Adaptive neural control for non-strict feedback stochastic nonlinear systems with input delay

Abstract

In this paper, the problem of adaptive control is investigated for a class of non-strict feedback stochastic nonlinear systems with input delay. First, the effect of the input delay is eliminated by constructing an appropriate auxiliary system with the same order as the considered system. Then, with the help of the backstepping technique and the structural characteristics of the radial basis function (RBF) neural network (NN), an adaptive neural control scheme is extended to non-strict feedback stochastic nonlinear systems with input delay, in which uncertain nonlinear functions are approximated by RBF NN. Furthermore, the proposed adaptive controller ensures that all the closed-loop signals remain bounded in probability. Finally, two examples are provided to confirm the effectiveness of the designed strategy.

Introduction

In practice, such as in chemical systems and power systems, stochastic uncertainties exist widely, which often lead to performance degradation and instability of closed-loop systems. During the past decades, the control of stochastic nonlinear systems has attracted extensive attention. Specially, the control scheme for deterministic nonlinear systems was successfully extended to stochastic cases by the aid of backstepping technique. A controller was proposed based on quadratic Lyapunov function and risk-sensitive criteria to ensure that the closed-loop system maintains global asymptotic stability (Pan and Basar, 1999). Deng and Krstić (1997a, 1997b) and Deng et al. (2001) used the biquadratic Lyapunov function to design a backstepping method for stochastic nonlinear systems, which was further introduced to inverse optimal control and so on (Duan and Xie, 2011; Liu and Zhu, 2021; Su et al., 2021; Wang et al., 2022; Zhang et al., 2022; Zhao et al., 2022). In addition, for the system with unknown nonlinear function, with the help of the radial basis function (RBF) neural network (NN) and fuzzy logic system approximation ability, an adaptive fuzzy/neural controller was presented to secure the stability of the closed-loop system (Fu et al., 2022; Jiang et al., 2022; Li et al., 2017, 2019, 2022; Liu et al., 2021; Ma et al., 2018; Qiu et al., 2022; Tehrani et al., 2019; Wang et al., 2020, 2022; Yang et al., 2020). In the design of these control strategies, NN and fuzzy logic systems were implemented to estimate unknown nonlinear functions, and ideal controllers were constructed by means of the backstepping method and the Lyapunov stability theorem. Although these control approaches have been successfully applied to practical systems, they were only suitable for strict feedback forms.

It is well known that non-strict feedback systems, such as ball-beam system and hyperchaotic LC oscillating circuit system, are more universal in practical engineering applications (Tsafack et al., 2020; Yu, 2009). Moreover, the function of the non-strict feedback system is related to all the system states, which makes it difficult to design the control strategy using the backstepping technique. Therefore, it is very important to study the control problem of non-strict feedback system. Using a monotone increasing function as boundary function, the variable separation method was proposed (Chen et al., 2012), and the adaptive fuzzy control scheme was introduced to the non-strict feedback form. Then, using the same thought, an adaptive control strategy based on NN was presented for stochastic nonlinear systems with non-strict feedback (Wang et al., 2014) and has been widely promoted (Li and Tong, 2017; Wang et al., 2015a, 2015b). In addition, according to the structural characteristics of RBF NN, the backstepping method was extended to stochastic nonlinear systems with non-strict feedback, and the complexity of the controller design was significantly reduced (Sun et al., 2016a, 2016b; Wang et al., 2020; Wu et al., 2022). However, the effect of input delay is not considered in the above literature. As is known to all, input delay is extremely common in engineering applications.

In practical engineering applications, the input delay is inevitable because of the time needed for signal transmission and material transportation, which often results in the instability of the system (Ai et al., 2016). To solve this issue, some meaningful control approaches were put forward for the nonlinear systems with input delay. Nguyen (2023) presented an adaptive delay compensation scheme, and a novel finite-time delay compensation mechanism was proposed to reduce the impact of input delay (Lin et al., 2023). A control method based on the predictor was constructed (González and García, 2021; Lhachemi and Prieur, 2022; Liu and Lin, 2022; Sun et al., 2022; Wu and Lian, 2022), while Yang et al. (2020) and Li et al. (2017, 2019, 2022) presented a coordinate transformation scheme combined with the Pade approximation to compensate for the input delay. An adaptive fuzzy control strategy for nonlinear input delay systems was designed (Yang et al., 2020), in which the problem of input delay was solved by fuzzy Laplace transform. Niu and Li (2018) dealt with the effect of input delay by adding an integral compensation term to the error transform. And Ma et al. (2018), Wang et al. (2020), Liu et al. (2019), and Zhai et al. (2023) proposed an adaptive neural control method by introducing an appropriate auxiliary system with the same order as the considered system to address the influence of input delay. However, to the author’s knowledge, there were few literatures on the use of auxiliary systems to solve the input delay problem. In addition, most of the existing methods were focused on the deterministic strict feedback system, and few involved non-strict feedback cases. This means that the research of non-strict feedback stochastic nonlinear systems with input delay is a valuable topic, which is also the motivation of this paper.

Motivated by the above considerations, in this paper, an NN-based adaptive control approach is proposed for non-strict feedback stochastic nonlinear systems with input delay. The main contributions of this paper are listed as follows:

1. Compared with the existing results, a more general non-strict feedback stochastic nonlinear input delay system is considered. In the design process of controller, RBF NN is used to model unknown nonlinear functions. In addition, with the aid of the structural characteristics of RBF NN, the backstepping method is extended to the non-strict feedback form, which significantly reduces the complexity of the controller design.

2. A suitable auxiliary system with the same order as the considered system is utilized to compensate for the influence of the input delay.

3. With the help of backstepping technique and the Lyapunov stability theorem, an NN-based adaptive controller is introduced to a class of non-strict feedback stochastic nonlinear systems with input delay. The presented controller in this paper guarantees that all signals in the closed-loop system are bounded in probability.

The remainder of this paper is organized as follows: Some preliminaries are given in section “Preliminary.” Section “Main Results” discusses the design and stability analysis of the controller. Two simulation examples are provided in section “Simulation.” And section “Conclusion” draws the conclusion.

Preliminary

In this paper, a class of non-strict feedback stochastic nonlinear systems with input delay is considered as follows

$$\{\begin{array}{l}\mathrm{d}{x}_i=\left(x_{i+1}+f_i\left(x\right)\right)\mathrm{d}t+h_i^{\top }\left(x\right)\mathrm{d}\omega \hfill \\ i=1,\dots ,n-1,\hfill \\ \mathrm{d}{x}_n=\left(u(t-\tau )+f_n\left(x\right)\right)\mathrm{d}t+h_n^{\top }\left(x\right)\mathrm{d}\omega \hfill \\ y=x_1\hfill \end{array}$$

(1)

where $x=[x_1,\dots ,x_n{]}^{\top }\in {\mathbb{R}}^n$ is the state variable, $u(t-\tau )\in \mathbb{R}$ represents the input signal, $\tau >0$ is a known delay, and $y\in \mathbb{R}$ denotes the system output. $\omega$ is a q-dimensional standard Brownian motion that would be specified on the complete probability space $\{\mathrm{\Omega },F,\{F_t{\}}_{t\ge 0},P\}$ , and $f_i(·):{\mathbb{R}}^n\to \mathbb{R}$ and $h_i(·):{\mathbb{R}}^n\to {\mathbb{R}}^q$ are smooth nonlinear functions.

