Adaptive neural network terminal sliding mode tracking control for uncertain nonlinear systems with time-varying state constraints

Abstract

Exploring a novel adaptive asymmetric sliding mode control methodology with time-varying state constraints (TVSCs), we address trajectory tracking issues in uncertain nonlinear systems. The asymmetric barrier Lyapunov functions (ABLFs) and neural networks is employed within each subsystem’s virtual control design process using back-stepping control (BSC) method. This ensures the imposition of TVSCs and effectively addresses challenges posed by system uncertainties. Additionally, to enhance the convergence of tracking deviations within small zero neighborhoods, a nonsingular integral terminal sliding mode control (NITSMC) method is incorporated into the actual control algorithm design. This method illustrates that, the system states consistently stay within the specified boundaries, tracking errors rapidly converge to a confined range. All signals within the system remain bounded. Simulation findings affirm the efficacy of the suggested control strategy.

Introduction

Background and motivation

In recent years, the rapid advancement of adaptive control theory, particularly through the utilization of fuzzy-logic system (FLS) and neural networks system (NNS), has led to the exploration of various control methodologies aimed at addressing numerous control challenges. Remarkable progress has been made in this domain.^1–4 FLS and NNS are widely recognized for their effectiveness in approximating uncertainty, while the back-stepping control (BSC) method is renowned for its efficacy in managing cascaded nonlinear systems. Consequently, the integration of adaptive BSC methods is becoming prominent topic in addressing tracking control problems in unknown nonlinear systems.^5–8

It is universally acknowledged that practical systems often encounter diverse nonlinear uncertainties and operational constraints. Consequently, the control theory community has increasingly concentrated on addressing the challenge of controlling nonlinear systems with state constraints. Referring from authors’ knowledge and as evidenced by existing literature, various types of barrier Lyapunov functions (BLFs), comprising log-type BLFs, tangent-type BLFs, and integral BLFs, along with barrier function-based approaches ⁹ have been extensively utilized to address output or state constraints.^10–13 An adaptive neural network (ANN) BSC controller was constructed to stabilize nonlinear strict-feedback systems (NSFS) with uncertain dynamics and complete state constraints in a research by Liu et al. ¹⁴ In the research by He et al., ¹⁵ a control strategy was introduced with the objective of stabilizing uncertain time-delayed systems with time-varying full-state constraints (FSCs). Authors of a research in 2020 ¹⁶ presented ANN control techniques employing ABLFs based on finite-time for a specific class of NSFS with time-varying FSCs. Furthermore, the ABLFs methodology has demonstrated successful application in real-world engineering scenarios, including control of robot manipulator constraints and permanent magnet synchronous motor systems.^17–20

While significant progress has been made in the research on state constraints control schemes as evidenced by the aforementioned papers, it is noteworthy that they do not explicitly address tracking dynamic performance metrics such as tracking convergence rate and robustness performance. Drawing from the foundation of adaptive state constraints control theory, recent research has delved into the concepts of finite-time control and robust control methodologies, including sliding mode control (SMC) ²¹ and finite time control ²² approaches. SMC is renowned for its resilience against external disturbances and unmodeled system dynamics. To address this challenge, this study proposes the utilization of the NITSMC method, which integrates ABLs to formulate the controller. ²³ Deepika et al. ²⁴ introduced an integral terminal SMC strategy along with UDE to track outputs in constrained settings for mismatched uncertain nonlinear systems, albeit limited to constant constraints. Li ²⁵ adopted an adaptive SMC technique leveraging neural networks and BLFs to ensure compliance with time-varying constraints for manipulator outputs, albeit without consideration for other state constraints. Meanwhile, finite-time control schemes with state constraints have emerged as a focal point within this research domain.^26–33

While significant progress has been made in addressing unknown nonlinear systems with constraints, it is noteworthy that, in broader context of such systems, certain drawbacks persist within existing control schemes. These limitations include the substantial computational load associated with RBF-NNs, “explosion of complexity” issue in BSC techniques, and the intricacies involved in implementing dynamic surface control (DSC) strategies, among others. Motivated by these observations, it becomes apparent that there is a paucity of research focusing on adaptive finite-time convergence nonsingular SMC with asymmetric TVSCs, representing a more universal and complex scenario.

Contributions of the paper

Motivated by the insights gained from the aforementioned observations, this paper delves into the exploration of an ANN-based NITSMC approach tailored for a subset of unknown nonlinear systems featuring TVSCs. Through this endeavor, the controller design pertaining to the TVSCs is rendered more practical, while concurrently mitigating the stringency of constraint limitations by integrating asymmetric time-varying BLFs. ¹⁷

Primary contributions are summarized as below.

• (1) Unlike barrier functions SMC method used to address output control problems as demonstrated in the research by Deepika et al. ²⁴ and symmetric SMC BLFs method, ²⁶ which are primarily effective for managing static constraints, this paper diverges by leveraging asymmetric constraints and neural networks to tackle the complexities inherent in nonlinear systems.

• (2) Special unknown functions for each subsystem are constructed, approximated through adaptive RBF-NNs. Neural networks are incorporated into the controller design to mitigate computational burdens and circumvent the complexities associated with the “explosion of complexity” in the BSC procedure, in contrast to existing works.^13,20,34

• (3) To expedite the convergence of trajectory tracking in asymmetric TVSCs, the research introduces a finite-time NITSMC strategy in the final subsystem to formulate the actual control algorithm, distinguishing itself from prior research efforts.

Structure of the paper

Rest research is structured as: Sectio Section 3 details control algorithms design utilizing adaptive BSC-ABLFs and NITSMC theory, followed by the presentation of a control theorem for uncertain nonlinear systems. As for Section 4, numerical example was presented to validate suggested controller’s efficacy. Lastly, Section 5 provides the conclusion with a summary of findings and discussions on future directions.

System description and preliminaries

System overview and problem formulation

Take into account the subsequent strict-feedback nonlinear system featuring TVSCs:

$$\{\begin{array}{c}{\stackrel{·}{x}}_i=g_i({\overline{x}}_i)x_{i+1}+f_i({\overline{x}}_i),i=1,2,\cdots ,n-1\hfill \\ {\stackrel{·}{x}}_n=g_n({\overline{x}}_n)u+f_n({\overline{x}}_n)\hfill \\ y=x_1\hfill \end{array}$$

(1)

given that ${\overline{x}}_i=(x_1,x_2,\cdots ,x_i{)}^{\mathrm{T}}\in {\Re }^i(i=1,2,\cdots ,n)$ represents system state variables vectors, $u\in R$ , $y\in R$ denote input and output of system control, respectively. $f_i({\overline{x}}_i)\in R^n,i=1,2,\cdots ,n$ are unknown nonlinear smooth functions and cannot be linearly parameterized ( $f_i({\overline{x}}_i)\ne {\mathbf{\theta }}^T{\psi }_i({\overline{x}}_i)$ ); $g_i({\overline{x}}_i)\ne 0$ , $i=1,2,\cdots ,n$ were bounded unknown nonlinear piece-wise continuous functions, designating system’s unknown control direction. Since $f_i({\overline{x}}_i),g_i({\overline{x}}_i),i=1,2,\cdots ,n$ are undefined, it’s impossible for them to be directly implemented to build control algorithms, which bring the difficulty to the systems control design.

Study aim is designing adaptive NITSMC neural network control for nonlinear system (1) with TVSCs, so as to achieve the following goals:

• (1)  $y(t)$ (system output) is necessary for satisfying the TVSC condition that ${\underset{¯}{k}}_1(t)<y(t)<{\overline{k}}_1(t)$ , when ${\underset{¯}{k}}_1(t),{\overline{k}}_1(t)$ presented to be continuous time varying functions, as well as precisely tracked ideal signal $y_d$ in a fast convergence.

• (2) System states $x_i(i=1,\cdots ,n-1)$ were also necessary for satisfying asymmetric TVSCs that ${\underset{¯}{k}}_i(t)<x_i<{\overline{k}}_i(t)$ , $(i=1,\cdots ,n-1)$ with ${\underset{¯}{k}}_i(t)<{\overline{k}}_i(t),\forall t\ge 0$ , tracking error ensures compliance with the constraints imposed on the system states.

• (3) Closed loop signals’ boundedness is guaranteed in finite time meaning.

Before proceeding with design process for aforementioned control objects, it is essential to clarify following assumptions and preliminaries.

• Assumption 1. (Ren et al., 2010 ³⁵ ). The desired trajectory tracking signal $y_d(t)$ and corresponding ith order derivatives are smooth and bounded, satisfying

$$|y_d(t)|\le A_0<k_{c1},|{y_d}^{(i)}(t)|\le A_i,i=1,2,\cdots ,n$$

(2)

where $A_0,A_1,\cdots ,A_n$ are positive constants.

• Assumption 2. (Ren et al., 2010 ³⁵ ) Control coefficients signs $g_i({\overline{x}}_i),(i=1,2,\cdots ,n)$ are known, that is, $g_{i0}$ (positive constants) and the range went like $0<g_{i0}<|g_i({\overline{x}}_i)|<\infty$ for $i=1,2,\cdots ,n$ . Study assumes that, positive constants $g_{i0}$ went like $0<g_{i0}<|g_i({\overline{x}}_i)|<\infty$ without losing generality.

• Remark 1. Assumption 1 is standard and in line with practical feasibility assumption, which can be seen in adaptive control literature such as.^{11–14,16–19,36} Assumption 2 is also quite common and realistic in the existed literature (e.g. see Refs.^{11–14,26–30}). Moreover, the representation of system control directions is outlined ( $g_i({\overline{x}}_i)$ ), with Assumption 2 stipulating that the signs of $g_i(•)$ are strictly either positive or negative and bounded. Meanwhile, Assumption 2 ensures that $g_i({\overline{x}}_i)$ are piece-wise smooth and bounded, thereby ensuring that $g_i({\overline{x}}_i)\ne 0$ for all time, which serves as required condition for the system (1) to be controllable.

