [4] and stochastic nonlinear systems [5]. Many solutions have been put forward, such as Bartels–Stewart algorithm, homogeneous approach, terminal sliding mode control approach and so on [6].

These methods can get an ideal result with dealing with small dimensions Sylvester equations. But when the dimensions of Sylvester equations get larger, they will have a poor efficiency. With the development of technology, the neural-dynamic solutions have been applied in practice. Gradient neural networks (GNN) is one of the methods. However, the evolution speed of GNN may not synchronize with the change speed of time-variant coefficient. To solve this problem, zero neural network (ZNN), a kind of RNN, is proposed. ZNN owns many desired features, for instance higher accuracy, disturbance rejection features and faster convergent rate [7, 8]. Particularly, the main way for analysis of preassigned-time stability in the nonlinear situation is the Lyapunov function method [7,8,9,10]. With the purpose of accelerating the convergence, Weibing Li first used a sign-bi-power function stimulated ZNN that owns finite-time convergence [1,2,3]. Recently, ZNNs have been broadly applied to solve many time-variant questions under finite-time [11,12,13,14].

The ZNN solutions involve a design convergence parameter (CP) that influences the ZNNs convergent performance [15,16,17,18]. The CPs of conventional ZNNs are limited to be constants [19,20,21]. To obtain a better convergent performance, different variant-parameter ZNNs (VP-ZNNs) with time-variant CPs have been obtained [21,22,23]. The time-varying CPs shown in [24,25,26] can expedite the convergence of ZNNs, but the current CPs design results in two problems: (1) the CPs increase over time in the course of this solving process; and (2) it occupies too much computing resources.

Fixed-time convergence/stability was firstly proposed by Polyakov in 2012 to eliminate the dependence of the convergent time on initial values [27, 28]. This elimination has great advantages and convenience in applications compared to finite time stability, as the initial values of many real systems may not be accessible in advance [27,28,29,30]. Therefore, in the fields of power systems and space technology, fixed-time stability has received a lot of attention and become a hot research field [27,28,29,30,31]. Compared with the finite-time convergence, preassigned-time convergence does not depend on the initial states of ZNN; based on the known design parameters, the convergence-time’s upper bound can be calculated beforehand [32,33,34]. Once preassigned-time convergence is reached, we also call it preassigned-time stable [33,34,35,36].

To solve the above problems, we propose the hyperbolic tangent-type variant-parameter robust ZNN (HTVPR-ZNN) solutions for resolving the time-variant Sylvester equations. The primary contributions of our work are as follow:

  1. (1)

    To resolve the complex time-variant Sylvester equations, two new HTVPR-ZNN solutions are designed. Differing from the existing VP-ZNNs and FP-ZNNs, the HTVPR-ZNN involves hyperbolic tangent-type parameters that can be changed over time and also converge to a constant once the HTVPR-ZNN is convergent in preassigned time.

  2. (2)

    Unlike other ZNN solutions, the proposed HTVPR-ZNN solutions employ two new nonlinear activation functions that own various variable parameters as well. The stability and robustness of the HTVPR-ZNN solutions are proven. Moreover, the preassigned-time convergence upper bound of the HTVPR-ZNN is obtained, which is smaller than other convergence ZNN solutions.

  3. (3)

    Comparative numerical simulations are done to demonstrate the excellent convergent characters of the HTVPR-ZNN solutions. Many stimulation functions are comparatively used to stimulate the HTVPR-ZNN solutions. The results theoretically illustrate that two HTVPR-ZNN solutions are effective to tackle the TVSEs under the interference of different noises (for instance the time-variant bounded constant noise). Moreover the preassigned time is calculated as a priority and independent of initial situations.

The structure of this article is as follows. In Sect. 2, we introduce the problem formulation and preliminary results. Section 3, we develop HTVPR-ZNN Solutions. Main theoretical analysis is shown in Sect. 4. One numerical simulation example is presented to prove the validity of solutions in Sect. 5. At last, a conclusion is put in Sect. 6.

2 Problem Formulation and Preliminaries

The time-variant Sylvester problem formulation and its preliminaries are described in this subsection. The design procedures of HTVPR-ZNN solutions and regularity condition (RC) for the solution of the TVSEs are illustrated as well [21, 22].

We consider a high smooth and rank matrix \(A(t)\in \mathbb {R}^{m\times m}\), the problem of the TVSE is defined as

$$\begin{aligned} A(t)X(t)-X(t)B(t)= & {} -C(t),\quad {\forall }t\in [0, +\infty ) \end{aligned}$$
(1)
$$\begin{aligned} M(t)X(t)= & {} L(t), \end{aligned}$$
(2)

while \(X(t)\in \mathbb {R}^{m\times n}\) is an uncertain time-variant state matrix that neeeds to be resolved and t denotes the time variable. Then the time-variant coefficient matrices \(A(t)=(a_{ij}(t))_{m\times m}\), \(B(t) = (b_{ij}(t))_{n\times n}\), and \(C(t) = (c_{ij}(t))_{m\times n}\) and the time derivatives matrices \(\dot{A(t)}\), \(\dot{B(t)}\) and \(\dot{C(t)}\) are settled to be estimated. If this Sylvester equation is true, then the corresponding scheme \(X(t) = X^*(t)\in \mathbb {R}^{m\times n}\) of the Sylvester equation holds, its reliable to figure out the right solution quickly.

