Adaptive Asymptotic Regulation for Uncertain Nonlinear Stochastic Systems with Time-Varying Delays

Abstract

In this paper, for a class of uncertain stochastic nonlinear systems with input time-varying delays, an adaptive neural dynamic surface control (DSC) method is proposed. To approximate the unknown continuous functions online, the neural network approximation technique was applied, and based on the DSC scheme, the desired controller was constructed. A compensation system is presented to compensate for the effect of the input delay. The Lyapunov–Krasovskii functionals (LKFs) were employed to compensate for the effect of the state delay. Compared with the existing works, based on using the DSC scheme with the nonlinear filter and stochastic Barbalat’s lemma, the asymptotic regulation performance of this closed-loop system can be guaranteed under the developed controller. To certify the availability for the designed control method, some simulation results are presented.

1. Introduction

In recent decades, due to the wide existence of random disturbances in real engineering applications, a large amount of achievements, with respect to the adaptive control problem for stochastic systems, have been reported [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. The adaptive control problems of stochastic nonlinear systems are addressed in [1,2,3,4] by using the output–feedback control method. For uncertain high-order stochastic nonlinear systems, the adaptive control problems are studied in [5,6,7], and the system stability can be ensured under the desired state feedback controller. For the uncertain stochastic switched systems in [10,11], the adaptive control schemes are established based on the fuzzy approximation method. The adaptive neural tracking problem for the uncertain stochastic nonlinear systems is presented in [12,13] where the unknown hysteresis is presented. For a class of stochastic interconnected non-strict feedback systems with dead zones, an adaptive neural DSC method is proposed in [15], and the “explosion of complexity” is avoided.

In addition, time delays frequently occur in real control systems, and will degrade the system performance. Therefore, many adaptive control achievements for the systems with time delays are presented in [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. In terms of where the time delays occur, the adaptive tracking control problem for uncertain input-delayed systems and state-delayed systems are considered in [17,18,19,20,21,22,23,24], respectively. The adaptive control problems for an uncertain discrete-time system and continuous-time system are separately addressed in [25,26,27,28,29,30]. For systems with time varying delays, some interesting results are shown in [30,31,32,33]. Moreover, by designing a compensation system and employing the Lyapunov–Krasovskii functionals (LKFs) to compensate for the effect of the input delay and state delay, respectively, some useful control methods are presented in [18,27]. This provides us with effective schemes to solve the adaptive control problem for the systems with time delays.

Based on the above research results, for the stochastic systems with time delays, some significant achievements are reported in [34,35,36,37,38,39,40,41,42,43,44]. For nonlinear stochastic full state constraint systems with unknown constant time delays, an adaptive NN controller is proposed based on the use of LKFs and the DSC method, under controlling the constructed controller, the bounded stability of this closed-loop signal is guaranteed in [34]. For nonlinear stochastic systems with time-varying delays, the adaptive output–feedback control problems are presented in [35,36,37]. Based on the LKF method, the global asymptotic stabilization problem for stochastic systems with state time-delays is solved in [41,44].

We should note that the traditional adaptive backstepping method is employed for stochastic time-delayed systems in [35,36,37,38,39,40,41,42,43,44], and the repeated differentiations of the virtual controller is essential in the process of designing the desired controller. Therefore, increasing the system order, the controller designing process will be more complicated, i.e., the “explosion of complexity” problem will happen. Moreover, the literature [34,35,36,37,38,39,40,41,42,43,44] just focuses on the systems with state time-invariant delays and time-vary delays and, therefore, for a class of uncertain stochastic nonlinear system with state and input time varying delays, an adaptive neural DSC scheme will be established in this paper, The main contributions of this paper are as follows:

(i) This paper addresses the adaptive neural DSC problem for a class of uncertain stochastic nonlinear systems with state and input time-varying delays, firstly. Compared to the existing works [34,35,36,37,38,39,40,41,42,43,44], the asymptotic stability of the system output signal is achieved under the proposed control scheme.

(ii) To achieve the control objective, a DSC scheme with the nonlinear filter is presented to develop the adaptive neural controller for the uncertain stochastic nonlinear systems. In addition, the LKFs and a compensation system are employed to compensate for the effect of the time-varying delays.

(iii) The semi-global boundedness in probability of all the closed-loop system signals are guaranteed, in particular, the system output signal is asymptotically stable in probability based on the use of the stochastic Barbalat lemma.

2. Problem Statement and Preliminary

First, the tables of abbreviations (Table 1) and symbols (Table 2) are presented as follows.

Table 1. Abbreviations.

Room Form	Abbreviations
dynamic surface control	DSC
Lyapunov–Krasovskii functionals	LKFs
neural network	NN
radial basis function neural networks	RBF NNs

Table 2. Symbols description.

Symbols	Description
${\overline{x}}_i,u$	system state, control input
$\tau \left(t\right),d\left(t\right)$	time-varying delays
$f_i,h_i,g_i,$	unknown functions
$h_{i,d},g_{i,d}$	unknown functions with time-varying delays
$\omega$	independent standard Wiener process
$W_i,Z_i\subset R^m,m$	weight vector, input vector, NN node number
${\eta }_i\left(Z_i\right)$	NN inherent bounded approximation error
${\eta }_i^{*}$	unknown parameter
$S\left(Z_i\right),S_j\left(Z_i\right)$	smooth vector function, Gaussian function
${\chi }_j,\Gamma$	the center of RBFNN, Gaussian function’s width
${\lambda }_i,p_i$	compensation system state, known constant
$z_i$	state coordinate transformation error
${\alpha }_i,s_i,e_i$	virtual controller, filtered virtual controller, boundary layer error
${\overline{\alpha }}_i,\overline{u}$	equivalent virtual units
${\varrho }_i$	filter time constant
${\theta }_i,D_j,H_j$	unknown parameters
${\widehat{\theta }}_i,{\varnothing }_i,{\widehat{D}}_j,{\widehat{H}}_j$	adaptive laws
${\tilde{\theta }}_i,{\tilde{D}}_j,{\tilde{H}}_j$	estimate errors
$k_i,{\Re }_i,{\beta }_i,{\varsigma }_j,{\xi }_j,\gamma$	positive constants

This paper considers the following stochastic nonlinear system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}d{x}_i=\hfill & (x_{i+1}+f_i\left({\overline{x}}_i\right)+h_i\left({\overline{x}}_i(t-d\left(t\right))\right))dt\hfill \\ \hfill & +g_i^T\left({\overline{x}}_i(t-d\left(t\right))\right)d\omega ,i=1,\cdots ,n-1\hfill \\ d{x}_n=\hfill & (u(t-\tau \left(t\right))+f_n\left({\overline{x}}_n\right)+h_n\left({\overline{x}}_n(t-d\left(t\right))\right))dt\hfill \\ \hfill & +g_n^T\left({\overline{x}}_n(t-d\left(t\right))\right)d\omega ,\hfill \\ y=x_1,\hfill \end{array}\right.\end{array}$$

(1)

where ${\overline{x}}_i={[x_1,\cdots ,x_i]}^T\in R^i$ means the system state, $u(t-\tau (t\left)\right)$ denotes the control input with known non-negative time-varying delay $\tau \left(t\right)$, $y\in R$ is the system output, $f_i\left({\overline{x}}_i\right)$ is the unknown function with $f_i\left(0\right)=0$, $d\left(t\right)$ is the Borel measurable non-negative time varying delay, ${\overline{x}}_i(t-d\left(t\right))={[x_1(t-d\left(t\right)),\cdots ,x_i(t-d\left(t\right))]}^T$ denotes the system state with time-varying delay, $h_i\left({\overline{x}}_i(t-d\left(t\right))\right)\in R$ and $g_i\left({\overline{x}}_i(t-d\left(t\right))\right)\in R_r$ are unknown locally Lipschitz functions with $h_i\left(0\right)=g_i\left(0\right)=0$, $\omega \in R^r$ means the $r-$dimensional independent standard Wiener process.

Control objective: The developed adaptive NN controller guarantees that all closed-loop system signals remain semi-globally bounded in probability, and the system output signal is asymptotically stable in probability.

The following assumption on the system (1) is necessary to achieve the control objective. For simplicity, $h_i\left({\overline{x}}_i(t-d\left(t\right))\right)$ and $g_i\left({\overline{x}}_i(t-d\left(t\right))\right)$ are written as $h_{i,d}$ and $g_{i,d}$.

Assumption 1. 