Preliminaries

RBF-NNs prove highly effective in addressing system uncertainties within the nonlinear control domain due to their adeptness in approximating uncertainties. In this investigation, RBF-NNs are devised, comprising two-layer networks. Herein, hidden layer functions as a predetermined nonlinear transformation devoid of adjustable parameters, while output layer linearly delivers outputs within subsequent space utilizing self-adjusting weight vectors. Consequently, RBF-NNs were employed to assess the unfamiliar functions $f_i({\overline{x}}_i),g_i({\overline{x}}_i)(i=1,2,\cdots ,n)$ within system (1).

• Definition 1. (Sanner and Slotine ³⁷ ) introduced the concept of input $Z\in {\mathrm{\Omega }}_z\subset R^n$ , weight vector $\mathit{W}=[w_1,w_2,\cdots ,w_l{]}^{\mathrm{T}}\in R^l$ , $l>1$ representing nodes count in RBF-NNs hidden layer, and $H(Z)=[h_1(Z),h_2(Z),\cdots ,h_l(Z){]}^{\mathrm{T}}$ with $h_i(Z)$ set as commonly applied Gaussian functions, being described by

$$h_i(Z)=\exp \left[\frac{-{(Z-c_i)}^{\mathrm{T}}(Z-c_i)}{{\upsilon }_i^2}\right],\begin{array}{c}i=1,2,\cdots ,l\hfill \end{array}$$

(3)

given that $c_i=[c_{i1},c_{i2},\cdots ,c_{\mathrm{in}}{]}^{\mathrm{T}}$ is receptive field center and ${\upsilon }_i$ is Gaussian function width. Then RBF-NNs output was utilized to model continuous function $f(Z):R^m\to R$ , which had the form as

$$f_{\mathrm{nn}}(Z)=W^{\mathrm{T}}H(Z)$$

(4)

• Lemma 1. (Ge and Wang ³⁴ ). Choosing the optimal $W^{*}$ (constant weight vector), using the proper structure of RBF-NNs as (5) in Definition 1, it is capable of representing any continuous function within bounded domain ${\mathrm{\Omega }}_z\subset R^n$ to arbitrary any accuracy as

$$f(Z)={W^{*}}^{\mathrm{T}}H(Z)+\epsilon (Z)$$

(5)

where $\epsilon (Z)$ was unascertained approximation error remains bounded within the confined range. The proof of Lemma 1 had been proven in Ref. ³⁷ Optimal $W^{*}$ (weight vector) was defined as

$$W^{*}:=\underset{W\in R^l}{\arg }min\left\{\underset{Z\in {\mathrm{\Omega }}_z}{sup}|f(Z)-{\hat{W}}^{\mathrm{T}}H(Z)|\right\}$$

(6)

In order to alleviate the significant computational burden associated with RBF-NN weight vectors online estimation, ${\mathit{\theta }}_i=\Vert {\mathit{W}}_i^{*}{\Vert }^2$ replaced the role of vector ${\mathit{W}}_i^{*}$ for $i=1,2,\cdots ,n$ , therefore, only n are updated to nonlinear closed loop systems. Moreover, we make ${\hat{\mathit{\theta }}}_i$ as estimation of ${\mathit{\theta }}_i$ , specify the error in estimating the weights of the constructed NN ${\tilde{\mathit{\theta }}}_i$ as ${\tilde{\mathit{\theta }}}_i={\hat{\mathit{\theta }}}_i-{\mathit{\theta }}_i$ .

• Lemma 2. (Ge and Wang ³ ) Considering the initial condition $V(0)$ remains within a bounded range, for any $t\in [0,t_f]$ , if $V(t)\ge 0$ and it is continuous, moreover it satisfies

$$\stackrel{·}{V}(t)\le -\mu V(t)+\eta$$

(7)

where $\mu >0,\eta >0$ , the candidate Lyapunov function $V(t)$ will be bounded on $[0,t_f]$ .

• Lemma 3. (Tee et al. ¹² ). For any $\left|\xi \right|<1$ and p (positive integer), following inequality holds.

$$\log \left(\frac{1}{1-{\xi }^{2p}}\right)\le \frac{{\xi }^{2p}}{1-{\xi }^{2p}}$$

(8)

• Lemma 4. (Witkowski ³⁸ ). If $p\ge 1,q\ge 1$ and $\frac{1}{p}+\frac{1}{q}=1$ , for any real number $x,y\in R$ and $\beta >0$ . Thus,

$$\mathrm{xy}\le \beta {\left|x\right|}^p+\frac{1}{q}(p\beta {)}^{-\frac{q}{p}}{\left|y\right|}^q.$$

(9)

• Lemma 5. (Tee et al. ¹² ) For time varying functions $k_{\mathrm{ai}}(t)$ , $k_{\mathrm{bi}}(t)$ and any $z_i\in R$ , define $q(z_i)=\{\begin{array}{c}1,\mathrm{if}\begin{array}{c}{z}_i>0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}\begin{array}{c}{z}_i<0\hfill \end{array}}\end{array}$ , $e_{\mathrm{ai}}=\frac{z_i}{k_{\mathrm{ai}}(t)},e_{\mathrm{bi}}=\frac{z_i}{k_{\mathrm{bi}}(t)}$ and $e_i=q(z_i)e_{\mathrm{bi}}+(1-q(z_i))e_{\mathrm{ai}}$ , if $-k_{\mathrm{ai}}(t)<z_i<k_{\mathrm{bi}}(t)$ , $i=1,2,\cdots ,n$ . then $\left|e_i\right|<1$ holds.

Adaptive NITSMC state constraints controller design

States coordinate transformation

Before proceeding with the detailed design procedure, we perform a coordinate transformation of the system states. To enhance simplicity and facilitate clarity in description, system time variations were disregarded throughout the research without introducing ambiguity. Following state coordinate transformation is implemented as

$$\{\begin{array}{c}{z}_1=x_1-y_d,\hfill \\ z_i=x_i-{\alpha }_{i-1},i=2,3,\dots ,n.\hfill \end{array}$$

(10)

where $y_d$ serves as desired trajectory, $z_1$ denotes system output tracking error, $z_i(i=2,3,\dots ,n)$ represents states tracing error between system states $x_i(i=2,3,\cdots ,n)$ and the desired states trajectory. ${\alpha }_{i-1}$ signifies virtual control law (VCL) being designed within every subsystem until final step while actual control law (ACL) $u(t)$ would be appeared.

Controller design process

To address TVSCs, accelerate system convergence rates, and improve control robustness, a novel NITSMC strategy is devised, integrating the BSC methodology using asymmetric time-varying BLFs. To elucidate the design philosophy and strategy, and to comprehend the overall design concept, the control structure of the composite control strategy is depicted in Figure 1. The design process for the control algorithms will be delineated in several steps, which will be deduced incrementally.

• Step 1: Defining initial subsystem’s tracking error as $z_1=x_1-y_d$ . We can differentiate it with respect to time to acquire

$${\stackrel{·}{z}}_1={\stackrel{·}{x}}_1-{\stackrel{·}{y}}_d=f_1(x_1)+g_1(x_1)x_2-{\stackrel{·}{y}}_d.$$

(11)

Figure 1.

Control schematic diagram.

In convention BSC technique, one can design VCL for the first subsystem using signal with $f_1(x_1),g_1(x_1)$ . However, due to the unknown functions of $f_1(x_1),g_1(x_1)$ , they weren’t directly utilized in practical applications. Therefore, researchers characterize the elusive function as

$$F_1(Z_1)=f_1(x_1)-{\stackrel{·}{y}}_d.$$

(12)

Given that $Z_1=[x_1,{\stackrel{·}{y}}_d{]}^{\mathrm{T}}\in {\mathrm{\Omega }}_z$ , therefore, $F_1(Z_1)$ is approximated by RBF-NNs as

$$F_1(Z_1)={\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)+{\epsilon }_1(Z_1).$$

(13)

where $W_1^{*\mathrm{T}}$ is the optimal weight vectors and $H_1(Z_1)\in R^{l_1}$ is the Gaussian functions vector, $l_1>0$ is the RBF-NNs hidden layer node number, ${\epsilon }_1(Z_1)$ is the approximation error, respectively. Suppose that $|{\mathbf{W}}_1^{\ast }\mathrm{|}\mathrm{<}{\overline{W}}_1$ , $|{\epsilon }_1(Z_1)|<{\overline{\epsilon }}_1$ , and ${\overline{W}}_1>0$ , ${\overline{\epsilon }}_1>0$ .

Thus, based on (10) and (13), equation (11) is changed as

$${\stackrel{·}{z}}_1={\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)+{\epsilon }_1(Z_1)+g_1(x_1)(z_2+{\alpha }_1),$$

(14)

Because of the output state constraints is asymmetric time-varying, thus, we design the $x_1$ -subsystems asymmetric barrier Lyapunov function as

$$\{\begin{array}{c}{V}_1=V_{L1}+\frac{1}{2}{g}_{10}{\tilde{\theta }}_1^2\hfill \\ V_{L1}=\frac{q(z_1)}{2p}\log \left(\frac{k_{b1}^{2p}(t)}{k_{b1}^{2p}(t)-z_1^{2p}}\right)+\frac{1-q(z_1)}{2p}\log \left(\frac{k_{a1}^{2p}(t)}{k_{a1}^{2p}(t)-z_1^{2p}}\right)\hfill \end{array}.$$

(15)

where p(positive integer) satisfied $2p\ge n$ in order to validate the $i-\mathrm{th}$ differentiation of virtual control algorithms ${\alpha }_i$ , $i=1,2,\cdots ,n-1$ existed. Throughout whole paper, except for special instructions, we shall define $(\widetilde{•})=(\widehat{•})-(•)$ . Thus, ${\tilde{\theta }}_1={\hat{\theta }}_1-{\theta }_1$ , and ${\hat{\theta }}_1>0$ is utilized to estimate ${\theta }_1$ , which denotes ${\theta }_1=g_{10}^{-1}{{\overline{W}}_1}^2$ . $k_{a1}(t),k_{b1}(t)$ (time-varying barrier functions) are positive definite smooth bounded functions defined as

$$\{\begin{array}{c}{k}_{a1}(t)\mathrm{:=}{y}_d(t)-{\underset{¯}{k}}_{c_1}(t)\hfill \\ k_{b1}(t)\mathrm{:=}{\overline{k}}_{c_1}(t)-y_d(t)\hfill \end{array},$$