The goal of this paper is to create an adaptive NN control law for system (1) so that all the closed-loop signals remain bounded in probability. To this end, some preliminary information on stochastic systems is presented below. Consider the following nonlinear stochastic system

$$\mathrm{d}\overline{x}=\overline{f}\left(\overline{x}\right)\mathrm{d}t+\overline{h}\left(\overline{x}\right)\mathrm{d}\omega$$

(2)

where $\overline{x}\in {\mathbb{R}}^n$ denotes the state vector, and $\overline{f}(·)$ , $\overline{h}(·)$ that satisfy $\overline{f}(0)=0$ , $\overline{h}(0)=0$ are locally Lipschitz functions in $\overline{x}$ .

• Lemma 1. (Liu and Zhang, 2004; Pan and Basar, 1999) If there exist constants ${\alpha }_0>0$ , ${\beta }_0>0$ and a Lyapunov function $V:{\mathbb{R}}^n\to {\mathbb{R}}_{+}$ , which is radially unbounded, positive definite, twice continuously differentiable, and satisfies the following inequality

$$\mathcal{L}V(x)\le -{\alpha }_{0}V(x)+{\beta }_0$$

(3)

Then stochastic system (2) has a unique solution almost surely and is said to be bounded in probability.

In this paper, the continuous function $f(X):{\mathbb{R}}^n\to \mathbb{R}$ will be modeled by employing the following RBF NN

$$f(X)=W^{\top }S(X)$$

(4)

where $X\in {\mathrm{\Omega }}_X\subset {\mathbb{R}}^q$ stands for input vector, $W=[w_1,\dots ,w_m{]}^{\top }\in {\mathbb{R}}^m$ represents weight vector with $m>1$ , and $S(X)=[s_1(X),\dots ,s_m(X){]}^{\top }$ denotes the RBF vector with $s_i(X)$ being chosen as the Gaussian function

$$s_i(X)=\exp [-\frac{{(X-{\mu }_i)}^{\top }(X-{\mu }_i)}{{\sigma }^2}]$$

(5)

where ${\mu }_i=[{\mu }_{i1},\dots ,{\mu }_{\mathrm{iq}}{]}^{\top }(i=1,\dots ,m)$ stands for the center of the receptive domain and $\sigma$ represents the width of the Gaussian function.

For any given accuracy $\delta >0$ , over a compact set ${\mathrm{\Omega }}_X\subset {\mathbb{R}}^q$ , continuous function $f(X)$ could be approximated by the RBF NN (4) (Sanner and Slotine, 1992) so that

$$f(X)=W^{*\top }S(X)+\mathrm{\Delta }(X),\parallel \mathrm{\Delta }(X)\parallel \le \delta$$

(6)

where $\mathrm{\Delta }(X)$ represents the approximation error, and $W^{*}$ is the ideal weight vector and constructed as

$$W^{*}=\arg \underset{W\in {\mathbb{R}}^m}{min}\underset{X\in {\mathrm{\Omega }}_X}{sup}|f(X)-W^{\top }S(X)|$$

(7)

Lemma 2. (Sun et al., 2016a; Wang et al., 2020) Assuming that $X_l=[x_1,\dots ,x_l{]}^{\top }$ is the input vector and $S(X_l)=[S_1(x_l),\dots ,S_m(x_l){]}^{\top }$ denotes the basis function of the RBF NN. Then, for any given positive integer $l\ge k$ , the inequality holds as follows

$$\parallel S(X_l){\parallel }^2\le \parallel S(X_k){\parallel }^2$$

(8)

Main results

Controller design

First, the following auxiliary system is introduced to address the issues raised by input delay (Ma et al., 2018; Wang et al., 2020)

$$\{\begin{array}{c}{\stackrel{·}{\lambda }}_i={\lambda }_{i+1}-p_i{\lambda }_i,i=1,2,\dots ,n-1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\lambda }}_n=-p_n{\lambda }_n+\mathrm{\Delta }u}\end{array}$$

(9)

where $\mathrm{\Delta }u=u(t-\tau )-u(t)$ , the initial condition of this system is $\lambda (0)=0$ , and $p_1>1/2$ , $p_i>1(i=2,\dots ,n)$ are design parameters.

• Remark 1. The signal ${\lambda }_i$ is a filtered version of the effect of the input delay on the variable being controlled. If stochastic nonlinear system (1) is the delay-free $(\tau =0)$ system, it is easy to prove that the variables ${\lambda }_i$ in equation (9) will remain zero when ${\lambda }_i(0)=0$ .

A controller based on NN is designed in this part, which is realized by the coordination transformation as follows

$$\{\begin{array}{c}{z}_1=x_1-{\lambda }_1\hfill \\ \multicolumn{1}{c}{z_i=x_i-{\alpha }_{i-1}-{\lambda }_i,i=2,\dots ,n}\end{array}$$

(10)

where ${\alpha }_{i-1}$ is the virtual control signal to be designed later.

Remark 2. With the error between the control input with delay and without delay as the input of the constructed auxiliary system, several signals are generated to compensate for the influence of input delay. Taking the error $\mathrm{\Delta }u$ as the input of the constructed system has no effect on $z_i$ . Therefore, it will not affect the design of the controller. Then, using the standard backstepping method to propose a suitable adaptive law can secure the boundedness of all closed-loop signals.

The design procedure is shown in the following steps.

Step 1. From equation (10), one has

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{d}{z}_1=\mathrm{d}{x}_1-\mathrm{d}{\lambda }_1}\\ \multicolumn{1}{c}{=(x_2+f_1-{\lambda }_2+p_1{\lambda }_1)\mathrm{d}t+h_1^{\top }(x)\mathrm{d}\omega }\\ \multicolumn{1}{c}{=(z_2+{\alpha }_1+{\lambda }_2+f_1-{\lambda }_2+p_1{\lambda }_1)\mathrm{d}t+h_1^{\top }(x)\mathrm{d}\omega }\\ \multicolumn{1}{c}{=(z_2+{\alpha }_1+f_1+p_1{\lambda }_1)\mathrm{d}t+h_1^{\top }(x)\mathrm{d}\omega }\end{array}$$

(11)

Consider the following Lyapunov function $V_1$

$$V_1=\frac{1}{4}{z}_1^4+\frac{1}{2r_1}{\tilde{\theta }}_1^2$$

(12)

where $r_1>0$ , ${\tilde{\theta }}_1={\theta }_1-{\hat{\theta }}_1$ denotes the estimation error, and ${\hat{\theta }}_1$ represents an approximation of the uncertain constant ${\theta }_1$ , which would be discussed later. Then using the It $\hat{\mathrm{o}}$ differential formula, one obtains

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_1=z_1^3(z_2+{\alpha }_1+f_1+p_1{\lambda }_1)-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1+\frac{3}{2}{z}_1^2{h}_1^{\top }{h}_1}\\ \multicolumn{1}{c}{=z_1^3({\alpha }_1+f_1+p_1{\lambda }_1)-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1+\frac{3}{2}{z}_1^2{h}_1^{\top }{h}_1+z_1^3{z}_2}\end{array}$$

(13)

According to Young’s inequality, the inequalities can be obtained as follows

$$\frac{3}{2}{z}_1^2{h}_1^{\top }{h}_1\le \frac{3}{4l_1^2}{z}_1^4\parallel h_1{\parallel }^4+\frac{3}{4}{l}_1^2$$

(14)

$$z_1^3{z}_2\le \frac{3}{4}{z}_1^4+\frac{1}{4}{z}_2^4$$

(15)

where $l_1$ is a positive constant. Substituting equations (14) and (15) into equation (13) yields

$$\mathcal{L}{V}_1\le z_1^3({\alpha }_1+p_1{\lambda }_1+{\overline{f}}_1)-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1+\frac{3}{4}{l}_1^2+\frac{1}{4}{z}_2^4$$