Before the new HTVPR-ZNN solutions are raised to solve the upper Sylvester equation (1), two important lemmas are provided as follows.

Lemma 1

[37, 38] We consider a nonlinear systems of the nether differential equations:

$$\begin{aligned} \dot{x}(t) = h (t ,x), x(0)=x_0 \end{aligned}$$
(3)

where \(h(0)=0\), and \(x=[x_1, x_2,\ldots ,x_n]^{T}\), \(h(t,x): R^{n}\rightarrow R^{n}\) is a nonlinear function.

If there is a unbounded and radially continuous function \(Q:R^{n}\rightarrow R^{n}\) with a result that

(1) for every \(x\in \mathbb {R}^{n} \backslash \{0\}\) there is \(x\in R_+\) so that \(Q(x)=0\);

(2) then any solution x(t) of (3) satisfies the inequality:

$$\begin{aligned} \dot{Q}(x(t)) \le -\alpha _1Q^\mu (x(t)) -\beta _2Q^\nu (x(t)), \end{aligned}$$

where parameters \(\alpha _1>0\), \(\beta _2>0\), \(0\le \mu <1\) and \(\nu >1\). The system is globally preassigned-time stable, then the upper bound of the settling time is:

$$\begin{aligned} T(x_0) = \frac{1}{\alpha _1(1-\mu )}+\frac{1}{\beta _2(\nu -1)}, {\forall x_0} \in \mathbb {R}^{n}. \end{aligned}$$

.

Lemma 2

If any solution x(t) of (3) satisfies the inequality:

$$\begin{aligned} \dot{Q}(x(t)) \le -\beta _1Q^p (x(t)) -\beta _2Q^q (x(t))-\beta _3Q^r (x(t)), \end{aligned}$$

where parameters \(\beta _1, \beta _2, \beta _3>0\), \(0\le q <1\), \(p, r >1\). The system is globally preassigned-time stable, and the upper boundary of the settling time is:

$$\begin{aligned} \begin{aligned} T(x_0)&= Max\left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} +\frac{1}{\beta _2(1-q)}. \end{aligned} \end{aligned}$$

Proof

For any x(t) so that Case I: \(Q(x(0))>1\), the last inequality

$$\begin{aligned} \dot{Q}(x(t)) \le -\beta _1 Q^p(x(t))-\beta _3 Q^r(x(t)). \end{aligned}$$

By classification, we have

$$\begin{aligned} \begin{aligned} \dot{Q}(x(t)) \le \left\{ \begin{array}{ll} -(\beta _1+\beta _3)Q^p(x(t)), &{}\quad p=r; \;(a)\\ -\beta _1Q^p(x(t))-\beta _3Q^1(x(t)),&{}\quad p>r>1; \;(b)\\ -\beta _1Q^1(x(t))-\beta _3Q^r(x(t)),&{}\quad r>p>1. \;(c) \end{array} \right. \end{aligned} \end{aligned}$$

Since situation (a) is similar to the situation in [29], we get

$$\begin{aligned} T(x_0) = \frac{1}{(\beta _1+\beta _3)(1-p)},\quad {\forall x_0} \in \mathbb {R}^{n}. \end{aligned}$$

Then we focus on case situation (b) and multiply (b) by \(\exp (\beta _3 t)\) yields,

$$\begin{aligned} e^{\beta _3 t}\dot{Q}(x(t))+e^{\beta _3 t}\beta _3 Q(x(t)) \le -e^{\beta _3 t}\beta _1 Q^p(x(t)). \end{aligned}$$

Then,

$$\begin{aligned} \frac{\textrm{d}e^{\beta _3 t}Q (t)}{(e^{\beta _3 t}Q (t))^p} \le -\beta _1 e^{(1-p)\beta _3 t} {\textrm{d}t}. \end{aligned}$$

Integrating the above differential inequality from 0 to t, we can obtain

$$\begin{aligned} Q(x(t))\ge e^{-\beta _3 t}\left[ Q(x(0))^{1-p}+\frac{\beta _1}{\beta _3}-\frac{\beta _1}{\beta _3}e^{(1-p)\beta _3 t}\right] ^{\frac{1}{1-p}}, \end{aligned}$$

let the upper right part equal to 1, we get \(t^{\prime }_1\),

$$\begin{aligned} t^{\prime }_1=\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)}, \end{aligned}$$

such that \(Q(x)\le 1\),\(t\ge t^{\prime }_1\).

Similarly, with regard to (c) situation, we get

$$\begin{aligned} Q(x(t))\ge e^{-\beta _1 t}\left[ Q(x(0))^{1-r}+\frac{\beta _3}{\beta _1}-\frac{\beta _3}{\beta _1}e^{(1-r)\beta _1 t}\right] ^{\frac{1}{1-r}}, \end{aligned}$$

let the upper right part equal to 1, we get \(t^{\prime \prime }_1\),

$$\begin{aligned} t^{\prime \prime }_1=\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}, \end{aligned}$$

while \(Q(x)\le 1\), \(t\ge t^{\prime \prime }_1\).