(16)

Based on Assumption 2, positive constants ${\underset{¯}{K}}_{a1},{\overline{K}}_{a1},{\underset{¯}{K}}_{b1},{\overline{K}}_{b1}$ are existed satisfying

$${\underset{¯}{K}}_{a1}\le k_{a1}(t)\le {\overline{K}}_{a1},{\underset{¯}{K}}_{b1}\le k_{b1}(t)\le {\overline{K}}_{b1}.$$

(17)

$q(•)$ is a piecewise function given by

$$q(•)\mathrm{:=}\{\begin{array}{c}1,\begin{array}{c}\mathrm{if}•>0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{0,\begin{array}{c}\mathrm{if}•\le 0\hfill \end{array}}\end{array}$$

(18)

For the simplicity of the formula and expression convenience, following states tracing error coordinate transformation is made as

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{e}_{\mathrm{ai}}=\frac{z_i}{k_{\mathrm{ai}}(t)},e_{\mathrm{bi}}=\frac{z_i}{k_{\mathrm{bi}}(t)}\begin{array}{c}i=1,2,\cdots ,n-1\hfill \end{array}\hfill \\ e_i=q(z_i)e_{\mathrm{bi}}+(1-q(z_i))e_{\mathrm{ai}}\hfill \end{array}}\end{array}$$

(19)

$k_{a1}(t),k_{b1}(t)$ are shown as (16), $k_{\mathrm{ai}}(t),k_{\mathrm{bi}}(t),i=2,3,\cdots ,n-1$ will be provided subsequently. According to (15) and (19), (15) can be modified as

$$V_1=\frac{1}{2p}\log \left(\frac{1}{1-e_1^{2p}}\right)+\frac{1}{2}{g}_{10}{\tilde{\theta }}_1^2$$

(20)

Furthermore, first derivative of $V_1$ with respect to time in (20) is computed by

$$\begin{array}{c}{\stackrel{·}{V}}_1=\frac{q(z_1)e_{b1}^{2p-1}}{k_{b1}(t)(1-e_{b1}^{2p})}({\stackrel{·}{z}}_1-z_1\frac{{\stackrel{·}{k}}_{b1}(t)}{k_{b1}(t)})\hfill \\ +\frac{(1-q(z_1))e_{a1}^{2p-1}}{k_{a1}(t)(1-e_{a1}^{2p})}({\stackrel{·}{z}}_1-z_1\frac{{\stackrel{·}{k}}_{a1}(t)}{k_{a1}(t)})+g_{10}{\tilde{\theta }}_1\stackrel{·}{\widehat{{\theta }_1}}\hfill \end{array}$$

(21)

Using (14), (21) gets

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\frac{q(z_1)e_{b1}^{2p-1}}{k_{b1}(t)(1-e_{b1}^{2p})}}\\ ({\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)+{\epsilon }_1(Z_1+g_1(x_1)(z_2+{\alpha }_1)-z_1\frac{{\stackrel{·}{k}}_{b1}(t)}{k_{b1}(t)})\hfill \\ +\frac{(1-q(z_1))e_{a1}^{2p-1}}{k_{a1}(t)(1-e_{a1}^{2p})}({\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)+{\epsilon }_1(Z_1)\hfill \\ +g_1(x_1)(z_2+{\alpha }_1)-z_1\frac{{\stackrel{·}{k}}_{a1}(t)}{k_{a1}(t)})+g_{10}{\tilde{\theta }}_1{\stackrel{·}{\hat{\theta }}}_1\hfill \end{array}$$

(22)

To simplify the formation of (22), we define

$${\xi }_i(t)=\frac{q(z_i)}{k_{\mathrm{bi}}^{2p}(t)-z_i^{2p}}+\frac{1-q(z_i)}{k_{\mathrm{ai}}^{2p}(t)-z_i^{2p}},\begin{array}{c}i=1,2,\cdots ,n-1\hfill \end{array}$$

(23)

Then, (23) can be simplified as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1={\xi }_1{z}_1^{2p-1}({\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)+{\epsilon }_1(Z_1))+{\xi }_1{z}_1^{2p-1}{g}_1(x_1)(z_2+{\alpha }_1)}\\ +{\xi }_1{z}_1^{2p-1}[q(z_1)\frac{{\stackrel{·}{k}}_{b1}(t)}{k_{b1}(t)}+(1-q(z_1))\frac{{\stackrel{·}{k}}_{a1}(t)}{k_{a1}(t)}]z_1\hfill \end{array}$$

(24)

According to Young’s inequality, items in (24) would be amplified as

$${\xi }_1{z}_1^{2p-1}{\mathit{W}}_1^{*\mathrm{T}}{H}_1(Z_1)\le \frac{1}{2}{b}_1^2+\frac{{\xi }_1^2{z}_1^{4p-2}{\overline{\mathit{W}}}_1^2{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}$$

(25)

$${\xi }_1{z}_1^{2p-1}{\epsilon }_1(Z_1)\le \frac{1}{2}{g}_{10}{{\xi }_1}^2{z}_1^{4p-2}+\frac{{\overline{\epsilon }}_1^2}{2g_{10}}$$

(26)

where $b_1$ serves as predesignated positive constant. Referring from Lemma 2, virtual stabilizing function ${\alpha }_1(t)$ of $x_1$ -subsystem was designed as

$$\begin{array}{l}{\alpha }_1=-({\kappa }_1+{\stackrel{⌢}{K}}_{z1}(t)+{\gamma }_1)z_1-\frac{{\xi }_1{z}_1^{2p-1}{\widehat{\theta }}_1{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}\\ -\frac{{\xi }_1{z}_1^{2p-1}}{2}\end{array}$$

(27)

where ${\kappa }_1>0$ is a constant to be designed and ${\gamma }_1$ is a related constant with p as ${\gamma }_1=(2p-1)/2p$ . ${\stackrel{⌢}{K}}_{z1}(t)$ is the time-variable gain function defined by

$$\begin{array}{l}{\stackrel{⌢}{K}}_{zi}(t)=\\ \sqrt{q(z_i){\left(\frac{{\stackrel{·}{k}}_{bi}(t)}{k_{bi}(t)}\right)}^2+(1-q(z_i)){\left(\frac{{\stackrel{·}{k}}_{ai}(t)}{k_{ai}(t)}\right)}^2+{\beta }_i},i=1,2,...,n-1\end{array}$$

(28)

where $k_{\mathrm{ai}}\mathrm{:=}{\alpha }_{i-1}-{\underset{¯}{k}}_{\mathrm{ci}}(t)$ , $k_{\mathrm{bi}}\mathrm{:=}{\overline{k}}_{\mathrm{ci}}(t)-{\alpha }_{i-1}$ , β serves as positive constant. Thus, it ensured ${\alpha }_i$ differentiable, as well as derivative over time is bounded, even if ${\stackrel{·}{k}}_{\mathrm{ai}}$ and ${\stackrel{·}{k}}_{\mathrm{bi}}$ are zero because of this constant β.

From (18), it gets that when $z_i>0$ , then $q(z_i)=1$ , ${\stackrel{⌢}{K}}_{zi}(t)+{\stackrel{·}{k}}_{bi}(t)/k_{bi(t)}\ge 0$ . When $z_i<0$ , then $q(z_i)<0$ and ${\stackrel{⌢}{K}}_{zi}(t)+{\stackrel{·}{k}}_{ai}(t)/k_{ai(t)}\ge 0$ holds. Thus, one can obtain the following inequality:

$${\stackrel{⌢}{K}}_{zi}(t)+q(z_i)\frac{{\stackrel{·}{k}}_{bi}(t)}{k_{bi}(t)}+(1-q(z_i))\frac{{\stackrel{·}{k}}_{ai}(t)}{k_{ai}(t)}\ge 0$$

(29)

Substituting (29) in the item ${\xi }_1{z}_1^{2p-1}{g}_1(x_1){\alpha }_1$ , it gets

$$\begin{array}{l}{\xi }_1{z}_1^{2p-1}{g}_1(x_1){\alpha }_1=-{\xi }_1{z}_1^{2p-1}{g}_1(x_1){\stackrel{⌢}{K}}_{z1}(t)z_1\\ -\frac{{\xi }_1^2{z}_1^{4p-2}{\widehat{\theta }}_1{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}{g}_1(x_1)-{\kappa }_1{\xi }_1{z}_1^{2p}{g}_1(x_1)\\ -\frac{{\xi }_1^2{z}_1^{4p-2}}{2}{g}_1(x_1)-{\gamma }_1{\xi }_1{z}_1^{2p}{g}_1(x_1)\end{array}$$

(30)

Because $g_1({\overline{x}}_1)$ is unknown and according to Assumption 2 with $g_i({\overline{x}}_i)\ge g_{i0}$ , one would get

$$\{\begin{array}{l}-{\kappa }_1{\xi }_1{z}_1^{2p}{g}_1(x_1)\le -{\kappa }_1{\xi }_1{z}_1^{2p}{g}_{10}\\ -\frac{{\xi }_1^2{z}_1^{4p-2}}{2}{g}_1(x_1)\le -\frac{1}{2}{\xi }_1^2{z}_1^{4p-2}{g}_{10}\\ -\frac{{\xi }_1^2{z}_1^{4p-2}{\widehat{\theta }}_1{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}{g}_1(x_1)\le -\frac{{\xi }_1^2{z}_1^{4p-2}{\widehat{\theta }}_1{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}{g}_{10}\end{array}$$

(31)