(16)

where ${\overline{f}}_1=f_1+(\frac{3}{4l_1^2})z_1\parallel h_1{\parallel }^4+(3/4)z_1$ . Since ${\overline{f}}_1$ is unknown, an NN $W_1^{*\top }{S}_1(X_1)$ is used to approximate it so that for any ${\delta }_1>0$

$${\overline{f}}_1=W_1^{*\top }{S}_1(X_1)+{\mathrm{\Delta }}_1(X_1),|{\mathrm{\Delta }}_1(X_1)|\le {\delta }_1$$

(17)

where ${\mathrm{\Delta }}_1$ is the approximation error and $X_1=[x^{\top },{\lambda }_1{]}^{\top }$ . Then, with the help of Lemma 2 and Young’s inequality, the following inequality holds

$$\begin{array}{c}\multicolumn{1}{c}{z_1^3{\overline{f}}_1=z_1^3(W_1^{*\top }{S}_1(X_1)+{\mathrm{\Delta }}_1(X_1))}\\ \multicolumn{1}{c}{\le |z_1{|}^3(\parallel W_1^{*}\parallel \parallel S_1(X_1)\parallel +{\delta }_1)}\\ \multicolumn{1}{c}{\le |z_1{|}^3(\parallel W_1^{*}\parallel \parallel S_1(Z_1)\parallel +{\delta }_1)}\\ \multicolumn{1}{c}{\le \frac{1}{2{ϵ}_1^2}{z}_1^6{\theta }_1{S}_1^{\top }(Z_1)S_1(Z_1)+\frac{1}{2}{ϵ}_1^2+\frac{3}{4}{z}_1^4+\frac{1}{4}{\delta }_1^4}\end{array}$$

(18)

where ${\theta }_1=\parallel W_1^{*}{\parallel }^2$ , $Z_1=[x_1,{\lambda }_1{]}^{\top }$ , and ${ϵ}_1>0$ is a constant. Then, substituting equation (18) into equation (16) gives

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_1\le z_1^3({\alpha }_1+p_1{\lambda }_1+\frac{1}{2{ϵ}_1^2}{z}_1^3{\theta }_1{S}_1^{\top }(Z_1)S_1(Z_1))}\\ \multicolumn{1}{c}{+\frac{1}{2}{ϵ}_1^2+\frac{3}{4}{z}_1^4+\frac{1}{4}{\delta }_1^4+\frac{3}{4}{l}_1^2+\frac{1}{4}{z}_2^4-\frac{1}{r_1}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1}\end{array}$$

(19)

Next, choose the following virtual control signal ${\alpha }_1$ and adaptive law ${\stackrel{·}{\hat{\theta }}}_1$

$${\alpha }_1=-\left(k_1+\frac{3}{4}\right)z_1-p_1{\lambda }_1-\frac{1}{2{ϵ}_1^2}{z}_1^3{\widehat{\theta }}_1{S}_1^{\top }(Z_1)S_1(Z_1)$$

(20)

$${\stackrel{·}{\hat{\theta }}}_1=\frac{r_1}{2{ϵ}_1^2}{z}_1^6{S}_1^{\top }(Z_1)S_1(Z_1)-{\eta }_1{\hat{\theta }}_1$$

(21)

where $k_1>0$ and ${\eta }_1>0$ . Substituting equations (20) and (21) into equation (19) produces

$$\mathcal{L}{V}_1\le -k_1{z}_1^4+\frac{1}{4}{z}_2^4+\frac{1}{2}{ϵ}_1^2+\frac{1}{4}{\delta }_1^4+\frac{3}{4}{l}_1^2+\frac{{\eta }_1}{r_1}{\tilde{\theta }}_1{\hat{\theta }}_1$$

(22)

• Step i $(2\le i\le n-1)$ . Equation (10) yields

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{d}{z}_i=(x_{i+1}+f_i-{\lambda }_{i+1}+p_i{\lambda }_i-L{\alpha }_{i-1})\mathrm{d}t+{(h_i-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}{h}_j)}^{\top }\mathrm{d}\omega }\\ \multicolumn{1}{c}{=(z_{i+1}+{\alpha }_i+f_i+p_i{\lambda }_i-L{\alpha }_{i-1})\mathrm{d}t+{(h_i-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}{h}_j)}^{\top }\mathrm{d}\omega }\end{array}$$

(23)

where

$$\begin{array}{c}\multicolumn{1}{c}{L{\alpha }_{i-1}=\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}(f_j+x_{j+1})+\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\lambda }_j}{\stackrel{·}{\lambda }}_j}\\ \multicolumn{1}{c}{+\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\hat{\theta }}_j}{\stackrel{·}{\hat{\theta }}}_j+\frac{1}{2}\sum _{p,q=1}^{i-1}\frac{{\partial }^2{\alpha }_{i-1}}{\partial x_p\partial x_q}{h}_p^{\top }{h}_q}\end{array}$$

(24)

Let ${\tilde{h}}_i=h_i-\sum _{j=1}^{i-1}(\partial {\alpha }_{i-1}/\partial x_j)h_j$ . Then, the Lyapunov function $V_i$ can be chosen as follows

$$V_i=V_{i-1}+\frac{1}{4}{z}_i^4+\frac{1}{2r_i}{\tilde{\theta }}_i^2$$

(25)

where $r_i>0$ , ${\tilde{\theta }}_i={\theta }_i-{\hat{\theta }}_i$ is the approximation error, and ${\hat{\theta }}_i$ denotes the estimation of the unknown parameter ${\theta }_i$ , which will be defined later. Based on the It00F4 differential formula, one has

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_i=\mathcal{L}{V}_{i-1}+z_i^3({\alpha }_i+f_i+p_i{\lambda }_i-L{\alpha }_{i-1})-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i}\\ \multicolumn{1}{c}{+\frac{3}{2}{z}_i^2{\tilde{h}}_i^{\top }{\tilde{h}}_i+z_i^3{z}_{i+1}}\end{array}$$

(26)

Applying Young’s inequality to the final two terms in equation (26) demonstrates that

$$\frac{3}{2}{z}_i^2{\tilde{h}}_i^{\top }{\tilde{h}}_i\le \frac{3}{4l_i^2}{z}_i^4\parallel {\tilde{h}}_i{\parallel }^4+\frac{3}{4}{l}_i^2$$

(27)

$$z_i^3{z}_{i+1}\le \frac{3}{4}{z}_i^4+\frac{1}{4}{z}_{i+1}^4$$

(28)

with $l_i$ being a positive constant. Substituting equations (27) and (28) into equation (26) obtains

$$\mathcal{L}{V}_i\le \mathcal{L}{V}_{i-1}+z_i^3({\alpha }_i+p_i{\lambda }_i+{\overline{f}}_i)-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i+\frac{3}{4}{l}_i^2+\frac{1}{4}{z}_{i+1}^4$$

(29)

where ${\overline{f}}_i=f_i-L{\alpha }_{i-1}+(3/4l_i^2)z_i\parallel {\tilde{h}}_i{\parallel }^4+(3/4)z_i$ . Then, an NN $W_i^{*\top }{S}_i(X_i)$ is utilized to model the unknown function ${\overline{f}}_i$ so that for any ${\delta }_i>0$

$${\overline{f}}_i=W_i^{*\top }{S}_i(X_i)+{\mathrm{\Delta }}_i(X_i),|{\mathrm{\Delta }}_i(X_i)|\le {\delta }_i$$