Let \(t_1=Max(t^{\prime }_1,t^{\prime \prime }_1)\),

$$\begin{aligned} t_1=Max\left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} . \end{aligned}$$

Then, for any x(t) such that \(Q(x(t^{\prime }))\le 1\), the before inequality

$$\begin{aligned} \dot{Q}(x(t^{\prime }))\le & {} -\beta _2 Q^q(x(t^{\prime })),\quad t^{\prime }\ge t_1, \\ t_2= & {} \frac{Q^{1-q}(t_1 )}{\beta _2(1-q)}=\frac{1}{\beta _2(1-q)}. \end{aligned}$$

So, we make \(Q(x(t))=0\), for

$$\begin{aligned} t \ge t_1+t_2=\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)}+\frac{1}{\beta _2(1-q)}. \end{aligned}$$

Case II: Once \(Q(x(0))\le 1\), the inequality

$$\begin{aligned} \dot{Q}(x(t)) \le -\beta _2 Q^q(x(t)), \end{aligned}$$

holds, indicates

$$\begin{aligned} t\ge \frac{1}{\beta _2(1-q)}, Q(x(t))=0. \end{aligned}$$

Finally, \(Q(x_0)\ge 1\) and \(Q(x(t))=0\) for

$$\begin{aligned} \begin{aligned}&{\forall }t\ge T(x_0)=Max\left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} +\frac{1}{\beta _2(1-q)}. \end{aligned} \end{aligned}$$

Therefore, the system has globally predefined-time convergence/stability and its settling time satisfies the upper boundary. \(\square \)

3 HTVPR-ZNN Solutons

In this part, through using two nonlinear activation functions we design and develop the HTVPR-ZNN solution for the specific Sylvester problem (1) [32,33,34,35,36].

Step 1: To survey the time-variant process, we define a matrix-type error function (EF) as

$$\begin{aligned} E(t)=A(t)X(t)-X(t)B(t)+C(t),\quad {\forall }t\in [0, +\infty ). \end{aligned}$$
(4)

The time-variant unique solution \(X^*(t)\) of the TVSE (1) can be obtained once the error function (4) equals or converges to zero. Thus it successfully turns the initial problem (1) into insuring E(Q(t), t) equals to 0.

Step 2: With the purpose of ensuring each element \(e_{ij}(t)\) (\(i =1,2,\ldots ,m\), \(j =1,2,\ldots ,n\)) of E(Q(t), t) convergent to 0, the EF derivation \(\dot{E}(Q(t),t)\) is expressed as:

$$\begin{aligned} \frac{\text {d}E(Q(t) ,t)}{\text {d}t}=-\varLambda (t) F (E(Q(t),t)), \end{aligned}$$
(5)

where \(F (\cdot ):\mathbb {R}^{m\times n}\rightarrow \mathbb {R}^{m\times n}\) is a mapping of a matrix and every unit is a monotone increasing odd function. The \(\varLambda \) is a positive-defined matrix that influences the convergent speed of (5). For the simplicity of theoretical analysis, \(\varLambda \) is set to \(\gamma I\), where \(\gamma >0\) and I is an identity matrix. Hence, each unit of E(Q(t), t) owns the coincident convergence speed adjusted by parameter \(\gamma \).

$$\begin{aligned} \varLambda (t)= & {} \left[ \begin{matrix} \lambda _1(t)&{}\quad 0&{}\quad \cdots &{}\quad 0\\ 0&{}\quad \lambda _2(t)&{}\quad \ddots &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad 0\\ 0&{}\quad 0&{}\quad \cdots &{}\quad \lambda _m(t)\\ \end{matrix} \right] \in \mathbb {R}^{m\times n}.\\ \lambda _{i}(t)= & {} \gamma \tanh (\left| e(t) \right| /\epsilon ) \end{aligned}$$

Step 3: Combining Steps 1 and 2, then we obtain one consistent HTVP-ZNN solution for resolving Sylvester equation (1):

$$\begin{aligned} \begin{aligned}&A(t)\dot{X}( t )+\dot{X}(t) B( t )= -\dot{A}( t ) X( t )-X( t ) \dot{B}( t )-\dot{C}( t )\\&\quad -\varLambda (t) F (A(t)X(t)+X(t)B(t)+C(t)). \end{aligned} \end{aligned}$$
(6)

where \(F(\cdot )\) denotes 2 new stimulation functions (i.e., PpSAF and NSBPAF).

With the injection of noise, the perturbed HTVPR-ZNN solution is obtained as follows:

$$\begin{aligned} \begin{aligned}&A( t )\dot{X}( t )+\dot{X}(t) B( t )=-\dot{A}( t ) X( t )-X( t ) \dot{B}( t )-\dot{C}( t )\\&\quad -\varLambda (t) F (A(t)X(t)+X(t)B(t)+C(t))+N(t). \end{aligned} \end{aligned}$$
(7)

where \(N(t)\in \mathbb {R}^{m\times n}\) denotes external noise. After dealing with the upper equation, the column vectorization of HTVPR-ZNN solution (7) can be deduced as

$$\begin{aligned} \begin{aligned} M(t)\dot{X}(t)=-\dot{M}(t)X(t)-\dot{C}(t)-\varLambda (t) F(E(t))+N(t), \end{aligned} \end{aligned}$$
(8)

where \(M(t):=I^{m\times m}\otimes A\left( t \right) +B^T(t )\otimes I^{n\times n}\), \(X(t):=\text {vec}(\dot{X}(t))\), \(e(t):=M(t)X(t)-\text {vec}(C(t))\), and \(N(t):=\text {vec}(N(t)).\) Our goal of this paper is to propose two new solutions for systems (1) so that the corresponding closed-loop system owns fast preassigned-time convergence.