Meanwhile, we utilize Young’s inequality to obtain that

$$\begin{array}{c}{\xi }_1{z}_1^{2p-1}{g}_1(x_1)z_2\le {\xi }_1{g}_1(x_1)(\frac{2p-1}{2p}{z}_1^{2p}+\frac{1}{2p}{z}_2^{2p})\hfill \\ ={\xi }_1{g}_1(x_1)({\gamma }_1{z}_1^{2p}+z_2^{2p}/2p)\hfill \end{array}$$

(32)

Substituting (25), (26), (30), (31), and (32) into (24) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\le -{\kappa }_1{\xi }_1{g}_{10}{z}_1^{2p}-g_{10}{\tilde{\theta }}_1(\frac{{\xi }_1^2{z}_1^{4p-2}{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}-\stackrel{·}{\widehat{{\theta }_1}})}\\ +\frac{{\overline{\epsilon }}_1^2}{2g_{10}}+\frac{b_1^2}{2}+\frac{{\xi }_1{g}_1({\overline{x}}_1)}{2p}{z}_2^{2p}\hfill \end{array}$$

(33)

Then, we define the $x_1$ -subsystem RBF-NNs weight vector adaption laws from (33) as following

$$\stackrel{·}{\widehat{{\theta }_1}}=\frac{{\xi }_1^2{z}_1^{4p-2}{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}-{\sigma }_1{\hat{\theta }}_1$$

(34)

where ${\sigma }_1>0$ serves as small positive constant that will be determined through design, the item ${\sigma }_1{\hat{\theta }}_1$ in equation (34) is called $\sigma -$ modification, which is employed to improve robustness performance and prevent weight parameters from drifting toward excessively large values in approximation errors approximation error ${\tilde{\theta }}_1$ . ³⁹ Substituting equation (34) back into (33), it can obtain

$${\stackrel{·}{V}}_1\le -{\kappa }_1{\xi }_1{g}_{10}{z}_1^{2p}-{\sigma }_1{g}_{10}{\tilde{\theta }}_1{\hat{\theta }}_1+\frac{{\overline{\epsilon }}_1^2}{2g_{10}}+\frac{b_1^2}{2}+\frac{{\xi }_1{g}_1({\overline{x}}_1)}{2p}{z}_2^{2p}$$

(35)

where the coupling term ${\xi }_1{g}_1({\overline{x}}_1)z_2^{2p}/2p$ will be nullified within subsequent stage.

• Step i $(2\le i\le n-1)$ : Take $z_i$ time derivative with $z_i=x_i-{\alpha }_{i-1}$ , it yields

$${\stackrel{·}{z}}_i={\stackrel{·}{x}}_i-{\stackrel{·}{\alpha }}_{i-1}=f_i({\overline{x}}_i)+g_i({\overline{x}}_i)x_{i+1}-{\stackrel{·}{\alpha }}_{i-1}$$

(36)

given that ${\alpha }_{i-1}$ is called the $x_i$ -subsystem virtual stabilization function with the variants of ${\overline{\mathit{x}}}_{i-1}$ , $y_d$ , $y_d^{(i-1)}$ , ${\hat{\theta }}_{i-1}$ . Therefore, ${\stackrel{·}{\alpha }}_{i-1}$ is computed by

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\alpha }}_{i-1}=\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}[f_j({\overline{x}}_j)+g_j({\overline{x}}_j)x_{j+1}]+\sum _{j=0}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial y_d^{(j)}}{y}_d^{(j+1)}}\\ +\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\hat{\theta }}_j}\hfill \end{array}$$

(37)

The unknown functions are defined as

$$F_i(Z_i)=f_i({\overline{x}}_i)-{\stackrel{·}{\alpha }}_{i-1}+\frac{{\xi }_{i-1}}{2p{\xi }_i}{g}_{i-1}({\overline{x}}_{i-1})z_i$$

(38)

where $Z_i=[x_1,\cdots ,x_i,y_d,{\stackrel{·}{y}}_d,\cdots ,y_d^{(i)},{\hat{\theta }}_1,\cdots ,{\hat{\theta }}_{i-1}{]}^{\mathrm{T}}\in {\mathrm{\Omega }}_i$ . The RBF-NNs are utilized to approximate $F_i(Z_i)$ as

$$F_i(Z_i)={\mathit{W}}_i^{*\mathrm{T}}{H}_i(Z_i)+{\epsilon }_i(Z_i)$$

(39)

Given $W_i^{*}$ as optimal weight vectors, $H_i(Z_i)\in R^{l_i}$ is the Gaussian functions vector, $l_i>0$ is the RBF-NNs hidden layer node number, ${\epsilon }_i(Z_i)$ is $x_i$ -subsystem approximation error, respectively. Suppose that $\Vert {{\mathit{W}}_i}^{*}\Vert <{\overline{W}}_i$ , $|{\epsilon }_i(Z_i)|<{\overline{\epsilon }}_i$ , and ${\overline{W}}_i>0$ , ${\overline{\epsilon }}_i>0$ .

Define the $x_i$ -subsystem time-varying asymmetric BLFs as

$$\{\begin{array}{c}{V}_i=V_{\mathrm{Li}}++\frac{1}{2}{g}_{i0}{\tilde{\theta }}_i^2+V_{i-1}\hfill \\ V_{\mathrm{Li}}=\frac{q(z_i)}{2p}\log \left(\frac{k_{\mathrm{bi}}^{2p}(t)}{k_{\mathrm{bi}}^{2p}(t)-z_i^{2p}}\right)+\frac{1-q(z_i)}{2p}\log \left(\frac{k_{\mathrm{ai}}^{2p}(t)}{k_{\mathrm{ai}}^{2p}(t)-z_i^{2p}}\right)\hfill \end{array}$$

(40)

where and ${\hat{\theta }}_i>0$ is utilized to estimate ${\theta }_i$ , which denotes ${\theta }_i=g_{i0}^{-1}{{\overline{W}}_i}^2$ . $e_{\mathrm{ai}},e_{\mathrm{bi}}$ are defined as (19).

Take time derivative of $V_i$ on ${\mathrm{\Omega }}_i$ , it gets

$$\{\begin{array}{c}{\stackrel{·}{V}}_i={\stackrel{·}{V}}_{\mathrm{Li}}+g_{i0}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i+{\stackrel{·}{V}}_{i-1}\hfill \\ {\stackrel{·}{V}}_{\mathrm{Li}}=\frac{q(z_i)e_{\mathrm{bi}}^{2p-1}}{k_{\mathrm{bi}}(t)(1-e_{\mathrm{bi}}^{2p-1})}({\stackrel{·}{z}}_i-z_i\frac{{\stackrel{·}{k}}_{\mathrm{bi}}(t)}{k_{\mathrm{bi}}(t)})+\frac{(1-q(z_i))e_{\mathrm{ai}}^{2p-1}}{k_{\mathrm{ai}}(t)(1-e_{\mathrm{ai}}^{2p})}\hfill \\ ({\stackrel{·}{z}}_i-z_i\frac{{\stackrel{·}{k}}_{\mathrm{ai}}(t)}{k_{\mathrm{ai}}(t)})\hfill \end{array}$$

(41)

Based on (36), (38), and (39), one would obtain that

$${\stackrel{·}{z}}_i={\mathit{W}}_i^{*\mathrm{T}}{H}_i(Z_i)+{\epsilon }_i(Z_i)-\frac{{\xi }_{i-1}}{2p{\xi }_i}{g}_{i-1}({\overline{x}}_{i-1})z_i{g}_i({\overline{x}}_i)x_{i+1}$$

(42)

Substituting (42) into (41), it yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_i={\xi }_i{z}_i^{2p-1}({\mathit{W}}_i^{*\mathrm{T}}{H}_i(Z_i)+{\epsilon }_i(Z_i))+{\xi }_i{z}_i^{2p-1}{g}_i({\overline{x}}_i)(z_{i+1}+{\alpha }_i)}\\ +{\xi }_i{z}_i^{2p-1}[q(z_i)\frac{{\stackrel{·}{k}}_{\mathrm{bi}}(t)}{k_{\mathrm{bi}}(t)}+(1-q(z_i))\frac{{\stackrel{·}{k}}_{\mathrm{ai}}(t)}{k_{\mathrm{ai}}(t)}]z_i\hfill \\ +g_{i0}{\tilde{\theta }}_i{\stackrel{·}{\hat{\theta }}}_i-\frac{{\xi }_{i-1}}{2p}{g}_{i-1}({\overline{x}}_{i-1})z_i^{2p}+{\stackrel{·}{V}}_{i-1}\hfill \end{array}$$

(43)

Youngs inequality is utilized to obtain that

$$\{\begin{array}{c}{\xi }_i{z}_i^{2p-1}{\mathit{W}}_i^{*\mathrm{T}}{H}_i(Z_i)\le \frac{1}{2}{b}_i^2+\frac{{{\xi }_i}^2{z}_i^{4p-2}{\overline{\mathit{W}}}_i^2{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}\hfill \\ {\xi }_i{z}_i^{2p-1}{\epsilon }_i(Z_i)\le \frac{1}{2}{g}_{i0}{{\xi }_i}^2{z}_i^{4p-2}+\frac{{\overline{\epsilon }}_i^2}{2g_{i0}}\hfill \end{array}$$

(44)

where $b_i$ serves as predesigned positive constant. Referring from Lemma 2, virtual stabilizing function ${\alpha }_i(t)$ of $x_i$ -subsystem was designed similarly like the first subsystem as

$${\alpha }_i=-({\kappa }_i+{\stackrel{⌢}{K}}_{zi}(t)+{\gamma }_i)z_i-\frac{{\xi }_i{z}_i^{2p-1}{\widehat{\theta }}_i{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}-\frac{{\xi }_i{z}_i^{2p-1}}{2}$$

(45)

where ${\kappa }_i>0$ was set constantly and ${\gamma }_i={\gamma }_1=(2p-1)/2p$ . ${\stackrel{⌢}{K}}_{zi}(t)$ is the time-variable gain function defined as (28).