(30)

where $X_i=[x^{\top },{\lambda }_1,\dots ,{\lambda }_i,{\hat{\theta }}_1,\dots ,{\hat{\theta }}_{i-1}{]}^{\top }$ and ${\mathrm{\Delta }}_i$ denotes the approximation error. Similar to equation (18), we have

$$z_i^3{\overline{f}}_i\le \frac{1}{2{ϵ}_i^2}{z}_i^6{\theta }_i{S}_i^{\top }(Z_i)S_i(Z_i)+\frac{1}{2}{ϵ}_i^2+\frac{3}{4}{z}_i^4+\frac{1}{4}{\delta }_i^4$$

(31)

where ${\theta }_i=\parallel W_i^{*}{\parallel }^2$ , $Z_i=[x_1,\dots ,x_i,{\lambda }_1,\dots ,{\lambda }_i,{\hat{\theta }}_1,\dots ,{\hat{\theta }}_{i-1}{]}^{\top }$ and ${ϵ}_i>0$ is a design parameter. Then, substituting equation (31) into equation (29) produces

$$\begin{array}{l}ℒV_i\le ℒV_{i-1}+z_i^3({\alpha }_i+p_i{\lambda }_i+\frac{1}{2{ϵ}_i^2}{z}_i^3{\theta }_i{S}_i^{\top }(Z_i)S_i(Z_i))\\ +\frac{1}{2}{ϵ}_i^2+\frac{3}{4}{z}_i^4+\frac{1}{4}{\delta }_i^4+\frac{3}{4}{l}_i^2+\frac{1}{4}{z}_{i+1}^4-\frac{1}{r_i}{\tilde{\theta }}_i{\stackrel{·}{\widehat{\theta }}}_i\end{array}$$

(32)

Next, the virtual control signal ${\alpha }_i$ and adaptive law ${\stackrel{·}{\hat{\theta }}}_i$ are chosen as follows

$${\alpha }_i=-(k_i+1)z_i-p_i{\lambda }_i-\frac{1}{2{ϵ}_i^2}{z}_i^3{\hat{\theta }}_i{S}_i^{\top }(Z_i)S_i(Z_i)$$

(33)

$${\stackrel{·}{\hat{\theta }}}_i=\frac{r_i}{2{ϵ}_i^2}{z}_i^6{S}_i^{\top }(Z_i)S_i(Z_i)-{\eta }_i{\hat{\theta }}_i$$

(34)

where $k_i>0$ and ${\eta }_i>0$ . Substituting equations (33) and (34) into equation (32) gives

$$\begin{array}{l}ℒV_i\le ℒV_{i-1}-k_i{z}_i^4+\frac{1}{4}{z}_{i+1}^4+\frac{1}{2}{ϵ}_i^2+\frac{1}{4}{\delta }_i^4+\frac{3}{4}{l}_i^2+\frac{{\eta }_i}{r_i}{\tilde{\theta }}_i{\widehat{\theta }}_i\\ \le -\sum _{j=1}^i{k}_j{z}_j^4+\frac{1}{2}\sum _{j=1}^i{ϵ}_j^2+\frac{1}{4}\sum _{j=1}^i{\delta }_j^4+\frac{3}{4}\sum _{j=1}^i{l}_j^2\\ +\sum _{j=1}^i\frac{{\eta }_j}{r_j}{\tilde{\theta }}_j{\widehat{\theta }}_j+\frac{1}{4}{z}_{i+1}^4\end{array}$$

(35)

Step n. Based on equation (10), we have

$$\mathrm{d}{z}_n=(f_n+p_n{\lambda }_n+u(t)-L{\alpha }_{n-1})\mathrm{d}t+{(h_n-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}{h}_j)}^{\top }\mathrm{d}\omega$$

(36)

where

$$\begin{array}{c}\multicolumn{1}{c}{L{\alpha }_{n-1}=\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}(f_j+x_{j+1})+\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\lambda }_j}{\stackrel{·}{\lambda }}_j}\\ \multicolumn{1}{c}{+\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\hat{\theta }}_j}{\stackrel{·}{\hat{\theta }}}_j+\frac{1}{2}\sum _{p,q=1}^{n-1}\frac{{\partial }^2{\alpha }_{n-1}}{\partial x_p\partial x_q}{h}_p^{\top }{h}_q}\end{array}$$

(37)

Let ${\tilde{h}}_n=h_n-\sum _{j=1}^{n-1}(\partial {\alpha }_{n-1}/\partial x_j)h_j$ . Then, consider a Lyapunov function as

$$V_n=V_{n-1}+\frac{1}{4}{z}_n^4+\frac{1}{2r_n}{\tilde{\theta }}_n^2$$

(38)

where $r_n>0$ , ${\tilde{\theta }}_n={\theta }_n-{\hat{\theta }}_n$ represents the estimation error, and ${\hat{\theta }}_n$ denotes the approximation of the unknown constant ${\theta }_n$ which will be constructed later. According to the Itô differential formula, the result is obtained as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_n=\mathcal{L}{V}_{n-1}+z_n^3(f_n+p_n{\lambda }_n-L{\alpha }_{n-1}+u(t))-\frac{1}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n}\\ \multicolumn{1}{c}{+\frac{3}{2}{z}_n^2{\tilde{h}}_n^{\top }{\tilde{h}}_n}\end{array}$$

(39)

With the help of Young’s inequality, we have

$$\frac{3}{2}{z}_n^2{\tilde{h}}_n^{\top }{\tilde{h}}_n\le \frac{3}{4l_n^2}{z}_n^4\parallel {\tilde{h}}_n{\parallel }^4+\frac{3}{4}{l}_n^2$$

(40)

where $l_n>0$ is a constant, and substituting equation (40) into equation (39) yields

$$\mathcal{L}{V}_n\le \mathcal{L}{V}_{n-1}+z_n^3(p_n{\lambda }_n+u(t)+{\overline{f}}_n)-\frac{1}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+\frac{3}{4}{l}_n^2$$

(41)

where ${\overline{f}}_n=f_n-L{\alpha }_{n-1}+(3/4l_n^2)z_n\parallel {\tilde{h}}_n{\parallel }^4$ . Again, an NN is employed to model the unknown function ${\overline{f}}_n$ ; similar to equation (30), for given ${\delta }_n>0$ , one has

$${\overline{f}}_n=W_n^{*\top }{S}_n(X_n)+{\mathrm{\Delta }}_n(X_n),|{\mathrm{\Delta }}_n(X_n)|\le {\delta }_n$$

(42)

with $X_n=[x^{\top },{\lambda }_1,\dots ,{\lambda }_n,{\hat{\theta }}_1,\dots ,{\hat{\theta }}_{n-1}{]}^{\top }$ and ${\mathrm{\Delta }}_n$ being the approximation error. Using the same method as equation (31), the following expression holds

$$z_n^3{\overline{f}}_n\le \frac{1}{2{ϵ}_n^2}{z}_n^6{\theta }_n{S}_n^{\top }(Z_n)S_n(Z_n)+\frac{1}{2}{ϵ}_n^2+\frac{3}{4}{z}_n^4+\frac{1}{4}{\delta }_n^4$$

(43)

where ${\theta }_n=\parallel W_n^{*}{\parallel }^2$ , $Z_n=[x^{\top },{\lambda }_1,\dots ,{\lambda }_n,{\hat{\theta }}_1,\dots ,{\hat{\theta }}_{n-1}{]}^{\top }$ , and ${ϵ}_n>0$ is a design parameter. Then, substituting equation (43) into equation (41) gives