(see [19]). For the state time delay, it satisfies $d\left(t\right)\le \Delta$ and the time derivative of $d\left(t\right)$ satisfies $\dot{d}\left(t\right)\le \gamma <1$, where Δ and γ are positive constants. In addition, the input time delay $\tau \left(t\right)$ is bounded.

In the following section, some preliminaries are presented to construct the controller and analyze the closed-loop control system stability.

Lemma 1 (Young’s Inequality).

(see [45]). For $\forall (p,q)\in R^2$, one has

$$\begin{array}{c}\hfill pq\le \frac{{\varphi }^a}{a}{\left|p\right|}^a+\frac{1}{b{\varphi }^b}{\left|q\right|}^b,\end{array}$$

(2)

where $\varphi ,a,b$ are positive constants, and $(a-1)(b-1)=1$.

Lemma 2. 

(see [46]). For $\vartheta \in R$ and any positive constant ξ, the following inequality can be obtained

$$\begin{array}{c}\hfill 0\le \left|\vartheta \right|-\frac{{\vartheta }^2}{\sqrt{{\vartheta }^2+{\xi }^2}}<\xi .\end{array}$$

(3)

Lemma 3. 

(see [47]). Let ${\nu }_i\ge 0$, $b>1$, the following inequalities hold

$$\begin{array}{c}\hfill \sum _{i=1}^k{\nu }_i^b\ge k^{1-b}{\left(\sum _{i=1}^k{\nu }_i\right)}^b,i=1,2,\cdots ,k.\end{array}$$

(4)

In addition, some RBF NNs are used to solve the problem of unknown continuous function ${\overline{f}}_i\left(Z_i\right),i=1,2,\cdots ,n$ in the process of controller designing, that is

$$\begin{array}{c}\hfill {\overline{f}}_i\left(Z_i\right)=W_i^T{S}_i\left(Z_i\right)+{\eta }_i\left(Z_i\right),\end{array}$$

(5)

where $W_i\in R^m$ denotes the weight vector, $Z_i\in {\Omega }_i\subset R^m$ denotes the input vector, $m>1$ means the NN node number, ${\eta }_i\left(Z_i\right)\le {\eta }_i^{*}$ with a unknown parameter ${\eta }_i^{*}$, and ${\eta }_i\left(Z_i\right)$ denotes the NN inherent bounded approximation error. $S\left(Z_i\right)={[S_1\left(Z_i\right),\cdots ,S_m\left(Z_i\right)]}^T:{\Omega }_i\to R^m$ means the known smooth vector function, and the Gaussian function $S_j\left(Z_i\right)$ is designed as

$$\begin{array}{c}\hfill S_j\left(Z_i\right)=exp\left[\frac{-{(Z_i-{\chi }_j)}^T(Z_i-{\chi }_j)}{{\Gamma }^2}\right],j=1,2,\cdots ,m,\end{array}$$

(6)

where ${\chi }_j={[{\chi }_{j1},{\chi }_{j2},\cdots ,{\chi }_{jl}]}^T$ means the center, the Gaussian function’s width is $\Gamma >0$. The optimal weight satisfies

$$\begin{array}{c}\hfill W_i=arg\underset{{\widehat{W}}_i\in R^m}{min}\underset{Z_i\in {\Omega }_i}{sup}|{\overline{f}}_i-{\widehat{W}}_i^T{S}_i\left(Z_i\right)|,\end{array}$$

(7)

where ${\widehat{W}}_i$ is the estimate of $W_i$.

In the following section, the stochastic nonlinear system is presented as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}dx\left(t\right)=\phi (x,t)dt+\psi (x,t)d\omega \left(t\right)\hfill \\ x\left(t_0\right)=x_0\in R^n,\hfill \end{array}\right.\end{array}$$

(8)

where $\phi (0,t)=0$, $\psi (0,t)=0$, $\phi$ and $\psi$ are continuous and locally Lipschitz functions.

Definition 1. 

(see [48]). Consider a given positive function $V\left(x\right)\in C^{2,1}$, and the differential operator $\mathit{L}$ is defined as

$$\begin{array}{c}\hfill \mathit{L}\left[V\left(x\right)\right]=\frac{\partial V}{\partial x}\phi +\frac{1}{2}\mathit{Tr}\left\{{\psi }^T\frac{{\partial }^{2}V}{\partial x^2}\psi \right\}\end{array}$$

(9)

with the matrix trace $\mathit{Tr}(·)$.

This paper uses the gain suppressing inequality method. For $t\in [0,t_W]$, $V\left(t\right)$ is a non-negative function, ${\varnothing }_j\left(t\right)$ is a smooth function, and ${\varnothing }_j\left(0\right)(j=1,2,\cdots ,n)$ is bounded. Then the following function is defined as [49]

$$\begin{array}{c}\hfill I_{st}\left({\varnothing }_j\right)=\left(e^{\frac{1}{2}{\varnothing }_j^2}{\varnothing }_j^2+2e^{\frac{1}{2}{\varnothing }_j^2}\right)sin\left({\varnothing }_j\right).\end{array}$$

(10)

Lemma 4. 

(see [49]). For $t\in [0,t_W]$, if the following inequality is satisfied

$$\begin{array}{cc}\hfill \mathit{L}\left[V\right(t\left)\right]\le & -\breve{p}V\left(t\right)-\sum _{j=1}^n{c}_j{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right)\hfill \\ \hfill & +\sum _{j=1}^n{\dot{\varnothing }}_j\left(t\right)+\breve{q},\hfill \end{array}$$

(11)

where $\breve{p}>0$ is a parameter, $c_j>0$ denotes an unknown bounded constant, and $\breve{q}>0$ means a constant variable. Then for $t\in [0,t_W]$, we can obtain that the signals involved in (11) are bounded in probability; that is, the boundedness of ${\varnothing }_j\left(t\right)$ and $V\left(t\right)$ in probability can be achieved.

3. Adaptive NN Control Scheme Design and Stability Analysis

The adaptive neural DSC scheme and the process of proving system stability are presented in this section, and developed around the following parts.

3.1. Adaptive NN Control Scheme Design

This subsection involves developing an adaptive neural DSC scheme via the backstepping technique, and the following n steps are involved. The estimate error $\tilde{\chi }$ is defined as $\tilde{\chi }=\chi -\widehat{\chi }$, and the estimate of $\chi$ is shown as $\widehat{\chi }$.

First, we design a system such that it can solve the problem of input time-delay as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\lambda }}_i={\lambda }_{i+1}-p_i{\lambda }_i,i=2,\cdots ,n-1\hfill \\ {\dot{\lambda }}_n=-p_n{\lambda }_n+u(t-\tau \left(t\right))-u\left(t\right),\hfill \end{array}\right.\end{array}$$

(12)

where $p_i>1,i=2,\cdots ,n$ is a known constant, the initial condition of the system is $\lambda \left(0\right)=0$.

Furthermore, the following state coordinate transformation is introduced as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{z}_1=x_1\hfill \\ z_i=x_i-{\lambda }_i-s_{i-1},i=2,\cdots ,n-1\hfill \\ z_n=x_n-{\lambda }_n-s_{n-1},\hfill \end{array}\right.\end{array}$$

(13)

where $s_i:={\alpha }_i+e_i$ denotes the filtered virtual controller, ${\alpha }_i$ means the virtual controller, and the construction of ${\alpha }_i$ is presented in the following part, $e_i$ represents the i-th boundary layer error.

step 1: Based on (1) and (13), one has

$$\begin{array}{cc}\hfill d{z}_1=& (x_2+f_1\left(x_1\right)+h_1\left(x_1(t-d\left(t\right))\right))dt\hfill \\ \hfill & +g_1^T\left(x_1(t-d\left(t\right))\right)d\omega \hfill \\ \hfill =& (z_2+{\lambda }_2+{\alpha }_1+e_1+f_1\left(x_1\right)+h_{1,d})dt\hfill \\ \hfill & +g_{1,d}^{T}d\omega .\hfill \end{array}$$

(14)

Then, we choose the first symmetric positive-definite Lyapunov function as

$$\begin{array}{c}\hfill V_1=\frac{1}{4}{z}_1^4+\frac{1}{2}{\tilde{\theta }}_1^2,\end{array}$$

(15)

where ${\tilde{\theta }}_1$ is the estimate error of ${\theta }_1$, ${\theta }_1:=W_1^T{W}_1$.