Substituting (45) in the item ${\xi }_i{z}_i^{2p-1}{g}_i({\overline{x}}_i){\alpha }_i$ , it gets

$$\begin{array}{l}{\xi }_i{z}_i^{2p-1}{g}_i({\overline{x}}_i){\alpha }_i=-{\xi }_i{z}_i^{2p-1}{g}_i({\overline{x}}_i){\stackrel{⌢}{K}}_{zi}(t)z_i\\ -\frac{{\xi }_i^2{z}_i^{4p-2}{\widehat{\theta }}_i{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}{g}_i({\overline{x}}_i)-{\kappa }_i{\xi }_i{z}_i^{2p}{g}_i({\overline{x}}_i)\\ -\frac{{\xi }_i^2{z}_i^{4p-2}}{2}{g}_i({\overline{x}}_i)-{\gamma }_i{\xi }_i{z}_i^{2p}{g}_i({\overline{x}}_i)\end{array}$$

(46)

Because $g_i({\overline{x}}_i)$ is unknown and according to Assumption 2 with $g_i({\overline{x}}_i)\ge g_{i0}$ , one would get

$$\{\begin{array}{c}-{\kappa }_i{\xi }_i{z}_i^{2p}{g}_i({\overline{x}}_i)\le -{\kappa }_i{\xi }_i{z}_i^{2p}{g}_{i0}\\ \begin{array}{l}-\frac{{\xi }_i^2{z}_i^{4p-2}}{2}{g}_i({\overline{x}}_i)\le -\frac{1}{2}{\xi }_i^2{z}_i^{4p-2}{g}_{i0}\\ -\frac{{\xi }_i^2{z}_i^{4p-2}{\widehat{\theta }}_i{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}{g}_i({\overline{x}}_i)\le -\frac{{\xi }_i^2{z}_i^{4p-2}{\widehat{\theta }}_i{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}{g}_{i0}\end{array}\end{array}$$

(47)

By exploiting Young’s inequality, it would get

$$\begin{array}{c}\multicolumn{1}{c}{{\xi }_i{z}_i^{2p-1}{g}_i({\overline{x}}_i)z_{i+1}\le {\xi }_i{g}_i({\overline{x}}_i)(\frac{2p-1}{2p}{z}_i^{2p}+\frac{1}{2p}{z}_{i+1}^{2p})}\\ ={\xi }_i{g}_i({\overline{x}}_i)({\gamma }_i{z}_i^{2p}+z_{i+1}^{2p}/2p)\hfill \end{array}$$

(48)

Substituting (46)–(48) into (43) leads to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_i\le -{\kappa }_i{\xi }_i{g}_{i0}{z}_i^{2p}-g_{i0}{\tilde{\theta }}_i(\frac{{\xi }_i^2{z}_i^{4p-2}{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}-{\stackrel{·}{\hat{\theta }}}_i)+\frac{{\xi }_i{g}_i({\overline{x}}_i)}{2p}{z}_{i+1}^{2p}}\\ -\frac{{\xi }_{i-1}{g}_{i-1}({\overline{x}}_{i-1})}{2p}{z}_i^{2p}+\frac{{\overline{\epsilon }}_i^2}{2g_{i0}}+\frac{b_i^2}{2}+{\stackrel{·}{V}}_{i-1}\hfill \end{array}$$

(49)

Then, we define the $x_i$ -subsystem RBF-NNs weight vector adaption laws from (33) as following

$$\stackrel{·}{\widehat{{\theta }_i}}=\frac{{\xi }_i^2{z}_i^{4p-2}{\Vert H_i(Z_i)\Vert }^2}{2b_i^2}-{\sigma }_i{\hat{\theta }}_i$$

(50)

where ${\sigma }_i>0$ is a small constant to be designed. Substituting equation (50) back into (49), it can obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_i\le -{\kappa }_i{\xi }_i{g}_{i0}{z}_i^{2p}-{\sigma }_i{g}_{i0}{\tilde{\theta }}_i{\hat{\theta }}_i+\frac{{\overline{\epsilon }}_i^2}{2g_{i0}}+\frac{b_i^2}{2}+\frac{{\xi }_i{g}_i({\overline{x}}_i)}{2p}{z}_{i+1}^{2p}}\\ -\frac{{\xi }_{i-1}{g}_{i-1}({\overline{x}}_{i-1})}{2p}{z}_i^{2p}+{\stackrel{·}{V}}_{i-1}\hfill \end{array}$$

(51)

Based on the recursion procedure, ${\stackrel{·}{V}}_{i-1}$ is described as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{i-1}\le -\sum _{j=1}^{i-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^{i-1}{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^{i-1}\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}}\\ +\sum _{j=1}^{i-1}\frac{b_j^2}{2}+\frac{{\xi }_1{g}_1({\overline{x}}_1)}{2p}{z}_2^{2p}+\frac{{\xi }_{i-1}{g}_{i-1}({\overline{x}}_{i-1})}{2p}{z}_i^{2p}\hfill \end{array}$$

(52)

Substituting (52) back into (51), one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_i\le -\sum _{j=1}^i{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^i{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^i\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}}\\ +\sum _{j=1}^i\frac{b_j^2}{2}+\frac{{\xi }_i{g}_i({\overline{x}}_i)}{2p}{z}_{i+1}^{2p}\hfill \end{array}$$

(53)

• Step n: In order to accelerate the convergence rate to predefined origin and improve control performance’s robustness, in the last subsystem, we design the integrate nonsingular terminal SMC strategies to control the whole system. Take time derivative of $z_n=x_n-{\alpha }_{n-1}$

$${\stackrel{·}{z}}_n={\stackrel{·}{x}}_n-{\stackrel{·}{\alpha }}_{n-1}=f_n({\overline{x}}_n)+g_n({\overline{x}}_n)u-{\stackrel{·}{\alpha }}_{n-1}$$

(54)

where

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\alpha }}_{i-1}=\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}[f_j({\overline{x}}_j)+g_j({\overline{x}}_j)x_{j+1}]}\\ +\sum _{j=0}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial y_d^{(j)}}{y}_d^{(j+1)}+\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\hat{\theta }}_j}\hfill \end{array}$$

(55)

An integral non-singular terminal sliding manifold surface is designed as

$$s=z_n+\eta {\int }_0^t{\left|z_n\right|}^{\lambda }d\tau$$

(56)

$\stackrel{·}{s}$ can be obtained by substituting (54) as

$$\stackrel{·}{s}={\stackrel{·}{z}}_n+\eta {\left|z_n\right|}^{\lambda }=f_n({\overline{x}}_n)+g_n({\overline{x}}_n)u-{\stackrel{·}{\alpha }}_{n-1}+\eta {\left|z_n\right|}^{\lambda }$$

(57)

where $1<\lambda <2$ is a selected parameter. Define the unknown parts in (57) as

$$F_n(Z_n)=f_n({\overline{x}}_n)-{\stackrel{·}{\alpha }}_{n-1}+\frac{{\xi }_{n-1}}{2\mathrm{ps}}{g}_{n-1}({\overline{x}}_{n-1})z_n^{2p}$$

(58)

It can be observed from (55) and (58) that $F_n(Z_n)$ is a function with $[{\overline{x}}_n^{\mathrm{T}},{\overline{y}}_d^{(n{)}^{\mathrm{T}}},\overline{\widehat{{\theta }_{n-1}^{\mathrm{T}}}}]$ , therefore, $Z_n=[{\overline{x}}_n^{\mathrm{T}},{\overline{y}}_d^{(n{)}^{\mathrm{T}}},\overline{\widehat{{\theta }_{n-1}^{\mathrm{T}}}}]\in {\mathrm{\Omega }}_n$ and $F_n(Z_n)$ is approximated by RBF-NNs as

$$F_n(Z_n)={\mathit{W}}_n^{*\mathrm{T}}{H}_n(Z_n)+{\epsilon }_n(Z_n)$$

(59)

given that ${\mathit{W}}_n^{*\mathrm{T}}$ serves as optimal weight vector, ${\epsilon }_n(Z_n)$ is the approximation error and $H_n(Z_n)\in R^{l_n}$ is the Gaussian functions vector, $l_n>0$ is the RBF-NNs hidden layer node number and $\Vert W_n^{*}\Vert \le {\overline{W}}_n$ , $\Vert {\epsilon }_n(Z_n)\Vert \le {\overline{\epsilon }}_n$ with constants ${\overline{W}}_n,{\overline{\epsilon }}_n$ of $x_n$ subsystem. From (58) and (59), (57) can be written by

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{s}={\stackrel{·}{z}}_n+\eta {\left|z_n\right|}^{\lambda }={\mathit{W}}_n^{*\mathrm{T}}{H}_n(Z_n)+{\epsilon }_n(Z_n)}\\ -\frac{{\xi }_{n-1}}{2\mathrm{ps}}{g}_{n-1}({\overline{x}}_{n-1})z_n^{2p}+\eta {\left|z_n\right|}^{\lambda }+g_n({\overline{x}}_n)u\hfill \end{array}$$

(60)

The $x_n$ -subsystem Lyapunov function is candidate as

$$V_n=V_{n-1}+\frac{1}{2}{g}_{n0}{\tilde{\theta }}_i^2+\frac{1}{2}{s}^2$$

(61)

$V_n$ (time derivative) is

$${\stackrel{·}{V}}_n={\stackrel{·}{V}}_{n-1}+s\stackrel{·}{s}+g_{n0}{\tilde{\theta }}_n\stackrel{·}{\widehat{{\theta }_n}}$$

(62)

Substituting (60) into (62) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n=s[{\mathit{W}}_n^{*\mathrm{T}}{H}_n(Z_n)+{\epsilon }_n(Z_n)-\frac{{\xi }_{n-1}}{2\mathrm{ps}}{g}_{n-1}({\overline{x}}_{n-1})z_n^{2p}}\\ +\eta {\left|z_n\right|}^{\lambda }+g_n({\overline{x}}_n)u]+g_{n0}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+{\stackrel{·}{V}}_{n-1}\hfill \end{array}$$