3.1 Different Stimulation Functions

Stimulation functions (SFs) play very vital roles in the ZNN solutions. Different stimulation functions are adopted to ZNN solutions with many results. When the ZNN solutions stimulated by SBPAF achieving finite-time convergence [1, 2]. And their finite-time convergence is mainly influenced by the initial conditions of ZNN solution, which lead it very difficult to find out the upper boundary because of the undiscovered initial condition [26, 39, 40]. In this paper, four conventional SFs and two new SFs have been adopted to stimulate the proposed ZNN solutions:

(1) Linear function (LF): \(F(x)=d\cdot x\).

(2) Smooth power-sigmoid function (SPSF):

$$\begin{aligned} F(x)= & {} \frac{1}{2}\left( \frac{1-\exp (-bx )}{1+\exp (-bx)} \cdot \frac{1+\exp (-b )}{1-\exp (-b)+x^a} \right) , \\{} & {} \quad \text {where}\; a\ge 3, \quad b>2. \end{aligned}$$

(3) Power-sigmoid function (PSF):

$$\begin{aligned} \begin{aligned} F(x)= \left\{ \begin{array}{ll} x^p, &{}\quad if\left| x \right| \ge 1,\\ \frac{1+\exp ( -b )}{1-\exp ( -b )} \cdot \frac{1-\exp ( -bx )}{1+\exp ( -bx )}, &{}\quad otherwise,\\ \end{array} \right. \end{aligned} \end{aligned}$$

(4) Sign-bi-Power function (SBPF):

$$\begin{aligned} F(x)= & {} \frac{1}{2} \left| x \right| ^\iota \text {sign}(x)+\frac{1}{2} \left| x \right| ^\frac{1}{\iota } \text {sign}(x) \\{} & {} \quad \text {where}\;0<\iota <1. \end{aligned}$$

Consider the features of different SFs, let’s induce it into the HTVPR-ZNN solution to resolve the TVSEs achieving the preassigned-time convergence. The expression of PpSAF can be:

$$\begin{aligned} F(u) =\left( \beta _1\text {sign}(u)+\beta _2 u^p \right) , \end{aligned}$$
(9)

When \(\beta _1=1\) and \(\beta _2=1\), (9) will be \(F(u) = \text {sign}(u)+u^p \), odd \(p>=3\).

Referring to previous works [34,35,36, 40], we come up with a new sign-bi-Power activation function (NSBPAF), which will be convergent in a preassigned time. Its formula is presented as follow:

$$\begin{aligned} F_1(u) =\left( \beta _1\left| u \right| ^p+\beta _2\left| u \right| ^q \right) \text{ sign }(u) +\beta _3 u^r+\beta _4 \text {sign}(u), \end{aligned}$$
(10)

where design parameters \(\beta _1>0\), \(\beta _2>0\), \(\beta _3\ge 0\), \(\beta _4\ge 0\), \(p>1\), \(0<q<1\), \(r=1,3,5\) and \(\text {sign}(\cdot )\) represents signum function. Initial parts \(( \beta _1| u |^p+\beta _2| u |^q) \text{ sign }(u )\) are used to obtain the preassigned-time convergence; Then time-variant boundary constant noises can be repressed by the intermittent items \(\beta _3u^r\) and \(\beta _4 \text {sign}(u )\).

For convenience, the HTVPR-ZNN solutions stimulated by PpSAF and NSBPAF are respectively called HTVPR-ZNN1 and HTVPR-ZNN2 solution.

4 Main Theoretical Analysis

In this subsection, we will address the preassigned-time convergence of proposed HTVPR-ZNN solutions in 2 situations. By analysis, the preassigned-time convergence and also the robustness of HTVPR-ZNN solutions for solving TVSEs is illustrated rigorously.

4.1 Pure-Case1: Without Noises

Under the pure-case1: with no noises, we find the preassigned-time and robust convergence of the HTVPR-ZNN solutions for solving time-variant and non-linear equations.

Theorem 1

Under pure environment, when PpSAF is adopted in HTVPR-ZNN1 solution, the state matrix P(t) of HTVPR-ZNN1 solution beginning from random \(P(0)\in \mathbb {R}^{m1 \times n1}\) can converge to theoretical \(P^*(t)\) in a preassigned-time T gradually:

$$\begin{aligned} T\le \frac{1}{\gamma \beta _1(1-q)}+\frac{1}{\gamma \beta _2(p-1)}. \end{aligned}$$