Based on Definition 1 and (14) and (15), the following equation can be obtained

$$\begin{array}{cc}\hfill \mathit{L}{V}_1=& z_1^3(z_2+{\lambda }_2+{\alpha }_1+e_1+f_1\left(x_1\right)+h_{1,d})\hfill \\ \hfill & +\frac{3}{2}{z}_1^2{\left|g_{1,d}\right|}^2-{\tilde{\theta }}_1{\dot{\widehat{\theta }}}_1.\hfill \end{array}$$

(16)

Let ${\overline{f}}_1\left(Z_1\right)=f_1\left(x_1\right)+{\lambda }_2+{\varphi }_1$, then the RBF NN approximation property is used such that

$$\begin{array}{c}\hfill {\overline{f}}_1\left(Z_1\right)=W_1^T{S}_1\left(Z_1\right)+{\eta }_1\left(Z_1\right),\end{array}$$

(17)

where $Z_1={[x_1,{\lambda }_2]}^T$, $|{\eta }_1\left(Z_1\right)|\le {\eta }_1^{*}$, ${\varphi }_1=\frac{{\delta }_1}{1-\gamma }exp(\Delta )z_1[1/4q_{11}^4\left(x_1\right)+1/4{\mu }_{11}^4\left(x_1\right)]$.

In view of Lemma 1, we obtain the following inequalities

$$\begin{array}{c}\hfill z_1^3{z}_2\le \frac{3}{4}{z}_1^4+\frac{1}{4}{z}_2^4,\end{array}$$

(18)

$$\begin{array}{c}\hfill z_1^3{e}_1\le \frac{3}{4}{z}_1^4+\frac{1}{4}{e}_1^4,\end{array}$$

(19)

$$\begin{array}{cc}\hfill & z_1^3[W_1^T{S}_1\left(Z_1\right)+{\eta }_1\left(Z_1\right)]\hfill \\ \hfill \le & \frac{1}{2}+\frac{1}{2}{z}_1^6{\theta }_1{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)+\frac{1}{2}{z}_1^6+\frac{1}{2}{\eta }_1^{*2},\hfill \end{array}$$

(20)

$$\begin{array}{c}\hfill z_1^3{h}_{1,d}\le \frac{3}{4}{z}_1^4+\frac{1}{4}{h}_{1,d}^4,\end{array}$$

(21)

$$\begin{array}{c}\hfill \frac{3}{2}{z}_1^2{\left|g_{1,d}\right|}^2\le \frac{9}{4}{z}_1^4+\frac{1}{4}{g}_{1,d}^4.\end{array}$$

(22)

Then the first virtual control unit ${\alpha }_1$ is constructed as

$$\begin{array}{c}\hfill {\alpha }_1=I_{st}\left({\varnothing }_1\right){\overline{\alpha }}_1,\end{array}$$

(23)

where ${\varnothing }_1$ will be designed in (26), and the equivalent virtual unit ${\overline{\alpha }}_1$ is constructed as

$$\begin{array}{cc}\hfill {\overline{\alpha }}_1=& -k_1{z}_1-(\frac{3}{2}+3)z_1\hfill \\ \hfill & -\frac{1}{2}{z}_1^3{\widehat{\theta }}_1{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)-\frac{1}{2}{z}_1^3,\hfill \end{array}$$

(24)

where $k_1>0$ is a parameter.

Next, the adaptive laws of ${\widehat{\theta }}_1$ and ${\varnothing }_1$ are shown as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_1=\frac{1}{2}{z}_1^6{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)-{\beta }_1{\widehat{\theta }}_1,\end{array}$$

(25)

$$\begin{array}{c}\hfill {\dot{\varnothing }}_1=-{\Re }_1{z}_1^3{\overline{\alpha }}_1\end{array}$$

(26)

with a constant ${\Re }_1>0$. Based on Lemma 1, we have

$$\begin{array}{c}\hfill {\tilde{\theta }}_1{\widehat{\theta }}_1\le -\frac{1}{2}{\tilde{\theta }}_1^2+\frac{1}{2}{\theta }_1^2.\end{array}$$

(27)

Based on (16)–(27), the following inequality can be obtained

$$\begin{array}{cc}\hfill \mathit{L}{V}_1\le & -k_1{z}_1^4-\frac{{\beta }_1}{2}{\tilde{\theta }}_1^2+\frac{1}{{\Re }_1}{\dot{\varnothing }}_1\left(t\right)-\frac{1}{{\Re }_1}{I}_{st}\left({\varnothing }_1\left(t\right)\right){\dot{\varnothing }}_1\left(t\right)\hfill \\ \hfill & +\frac{1}{4}{e}_1^4+\frac{1}{4}{z}_2^4-z_1^3{\varphi }_1+\frac{1}{4}[h_{1,d}^4+g_{1,d}^4]\hfill \\ \hfill & +\frac{1}{2}+\frac{1}{2}{\eta }_1^{*2}+\frac{{\beta }_1}{2}{\theta }_1^2.\hfill \end{array}$$

(28)

The following nonlinear filter is designed

$$\begin{array}{cc}\hfill {\varrho }_1{\dot{s}}_1=& -e_1-\frac{{\varrho }_1{\widehat{D}}_1^2{e}_1^3}{\sqrt{{\widehat{D}}_1^2{e}_1^6+{\varpi }^2}}\hfill \\ \hfill & -\frac{3}{2}{\varrho }_1\frac{{\widehat{H}}_1^2{e}_1}{\sqrt{{\widehat{H}}_1^2{e}_1^4+{\varpi }^2}}-\frac{1}{4}{\varrho }_1{e}_1,\hfill \end{array}$$

(29)

$$\begin{array}{c}\hfill s_1\left(0\right)={\alpha }_1\left(0\right),\end{array}$$

(30)

where ${\varrho }_1$ means a filter time constant, the first boundary layer error is $e_1=s_1-{\alpha }_1$, the estimates of $D_1$ and $H_1$ are ${\widehat{D}}_1$ and ${\widehat{H}}_1$; they will be constructed later.

step $\mathit{i}(\mathbf{2}\le \mathit{i}\le \mathit{n}-\mathbf{1})$: the i-th state coordinate transformation is designed as $z_i=x_i-{\lambda }_i-s_{i-1}$ and based on (1), one has

$$\begin{array}{cc}\hfill d{z}_i=& (x_{i+1}+f_i\left({\overline{x}}_i\right)+h_{i,d}-{\lambda }_{i+1}+p_i{\lambda }_i-{\dot{s}}_{i-1})dt+g_{i,d}^{T}d\omega \hfill \\ \hfill =& (z_{i+1}+{\alpha }_i+e_i+p_i{\lambda }_i-{\dot{s}}_{i-1}+f_i\left({\overline{x}}_i\right)\hfill \\ \hfill & +h_{i,d})dt+g_{i,d}^{T}d\omega .\hfill \end{array}$$

(31)

The symmetric positive-definite Lyapunov function $V_i$ is presented as

$$\begin{array}{c}\hfill V_i=V_{i-1}+\frac{1}{4}{z}_i^4+\frac{1}{2}{\tilde{\theta }}_i^2,\end{array}$$

(32)

where ${\tilde{\theta }}_i$ is the estimate error of ${\theta }_i$, ${\theta }_i:=W_i^T{W}_i$.

According to Definition 1 and (31) and (32), we have

$$\begin{array}{cc}\hfill \mathit{L}{V}_i=& \mathit{L}{V}_{i-1}+z_i^3(z_{i+1}+{\alpha }_i+e_i+p_i{\lambda }_i-{\dot{s}}_{i-1}\hfill \\ \hfill & +f_i\left({\overline{x}}_i\right)+h_{i,d})+\frac{3}{2}{z}_i^2{\left|g_{i,d}\right|}^2-{\tilde{\theta }}_i{\dot{\widehat{\theta }}}_i.\hfill \end{array}$$

(33)

Let ${\overline{f}}_i\left(Z_i\right)=p_i{\lambda }_i-{\dot{s}}_{i-1}+f_i\left({\overline{x}}_i\right)+{\varphi }_i$, then the following equation can be obtained via the RBF NN approximation theory

$$\begin{array}{c}\hfill {\overline{f}}_i\left(Z_i\right)=W_i^T{S}_i\left(Z_i\right)+{\eta }_i\left(Z_i\right),\end{array}$$

(34)

where $Z_i={[x_1,\cdots ,x_i,{\lambda }_2,\cdots ,{\lambda }_i,s_1,\cdots ,s_{i-1},{\dot{s}}_1,\cdots ,{\dot{s}}_{i-1},{\widehat{\theta }}_1,\cdots ,{\widehat{\theta }}_{i-1},{\varnothing }_1,\cdots ,{\varnothing }_{i-1}]}^T$, $|{\eta }_i\left(Z_i\right)|\le {\eta }_i^{*}$, ${\varphi }_i=exp(\Delta ){\sum }_{j=1}^i\frac{{\delta }_i}{1-\gamma }{z}_j[1/4q_{ij}^4\left({\overline{x}}_j\right)+1/4{\mu }_{ij}^4\left({\overline{x}}_j\right)]$.