(63)

By exploiting Young’s inequality, one gets

$$\{\begin{array}{c}s{\epsilon }_n(Z_n)\le \frac{1}{2}{g}_{n0}{s}^2+\frac{{\overline{\epsilon }}_n^2}{2g_{n0}}\hfill \\ s\eta {\left|z_n\right|}^{\lambda }\le \frac{1}{2g_{n0}}+\frac{g_{n0}{s}^2{\eta }^2{\left|z_n\right|}^{2\lambda }}{2}\hfill \\ s{\mathit{W}}_n^{*\mathrm{T}}{H}_n(Z_n)\le \frac{1}{2}{b}_n^2+\frac{s^2{{\overline{\mathit{W}}}_n}^2{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}\hfill \end{array}$$

(64)

Substituting (64) into (63), it yields

$$\{\begin{array}{c}{\stackrel{·}{V}}_n\le \frac{1}{2}{b}_n^2+\frac{s^2{{\overline{\mathit{W}}}_n}^2{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}+\frac{1}{2}{g}_{n0}{s}^2+\frac{{\overline{\epsilon }}_n^2}{2g_{n0}}+\frac{1}{2g_{n0}}\hfill \\ +\frac{g_{n0}{s}^2{\eta }^2{\left|z_n\right|}^{2\lambda }}{2}+s[-\frac{{\xi }_{n-1}}{2\mathrm{ps}}{g}_{n-1}({\overline{x}}_{n-1})z_n^{2p}+g_n({\overline{x}}_n)u]\hfill \\ +g_{n0}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n+{\stackrel{·}{V}}_{n-1}\hfill \\ {\stackrel{·}{V}}_{n-1}\le -\sum _{j=1}^{n-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^{n-1}{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^{n-1}\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}\hfill \\ +\sum _{j=1}^{n-1}\frac{b_j^2}{2}+\frac{{\xi }_i{g}_i({\overline{x}}_i)}{2p}{z}_{i+1}^{2p}\hfill \end{array}$$

(65)

Controller $u(t)$ is designed as

$$u=-{\rho }_{n}s-\frac{s{\hat{\theta }}_n{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}-\frac{s{\eta }^2{\left|z_n\right|}^{2\lambda }}{2}\mathrm{sign}(z_n)$$

(66)

where $k_n$ is a selected positive parameter and ${\rho }_n>1/2$ . Then, substituting (66) into the item $g_n({\overline{x}}_n)\mathrm{us}$ and using Assumption 2 it gets

$$g_n({\overline{x}}_n)\mathrm{us}\le -{\rho }_n{g}_{n0}{s}^2-\frac{g_{n0}{s}^2{\hat{\theta }}_n{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}-\frac{g_{n0}{s}^2{\eta }^2{\left|z_n\right|}^{2\lambda }}{2}$$

(67)

Then, substituting (67) into (65), it yields

$$\begin{array}{c}{\stackrel{·}{V}}_n\le -({\rho }_n-\frac{1}{2})g_{n0}{s}^2+\frac{s^2{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}({\overline{\mathit{W}}}_n^2-g_{n0}{\hat{\theta }}_n)+g_{n0}{\tilde{\theta }}_n{\stackrel{·}{\hat{\theta }}}_n\hfill \\ -\sum _{j=1}^{n-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^{n-1}{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^n\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}+\sum _{j=1}^n\frac{b_j^2}{2}+\frac{1}{2g_{n0}}\hfill \end{array}$$

(68)

Because of ${\tilde{\theta }}_n={\hat{\theta }}_n-{\theta }_n$ and ${\hat{\theta }}_n$ is used to estimate ${\theta }_n=g_{n0}^{-1}{\overline{W}}_n^2$ , then (68) can be written

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_n\le }& -({\rho }_n-\frac{1}{2})g_{n0}{s}^2+g_{n0}{\tilde{\theta }}_n(\frac{s^2{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}-{\stackrel{·}{\hat{\theta }}}_n)\hfill \\ \hfill & -\sum _{j=1}^{n-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^{n-1}{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^n\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}\hfill \\ \hfill & +\sum _{j=1}^n\frac{b_j^2}{2}+\frac{1}{2g_{n0}}\hfill \end{array}$$

(69)

Therefore, we design the $x_n$ -subsystem RBF-NNs weight adaption law as

$$\stackrel{·}{\widehat{{\theta }_n}}=-{\sigma }_n{\hat{\theta }}_n+\frac{s^2{\hat{\theta }}_n{\Vert H_n(Z_n)\Vert }^2}{2b_n^2}$$

(70)

After substituting (70) into (69) and let ${\chi }_n=({\rho }_n-\frac{1}{2})g_{n0}$ , one would obtain that

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_n\le {\chi }_n{s}^2-\sum _{j=1}^{n-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\sum _{j=1}^n{\sigma }_j{g}_{j0}{\tilde{\theta }}_j{\hat{\theta }}_j+\sum _{j=1}^n\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}}\\ +\sum _{j=1}^n\frac{b_j^2}{2}+\frac{1}{2g_{n0}}\hfill \end{array}$$

(71)

From (15), (40), and (69), it can be obtained that

$$\begin{array}{c}\multicolumn{1}{c}{V_n=\sum _{j=1}^{n-1}\frac{q(z_j)}{2p}\log \left(\frac{k_{\mathrm{bj}}^{2p}(t)}{k_{\mathrm{bj}}^{2p}(t)-z_j^{2p}}\right)+\sum _{j=1}^{n-1}\frac{1-q(z_j)}{2p}}\\ \multicolumn{1}{c}{\log \left(\frac{k_{\mathrm{aj}}^{2p}(t)}{k_{\mathrm{aj}}^{2p}(t)-z_j^{2p}}\right)+\frac{1}{2}\sum _{j=1}^n{g}_{j0}{\tilde{\theta }}_j^2+\frac{1}{2}{s}^2}\end{array}$$

(72)

By utilizing Young’s inequality, one gets

$$\begin{array}{cc}\multicolumn{1}{c}{-g_{j0}{\tilde{\theta }}_j{\sigma }_j{\hat{\theta }}_j}& =-g_{j0}{\sigma }_j{\tilde{\theta }}_j^2-g_{j0}{\sigma }_j{\tilde{\theta }}_j{\theta }_j\hfill \\ \hfill & \le -g_{j0}{\sigma }_j{\tilde{\theta }}_j^2+g_{j0}{\sigma }_j(\frac{1}{2}{\tilde{\theta }}_j^2+\frac{1}{2}{\theta }_j^2)\hfill \\ \hfill & \le -\frac{1}{2}{g}_{j0}{\sigma }_j{\tilde{\theta }}_j^2+\frac{1}{2}{g}_{j0}{\sigma }_j{\theta }_j^2\hfill \end{array}$$

(73)

Therefore, (71) can be transformed to

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_n}& \le -\sum _{j=1}^{n-1}{\kappa }_j{\xi }_j{g}_{j0}{z}_j^{2p}-\frac{1}{2}\sum _{j=1}^{n-1}{g}_{j0}{\sigma }_j{\tilde{\theta }}_j^2+\frac{1}{2}\sum _{j=1}^{n-1}{g}_{j0}{\sigma }_j{\theta }_j^2\hfill \\ \hfill & +\sum _{j=1}^n\frac{b_j^2}{2}+\sum _{j=1}^n\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}+{\chi }_n{s}^2+\frac{1}{2g_{n0}}\hfill \end{array}$$

(74)

According to Lemma 1, it would get

$$\begin{array}{l}-{\xi }_j{z}_j^{2p}=-\frac{q(z_j)k_{bj}^{2p}(t)}{k_{bj}^{2p}(t)-z_j^{2p}}-\frac{(1-q(z_j))k_{aj}^{2p}(t)}{k_{aj}^{2p}(t)-z_j^{2p}}\\ \begin{array}{ccc}& & =-q(z_j)\frac{e_{bj}^{2p}}{1-e_{bj}^{2p}}\end{array}-(1-q(z_j))\frac{e_{aj}^{2p}}{1-e_{aj}^{2p}}\\ \begin{array}{ccc}& & \le -q(z_j)\log \frac{1}{1-e_{bj}^{2p}}\end{array}-(1-q(z_j))\frac{1}{1-e_{aj}^{2p}}\end{array}$$

(75)

Let $\mu =min(2p{k}_j{g}_{j0,}{\sigma }_j,{\chi }_n)$ , $\eta =\frac{1}{2}\sum _{j=1}^{n-1}{g}_{j0}{\sigma }_j{\theta }_j^2+\sum _{j=1}^n\frac{b_j^2}{2}+\sum _{j=1}^n\frac{{\overline{\epsilon }}_j^2}{2g_{j0}}+\frac{1}{2g_{n0}}$ , then, (74) can be described as

$${\stackrel{·}{V}}_n\le -\mu V_n+\eta$$

(76)

According to Lemma 2, system in (1) under the proposed control algorithm would be stability and until now, it has finished the detailing designing procedure for suggested control algorithms. Following theorem showcases primary control methodology’s outcomes.

• Theorem 1. Take into account the nonlinear systems depicted in equation (1) with Assumptions 1–2 subject to asymmetric TVSCs of ${\underset{¯}{k}}_{\mathrm{ci}}(t)<x_i<{\overline{k}}_{\mathrm{ci}}(t),(i=1,2,\cdots ,n-1)$ . Through design of VCL(16), (32) and ACL (45), and RBF-NNs weights adaption laws (20), (36), and (47) with the existed sufficiently large compact sets ${\mathrm{\Omega }}_W,{\mathrm{\Omega }}_{\Theta }$ such that $\hat{W}\in {\mathrm{\Omega }}_W,\hat{\Theta }\in {\mathrm{\Omega }}_{\Theta }$ for all $t\ge 0$ , the designed approach guarantees the following properties: 1). all signals within closed-loop system remain within finite bounds; 2). the output signal and $x_i,(i=1,2\cdots ,n-1)$ of the system are never violated; 3). System output tracking error $z_1=x_1-y_d$ fast converges to ${\mathrm{\Omega }}_z$ ( compact sets), which will be defined subsequently.