Though Eqs. (4) and (5), we use an EF \(\dot{E}(t)=-\lambda F(E(t))\). Then the subelement \({\dot{\epsilon }}(t)=-\lambda f(\epsilon (t))\), and \(f(\cdot )\) is a common unit of \(F(\cdot )\) are got. Using the Lyapunov theory, we introduce a Lyapunov energy-function (LEF) \(Nov(t)=|\epsilon (t)|\). When PpSAF is invasive, the formula of \(\dot{N}ov (t)\) can be obtained as below:

$$\begin{aligned} \begin{aligned} \dot{N}ov (t)&=\frac{\text {d} Nov (t)}{\text {d}t}={\dot{\epsilon }}(t) \text {sign}(\epsilon (t))\\&=-\lambda f(\epsilon (t)) \text {sign}(\epsilon (t))\\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1 \text {sign}(\epsilon (t))+\beta _2 \epsilon (t)^p\right) \text {sign}(\epsilon (t))\\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1+\beta _2 \epsilon (t)^p \text {sign}(\epsilon (t)) \right) \\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1+\beta _2\left| \epsilon (t) \right| ^p \right) \\&\le -\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1\left| \epsilon \left( t \right) \right| ^q+\beta _2\left| \epsilon (t ) \right| ^p \right) \\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1 Nov ^q(t)+\beta _2 Nov ^p(t) \right) . \end{aligned} \end{aligned}$$

On account of the Lemma 1, the scheme for the upper LEF inequality can be

$$\begin{aligned} t\le \frac{1}{\gamma \tanh (\left| e( t )\right| /\epsilon )} \left( \frac{1}{\beta _1\left( p-1 \right) }+\frac{1}{\beta _2\left( 1-q \right) }\right) . \end{aligned}$$

Generally, the convergent time hinges on the trial parameters. Once the trial parameters are fixed, \(T=\text {max}(t)\) can be preassigned, the convergent time of HTVPR-ZNN1 solution

$$\begin{aligned} \begin{aligned} T_\text {max}(t)&\le \frac{1}{\gamma \tanh (\left| e( t )\right| /\epsilon )}\left( \frac{1}{\beta _1\left( 1-q \right) }+\frac{1}{\beta _2\left( p-1 \right) }\right) \\&\quad (0\le \tanh (\left| e(t)\right| /\epsilon )\le 1)\\&\le \frac{1}{\gamma }\left( \frac{1}{\beta _1\left( 1-q \right) }+\frac{1}{\beta _2\left( p-1 \right) }\right) . \end{aligned} \end{aligned}$$
(11)

make parameters \(\beta _1=1\), \(\beta _2=1\); \(\gamma =0.5,1,2,3,\ldots \), \(p=3\) and \(q=0\); we have

$$\begin{aligned} T_\textrm{max}(t)< \frac{1}{\gamma }+\frac{1}{2 \gamma }=\frac{1.5}{\gamma }. \end{aligned}$$

Finally, all the processes and steps show the preassigned-time convergence of HTVPR-ZNN1 solution stimulated by PpSAF for resolving the Sylvester equation (1).

Theorem 2

Under pure environment, when NSBPAF is invasive in HTVPR-ZNN2 solution, the matrix V(t) of HTVPR-ZNN2 solution beginning from \(V(0)\in \mathbb {R}^{m \times n}\) will converge to theoretical \(V^*(t)\) under preassigned-time T:

$$\begin{aligned} \begin{aligned} T \le Max\left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} +\frac{1}{\beta _2(1-q)} \end{aligned} \end{aligned}$$

Proof

In Eqs. (4) and (5), we let an EF \(\dot{E}(t)=-\gamma F(E(t))\). When NSBPAF is invasive, the formula of \(\dot{N}ov(t)\) can be obtained as below:

$$\begin{aligned} \begin{aligned} \dot{N}ov (t)&=\frac{\text {d} Nov \left( t \right) }{\text {d}t}={\dot{\epsilon }}(t) \text {sign}(\epsilon (t))=-\lambda (t) f(\epsilon (t)) \text {sign}(\epsilon (t))\\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1\left| \epsilon (t) \right| ^p+\beta _2\left| \epsilon (t) \right| ^q+\beta _3\left| \epsilon (t) \right| ^r+\beta _4 \right) \\&\le -\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1\left| \epsilon (t) \right| ^p+\beta _2\left| \epsilon (t) \right| ^q+\beta _3\left| \epsilon (t) \right| ^r \right) \\&=-\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1 Nov ^p(t)+\beta _2 Nov ^q(t)+\beta _3 Nov ^r(t) \right) . \end{aligned} \end{aligned}$$

Based on the Lemma 2, then we get

$$\begin{aligned} \begin{aligned} t&\le \frac{1}{\gamma \tanh (\left| e(t)\right| /\epsilon )}\\&\quad \left( Max \left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} +\frac{1}{\beta _2(1-q)}\right) \\&\le \frac{1}{\gamma }\left( Max\left\{ \frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _3Q(x(0))^{1-p}+\beta _1}\right] }{\beta _3(p-1)},\frac{\ln \left[ \frac{\beta _1+\beta _3}{\beta _1Q(x(0))^{1-r}+\beta _3}\right] }{\beta _1(r-1)}\right\} +\frac{1}{\beta _2(1-q)}\right) . \end{aligned} \end{aligned}$$
(12)

Evidently the upper threshold of convergent time is preassigned and relied on the trial parameters. Make the parameters \(\beta _1=\beta _2=\beta _3=1\); \(\gamma =0.5,1,2,\ldots \), \(p=r=3\), \(q=1{/}3\) and \(Q(x_0)\ge 1\); then, we have

$$\begin{aligned} \begin{aligned} t\le&\frac{1}{\gamma }\left( Max\left\{ \frac{\ln \left[ \frac{2}{Q(x(0))^(-2)+1}\right] }{2},\frac{\ln \left[ \frac{1}{Q(x(0))^(-2)+1}\right] }{2}\right\} +1.5\right) . \end{aligned} \end{aligned}$$
(13)