In view of Lemma 1, the following inequalities hold

$$\begin{array}{c}\hfill z_i^3{z}_{i+1}\le \frac{3}{4}{z}_i^4+\frac{1}{4}{z}_{i+1}^4,\end{array}$$

(35)

$$\begin{array}{c}\hfill z_i^3{e}_i\le \frac{3}{4}{z}_i^4+\frac{1}{4}{e}_i^4,\end{array}$$

(36)

$$\begin{array}{cc}\hfill & z_i^3[W_i^T{S}_i\left(Z_i\right)+{\eta }_i\left(Z_i\right)]\hfill \\ \hfill \le & \frac{1}{2}+\frac{1}{2}{z}_i^6{\theta }_i{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)+\frac{1}{2}{z}_i^6+\frac{1}{2}{\eta }_i^{*2},\hfill \end{array}$$

(37)

$$\begin{array}{c}\hfill z_i^3{h}_{i,d}\le \frac{3}{4}{z}_i^4+\frac{1}{4}{h}_{i,d}^4,\end{array}$$

(38)

$$\begin{array}{c}\hfill \frac{3}{2}{z}_i^2{\left|g_{i,d}\right|}^2\le \frac{9}{4}{z}_i^4+\frac{1}{4}{g}_{i,d}^4.\end{array}$$

(39)

Construct the i-th virtual control unit ${\alpha }_i$ as

$$\begin{array}{c}\hfill {\alpha }_i=I_{st}\left({\varnothing }_i\right){\overline{\alpha }}_i,\end{array}$$

(40)

where ${\varnothing }_i$ will be given in (43). The equivalent virtual unit ${\overline{\alpha }}_i$ is proposed as

$$\begin{array}{cc}\hfill {\overline{\alpha }}_i=& -k_i{z}_i-(\frac{3}{2}+\frac{1}{4}+3)z_i\hfill \\ \hfill & -\frac{1}{2}{z}_i^3{\widehat{\theta }}_i{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)-\frac{1}{2}{z}_i^3,\hfill \end{array}$$

(41)

where $k_i>0$ is a parameter.

The adaptive laws of ${\widehat{\theta }}_i$ and ${\tilde{\theta }}_i$ are designed as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_i=\frac{1}{2}{z}_i^6{S}_i^T\left(Z_i\right)S_i\left(Z_i\right)-{\beta }_i{\widehat{\theta }}_i\end{array}$$

(42)

$$\begin{array}{c}\hfill {\dot{\varnothing }}_i=-{\Re }_i{z}_i^3{\overline{\alpha }}_i,\end{array}$$

(43)

where the parameter ${\Re }_i>0$. By using Lemma 1, the following inequality holds

$$\begin{array}{c}\hfill {\tilde{\theta }}_i{\widehat{\theta }}_i\le -\frac{1}{2}{\tilde{\theta }}_i^2+\frac{1}{2}{\theta }_i^2.\end{array}$$

(44)

Based on (33)–(44), one has

$$\begin{array}{cc}\hfill \mathit{L}{V}_i\le & \mathit{L}{V}_{i-1}-k_i{z}_i^4-\frac{{\beta }_i}{2}{\tilde{\theta }}_i^2+\frac{1}{{\Re }_i}{\dot{\varnothing }}_i\left(t\right)\hfill \\ \hfill & -\frac{1}{{\Re }_i}{I}_{st}(\varnothing \left(t\right)){\dot{\varnothing }}_i\left(t\right)+\frac{1}{4}{e}_i^4+\frac{1}{4}{z}_{i+1}^4-z_i^3{\varphi }_i\hfill \\ \hfill & +\frac{1}{4}\sum _{j=1}^i[h_{j,d}^4+g_{j,d}^4]+\frac{1}{2}+\frac{1}{2}{\eta }_i^{*2}+\frac{{\beta }_i}{2}{\theta }_i^2\hfill \\ \hfill \le & -\sum _{j=1}^i{k}_j{z}_j^4-\sum _{j=1}^i\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2\hfill \\ \hfill & +\sum _{j=1}^i\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)-\sum _{j=1}^i\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right)\hfill \\ \hfill & +\sum _{j=1}^i\frac{1}{4}{e}_j^4+\frac{1}{4}{z}_{i+1}^4-\sum _{j=1}^i{z}_j^3{\varphi }_j+\frac{i}{2}+\sum _{j=1}^i\frac{1}{2}{\eta }_j^{*2}\hfill \\ \hfill & +\sum _{j=1}^i\frac{{\beta }_j}{2}{\theta }_j^2+\frac{1}{4}\sum _{j=1}^i[h_{j,d}^4+g_{j,d}^4].\hfill \end{array}$$

(45)

The nonlinear filtered virtual controller is designed as

$$\begin{array}{cc}\hfill {\varrho }_i{\dot{s}}_i=& -e_i-\frac{{\varrho }_i{\widehat{D}}_i^2{e}_i^3}{\sqrt{{\widehat{D}}_i^2{e}_i^6+{\varpi }^2}}\hfill \\ \hfill & -\frac{3}{2}{\varrho }_i\frac{{\widehat{H}}_i^2{e}_i}{\sqrt{{\widehat{H}}_i^2{e}_i^4+{\varpi }^2}}-\frac{1}{4}{\varrho }_i{e}_i,\hfill \end{array}$$

(46)

$$\begin{array}{c}\hfill s_i\left(0\right)={\alpha }_i\left(0\right),\end{array}$$

(47)

where ${\varrho }_i$ means a filter time constant, the i-th boundary layer error is presented as $e_i=s_i-{\alpha }_i$, the adaptive update laws of $D_i$ and ${\widehat{H}}_i$ will be designed later.

step $\mathit{n}$: the n-th state coordinate transformation is designed as $z_n=x_n-{\lambda }_n-s_{n-1}$ and based on (1), one has

$$\begin{array}{cc}\hfill d{z}_n=& (u+f_n\left({\overline{x}}_n\right)+h_{n,d}+p_n{\lambda }_n-{\dot{s}}_{n-1})dt+g_{n,d}^{T}d\omega .\hfill \end{array}$$

(48)

Design the symmetric positive-definite Lyapunov function $V_n$ as

$$\begin{array}{c}\hfill V_n=V_{n-1}+\frac{1}{4}{z}_n^4+\frac{1}{2}{\tilde{\theta }}_n^2,\end{array}$$

(49)

where ${\tilde{\theta }}_n$ is the estimate error of ${\theta }_n$, ${\theta }_n:=W_n^T{W}_n$.

Based on Definition 1 and (48) and (49), one has

$$\begin{array}{cc}\hfill \mathit{L}{V}_n=& \mathit{L}{V}_{n-1}+z_n^3(u+p_n{\lambda }_n-{\dot{s}}_{n-1}+f_n\left({\overline{x}}_n\right)+h_{n,d})\hfill \\ \hfill & +\frac{3}{2}{z}_n^2{\left|g_{n,d}\right|}^2-{\tilde{\theta }}_n{\dot{\widehat{\theta }}}_n.\hfill \end{array}$$

(50)

Let ${\overline{f}}_n\left(Z_n\right)=p_n{\lambda }_n-{\dot{s}}_{n-1}+f_n\left({\overline{x}}_n\right)+{\varphi }_n$, the following equation can be proposed via the RBF NN approximation theory

$$\begin{array}{c}\hfill {\overline{f}}_n\left(Z_n\right)=W_n^T{S}_n\left(Z_n\right)+{\eta }_n\left(Z_n\right),\end{array}$$

(51)

where $Z_n={[x_1,\cdots ,x_n,{\lambda }_2,\cdots ,{\lambda }_n,s_1,\cdots ,s_{n-1},{\dot{s}}_1,\cdots ,{\dot{s}}_{n-1},{\widehat{\theta }}_1,\cdots ,{\widehat{\theta }}_{n-1},{\varnothing }_1,\cdots ,{\varnothing }_{n-1}]}^T$, $|{\eta }_n\left(Z_n\right)|\le {\eta }_n^{*}$, ${\varphi }_n=exp(\Delta ){\sum }_{j=1}^n\frac{{\delta }_n}{1-\gamma }{z}_n[1/4q_{ij}^4\left({\overline{x}}_j\right)+1/4{\mu }_{ij}^4\left({\overline{x}}_j\right)]$.