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_n\le \mathcal{L}{V}_{n-1}+z_n^3({\alpha }_n+p_n{\lambda }_n+\frac{1}{2{ϵ}_n^2}{z}_n^3{\theta }_n{S}_n^{\top }(Z_n)S_n(Z_n))}\\ \multicolumn{1}{c}{+\frac{1}{2}{ϵ}_n^2+\frac{3}{4}{z}_n^4+\frac{1}{4}{\delta }_n^4+\frac{3}{4}{l}_n^2-\frac{1}{r_n}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n}\end{array}$$

(44)

Next, select the control signal $u(t)$ and adaptive law ${\stackrel{·}{\hat{\theta }}}_n$ as follows

$$u(t)=-(k_n+1)z_n-p_n{\lambda }_n-\frac{1}{2{ϵ}_n^2}{z}_n^3{\hat{\theta }}_n{S}_n^{\top }(Z_n)S_n(Z_n)$$

(45)

$${\stackrel{·}{\hat{\theta }}}_n=\frac{r_n}{2{ϵ}_n^2}{z}_n^6{S}_n^{\top }(Z_n)S_n(Z_n)-{\eta }_n{\hat{\theta }}_n$$

(46)

with $k_n>0$ and ${\eta }_n>0$ . Substituting equations (45) and (46) into equation (44) produces

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}{V}_n\le \mathcal{L}{V}_{n-1}-k_n{z}_n^4+\frac{1}{2}{ϵ}_n^2+\frac{1}{4}{\delta }_n^4+\frac{3}{4}{l}_n^2+\frac{{\eta }_n}{r_n}{\tilde{\theta }}_n{\hat{\theta }}_n}\\ \multicolumn{1}{c}{\le -\sum _{j=1}^n{k}_j{z}_j^4+\frac{1}{2}\sum _{j=1}^n{ϵ}_j^2+\frac{1}{4}\sum _{j=1}^n{\delta }_j^4+\frac{3}{4}\sum _{j=1}^n{l}_j^2+\sum _{j=1}^n\frac{{\eta }_j}{r_j}{\tilde{\theta }}_j{\hat{\theta }}_j}\end{array}$$

(47)

• Remark 3. Different from the standard backstepping approach, the signal ${\lambda }_i$ is introduced in each step of the above backstepping design, which is used to compensate for the effect of input delay. Thus, the input delay will not affect the design of controller in the last step.

Stability analysis

According to the above discussion, the main results are expressed as follows:

• Theorem 1. Consider stochastic nonlinear systems with input-delay (1), if the system controller is designed as equation (45) and virtual controllers are described as equations (20) and (33) and adaptive laws are constructed as equations (21), (34), and (46), then all the closed-loop signals remain bounded in probability.

Proof. Define the Lyapunov function V as follows

$$V=\frac{1}{4}\sum _{i=1}^n{z}_i^4+\sum _{i=1}^n\frac{1}{2r_i}{\tilde{\theta }}_i^2$$

(48)

Based on equation (47), it follows that

$$\mathcal{L}V\le -\sum _{i=1}^n{k}_i{z}_i^4+\frac{1}{2}\sum _{i=1}^n{ϵ}_i^2+\frac{1}{4}\sum _{i=1}^n{\delta }_i^4+\frac{3}{4}\sum _{i=1}^n{l}_i^2+\sum _{i=1}^n\frac{{\eta }_i}{r_i}{\tilde{\theta }}_i{\hat{\theta }}_i$$

(49)

Since

$${\tilde{\theta }}_i{\hat{\theta }}_i={\tilde{\theta }}_i({\theta }_i-{\tilde{\theta }}_i)\le -\frac{1}{2}{\tilde{\theta }}_i^2+\frac{1}{2}\parallel {\theta }_i{\parallel }^2$$

(50)

The inequality equation (49) can be further expressed as

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{L}V\le -\sum _{i=1}^n{k}_i{z}_i^4+\sum _{i=1}^n\frac{{\eta }_i}{r_i}(-\frac{1}{2}{\tilde{\theta }}_i^2+\frac{1}{2}\parallel {\theta }_i{\parallel }^2)}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i=1}^n{ϵ}_i^2+\frac{1}{4}\sum _{i=1}^n{\delta }_i^4+\frac{3}{4}\sum _{i=1}^n{l}_i^2}\\ \multicolumn{1}{c}{\le -(\sum _{i=1}^{n}4k_i\frac{1}{4}{z}_i^4+\sum _{i=1}^n{\eta }_i\frac{1}{2r_i}{\tilde{\theta }}_i^2)+\sum _{i=1}^n\frac{{\eta }_i}{2r_i}\parallel {\theta }_i{\parallel }^2}\\ \multicolumn{1}{c}{+\frac{1}{2}\sum _{i=1}^n{ϵ}_i^2+\frac{1}{4}\sum _{i=1}^n{\delta }_i^4+\frac{3}{4}\sum _{i=1}^n{l}_i^2}\end{array}$$

(51)

Let $a_0=min\{4k_i,{\eta }_i|1\le i\le n\}$ and $b_0=\sum _{i=1}^n(({\eta }_i/2r_i)\parallel {\theta }_i{\parallel }^2+(1/2){ϵ}_i^2+(1/4){\delta }_i^4+(3/4)l_i^2)$ . According to equation (51), one has

$$\mathcal{L}V\le -a_{0}V+b_0$$

(52)

Based on Lemma 1, it can be concluded that the closed-loop signals $z_i=x_i-{\alpha }_{i-1}-{\lambda }_i$ and ${\tilde{\theta }}_i$ are bounded. Then

$$\frac{\mathrm{d}E(V(t))}{\mathrm{d}t}\le -a_{0}E(V(t))+b_0$$

(53)

By integrating equation (53) over $[0,t]$ , the inequality can be obtained as follows:

$$E(V(t))\le V(0)e^{-a_{0}t}+\frac{b_0}{a_0}$$

(54)

And according to equation (48), we have

$$E(|z_1{|}^4)\le 4E(V(t))\le 4V(0)e^{-a_{0}t}+4\frac{b_0}{a_0}$$

Using the inequality $|x_1{|}^4\le 8|z_1{|}^4+8|{\lambda }_1{|}^4$ , we obtain

$$E(|x_1{|}^4)\le 32V(0)e^{-a_{0}t}+32\frac{b_0}{a_0}+8E(|{\lambda }_1{|}^4)$$

(55)

Then, this paper discusses the boundedness of ${\lambda }_i$ in auxiliary system (9) to secure the boundedness of the system variables $x_i$ . First, the Lyapunov functional, which is motivated by Wang et al. (2020), is defined as follows

$$V_{\lambda }=\frac{1}{2}\sum _{i=1}^n{\lambda }_i^2+\frac{1}{\kappa }{\int }_{t-\tau }^t{\int }_{\omega }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s\mathrm{d}\omega$$

(56)

where $\kappa >0$ is a constant. The following inequality is obtained by solving the derivative of $V_{\lambda }$

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\lambda }\le \sum _{i=1}^{n-1}{\lambda }_i({\lambda }_{i+1}-p_i{\lambda }_i)+{\lambda }_n(-p_n{\lambda }_n+u(t-\tau )-u(t))}\\ \multicolumn{1}{c}{+\frac{\tau }{\kappa }\parallel \stackrel{·}{u}(t){\parallel }^2-\frac{1}{\kappa }{\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s}\\ \multicolumn{1}{c}{\le -\sum _{i=1}^n{\overline{p}}_i{\lambda }_i^2-\frac{1}{\kappa }{\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s+\frac{\tau }{\kappa }\parallel \stackrel{·}{u}(t){\parallel }^2}\\ \multicolumn{1}{c}{+\frac{1}{2\xi }\parallel u(t-\tau )-u(t){\parallel }^2}\end{array}$$