\(\square \)

Fig. 1
figure 1

Theoretical and analytical trajectories of Sylvester (1), where blue lines represent the process resolution of HTVP-ZNN solution (6) stimulated by PpSAF and the red represent the theory resolution of Sylvester (1). (Color figure online)

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Fig. 2
figure 2

Theoretical and analytical trajectories of Sylvester (1), where blue lines represent the process resolution of HTVP-ZNN solution (6) stimulated by NSBPAF and the red represent the theory resolution of Sylvester (1). (Color figure online)

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Fig. 3
figure 3

Steady-state residual errors of HTVP-ZNN solution (6) with no noise and r \(=\) 1. a By PpSAF. b By NSBPAF

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4.2 Case2: Time-Variant Boundary-Constant Noise

When the boundary time-variant and constant noise is considered, every entry \(n_{ij}(t)\) satifies the inequation \(|n_{ij}(t)| \le N+\delta \), whie \(\delta \in (0,+\infty )\); and the convergent peculiarity of HTVPR-ZNN solutions under preassigned time is investigated.

Theorem 3

When the bounded time-variant and constant noise is interferential, if PpSAF is used in HTVPR-ZNN1 solution with \(\beta _1\gamma \tanh (\left| e(t)\right| /\epsilon ) \ge N+\delta \), the state matrix P(t) of HTVPR-ZNN1 solution beginning from random \(P(0)\in \mathbb {R}^{m1 \times n1}\) can slowly converge to theoretical \(P^*(t)\) under the preassigned-time T:

$$\begin{aligned} T\le \frac{1}{(\gamma \beta _1-(N+\delta ))\left( 1-q \right) }+\frac{1}{\gamma \beta _2\left( p-1 \right) }. \end{aligned}$$

Induce a LEF \(Nov(t)=|\epsilon (t)|^2\), let’s compute the \(\dot{N}ov (t)\):

$$\begin{aligned} \begin{aligned} \dot{N}ov (t)&=2\epsilon (t){\dot{\epsilon }}(t)=2\epsilon (t)(-\lambda f(\epsilon (t))+n(t))\\&=-2\lambda \epsilon (t)\left( \beta _1 \text {sign}(\epsilon (t))+\beta _2 \epsilon (t)^p\right) +2\epsilon (t)n(t)\\&=-2\gamma \tanh \left( \left| e(t)\right| /\epsilon \right) \left( \beta _1|\epsilon (t)|+\beta _2|\epsilon (t)|^{p+1}\right) +2\epsilon (t)n(t)\\&\le -2\gamma \tanh (\left| e(t)\right| /\epsilon ) \left( \beta _1|\epsilon (t)|+\beta _2|\epsilon (t)|^{p+1}\right) +2|\epsilon (t)|(N+\delta )\\&\quad \quad (\text {when}\;\beta _1\gamma \tanh (\left| e(t)\right| /\epsilon ) \ge N+\delta )\\&=-2\gamma \tanh (\left| e( t )\right| /\epsilon )\left( (\beta _1-(N+\delta )/\gamma )|\epsilon (t)|+\beta _2|\epsilon (t)|^{p+1}\right) \\&=-2\gamma \tanh (\left| e(t)\right| /\epsilon )\left( (\beta _1-(N+\delta )/\gamma ) Nov ^{\frac{q+1}{2}}(t)+\beta _2 Nov ^{\frac{q+1}{2}}(t)\right) . \end{aligned} \end{aligned}$$

Then, we get the restringent time of contaminated HTVPR-ZNN1 solution:

$$\begin{aligned} \begin{aligned} T_\text {max}(t)&\le \frac{1}{(\gamma \tanh (\left| e( t )\right| /\epsilon )\beta _1-(N+\delta ))\left( 1-q \right) }+\frac{1}{\gamma \tanh (\left| e(t)\right| /\epsilon ) \beta _2(p-1)}\\&\quad (0\le \tanh (\left| e(t)\right| /\epsilon )\le 1)\\&\le \frac{1}{(\gamma \beta _1-(N+\delta ))(1-q )}+\frac{1}{\gamma \beta _2(p-1)}. \end{aligned} \end{aligned}$$
(14)

Set \(\beta _1=1\), \(\beta _2=1\); \(\gamma =1,1.5,2,\ldots \), \(p=3\) and \(q=0\); there is

$$\begin{aligned} T_\text {max}(t)\le \frac{1}{(\gamma -(N+\delta ))}+\frac{1}{2 \gamma }. \end{aligned}$$
(15)

Theorem 4

When the noise is invasive, once NSbpAF is used in the polluted HTVPR-ZNN2 solution with \(\beta _4 \tanh (|e(t )|/\epsilon ) \ge \delta \), Q(t) of HTVPR-ZNN2 solution beginning from \(Q(0)\in \mathbb {R}^{m \times n}\) can converge to theoretical \(Q^*(t)\) under preassigned-time.