In view of Lemma 1, we can obtain

$$\begin{array}{cc}\hfill & z_n^3[W_n^T{S}_n\left(Z_n\right)+{\eta }_n\left(Z_n\right)]\hfill \\ \hfill \le & \frac{1}{2}+\frac{1}{2}{z}_n^6{\theta }_n{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)+\frac{1}{2}{z}_n^6+\frac{1}{2}{\eta }_n^{*2},\hfill \end{array}$$

(52)

$$\begin{array}{c}\hfill z_n^3{h}_{n,d}\le \frac{3}{4}{z}_n^4+\frac{1}{4}{h}_{n,d}^4,\end{array}$$

(53)

$$\begin{array}{c}\hfill \frac{3}{2}{z}_n^2{\left|g_{n,d}\right|}^2\le \frac{9}{4}{z}_n^4+\frac{1}{4}{g}_{n,d}^4.\end{array}$$

(54)

We construct the control signal u as

$$\begin{array}{c}\hfill u=I_{st}\left({\varnothing }_n\right)\overline{u},\end{array}$$

(55)

where ${\varnothing }_n$ will be designed in (58), and the equivalent virtual unit is proposed as

$$\begin{array}{cc}\hfill \overline{u}=& -k_n{z}_n-(\frac{1}{4}+3)z_n\hfill \\ \hfill & -\frac{1}{2}{z}_n^3{\widehat{\theta }}_n{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)-\frac{1}{2}{z}_n^3.\hfill \end{array}$$

(56)

Develop the update laws of ${\widehat{\theta }}_n$ and ${\hslash }_n$ as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_n=\frac{1}{2}{z}_n^6{S}_n^T\left(Z_n\right)S_n\left(Z_n\right)-{\beta }_n{\widehat{\theta }}_n,\end{array}$$

(57)

$$\begin{array}{c}\hfill {\dot{\varnothing }}_n=-{\Re }_n{z}_n^3\overline{u},\end{array}$$

(58)

where the parameter ${\Re }_n>0$. Then, one can obtain, based on Lemma 1

$$\begin{array}{c}\hfill {\tilde{\theta }}_n{\widehat{\theta }}_n\le -\frac{1}{2}{\tilde{\theta }}_n^2+\frac{1}{2}{\theta }_n^2.\end{array}$$

(59)

Based on (50)–(59), one has

$$\begin{array}{cc}\hfill \mathit{L}{V}_n\le & \mathit{L}{V}_{n-1}-k_n{z}_n^4-\frac{{\beta }_i}{2}{\tilde{\theta }}_n^2+\frac{1}{{\Re }_n}{\dot{\varnothing }}_n\left(t\right)\hfill \\ \hfill & -\frac{1}{{\Re }_n}{I}_{st}\left({\varnothing }_n\left(t\right)\right){\dot{\varnothing }}_n\left(t\right)-z_n^3{\varphi }_n\hfill \\ \hfill & +\frac{1}{4}[h_{n,d}^4+g_{n,d}^4]+\frac{1}{2}+\frac{1}{2}{\eta }_n^{*2}+\frac{{\beta }_n}{2}{\theta }_n^2\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{z}_j^4-\sum _{j=1}^n\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2+\sum _{j=1}^n\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)\hfill \\ \hfill & -\sum _{j=1}^n\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right)+\sum _{j=1}^{n-1}\frac{1}{4}{e}_j^4\hfill \\ \hfill & -\sum _{j=1}^n{z}_j^3{\varphi }_j+\frac{n}{2}+\sum _{j=1}^n\frac{1}{2}{\eta }_j^{*2}+\sum _{j=1}^n\frac{{\beta }_j}{2}{\theta }_j^2\hfill \\ \hfill & +\frac{1}{4}\sum _{j=1}^n[h_{j,d}^4+g_{j,d}^4].\hfill \end{array}$$

(60)

In view of ([49], Assumption 1) and Lemma 3, one holds

$$\begin{array}{cc}\hfill & \frac{1}{4}\sum _{j=1}^n[h_{j,d}^4+g_{j,d}^4]\hfill \\ \hfill \le & \frac{1}{4}\sum _{j=1}^n[{\left(\sum _{j=1}^i{z}_j(t-d\left(t\right))q_{ij}\left({\overline{x}}_j(t-d\left(t\right))\right)\right)}^4\hfill \\ \hfill & +{\left(\sum _{j=1}^i{z}_j(t-d\left(t\right)){\mu }_{ij}\left({\overline{x}}_j(t-d\left(t\right))\right)\right)}^4]\hfill \\ \hfill \le & \frac{1}{4}\sum _{j=1}^n\sum _{j=1}^i\frac{1}{i^{-3}}[z_j^4(t-d\left(t\right))q_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)\hfill \\ \hfill & +z_j^4(t-d\left(t\right)){\mu }_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)]\hfill \\ \hfill =& \sum _{j=1}^n\sum _{j=1}^i\frac{{\delta }_i}{4}[z_j^4(t-d\left(t\right))q_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)\hfill \\ \hfill & +z_j^4(t-d\left(t\right)){\mu }_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)],\hfill \end{array}$$

(61)

where ${\delta }_i=\frac{1}{i^{-3}}$, $q_{ij}(·)$ and ${\mu }_{ij}(·)$ are positive unknown smooth functions.

Consider the following symmetric positive-definite Lyapunov function ${\overline{V}}_n$

$$\begin{array}{c}\hfill {\overline{V}}_n=V_n+V_Q,\end{array}$$

(62)

where $V_Q$ is a Lyapunov–Krasovskii function, which is defined as

$$\begin{array}{cc}\hfill V_Q=& \frac{1}{1-\gamma }\sum _{i=1}^n\sum _{j=1}^i{\delta }_{i}exp(d\left(t\right)-t){\int }_{t-d\left(t\right)}^{t}exp\left(\tau \right)\frac{1}{4}{z}_j^4\left(\tau \right)\hfill \\ \hfill & [q_{ij}^4\left({\overline{x}}_j\left(\tau \right)\right)+{\mu }_{ij}^4\left({\overline{x}}_j\left(\tau \right)\right)]d\tau .\hfill \end{array}$$

(63)

The time derivative of (63) is given as

$$\begin{array}{cc}\hfill {\dot{V}}_Q=& (\dot{d}\left(t\right)-1)V_Q+\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{1-\gamma }exp\left(d\left(t\right)\right)z_j^4[1/4q_{ij}^4\left({\overline{x}}_j\right)\hfill \\ \hfill & +1/4{\mu }_{ij}^4\left({\overline{x}}_j\right)]-\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{1-\gamma }(1-\dot{d}\left(t\right))z_j^4(t-d\left(t\right))\hfill \\ \hfill & [1/4q_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)+1/4{\mu }_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)]\hfill \\ \hfill \le & (\gamma -1)V_Q+\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{1-\gamma }exp\left(d\left(t\right)\right)z_j^4[1/4q_{ij}^4\left({\overline{x}}_j\right)\hfill \\ \hfill & +1/4{\mu }_{ij}^4\left({\overline{x}}_j\right)]+\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{1-\gamma }(\gamma -1)z_j^4(t-d\left(t\right))\hfill \\ \hfill & [1/4q_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)+1/4{\mu }_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)]\hfill \\ \hfill =& -(1-\gamma )V_Q+\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{1-\gamma }exp\left(d\left(t\right)\right)z_j^4[1/4q_{ij}^4\left({\overline{x}}_j\right)\hfill \\ \hfill & +1/4{\mu }_{ij}^4\left({\overline{x}}_j\right)]-\sum _{i=1}^n\sum _{j=1}^i\frac{{\delta }_i}{4}{z}_j^4(t-d\left(t\right))\hfill \\ \hfill & [q_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)+{\mu }_{ij}^4\left({\overline{x}}_j(t-d\left(t\right))\right)].\hfill \end{array}$$

(64)

Along with (60)–(64) and the definition of ${\varphi }_i$, one implies

$$\begin{array}{cc}\hfill \mathit{L}{\overline{V}}_n=& \mathit{L}{V}_n+{\dot{V}}_Q\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{z}_j^4-\sum _{j=1}^n\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2-(1-\gamma )V_Q+\sum _{j=1}^n\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)\hfill \\ \hfill & -\sum _{j=1}^n\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right)+\sum _{j=1}^{n-1}\frac{1}{4}{e}_j^4+\frac{n}{2}\hfill \\ \hfill & +\sum _{j=1}^n\frac{1}{2}{\eta }_j^{*2}+\sum _{j=1}^n\frac{{\beta }_j}{2}{\theta }_j^2.\hfill \end{array}$$

(65)

3.2. Stability Analysis

The proof of system stability will be given in this section.