Proof: Multiplying both sides by $e^{\mu t}$ with (75), then it can be rewritten as $d(V_n(t)e^{\mu t})/\mathrm{dt}\le \eta e^{\mu t}$ . Making time integration over $[0,t]$ , it gets:

$$V_n(t)\le [V_n(0)-\eta /\mu ]e^{-\mu t}+\eta /\mu \le V_n(0)+\eta /\mu$$

(77)

From (61) and (77), we can easily obtain that $\log (1/(1-e_j^{2p}))$ and ${\tilde{\theta }}_j$ remain within finite bounds. Subsequent inequality can be derived

$$\begin{array}{c}\frac{1}{2p}\log \left(\frac{1}{1-e_j^{2p}}\right)\le V_n(t)\le \frac{1}{2p}\sum _{j=1}^n\log \left(\frac{1}{1-e_j^{2p}(0)}\right)\hfill \\ +\frac{1}{2}\sum _{j=1}^n{g}_{j0}{\tilde{\theta }}_j^2(0)+\frac{1}{2}s(0{)}^2\hfill \end{array}$$

(78)

Then, it can yields $e_j^{2p}\le 1-e^{-2\mu V_n(0)}$ and $e_j\le (1-e^{-2\mu V_n(0)}{)}^{1/2p}$ , where $e^{•}$ denotes the exponential function and $e_j^{•}$ was defined as (19). From (19), one can get $-{\underset{¯}{D}}_i(t)<z_i(t)<{\overline{D}}_i(t)$ , where ${\underset{¯}{D}}_i(t)=k_{\mathrm{ai}}(1-e^{-2\mu V_n(0)}{)}^{1/2p}$ and ${\overline{D}}_i(t)=k_{\mathrm{bi}}(1-e^{-2\mu V_n(0)}{)}^{1/2p}$ . Because of ${\tilde{\theta }}_j$ and ${\theta }_j$ are bounded, it gets ${\hat{\theta }}_j={\tilde{\theta }}_j+{\theta }_j$ are bounded. Meanwhile, let ${\Delta }_{j-1}=max\left|{\alpha }_{j-1}\right|$ , from $x_2=z_2+{\alpha }_1$ , then the following inequality ${\Delta }_1-k_{a2}\le x_2\le {\Delta }_1+k_{b2}$ holds. Since $k_{a2}={\Delta }_1-{\underset{¯}{k}}_{c2}$ and $k_{b2}={\overline{k}}_{c2}-{\Delta }_1$ , obviously it has ${\underset{¯}{k}}_{c2}<x_2<{\overline{k}}_{c2}$ , thus $x_2$ state is not transgressed the predefined boundary. Similarly, virtual control algorithms of ${\alpha }_i,(i=2,\cdots ,n-1)$ must be bounded, which are proved by the same proven process. According to recursive inference, if we choose the parameters satisfying the feasibility conditions ${\underset{¯}{k}}_{\mathrm{cj}}(t)<{\Delta }_{j-1}<{\overline{k}}_{\mathrm{cj}}(t)$ , it can be inferred that $x_i,(i=3,\cdots ,n-1)$ satisfying asymmetric TVSCs requirements.

Lastly, NITSMC methodology ensures whole system states deviation converges to zero within finite time and it manifests actual control signal’s boundedness based on the definition of $u(t)$ . Therefore, it can assure all signals of ${\hat{\theta }}_i,x_i,z_i$ and u are bounded.

Until now, this completes the proof.

• Remark 2. One can see that in every step design, although $f_i({\overline{x}}_i)$ , $g_i({\overline{x}}_i),(i=1,2,\cdots ,n)$ are both unknown, we tackle it by $f_i({\overline{x}}_i)-{\stackrel{·}{\alpha }}_{i-1}+\frac{{\xi }_{i-1}}{2p{\xi }_i}{g}_{i-1}({\overline{x}}_{i-1})z_i=F_i(Z_i)={\mathit{W}}_i^{*\mathrm{T}}{H}_i(Z_i)+{\epsilon }_i(Z_i)$ with RBF-NN methodology. There is no excessive computational burden because of the less dimensional increase parameters and input vectors of the RBF-NNs. Moreover, it avoids the defects of “complexity boast” by high order derivation of ${\stackrel{·}{\alpha }}_i$ .

Simulation and results

Simulation examples get investigated to verify feasibility and effectiveness of suggested control technique here.

Simulations and comparison

To further validate control method progressiveness proposed within this research, we conducted a comparative experiment with the method of Li. ²⁵ To ensure a fair comparison, all parameters and models were standardized to match the original literature. The simulation results are depicted in Figures 2 to 7. Based on the aforementioned control design, the simulation control law is presented in Table 1.

Table 1.

Control algorithm in practical applications and simulation.

Control algorithm of proposed methodology	Control algorithm of Li’s method
${\alpha }_1=-({\kappa }_1+{\stackrel{⌢}{K}}_{z1}(t)+{\gamma }_1)z_1-\frac{{\xi }_1{z}_1^{2p-1}{\widehat{\theta }}_1{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}-\frac{{\xi }_1{z}_1^{2p-1}}{2}$ ;	$u=-{\gamma }_{2}s-0.5s-z_1{z}_2/s-s{\hat{\theta }}_2{\phi }_2^T(X_2){\phi }_2(X_2)/4{\zeta }_2$ ;
$u=-{\rho }_{2}s-\frac{s{\hat{\theta }}_2{\Vert H_2(Z_2)\Vert }^2}{2b_2^2}-\frac{s{\eta }^2{\left|z_2\right|}^{2\lambda }}{2}\mathrm{sign}(z_2)$ ;	$\stackrel{·}{\widehat{{\theta }_2}}={\delta }_2(-{\sigma }_2{\hat{\theta }}_2+s{\phi }_2^T(X_2){\phi }_2(X_2)/4{\zeta }_2)$ ;
$\stackrel{·}{\widehat{{\theta }_1}}=\frac{{\xi }_1^2{z}_1^{4p-2}{\Vert H_1(Z_1)\Vert }^2}{2b_1^2}-{\sigma }_1{\hat{\theta }}_1$ ;	${\overline{\gamma }}_1(t)=\sqrt{q(z_1){\left(\frac{{\stackrel{·}{k}}_{b1}(t)}{k_{b1}(t)}\right)}^2+(1-q(z_1)){\left(\frac{{\stackrel{·}{k}}_{a1}(t)}{k_{a1}(t)}\right)}^2+{\gamma }_0}$ ;
$\stackrel{·}{\widehat{{\theta }_2}}=-{\sigma }_2{\hat{\theta }}_2+\frac{s^2{\hat{\theta }}_2{\Vert H_2(Z_2)\Vert }^2}{2b_2^2}$ ; $s=z_2+\eta {\int }_0^t{\left|z_2\right|}^{\lambda }d\tau$ .	$s=z_2+\eta {\int }_0^t{z}_2^{\lambda }d\tau$ .

• Example 1. ⁴⁰ To test suggested controller’s engineering application performance, following robotic manipulator system model described in Ref. ⁴⁰ was utilized as

$$J\stackrel{··}{q}(t)+B\stackrel{·}{q}(t)+\mathrm{MgL}\sin (q(t))=u(t)$$

(79)

given that $q(t)\in \Re$ , $\stackrel{·}{q}(t)\in \Re$ serve as angle and angular velocity of rigid link robot axis. $J\in \Re$ $B\in \Re$ represent servo motor’s axis rotation inertia and damping coefficient. L signifies length between joint axis and mass center, M serves as link mass, g serves as gravitational acceleration, respectively. According to research by Xing et al., ⁴⁰ simulation of the manipulator’s system is given as $J=1$ , $\mathrm{MgL}=10$ , $B=2$ . Control objective is applying designed controller for robot output $q(t)$ fast track required trajectory $q_d(t)=0.5\sin (t)(\mathrm{rad})$ with the TVSCs condition as

$$-0.8+0.2\cos (t)<q(t)-q_d(t)<1+0.4\cos (t)$$

(80)

Figure 2.

Trajectory of robot states subject to time varying constraint:  (a) angle position tracking curves of robot with control algorithms and (b) Angular velocity tracking curves of robot with control algorithms

Figure 3.

Trajectory tracking error of states subject to time varying constraint.

Figure 4.

Phase portrait of tracking error and its derivation: (a) phase portrait of $q_e,{\stackrel{·}{q}}_e$ with proposed algorithms and (b) phase portrait of $q_e,{\stackrel{·}{q}}_e$ by Li’s algorithms

Figure 5.

Weight value estimation of RBFNN of robot.

Figure 6.

NITSMC manifold of robotic manipulator.

Figure 7.

Control signal u(t) of robotic manipulator.

Initial states values serve as $q(0)=0.3(\mathrm{rad})$ , $\stackrel{·}{q}(0)=-0.1(\mathrm{rad}/s)$ , ${\hat{\theta }}_1(0)=0.1$ and ${\hat{\theta }}_2(0)=0.15$ . Controller parameters were chosen as $p=2$ , $k_1=20$ , ${\rho }_2=24$ , ${\gamma }_1=0.75$ , $b_1=1.0$ , $b_2=3.0$ , ${\sigma }_1=7$ , ${\sigma }_2=3$ , ${\beta }_1=0.5$ , ${\beta }_2=0.5$ , $\eta =0.5$ , $\gamma =0.9$ . Figures 2 to 7 illustrated simulating outcomes. According to the above control design, the simulation control laws are obta-ined in Table 1. RBF parameters are selected as the example 2.

Figure 2 shows that both the control algorithms can make the robot output signals precisely track ideal trajectory subjected to the TVSCs domain. Figure 2(b) manifests that the robot speed signal tracks the desired virtual control signal of ${\alpha }_1$ under the proposed controller. From the signal curves, it is evident that, suggested algorithm exhibits superior dynamic and steady performance when compared to Li’s method.