(57)

where ${\overline{p}}_1=p_1-(1/2)$ , ${\overline{p}}_i=p_i-1,(i=2,\dots ,n-1)$ , ${\overline{p}}_n=p_n-(1/2)-(\xi /2)$ , and $\xi$ denotes a positive constant. The following result may be reached employing the Jensen’s inequality (Fridman, 2014)

$$\begin{array}{l}\frac{\tau }{2\xi }{\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s\ge \frac{1}{2\xi }{\int }_{t-\tau }^t{\stackrel{·}{u}}^{\top }(s)\mathrm{d}s{\int }_{t-\tau }^t\stackrel{·}{u}(s)\mathrm{d}s\\ \ge \frac{1}{2\xi }\parallel u(t-\tau )-u(t){\parallel }^2\end{array}$$

(58)

Then, substituting equation (58) into equation (57) yields

$${\stackrel{·}{V}}_{\lambda }\le -\sum _{i=1}^n{\overline{p}}_i{\lambda }_i^2-(\frac{1}{\kappa }-\frac{\tau }{2\xi }){\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s+\frac{\tau }{\kappa }\parallel \stackrel{·}{u}(t){\parallel }^2$$

(59)

Then the term $(\tau /\kappa )\parallel \stackrel{·}{u}(t){\parallel }^2$ will be dealt with. From equations (10), (20), (21), (33), and (45), it is easy to prove that $u(t)$ and $\stackrel{·}{u}(t)$ may be expressed as

$$u(t)=g_1(z,\hat{\theta })+\sum _{i=1}^n{g}_{4i}(z,\hat{\theta }){\lambda }_i$$

(60)

$$\stackrel{·}{u}(t)=g_2(z,\hat{\theta })+\sum _{i=1}^n{g}_{5i}(z,\hat{\theta }){\lambda }_i+g_3(z,\hat{\theta })u(t-\tau )$$

(61)

where $g_1$ , $g_2$ , $g_3$ , $g_{4i}$ , and $g_{5i}(i=1,\dots ,n)$ are $C^1$ functions, $z=[z_1,\dots ,z_n{]}^{\top }$ , and $\hat{\theta }=[{\hat{\theta }}_1,\dots ,{\hat{\theta }}_n{]}^{\top }$ . Since $z_i(i=1,\dots ,n)$ and ${\hat{\theta }}_i(i=1,\dots ,n)$ are bounded, the following results can be easily obtained

$$\parallel g_i\parallel \le {\beta }_i,i=1,2,3$$

(62)

$$\parallel g_{\mathrm{ij}}\parallel \le {\beta }_{\mathrm{ij}},i=4,5;j=1,\dots ,n$$

(63)

where ${\beta }_i$ and ${\beta }_{\mathrm{ij}}$ are positive constants. As a result, it is simple to demonstrate that

$$\begin{array}{c}\parallel u(t){\parallel }^2=\parallel g_1+\sum _{i=1}^n{g}_{4i}{\lambda }_i{\parallel }^2\le 2\parallel g_1{\parallel }^2+2n\sum _{i=1}^n\parallel g_{4i}{\parallel }^2{\lambda }_i^2\hfill \\ \multicolumn{1}{c}{\le 2{\beta }_1^2+2n\sum _{i=1}^n{\beta }_{4i}^2{\lambda }_i^2\le {\beta }_1^{\prime }+{\beta }_4^{\prime }\sum _{i=1}^n{\lambda }_i^2}\end{array}$$

(64)

where ${\beta }_1^{\prime }=2{\beta }_1^2$ and ${\beta }_4^{\prime }=max\{2n{\beta }_{4i}^2|i=1,\dots ,n\}$ . Then, one has

$$\parallel u(t-\tau ){\parallel }^2\le {\beta }_1^{\prime }+{\beta }_4^{\prime }\sum _{i=1}^n{\lambda }_i^2(t-\tau ).$$

Next

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\tau }{\kappa }\parallel \stackrel{·}{u}(t){\parallel }^2=\frac{\tau }{\kappa }\parallel g_2+\sum _{i=1}^n{g}_{5i}{\lambda }_i+g_{3}u(t-\tau ){\parallel }^2}\\ \multicolumn{1}{c}{\le 3\frac{\tau }{\kappa }(\parallel g_2{\parallel }^2+n\sum _{i=1}^n\parallel g_{5i}{\parallel }^2{\lambda }_i^2+\parallel g_3{\parallel }^2\parallel u(t-\tau ){\parallel }^2)}\\ \multicolumn{1}{c}{\le 3\frac{\tau }{\kappa }({\beta }_2^2+n\sum _{i=1}^n{\beta }_{5i}^2{\lambda }_i^2+{\beta }_3^2{\beta }_1^{\prime }+{\beta }_3^2{\beta }_4^{\prime }\sum _{i=1}^n{\lambda }_i^2(t-\tau ))}\\ \multicolumn{1}{c}{\le \frac{\tau }{\kappa }({\beta }_2^{\prime }+{\beta }_5^{\prime }\sum _{i=1}^n{\lambda }_i^2+{\beta }_3^{\prime }\sum _{i=1}^n{\lambda }_i^2(t-\tau ))}\end{array}$$

(65)

where ${\beta }_2^{\prime }=3{\beta }_2^2+3{\beta }_3^2{\beta }_1^{\prime }$ , ${\beta }_5^{\prime }=max\{3n{\beta }_{5i}^2|i=1,\dots ,n\}$ , and ${\beta }_3^{\prime }=3{\beta }_3^2{\beta }_4^{\prime }$ . Based on equation (65), inequality (59) can be further expressed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\lambda }\le -\sum _{i=1}^n({\overline{p}}_i-\frac{\tau }{\kappa }{\beta }_5^{\prime }){\lambda }_i^2-(\frac{1}{\kappa }-\frac{\tau }{2\xi }){\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s}\\ \multicolumn{1}{c}{+\frac{\tau }{\kappa }{\beta }_2^{\prime }+\frac{\tau {\beta }_3^{\prime }}{\kappa }\sum _{i=1}^n{\lambda }_i^2(t-\tau )}\end{array}$$

(66)

Subsequently, for auxiliary system (9), the Lyapunov functional is chosen as follows

$$V_{\lambda }^{\prime }=V_{\lambda }+\frac{\tau {\beta }_3^{\prime }}{\kappa }{\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s+\frac{1}{\rho }{\int }_{t-\tau }^t{\int }_{\omega }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s\mathrm{d}\omega$$

(67)

where $\rho$ is a positive constant. Then, the time-derivative of $V_{\lambda }^{\prime }$ is derived as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\lambda }^{\prime }\le -\sum _{i=1}^n{\tilde{p}}_i{\lambda }_i^2-(\frac{1}{\kappa }-\frac{\tau }{2\xi }){\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s}\\ \multicolumn{1}{c}{-\frac{1}{\rho }{\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s+\frac{\tau }{\kappa }{\beta }_2^{\prime }}\end{array}$$