The boundary layer errors $e_i=s_i-{\alpha }_i,i=1,\cdots ,n-1$ satisfy

$$\begin{array}{cc}\hfill d{e}_i=& [-\frac{e_i}{{\varrho }_i}-\frac{1}{4}{e}_i-\frac{{\widehat{D}}_i^2{e}_i^3}{\sqrt{{\widehat{D}}_i^2{e}_i^6+{\varpi }^2}}\hfill \\ \hfill & -\frac{3}{2}\frac{{\widehat{H}}_i^2{e}_i}{\sqrt{{\widehat{H}}_i^2{e}_i^4+{\varpi }^2}}+{\mathcal{C}}_i(z_1,\cdots ,z_{i+1},e_1,\cdots ,e_i,\hfill \\ \hfill & {\widehat{\theta }}_1,\cdots ,{\widehat{\theta }}_{i+1},{\widehat{D}}_1,\cdots ,{\widehat{D}}_i,{\widehat{H}}_1,\cdots ,{\widehat{H}}_i,\hfill \\ \hfill & {\varnothing }_1\left(t\right),\cdots ,{\varnothing }_i\left(t\right),{\dot{\varnothing }}_1\left(t\right),\cdots ,{\dot{\varnothing }}_i\left(t\right)\left)\right]dt+{\mathcal{B}}_i(z_1,\hfill \\ \hfill & \cdots ,z_{i+1},e_1,\cdots ,e_i,{\widehat{\theta }}_1,\cdots ,{\widehat{\theta }}_{i+1},{\widehat{D}}_1,\hfill \\ \hfill & \cdots ,{\widehat{D}}_i,{\widehat{H}}_1,\cdots ,{\widehat{H}}_i,{\varnothing }_1\left(t\right),\cdots ,{\varnothing }_i\left(t\right),\hfill \\ \hfill & {\dot{\varnothing }}_1\left(t\right),\cdots ,{\dot{\varnothing }}_i\left(t\right))d\omega ,\hfill \end{array}$$

(66)

where

$$\begin{array}{cc}\hfill {\mathcal{C}}_1(·)=& -\mathit{L}{\alpha }_1\hfill \\ \hfill =& -\frac{\partial {\alpha }_1}{\partial x_1}[x_2+f_1\left(x_1\right)+h_{1d}]-\frac{\partial {\alpha }_1}{\partial {\widehat{\theta }}_1}{\dot{\widehat{\theta }}}_1\hfill \\ \hfill & -\frac{\partial {\alpha }_1}{\partial {\hslash }_1\left(t\right)}{\dot{\hslash }}_1\left(t\right)-\frac{1}{2}\frac{{\partial }^2{\alpha }_1}{\partial x_1^2}{g}_{i,d}^T{g}_{i,d},\hfill \end{array}$$

(67)

$$\begin{array}{c}\hfill {\mathcal{B}}_1(·)=-\frac{\partial {\alpha }_1}{\partial x_1}{g}_{i,d}^T,\end{array}$$

(68)

$$\begin{array}{cc}\hfill {\mathcal{C}}_i(·)=& -\mathit{L}{\alpha }_i\hfill \\ \hfill =& -\frac{\partial {\alpha }_i}{\partial x_i}[x_{i+1}+f_i\left({\overline{x}}_i\right)+h_{i,d}]-\frac{\partial {\alpha }_i}{\partial {\widehat{\theta }}_j}{\dot{\widehat{\theta }}}_j-\frac{\partial {\alpha }_i}{\partial {\widehat{D}}_{i-1}}{\dot{\widehat{D}}}_{i-1}\hfill \\ \hfill & -\frac{\partial {\alpha }_i}{\partial {\widehat{H}}_{i-1}}{\dot{\widehat{H}}}_{i-1}-\frac{\partial {\alpha }_i}{\partial {\hslash }_i\left(t\right)}{\dot{\hslash }}_i\left(t\right)-\frac{1}{2}\frac{{\partial }^2{\alpha }_i}{\partial x_i^2}{g}_{i,d}^T{g}_{i,d},\hfill \end{array}$$

(69)

$$\begin{array}{c}\hfill {\mathcal{B}}_i(·)=-\frac{\partial {\alpha }_i}{\partial x_i}{g}_{i,d}^T\end{array}$$

(70)

are continuous functions.

The main result of this paper is summarized by the following theorem.

Theorem 1.

The closed-loop system is considered, including the plant (1), the control units (23), (40) and (55), the nonlinear filters virtual controller (29) and (46), and the adaptive update laws (25) and (26), (42) and (43), (57) and (58), (77) and (78). Then the following statements are guaranteed based on the Assumption 1 and some appropriate parameters $k_i$, ${\Re }_i$, ${\beta }_i$, $i=1,\cdots ,n$, ${\varsigma }_j$, ${\xi }_j$, $j=1,\cdots ,n-1$, and $p_l$, $l=2,\cdots ,n$

(i) The semi-global bounded stability in probability of all the signals in this system is achieved.

(ii) The system output signal is asymptotically stable in probability.

Proof. 

The compact set is designed as follows

$$\begin{array}{c}\hfill {\Omega }_1=\{V\left(t\right)\le p\}.\end{array}$$

(71)

Therefore, there exists positive constants $D_i,H_i$, such that $|{\mathcal{C}}_i(·)|\le D_i$ on ${\Omega }_1$ and $Tr\left\{{\mathcal{B}}_i^T{\mathcal{B}}_i\right\}\le H_i$, where $D_i$ and $H_i$ are unknown parameters. Then, the system stability analysis is presented as follows.

Choose the whole Lyapunov function V as follows

$$\begin{array}{c}\hfill V={\overline{V}}_n+\sum _{i=1}^{n-1}\frac{1}{4}{e}_i^4+\sum _{i=1}^{n-1}\frac{1}{2{\varsigma }_i}{\tilde{D}}_i^2+\sum _{i=1}^{n-1}\frac{1}{2{\xi }_i}{\tilde{H}}_i^2,\end{array}$$

(72)

with some constants ${\varsigma }_i$, ${\xi }_i>0,i=1,\cdots ,n-1$.

Then, one holds

$$\begin{array}{cc}\hfill \mathit{L}V=& \mathit{L}{\overline{V}}_n+\sum _{i=1}^{n-1}{e}_i^3[-\frac{e_i}{{\varrho }_i}-\frac{1}{4}{e}_i-\frac{{\widehat{D}}_i^2{e}_i^3}{\sqrt{{\widehat{D}}_i^2{e}_i^6+{\varpi }^2}}\hfill \\ \hfill & -\frac{3}{2}\frac{{\widehat{H}}_i^2{e}_i}{\sqrt{{\widehat{H}}_i^2{e}_i^4+{\varpi }^2}}+{\mathcal{C}}_i]+\frac{3}{2}\sum _{i=1}^{n-1}{e}_i^{2}Tr\left\{{\mathcal{B}}_i^T{\mathcal{B}}_i\right\}\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{1}{{\varsigma }_i}{\tilde{D}}_i{\dot{\widehat{D}}}_i-\sum _{i=1}^{n-1}\frac{1}{{\xi }_i}{\tilde{H}}_i{\dot{\widehat{H}}}_i\hfill \\ \hfill \le & -\sum _{j=1}^n{k}_j{z}_j^4-\sum _{j=1}^n\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2-\sum _{i=1}^{n-1}\frac{e_i^4}{{\varrho }_i}-(1-\gamma )V_Q\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{{\widehat{D}}_i^2{e}_i^6}{\sqrt{{\widehat{D}}_i^2{e}_i^6+{\varpi }^2}}-\frac{3}{2}\sum _{i=1}^{n-1}\frac{{\widehat{H}}_i^2{e}_i^4}{\sqrt{{\widehat{H}}_i^2{e}_i^4+{\varpi }^2}}\hfill \\ \hfill & +\sum _{i=1}^{n-1}{D}_i\left|e_i^3\right|+\frac{3}{2}\sum _{i=1}^{n-1}{e}_i^2{H}_i-\sum _{i=1}^{n-1}\frac{1}{{\varsigma }_i}{\tilde{D}}_i{\dot{\widehat{D}}}_i\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{1}{{\xi }_i}{\tilde{H}}_i{\dot{\widehat{H}}}_i+\frac{n}{2}+\sum _{j=1}^n\frac{1}{2}{\eta }_j^{*2}+\sum _{j=1}^n\frac{{\beta }_j}{2}{\theta }_j^2\hfill \\ \hfill & +\sum _{j=1}^n\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)-\sum _{j=1}^n\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right).\hfill \end{array}$$