Figure 3 further witness the control performance by the robot position and speed tracking deviation. Although two methods make the position tracking error not transgress off the constraint boundary, suggested control algorithm attains superior control accuracy. Figure 4 depicts the graph for phase portrait of $q_e,{\stackrel{·}{q}}_e$ , The proposed method achieves a smaller convergence range. The RBF-NNs weights adaption laws were depicted within Figure 5. Figure 6 showcases NITSMC manifold. Controller output signals are shown as Figure 7. Referring from aforementioned curves, it is evident all signals remain bounded, and corresponding control performance surpasses that of Li’s method.

• Example 2. ¹⁴ Examine following unknown nonlinear system, given that output must remain within predefined time-varying bound, trajectory tracking deviation should converge to compact set around 0 within a finite time.

$$\{\begin{array}{c}{\stackrel{·}{x}}_1=x_2{e}^{0.5x_1}+(1+x_1)x_2\hfill \\ {\stackrel{·}{x}}_2=x_1{x}_2^2+(3+\cos (x_1))u\hfill \\ y=x_1\hfill \end{array}$$

(81)

$x_1$ , $x_2$ serve as system states variables, u is input and y serves as system output, respectively. $g_1(x_1)=1+x_1$ , $g_2({\overline{x}}_2)=3+\cos (x_1)$ , $f_1(x_1)=x_2{e}^{0.5x_1}$ , $f_2({\overline{x}}_2)=x_1{x}_2^2$ . Example 1’s primary objective is to utilize suggested control methodology to attain targets below: (1) all the closed-loop system signals are bounded; (2) system state of $x_i,(i=1,2\cdots ,n-1)$ meet the TVSCs’ conditions as ${\underset{¯}{k}}_{c1}(t)\le y(t)=x_1(t)\le {\overline{k}}_{c1}(t)$ with ${\overline{k}}_{c1}=1+0.4\cos (t)$ , ${\underset{¯}{k}}_{c1}(t)=-0.8+0.2\cos (t)$ ; (3) Full-state system tracking errors converge to small neighborhood around origin within finite time with SMC methodology. Reference signal is $y_d(t)=0.2\cos (0.5t)+0.5\sin (t)$ .

Initial states values are $x_1(0)=0.1$ , $x_2(0)=1.5$ , ${\hat{\theta }}_1(0)=0.1$ , and ${\hat{\theta }}_2(0)=0.15$ . The controller parameters were set as $p=2$ $k_1=20$ , ${\rho }_2=100$ , $b_1=1.0$ , $b_2=3.0$ , ${\sigma }_1=7$ , ${\sigma }_2=3$ , ${\beta }_1=0.4,{\beta }_2=0.4$ , $\eta =0.5$ , $\gamma =0.9$ . The controller parameters are designed as the Table 1.

In this study, all the RBF-NNs have the uniform form such as (3). The RBF-NNs of ${\hat{W}}_1^{\mathrm{T}}{H}_1(Z_1),{\hat{W}}_2^{\mathrm{T}}{H}_2(Z_2)$ and are all contains 21 nodes (i.e. $l_1=l_2=21$ ), with centers $c_j$ evenly spaced in $[-5,5]$ and width ${\upsilon }_j=1,2,\cdots ,l_j$ , $(j=1,2)$ , in which $Z_1$ and $Z_2$ denote $Z_1=[x_1,y_d,{\stackrel{·}{y}}_d{]}^{\mathrm{T}}$ , $Z_2=[x_1,x_2,y_d,{\stackrel{·}{y}}_d,{\hat{\theta }}_1{]}^{\mathrm{T}}$ . Initial weights ${\hat{W}}_1,{\hat{W}}_2$ were set arbitrarily within range $[-1,1]$ . The controller algorithms are presented as Table 1 and simulating outcomes get illustrated in Figures 8 to 12.

Figure 8.

The tracing trajectory of the system states:  (a) the tracking trajectory $x_1(t)$ with constraints and (b) the tracking trajectory $x_2(t)$

Figure 9.

$z_1,z_2$ trajectories: (a) The trajectory of $z_1$ and (b) the trajectory of $z_2$

Figure 10.

Phase portrait of $z_1,z_2$ .

Figure 11.

Estimation parameter ${\hat{\theta }}_1,{\hat{\theta }}_2$ trajectory.

Figure 12.

NITSMC manifold.

Figure 8 depicted two curves closely coincide, indicating excellent tracking performance. Output-state $y=x_1$ closely follows ideal trajectory $y_d$ with both dynamic and steady performance. Furthermore, the system’s output signal remains strictly in domain of TVSCs. Figure 8(b) manifests that systems state of $x_2$ precisely tracks the desired virtual control signal of ${\alpha }_1$ under proposed controller.

Figure 9 depicts system’s states tracking errors under proposed control methodology. This figure further illustrates that, closed-loop system’s tracking errors do not exceed boundary, and the output trajectory error meet the conditions of $k_{a1}<z_1<k_{b1}$ . Figure 10 is provided to show the graph for phase portrait of $z_1$ , $z_2$ . The system states tracking errors of $z_1(t),z_2(t)$ have not transgressed the time-varying boundary and quickly converge to a small zero compact under the proposed control techniques, when its initial values satisfy the feasibility conditions. Obviously, we can draw the conclusion, that the system output-state is not transgressed off the time-varying boundary, a good tracking performance of the states are obtained, better convergence rate and transient performance can be acquired from Figures 8 to 10.

The RBF-NNs weights adaption laws are illustrated in Figure 11. We can get that the NNs adaption laws signal is bounded, what is more, the system uncertainty challenge is effectively managed by the NNs approximation. Furthermore, the algorithm not only simplifies the adaptation law but also ensures VCL existence and smoothness. Adaptation laws are designed to maintain boundedness for all signals throughout the entire closed-loop system. Figure 12 shows the integral nonsingular terminal sliding mode surface. Figure 13 is the control output signals ${\alpha }_1$ and u of the proposed controller. Figures above validate that, all control signals exhibit smoothness and remain within bounded limits.

Figure 13.

Controller output.

All in all, Figures 1 to 13 collectively demonstrate, all signals within closed-loop system maintain boundedness, and output state consistently adheres to constraint boundary at all times. Concurrently, asymptotically accurate tracking performance is achieved, and trajectory tracking errors converge within a finite time-frame. It’s evident that the signal from the adaptive NN controller signal of u remains moderate, affirming that our approach doesn’t amplify control energy to enhance control performance.

• Remark 3. According to the controller design process analysis and the simulation test, it is clear that (1) the uniform ultimate boundedness of all the signals are guaranteed by ${\kappa }_i$ . Moreover, increasing ${\kappa }_i$ will lead to accelerate the system convergence speed and smaller ${\overline{D}}_i(t)$ and ${\underset{¯}{D}}_i(t)$ , but it will affect the system state following oscillation. (2) ${\rho }_2$ are choosing properly to modify the convergence speed and the steady-state accuracy and η, λ are chosen to modify the sliding mode surface in order to avert singularity problems. (3) $p\in N^{+}$ and it must satisfy $2p\ge n+2$ to guarantee piece-wise Lyapunove function differentiable. (4) ${\sigma }_i$ are required to be chosen as small positive constants to prevent the NN weight estimation from drifting to a large values, which is called σ-modification as Ge and Wang. ¹³

Theoretical comparison

This study focus on the a new adaptive asymmetric sliding mode control methodology with TVSCs to address finite-time trajectory tracking issues in uncertain nonlinear systems. In comparison to the previous works presented in Refs.^23–26 and the latest findings in Refs.,^36,41 the main characters of the proposed control method are:

• (1) For the time varying states constraints, feasibility conditions are key factors in solving constraint problems. This research is conducted under the feasibility hypothesis of Lemma 5, thus it is constrained by the requirement that the initial error remains within the bounds of time-varying constraints. Fortunately, the literature^36,41 presents a constraint solution method that does not require feasibility conditions, which is implemented by the virtual controls with predefined bounds and utilized a bounded function in the output error variable to solve the feasibility condition problem intrinsically BLFs. Compared with Ref., ²³ feasibility conditions are removed by a new coordinate transformation based on a nonlinear state- dependent function method. In summarize, we will carry our future work by the above advanced methodology.

• (2) To enhance the robustness of the control performance, sliding mode control together with states constraint control assures the state consistently stay within the specified boundaries, tracking errors rapidly converge to a confined range. Compared with Refs.,^24–26 a barrier Lyapunov function approach is adopted with the states coordinate transformation and there also exists feasibility conditions, however, this paper can manage the time varying asymmetric states constraint conditions.

• (3) Compared with Refs.,^36,41 this paper only solve the adaptive neural network terminal sliding mode tracking control for uncertain nonlinear systems with TVSCs, but the system model in (1) not contains factors like measurement noise and more significant external disturbances and other nonlinerity like input deadzone and time delaying, therefore, in the next study, we will devote to above problems in Refs.^36,41

Conclusions

This paper introduces an adaptive asymmetric TVSCs SMC methodology aimed at addressing trajectory tracking issues in unknown nonlinear systems. This paper proposes a control methodology utilizing back-stepping technique, which is designed to address trajectory tracking problems in unknown nonlinear systems. The control algorithm encompasses three main aspects. firstly, the RBF-NNs approximation function is utilized to handle uncertain components in controller design, effectively managing unknown parts in nonlinear systems, secondly, the ABLFs are employed to uphold state constraints condition’s satisfaction, thirdly, to improve system states tracking dynamic performance, the NITSMC methodology is explored. This approach ensures that entire system’s states deviation converges to a small zero compact set within a finite time. Moreover, in accordance with ABLFs theorem, it has been demonstrated that all the closed-loop signals remain bounded, asymmetric TVSCs are not violated, tracking error rapidly converges to a small zero compact set. In conclusion, suggested control methodology’s applicability and effectiveness are validated through ideal simulating outcomes. Moving forward, our future efforts will be directed toward addressing full state constraints using SMC with BLFs.