(68)

where ${\tilde{p}}_i={\overline{p}}_i-(\tau /\kappa ){\beta }_5^{\prime }-(\tau /\rho )-(\tau {\beta }_3^{\prime }/\kappa )(i=1,\dots ,n)$ . By choosing appropriate parameters $\kappa$ and $\xi$ , the inequalities ${\tilde{p}}_i>0$ and $(1/\kappa )-(\tau /2\xi )>0$ can be secured. Furthermore, it is easy to demonstrate that

$$\begin{array}{c}\multicolumn{1}{c}{{\int }_{t-\tau }^t{\int }_{\omega }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s\mathrm{d}\omega \le \tau \underset{\omega \in [t-\tau ,t]}{sup}{\int }_{\omega }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s}\\ \multicolumn{1}{c}{=\tau {\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s}\\ \multicolumn{1}{c}{{\int }_{t-\tau }^t{\int }_{\omega }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s\mathrm{d}\omega \le \tau \underset{\omega \in [t-\tau ,t]}{sup}{\int }_{\omega }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s}\\ \multicolumn{1}{c}{=\tau {\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s}\end{array}$$

(69)

Substituting equation (69) into equation (68) gives

$$\begin{array}{l}{\stackrel{·}{V}}_{\lambda }^{\prime }\le -\sum _{i=1}^n{\tilde{p}}_i{\lambda }_i^2-\left(\frac{1}{\kappa }-\frac{\tau }{2\xi }\right){\int }_{t-\tau }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s+\frac{\tau }{\kappa }{\beta }_2^{\prime }\hfill \\ -\left(\frac{1}{\rho }-\frac{\tau {\beta }_3^{\prime }}{\kappa }\right){\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s-\frac{\tau {\beta }_3^{\prime }}{\kappa }{\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s\hfill \\ \le -\sum _{i=1}^n{\tilde{p}}_i{\lambda }_i^2-\left(\frac{1}{\tau }-\frac{\kappa }{2\xi }\right)\frac{1}{\kappa }{\int }_{t-\tau }^t{\int }_{\omega }^t\parallel \stackrel{·}{u}(s){\parallel }^2\mathrm{d}s\mathrm{d}\omega \hfill \\ -\left(\frac{1}{\tau }-\frac{\rho {\beta }_3^{\prime }}{\kappa }\right)\frac{1}{\rho }{\int }_{t-\tau }^t{\int }_{\omega }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s\mathrm{d}\omega -\frac{\tau {\beta }_3^{\prime }}{\kappa }{\int }_{t-\tau }^t\sum _{i=1}^n{\lambda }_i^2(s)\mathrm{d}s+\frac{\tau }{\kappa }{\beta }_2^{\prime }\hfill \\ \le -c_0{V}_{\lambda }^{\prime }+d_0\hfill \end{array}$$

(70)

where $c_0=\min \left\{2{\tilde{p}}_i,(1/\tau )-(\kappa /2\xi ),(1/\tau )-(\rho {\beta }_3^{\prime }/\kappa ),1,i=1,\dots ,n\right\}$ and $d_0=(\tau /\kappa ){\beta }_2^{\prime }$ . Similarly, it follows from Lemma 1 that the signal ${\lambda }_i$ is also bounded in probability. Since $z_1=x_1-{\lambda }_1$ and $z_i=x_i-{\alpha }_{i-1}-{\lambda }_i$ , it is possible to demonstrate that $x_i$ remains bounded. Therefore, it is concluded that all the closed-loop signals are bounded in probability.

Integrating the above differential inequality from $0$ to t, the following result can be obtained

$$V_{\lambda }^{\prime }\le V_{\lambda }^{\prime }(0)e^{-c_{0}t}+\frac{d_0}{c_0}$$

(71)

Similarly, based on equation (67), we obtain

$$E(|{\lambda }_1{|}^4)\le 8(V_{\lambda }^{\prime }(0){)}^2{e}^{-2c_{0}t}+8{\left(\frac{d_0}{c_0}\right)}^2$$

(72)

Therefore, it is obtained from equation (55) and equation (72) that

$$E(|x_1{|}^4)\le 32V(0)e^{-a_{0}t}+32\frac{b_0}{a_0}+64(V_{\lambda }^{\prime }(0){)}^2{e}^{-2c_{0}t}+64{\left(\frac{d_0}{c_0}\right)}^2$$

Then

$$\underset{t\to \infty }{lim}E(|x_1{|}^4)\le 32\frac{b_0}{a_0}+64{\left(\frac{d_0}{c_0}\right)}^2$$

This means that the system state $x_1$ eventually converges to a small region around the origin.

From the above discussion, according to Lemma 1, equation (52), and equation (70), it is easy to prove that the signals ${\tilde{\theta }}_i$ , $z_i$ and ${\lambda }_i$ are bounded in probability. Furthermore, from equations (20), (33), and (45), one can get that the signals ${\alpha }_{i-1}$ and u remain bounded in probability. Based on the boundedness of $z_i$ , ${\lambda }_i$ , ${\alpha }_{i-1}$ , and u, one can get that $x_i$ is also bounded in probability. Specially, the state $x_1$ can be bounded by a constant $32{a_0}^{-1}{b}_0+64({c_0}^{-1}{d}_0{)}^2$ , and we can properly adjust the design parameters $a_0$ , $b_0$ , $c_0$ , and $d_0$ to obtain better control results. Therefore, the proposed scheme ensures that all the closed-loop signals remain bounded in probability. The proof is finished.

The design scheme of the aforementioned adaptive controller could be visualized in Figure 1.

Figure 1.

The block diagram of controlled system.

Remark 4. It should be pointed out that the design process of the proposed control scheme is independent of input delay, and the stability of the closed-loop system can be guaranteed by setting appropriate design parameters and selecting suitable Lyapunov function. Therefore, a suitable auxiliary system can be constructed to deal with the case of known time-varying input delay.

Remark 5. Compared with the existing result (Wang et al., 2020), which only works on the non-strict feedback nonlinear system with input delay, system (1) is more general. The main idea of this research is to design an effective adaptive neural controller, which can secure that all the closed-loop signals remain bounded in probability.

Simulation

In order to demonstrate the effectiveness of the proposed control scheme, two examples are provided as follows:

Example 1. Consider the following non-strict feedback stochastic nonlinear systems with input delay used in Wang et al. (2020)

$$\begin{array}{c}\mathrm{d}{x}_1=(x_2+0.5x_2^2{x}_3(1+x_1^2))\mathrm{d}t+x_1{x}_2\mathrm{d}\omega \hfill \\ \multicolumn{1}{c}{\mathrm{d}{x}_2=(x_3+0.2x_1{x}_2\sin (x_3^2))\mathrm{d}t+x_2{x}_3\sin x_1\mathrm{d}\omega }\\ \multicolumn{1}{c}{\mathrm{d}{x}_3=(u(t-\tau )+x_2{x}_3^2\sin (x_1^2))\mathrm{d}t+x_2\sin x_3\mathrm{d}\omega }\\ \multicolumn{1}{c}{y=x_1}\end{array}$$

(73)

where $x_1$ , $x_2$ , and $x_3$ denote the system states, y represents the system output, $u(t-\tau )$ stands for the input, and $\tau =0.2$ is the time delay. The control goal is to design an adaptive NN control scheme so that all the closed-loop signals remain bounded in probability. And the following auxiliary system is designed

$$\begin{array}{c}{\stackrel{·}{\lambda }}_i={\lambda }_{i+1}-p_i{\lambda }_i,i=1,2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\lambda }}_3=-p_3{\lambda }_3+u(t-\tau )-u(t)}\end{array}$$

(74)

Based on Theorem 1, the virtual control ${\alpha }_i$ and the actual control u are constructed as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_1=-(k_1+\frac{3}{4})z_1-p_1{\lambda }_1-\frac{1}{2{ϵ}_1^2}{z}_1^3{\hat{\theta }}_1{S}_1^{\top }(Z_1)S_1(Z_1)}\\ \multicolumn{1}{c}{{\alpha }_2=-(k_2+1)z_2-p_2{\lambda }_2-\frac{1}{2{ϵ}_2^2}{z}_2^3{\hat{\theta }}_2{S}_2^{\top }(Z_2)S_2(Z_2)}\\ \multicolumn{1}{c}{u=-(k_3+1)z_3-p_3{\lambda }_3-\frac{1}{2{ϵ}_3^2}{z}_3^3{\hat{\theta }}_3{S}_3^{\top }(Z_3)S_3(Z_3)}\end{array}$$