(73)

The following inequalities are hold by using Lemma 2

$$\begin{array}{cc}\hfill D_i\left|e_i^3\right|=& {\widehat{D}}_i|e_i^3|+{\tilde{D}}_i\left|e_i^3\right|\hfill \\ \hfill \le & \frac{{\widehat{D}}_i^2{e}_i^6}{\sqrt{{\widehat{D}}_i^2{e}_i^6+{\varpi }^2}}+\varpi +{\tilde{D}}_i\left|e_i^3\right|\hfill \end{array}$$

(74)

and

$$\begin{array}{cc}\hfill H_i{e}_i^2=& {\widehat{H}}_i{e}_i^2+{\tilde{H}}_i{e}_i^2\hfill \\ \hfill \le & \frac{{\widehat{H}}_i^2{e}_i^4}{\sqrt{{\widehat{H}}_i^2{e}_i^4+{\varpi }^2}}+\varpi +{\tilde{H}}_i{e}_i^2.\hfill \end{array}$$

(75)

In view of (73)–(75), one has

$$\begin{array}{cc}\hfill \mathit{L}V\le & -\sum _{j=1}^n{k}_j{z}_j^4-\sum _{j=1}^n\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2-\sum _{i=1}^{n-1}\frac{e_i^4}{{\varrho }_i}-(1-\gamma )V_Q\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{1}{{\varsigma }_i}{\tilde{D}}_i({\dot{\widehat{D}}}_i-{\varsigma }_i|e_i^3\left|\right)-\sum _{i=1}^{n-1}\frac{1}{{\xi }_i}{\tilde{H}}_i({\dot{\widehat{H}}}_i-\frac{3}{2}{\xi }_i{e}_i^2)\hfill \\ \hfill & +\frac{5}{2}(n-1)\varpi +\frac{n}{2}+\sum _{j=1}^n\frac{1}{2}{\eta }_j^{*2}+\sum _{j=1}^n\frac{{\beta }_j}{2}{\theta }_j^2\hfill \\ \hfill & +\sum _{j=1}^n\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)-\sum _{j=1}^n\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right).\hfill \end{array}$$

(76)

The adaptive update laws of ${\widehat{D}}_i$ and ${\widehat{H}}_i$ are presented as follows

$$\begin{array}{c}\hfill {\dot{\widehat{D}}}_i={\varsigma }_i\left|e_i^3\right|-{\varsigma }_i{\widehat{D}}_i,\end{array}$$

(77)

and

$$\begin{array}{c}\hfill {\dot{\widehat{H}}}_i=\frac{3}{2}{\xi }_i{e}_i^2-{\xi }_i{\widehat{H}}_i.\end{array}$$

(78)

In view of Lemma 1, we can obtain that

$$\begin{array}{c}\hfill {\tilde{D}}_i{\widehat{D}}_i\le -\frac{1}{2}{\tilde{D}}_i^2+\frac{1}{2}{D}_i^2,\end{array}$$

(79)

and

$$\begin{array}{c}\hfill {\tilde{H}}_i{\widehat{H}}_i\le -\frac{1}{2}{\tilde{H}}_i^2+\frac{1}{2}{H}_i^2.\end{array}$$

(80)

Therefore, the following inequality holds

$$\begin{array}{cc}\hfill \mathit{L}V\le & -\sum _{j=1}^n{k}_j{z}_j^4-\sum _{j=1}^n\frac{{\beta }_j}{2}{\tilde{\theta }}_j^2-\sum _{i=1}^{n-1}\frac{e_i^4}{{\varrho }_i}-(1-\gamma )V_Q\hfill \\ \hfill & -\sum _{i=1}^{n-1}\frac{1}{2}{\tilde{D}}_i^2-\sum _{i=1}^{n-1}\frac{1}{2}{\tilde{H}}_i^2+\overline{\Lambda }\hfill \\ \hfill & +\sum _{j=1}^n\frac{1}{{\Re }_j}{\dot{\varnothing }}_j\left(t\right)-\sum _{j=1}^n\frac{1}{{\Re }_j}{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right).\hfill \end{array}$$

(81)

where $\overline{\Lambda }=\frac{5}{2}(n-1)\varpi +\frac{n}{2}+{\sum }_{j=1}^n\frac{1}{2}{\eta }_j^{*2}+{\sum }_{j=1}^n\frac{{\beta }_j}{2}{\theta }_j^2$.

In view of (81), one can obtain

$$\begin{array}{cc}\hfill \mathit{L}V\le & -cV+\frac{1}{{\Re }_j}\sum _{j=1}^n{\dot{\varnothing }}_j\left(t\right)\hfill \\ \hfill & -\frac{1}{{\Re }_j}\sum _{j=1}^n{I}_{st}\left({\varnothing }_j\left(t\right)\right){\dot{\varnothing }}_j\left(t\right)+\overline{\Lambda },\hfill \end{array}$$

(82)

where $c=min\{4k_i,{\beta }_i,1,\frac{4}{{\varrho }_j},{\varsigma }_j,{\xi }_j,1-\gamma |i=1,\cdots ,n,j=1,\cdots ,n-1\}$. Thus, the signals of this closed-loop system remain semi-global bounded in probability.

Next, the boundedness of ${\lambda }_i$ is proved as following.

It is easy to verify that $u(t-\tau \left(t\right))-u\left(t\right)\le {\sigma }_u$, where ${\sigma }_u>0$ is a parameter.

Choose the candidate Lyapunov function

$$\begin{array}{c}\hfill V_{\lambda }=\frac{1}{2}\sum _{i=2}^n{\lambda }_i^2,\end{array}$$

(83)

then the derivative of $V_{\lambda }$ is shown as

$$\begin{array}{cc}\hfill {\dot{V}}_{\lambda }\le & \sum _{j=2}^{n-1}{\lambda }_j({\lambda }_{j+1}-p_j{\lambda }_j)+{\lambda }_n-p_n{\lambda }_n+u(t-\tau \left(t\right))-u\left(t\right)\hfill \\ \hfill \le & -\sum _{j=1}^n{\overline{p}}_j{\lambda }_j^2+\frac{{\sigma }_u^2}{2},\hfill \end{array}$$

(84)

where ${\overline{p}}_2=p_2-\frac{1}{2}$, ${\overline{p}}_i=p_i-1,i=3,\cdots ,n-1$, ${\overline{p}}_n=p_n-1$.

Thus, the boundedness of ${\lambda }_i$ can be obtained.

Then, we set ${\widehat{\theta }}_i\left(0\right)>0$, such that ${\widehat{\theta }}_i\left(t\right)\ge 0$ based on ([49], Lemma 4). Hence, from (25), (42), (57), the following inequalities can be obtained

$$\begin{array}{c}\hfill \frac{1}{2}{z}_1^6{\widehat{\theta }}_1{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)\ge 0.\end{array}$$

(85)

Therefore, in view of (26), (43) and (58), one holds

$$\begin{array}{c}\hfill {\dot{\varnothing }}_i\left(t\right)\ge k_i{\Re }_i{z}_i^4.\end{array}$$

(86)

In view of (86), one has

$$\begin{array}{c}\hfill {\int }_0^t{k}_i{\Re }_i{z}_i^4(\ell )d\ell \le {\varnothing }_i\left(t\right)-{\varnothing }_i\left(0\right).\end{array}$$

(87)

In addition, based on (87) and Lemma 4, we can obtain that ${\varnothing }_i\left(0\right)$ is bounded, and the boundedness of ${\varnothing }_i\left(t\right)$ can be guaranteed. Therefore, the following inequality can be obtained

$$\begin{array}{c}\hfill E{\int }_0^t{k}_i{\Re }_i{z}_i^4(\ell )d\ell <\infty .\end{array}$$

(88)