Proof

Bring in a LEF \(Nov(t)=|\epsilon (t)|^2\), let’s compute the \(\dot{N}ov (t)\):

$$\begin{aligned} \begin{aligned} \dot{N}ov (t)&=2\epsilon (t){\dot{\epsilon }}(t)(t)=2\epsilon (t)(-\lambda f(\epsilon (t))+n(t))\\&=-2\gamma \left( \beta _1|\epsilon (t)|^{p+1}+\beta _2|\epsilon (t)|^{q+1}+\beta _3|\epsilon (t)|^{r+1}\right) \\&\quad +2(\epsilon (t)n(t)-\gamma \beta _4|\epsilon (t)|)\\&\le -2\gamma \left( \beta _1|\epsilon (t)|^{p+1}+\beta _2|\epsilon (t)|^{q+1}+\beta _3|\epsilon (t)|^{r+1}\right) \\&\quad +2((N+\delta )|\epsilon (t)|-\gamma \beta _4|\epsilon (t)|)\\&\le -2\gamma \left( \beta _1|\epsilon (t)|^{p+1}+\beta _2|\epsilon (t)|^{q+1}+\beta _3|\epsilon (t)|^{r+1}\right) \\&=-2\gamma \left( \beta _1 Nov ^{\frac{p+1}{2}}(t)+\beta _2 Nov ^{\frac{q+1}{2}}(t)+\beta _3 Nov ^{\frac{r+1}{2}}(t)\right) . \end{aligned} \end{aligned}$$

From Lemma 2, it can be concluded that the convergent time of the contaminated HTVPR-ZNN2 solution is also preassigned. \(\square \)

Fig. 4
figure 4

Errors of HTVP-ZNN1 solution (6) stimulated by PpSAF to resolve the time-variant Sylvester (1) with the six values of the parameter \(\gamma \)

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Fig. 5
figure 5

Errors of HTVP-ZNN2 solution (6) stimulated by NSBPAF to resolve the time-variant Sylvester (1) with the six values of the parameter \(\gamma \)

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Remark 1

It can be seen from Theorems 1 and 2 that there exists a preassigned-time convergent peculiarity of the HTVPR-ZNN1 and the HTVPR-ZNN2 solutions in pure environment, based on the known design parameters, the convergence-time’s upper bound can be calculated beforehand. Then, seen from the convergence time of two HTVPR-ZNN solutions presented in (11) and (13). Then the convergence speed of HTVPR-ZNN2 solution will be faster than that of HTVPR-ZNN1 solution.

Remark 2

Seen from the Theorems 3 and 4, the 2 polluted HTVPR-ZNN solutions can converge to the theoretical solution of TVSE (1) under preassigned time. And they also suppress the time-variant bounded constant noise very well under the noised environment.

Remark 3

From the above design procedure, we know that two HTVPR-ZNN solutions with PpSAF and NSBPAF bring lots of parameters: \(\beta _i\) (\(i=1,2,3,4\)), p, q and r; they determine the peculiarity of HTVPR-ZNN solutions. Theorems 1 and 2 illustrate that HTVPR-ZNN1 solution stimulated by PpSAF, its convergence time is contingent on \(\gamma \), \(\beta _1\), \(\beta _2\), p. When HTVPR-ZNN2 solution stimulated by NSBPAF, its convergence time is determined by the \(\gamma \), \(\beta _1\), \(\beta _2\), \(\beta _3\), \(\beta _4\), p, q and r. Seen from Theorems 3 and 4, coefficients \(\beta _3\) and \(\beta _4\) would suppress the invasive noises. In addition, their parameters \(\beta _1\) and \(\beta _2\) are used to tolerate various noises respectively. Increasing the value of \(\gamma \), all the preassigned-time T will be much smaller.

Fig. 6
figure 6

Errors of disturbed HTVPR-ZNN1 solution (7) stimulated by PpSAF to resolve the time-variant Sylvester (1) with 3 values of \(\gamma \)

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Fig. 7
figure 7

Errors of disturbed HTVPR-ZNN2 solution (7) stimulated by NSBPAF to resolve the time-variant nonlinear Sylvester (1) with the 3 values of \(\gamma \)

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5 Numerical Simulation Example

One example is provided in this section on dealing with the TVSE (1). We apply the 2 HTVPR-ZNN solutions to prove the validity of the our theorems.

Firstly, We bring in the coefficient matrices of Sylvester equation as below:

$$\begin{aligned} A(t)= & {} \left[ \begin{matrix} 3c^2/2-1&{}\quad -3s/4-1\\ -3s/4+1&{}\quad 3s^2/2-1\\ \end{matrix} \right] , \\ C(t)= & {} \left[ \begin{matrix} 2s-3sc^2&{}\quad -c(1-6s^2)/2\\ c(4-3c^2+3s^2)/2&{}\quad s(1-3s^2+3c^2)/2\\ \end{matrix} \right] , \end{aligned}$$

while \(s= sin(5t)\), \(c= cos(5t)\). The theoretical solution can be:

$$\begin{aligned} P^*(t) =\left[ \begin{matrix} \sin (5t)&{}\quad -\cos (5t)\\ \cos (5t)&{}\quad \sin (5t)\\ \end{matrix} \right] , \end{aligned}$$

which settles a basis to prove the availability of our proposed solutions.

$$\begin{aligned} N(t) =\left[ \begin{matrix} \sin (10t)&{}\quad 2+\cos (10t)\\ 1+\cos (10t)&{}\quad -\sin (10t)\\ \end{matrix} \right] , \end{aligned}$$

which is used as a invasive noise to verify the robustness of our proposed solutions.