(75)

where $z_1=x_1-{\lambda }_1$ , $z_2=x_2-{\lambda }_2-{\alpha }_1$ , $z_3=x_3-{\lambda }_3-{\alpha }_2$ , and the adaptive laws are selected as below

$${\stackrel{·}{\hat{\theta }}}_i=\frac{r_i}{2{ϵ}_i^2}{z}_i^6{S}_i^{\top }(Z_i)S_i(Z_i)-{\eta }_i{\hat{\theta }}_i,i=1,2,3$$

(76)

where, $Z_1=[x_1,{\lambda }_1{]}^{\top }$ , $Z_2=[x_1,x_2,{\lambda }_1,{\lambda }_2,{\hat{\theta }}_1{]}^{\top }$ , and $Z_3=[x_1,x_2,x_3,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\hat{\theta }}_1,{\hat{\theta }}_2{]}^{\top }$ . The design parameters in the simulation are chosen as follows: $k_1=0.01$ , $k_2=0.5$ , $k_3=7$ , $p_1=p_2=1.5$ , $p_3=2$ , $r_1=1$ , $r_2=12$ , $r_3=8$ , ${\eta }_1=0.04$ , ${\eta }_2=0.2$ , ${\eta }_3=0.001$ , and ${ϵ}_i=1$ , $(i=1,2,3)$ . The initial conditions are $[x_1(0),x_2(0),x_3(0){]}^{\top }=[0.5,0.25,0.25{]}^{\top }$ , $[{\lambda }_1(0),{\lambda }_2(0),{\lambda }_3(0){]}^{\top }=[0,0,0{]}^{\top }$ , and $[{\hat{\theta }}_1(0),{\hat{\theta }}_2(0),{\hat{\theta }}_3(0){]}^{\top }=[0,0,0{]}^{\top }$ .

The simulation results are illustrated in Figures 2–4. The state trajectories $x_1$ , $x_2$ , and $x_3$ are depicted in Figure 2. Figure 3(a) displays the control signal $u(t-\tau )$ and Figure 3(b) shows the control signal $u(t)$ . The responses of auxiliary system states are shown in Figure 4. From the simulation results, we can find that all the closed-loop signals remain bounded in probability, which confirms the validity of our presented control scheme.

Figure 2.

The trajectories of system states $x_1$ , $x_2$ and $x_3$ (Example 1).

Figure 3.

(a) The control signal $u(t-\tau )$ and (b) the control signal $u(t)$ (Example 1).

Figure 4.

The states ${\lambda }_1$ , ${\lambda }_2$ and ${\lambda }_3$ of the auxiliary system (Example 1).

Example 2. Consider the following practical one-link manipulator with input delay (Yu et al., 2018)

$$\begin{array}{c}D\stackrel{··}{q}+J\stackrel{·}{q}+N\sin (q)={\eta }^{\prime }+{\eta }_d\hfill \\ \multicolumn{1}{c}{M{\stackrel{·}{\eta }}^{\prime }+I{\eta }^{\prime }+K_m\stackrel{·}{q}=u+v_d}\end{array}$$

(77)

where $D=1\mathrm{kg}·{\mathrm{m}}^2$ represents the mechanical inertia, and $\stackrel{··}{q}$ , $\stackrel{·}{q}$ , and q are the link acceleration, velocity, and position, respectively. $J=1\mathrm{Nm}·\mathrm{s}/\mathrm{rad}$ denotes the viscous friction coefficient, $N=0.5$ is the gravity coefficient, and ${\eta }^{\prime }$ stands for the torque of electrical subsystem. $M=1H$ represents the armature inductance, $I=0.1\mathrm{\Omega }$ is the resistance, and $K_m=1\mathrm{Nm}/\mathrm{A}$ denotes the back electromotive force coefficient. ${\eta }_d=\stackrel{·}{q}\sin ({\eta }^{\prime })+q\sin (q)\stackrel{·}{\omega }$ stands for the model errors uncertainties, $v_d={\eta }^{\prime }\sin (q\stackrel{·}{q})+2q^2\stackrel{·}{\omega }$ represents the stochastic disturbance, and the electromechanical torque u is the control input with the time delay $\tau =0.4$ .

Let $x_1=q$ , $x_2=\stackrel{·}{q}$ , and $x_3={\eta }^{\prime }$ . Then the real system model can be rearranged as the following non-strict feedback forms

$$\begin{array}{c}\mathrm{d}{x}_1=x_2\mathrm{d}t\hfill \\ \multicolumn{1}{c}{\mathrm{d}{x}_2=(x_3+x_2\sin (x_3)-x_2-0.5\sin (x_1))\mathrm{d}t+x_1\sin x_1\mathrm{d}\omega }\\ \multicolumn{1}{c}{\mathrm{d}{x}_3=(u(t-\tau )+x_3\sin (x_1{x}_2)-0.1x_3-2x_2)\mathrm{d}t+2x_1^2\mathrm{d}\omega }\\ \multicolumn{1}{c}{y=x_1}\end{array}$$

(78)

For the above system, an appropriate controller is designed to ensure that all the closed-loop signals are bounded in probability. According to the above introduction, the corresponding auxiliary system, virtual control ${\alpha }_i$ , actual control u, and adaptive laws are adopted in the form of equations (74)–(76). And the design parameters in the simulation are chosen as follows: $k_1=0.05$ , $k_2=0.5$ , $k_3=9$ , $r_1=1$ , $r_2=r_3=2$ , and ${\eta }_i=0.5$ , ${ϵ}_i=1$ , $p_i=1.5$ , $(i=1,2,3)$ . The initial conditions are $[x_1(0),x_2(0),x_3(0){]}^{\top }=[0.5,0.5,0.5{]}^{\top }$ , ${[{\lambda }_1(0),{\lambda }_2(0),{\lambda }_3(0)]}^{\top }={[0,0,0]}^{\top }$ , and ${[{\widehat{\theta }}_1(0),{\widehat{\theta }}_2(0),{\widehat{\theta }}_3(0)]}^{\top }={[0,0,0]}^{\top }$ .

The simulation results are depicted in Figures 5–7. Similar to Example 1, the simulation results show that all the closed-loop signals are bounded in probability, which further proves the effectiveness of the proposed control strategy.

Figure 5.

The trajectories of system states $x_1$ , $x_2$ , and $x_3$ (Example 2).

Figure 6.

(a) The control signal $u(t-\tau )$ and (b) the control signal $u(t)$ (Example 2).

Figure 7.

The states ${\lambda }_1$ , ${\lambda }_2$ , and ${\lambda }_3$ of the auxiliary system (Example 2).

Conclusion

In this paper, an adaptive neural control strategy was designed for a class of non-strict feedback stochastic nonlinear systems with input delay, in which RBF NN was introduced to model the uncertain nonlinear functions. Then, a suitable auxiliary system was implemented to compensate for the effect of input delay. Moreover, by means of the structural characteristics of RBF NN, the controller’s design difficulty of non-strict feedback form was reduced. Meanwhile, the proposed adaptive controller secured that all the closed-loop signals remain bounded in probability. And the effectiveness of the constructed control scheme was further verified by two simulation examples.