Then, the following equation can be obtained via stochastic Barbalat’s lemma

$$\begin{array}{c}\hfill P\left\{\underset{t\to \infty }{lim}\sum _{i=1}^n{k}_i{\Re }_i{z}_i^4=0\right\}=1,\end{array}$$

(89)

which means that

$$\begin{array}{c}\hfill P\left\{\underset{t\to \infty }{lim}\right|x_1\left(t\right)|=0\}=1.\end{array}$$

(90)

That is, the system output signal is asymptotically stable in probability. □

4. Simulation Results

Consider the following stochastic nonlinear system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}d{x}_1=\hfill & (x_2+f_1\left(x_1\right)+h_1\left(x_1(t-d\left(t\right))\right))dt\hfill \\ \hfill & +g_1^T\left(x_1(t-d\left(t\right))\right)d\omega ,\hfill \\ d{x}_2=\hfill & (u(t-\tau \left(t\right))+f_2\left({\overline{x}}_2\right)+h_2\left({\overline{x}}_2(t-d\left(t\right))\right))dt\hfill \\ \hfill & +g_2^T\left({\overline{x}}_2(t-d\left(t\right))\right)d\omega ,\hfill \\ y=x_1,\hfill \end{array}\right.\end{array}$$

(91)

where $f_1\left(x_1\right)=-10sin\left(8x_1\right)cos\left(x_1\right)$, $h_1\left(x_1(t-d\left(t\right))\right)=2x_1^3(t-d\left(t\right))$, $g_1\left(x_1(t-d\left(t\right))\right)=x_1^2(t-d\left(t\right))$, $f_2\left({\overline{x}}_2\right)=x_1{x}_{2}sin\left(0.2x_1\right)cos\left(2x_1\right)$, $h_2\left({\overline{x}}_2(t-d\left(t\right))\right)=sin\left(x_1^2(t-d\left(t\right))\right)sin\left(0.02x_2(t-d\left(t\right))\right)$, $g_2\left({\overline{x}}_2(t-d\left(t\right))\right)=x_1(t-d\left(t\right)){exp}^{-x_2(t-d\left(t\right))}$, $d\left(t\right)=1+0.5sin\left(t\right)$, $\tau \left(t\right)=1+0.5cos\left(t\right)$.

We design the virtual controller ${\alpha }_1$ as

$$\begin{array}{c}\hfill {\alpha }_1=I_{st}\left({\varnothing }_1\right){\overline{\alpha }}_1,\end{array}$$

(92)

where

$$\begin{array}{cc}\hfill {\overline{\alpha }}_1=& -k_1{z}_1-(\frac{3}{2}+3)z_1\hfill \\ \hfill & -\frac{1}{2}{z}_1^3{\widehat{\theta }}_1{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)-\frac{1}{2}{z}_1^3.\hfill \end{array}$$

(93)

The update laws of ${\widehat{\theta }}_1$ and ${\hslash }_1$ are proposed as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_1=\frac{1}{2}{z}_1^6{S}_1^T\left(Z_1\right)S_1\left(Z_1\right)-{\beta }_1{\widehat{\theta }}_1,\end{array}$$

(94)

$$\begin{array}{c}\hfill {\dot{\varnothing }}_1=-{\Re }_1{z}_1^3{\overline{\alpha }}_1.\end{array}$$

(95)

The filtered virtual controller $s_1$ with the novel nonlinear filter is designed as

$$\begin{array}{cc}\hfill {\varrho }_1{\dot{s}}_1=& -e_1-\frac{{\varrho }_1{\widehat{D}}_1^2{e}_1^3}{\sqrt{{\widehat{D}}_1^2{e}_1^6+{\varpi }^2}}\hfill \\ \hfill & -\frac{3}{2}{\varrho }_1\frac{{\widehat{H}}_1^2{e}_1}{\sqrt{{\widehat{H}}_1^2{e}_1^4+{\varpi }^2}}-\frac{1}{4}{\varrho }_1{e}_1,\hfill \end{array}$$

(96)

where

$$\begin{array}{c}\hfill {\dot{\widehat{D}}}_1={\varsigma }_1\left|e_1^3\right|-{\varsigma }_1{\widehat{D}}_1,\end{array}$$

(97)

and

$$\begin{array}{c}\hfill {\dot{\widehat{H}}}_1=\frac{3}{2}{\xi }_1{e}_1^2-{\xi }_1{\widehat{H}}_1.\end{array}$$

(98)

Design control signal u as

$$\begin{array}{c}\hfill u=I_{st}\left({\varnothing }_2\right)\overline{u},\end{array}$$

(99)

where

$$\begin{array}{cc}\hfill \overline{u}=& -k_2{z}_2-(\frac{1}{4}+3)z_2\hfill \\ \hfill & -\frac{1}{2}{z}_2^3{\widehat{\theta }}_2{S}_2^T\left(Z_2\right)S_2\left(Z_2\right)-\frac{1}{2}{z}_2^3\hfill \end{array}$$

(100)

with the update laws of ${\widehat{\theta }}_2$ and ${\hslash }_2$ as

$$\begin{array}{c}\hfill {\dot{\widehat{\theta }}}_2=\frac{1}{2}{z}_2^6{S}_2^T\left(Z_2\right)S_2\left(Z_2\right)-{\beta }_2{\widehat{\theta }}_2,\end{array}$$

(101)

and

$$\begin{array}{c}\hfill {\dot{\varnothing }}_2=-{\Re }_2{z}_2^3\overline{u}.\end{array}$$

(102)

The nodes number of the first RBF NN is 9, the center is placing on $[-5,5]\times [-5,5]$, and we design the Gaussian functions width as $\Gamma =1.5$. The nodes number of the second RBF NN is 576, the center is placing on $[-5,5]\times [-5,5]\times [-5,5]\times [-5,5]\times [-5,5]\times [-5,5]\times [-5,5]$, and we design the Gaussian functions width as $\Gamma =1.6$.

The parameters are selected as $k_1=k_2=3$, ${\beta }_1=1$, ${\beta }_2=3$, ${\Re }_1=5$, ${\Re }_2=2$, ${\varsigma }_1=5$, ${\xi }_1=3$, $p_2=3$, $\varpi =0.5$, ${\varrho }_1=1$. For this closed-loop system, we design the initial values as $x_1\left(0\right)=0.1$, $x_2\left(0\right)=0.2$, ${\theta }_1\left(0\right)={\theta }_2\left(0\right)=0.3$, ${\varnothing }_1\left(0\right)=0.2$, ${\varnothing }_2\left(0\right)=0.01$, ${\lambda }_2\left(0\right)=0$, ${\widehat{D}}_1\left(0\right)=1$, ${\widehat{H}}_1\left(0\right)=0.5$.

Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 show the simulation results. Figure 1 presents the system state $x_1$ from which it can be seen that the asymptotic regulation is achieved. The input signal u is presented in Figure 2. The adaptive update laws of ${\widehat{\theta }}_1$, ${\widehat{\theta }}_2$, and ${\varnothing }_1$, ${\varnothing }_1$ are shown in Figure 3 and Figure 4, respectively. The curves of filter parameters ${\widehat{D}}_1$ and ${\widehat{H}}_1$ are displayed in Figure 5 and Figure 6 presents the compensation system state ${\lambda }_2$. According to the simulation results, we can ensure that the objective of this paper is achieved.

Figure 1. The system output signal y.

Figure 2. Control input signal u.

Figure 3. The adaptation laws ${\widehat{\theta }}_1$ and ${\widehat{\theta }}_2$.

Figure 4. The adaptation laws ${\varnothing }_1$ and ${\varnothing }_2$.

Figure 5. The adaptation parameters ${\widehat{H}}_1$ and ${\widehat{D}}_1$.

Figure 6. The compensation system state ${\lambda }_2$.

5. Conclusions

This paper addresses the adaptive neural DSC problem for the uncertain stochastic nonlinear systems with state and input time-varying delays. To compensate for the effect of time-varying delays, a compensation system and the Lyapunov–Krasovskii functional scheme are presented. The DSC method with nonlinear filter was developed to avoid the“explosion of complexity”. Compared with the existing works, under controlling of this developed adaptive neural controller, the asymptotic regulation performance of this closed-loop system can be guaranteed based on the use of the DSC scheme with the nonlinear filter and the stochastic Barbalat lemma. Specifically, all signals of this closed-loop system are semi-globally bounded in probability, in particularly, the output signal is asymptotically stable in probability. To certify the availability of the presented control scheme, simulation results were shown.