Fig. 8
figure 8

Errors of HTVP-ZNN solution (6) stimulated by various functions

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Under the pure environment, the convergence time of our proposed solutions can be predicted and preassigned. Figures 1 and 2 exhibit the moving trajectories of TVSE (1) respectively stimulated by PpSAF and NSBPAF. Seen from Fig. 3a, the solution (6) stimulated by PpSAF, \(\gamma =1\) the corresponding convergent time \(T \le 0.61\) s; and \(T \le 0.42\) s when stimulated by NSBPAF. The residual errors presented in following figures also verify the truth, which proves the effectiveness of the Theorems 1 and 2 further.

Fig. 9
figure 9

Errors of solution (6) with the adjustments of the ODE45 solver when \(\gamma =1\)

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Figure 4 shows the corresponding errors of HTVPR-ZNN1 solution stimulated by PpSAF to resolve the TVSE(1) with 6 different values of \(\gamma \). And Fig. 5 displays the corresponding errors of HTVPR-ZNN2 solution stimulated by NSBPAF to resolve the TVSE(1) with the six different values of \(\gamma \) (\(\gamma =0.5\), \(\gamma =1\), \(\gamma =1.5\), \(\gamma =2\), \(\gamma =3\) and \(\gamma =4\)). Moreover, seen from Figs. 4 and 5, HTVPR-ZNN2 solution has more quickly convergent speed than HTVPR-ZNN1 solution, in the matter of errors.

Figures 6 and 7 adopt the time-variant boundary constant noise refer to the inescapability of distruction. HTVPR-ZNN solutions own nearly the uniform convergence time. However, when stimulated by linear, smooth power-sigmoid, power-sigmoid and SBPAF, the convergent delays of time seriously. And the corresponding errors of HTVPR-ZNN solutions can converge to zero as usual, while the errors of stimulation functions such as linear, Smooth power-sigmoid, power-sigmoid and SBPAF cannot converge to 0 and merely provide the upper boundaries. Numerical experiments results prove the effectiveness of preassigned-time convergence theory from Theorems 1–4. Moreover, NSbpAF do better than others, which ulteriorly bears out the admirable noise-tolerance character of HTVPR-ZNN solutions.

In general, HTVPR-ZNN solutions’ parameters play important roles in the numerical experiment simulations. And the influence of 3 parameters in simulations are presented in the Figs. 6 and 7. Then, the error would draw to 0 bit by bit. By fixing parameters \(\beta _1=\beta _2=\beta _3=\beta _4=1\), \(p=4\), \(q=0.25\) and \(r=3\), and convergence time of the proposed solution (6) stimulated by PpSAF and NSbpAF are depend on \(\gamma \). Then in later simulations, we adjust 3 various values (\(\gamma =3\), \(\gamma =4\) and \(\gamma =5\)). Counting the convergent time, they will be smaller as we increase increase the value of the parameter \(\gamma \).

To be specific, to further present the remarkable peculiarity of the proposed solution (6) with PpSAF and NSBPAF, other AFs involving Linear AF, SPSAF, BSAF, PSAF and SBPAF are used in the simulations too. Seen from the Fig. 8, the errors of PpSAF or NSBPAF converge to 0 more quickly than others. Furthermore HTVPR-ZNN2 solution has the fastest speed while HTVPR-ZNN1 solution is in the second place.

To improve the precision of the experimental results, we can adjust and set the values of the ODE45 solver. From Figs. 5, 6, 7 and 8, the relative and absolute error tolerances of the ODE45 solver are respectively preset as default values of \(10^{-3}\) and \(10^{-6}\). Seen from the Fig. 9, the experimental results HTVPR-ZNN solutions errors become much smaller when decrease the RelTol and AbsTol values of the ODE45 solver (values of \(10^{-4}\) and \(10^{-8}\)). So the setting of ODE45 play important roles on the numerical experiment simulations. In this case, its convergence time is the same; its setting time is mainly related to the design parameters.

In summary, all the results of simulations show that the 2 HTVPR-ZNN solutions have outstanding advantages. Moreover, facing with time-variant boundary-constant noise, both the 2 HTVPR-ZNN solutions maintain valid.

6 Conclusion

In the paper, 2 HTVPR-ZNN solutions for the time-variant Sylvester equation are come up with. HTVPR-ZNN has hyperbolic tangent-type parameters that can change over time; they will be constant when the HTVPR-ZNN are convergent at last. HTVPR-ZNN owns faster convergence than others such as FP-ZNNs. Global and preassigned-time convergence properties of HTVPR-ZNN solutions are proved theoretically. A number of simulations substantiate the accuracy and efficacy of HTVPR-ZNN solutions. Besides, we also study the influences of the related parameters and stimulation functions in the HTVPR-ZNN related to the convergence. In addition, the HTVPR-ZNN solutions can achieve an expedited convergence speed and reduce the assigned time. What’s more, one example is further presented to indicate the effectiveness of our designed solutions intuitively. One of our future research efforts is to apply the HTVPR-ZNN solutions to some engineering situations such as robot manipulators’ control.