Adaptive tracking control for nonlinear input-delay systems with full state constraints and unmodeled dynamics

Abstract

An innovative adaptive tracking control strategy is proposed in this paper for a class of nonlinear systems, which considers input-delay, full state constraints, and unmodeled dynamics simultaneously. To address the system’s unknown nonlinear dynamics, the approximation ability of multi-dimensional Taylor network (MTN) is employed in the controller design process. The effect of input-delay is reduced through the application of Pade approximation. Additionally, the impact of state constraints is mitigated through the introduction of barrier Lyapunov functions (BLFs). To deal with unmodeled dynamics, a dynamic signal is formulated. By integrating the backstepping control strategy with Lyapunov stability theory, it is ensured that all signals in the closed-loop system remain bounded, the tracking error approaches a small region close to the origin, and the full state constraints are not violated. Finally, simulation results are provided to validate the proposed strategy’s effectiveness.

Keywords

Multi-dimensional Taylor network input-delay full state constraints unmodeled dynamics backstepping control strategy

Introduction

Nonlinear systems are prevalent in numerous practical engineering applications and frequently exhibit complex behaviors that can severely degrade system performance and induce instability. To tackle above challenges, various control strategies, including adaptive control (Kong et al., 2025; Krstić et al., 1992; Wang et al., 2025c; Yang et al., 2025), feedback control (Tsai et al., 2024; Wang et al., 2024a), sliding mode control (Mokhtare et al., 2022; Xiong and Chen, 2025), robust control (Peng et al., 2025; Wang et al., 2024b), fuzzy-model-based control (Qiu et al., 2023; Wang et al., 2024d), and sampled-data control (Mao et al., 2025, 2022, 2024; Ni et al., 2024), have been developed. Among these control strategies, neural networks (NNs) (Aly et al., 2022; Feng et al., 2024; Lv et al., 2025) and backstepping (Wang et al., 2024f; Zhang et al., 2025) are also widely used. However, these methods exhibit a critical limitation in the form of prohibitive computational complexity. Therefore, the multi-dimensional Taylor networks (MTNs) (Hao et al., 2025a, 2025b; Kharrat, 2024; Lu et al., 2024; Wang et al., 2023; Yan and Wang, 2023) have garnered increasing attention due to its low computational complexity, and numerous results have been reported. The authors Yan and Wang (2023) systematically examined the application of MTN to time-varying delay stochastic nonlinear systems. Hao et al. (2025a), Lu et al. (2024), and Kharrat (2024) investigated nonlinear systems under various challenging conditions, including input saturation (Hao et al., 2025a), prescribed performance (Lu et al., 2024), dead-zone output and unmodeled dynamics (Kharrat, 2024). Upon reviewing the existing studies, it is evident that there is still a research gap regarding nonlinear systems that is simultaneously constrained by input-delay, full state constraints, and unmodeled dynamics. Therefore, this paper aims to address this specific research issue.

Time-delay is one of the intricate behaviors in engineering applications, which has the potential to significantly deteriorate system performance and induce instability (Xia et al., 2022). As input-delay systems are a key subclass of time-delay systems, researchers have recently developed diverse adaptive control approaches for nonlinear systems with input-delay (Jain and Katiyar, 2023; Khanesar et al., 2015; Song et al., 2023; Yu et al., 2024; Wang et al., 2024c, 2024e). In Jain and Katiyar (2023), two robust feedback terms were proposed to counteract the input-delay. In Wang et al. (2024e), the impact of input-delay was mitigated through the utilization of an auxiliary signal. In Khanesar et al. (2015), Song et al. (2023), and Yu et al. (2024), the Pade approximation was proposed as another method to address input-delay issues, transforming the original system into a delay-free model. In Wang et al. (2024c), a novel variable designed via the Pade approximation and Laplace transform is introduced to dynamically address the uncertain length of input-delay. While progress has been made in studying nonlinear systems with input-delay, many challenges still persist. For instance, drawing upon existing research, a novel and efficient control strategy needs to be proposed to handle nonlinear systems that are characterized by input-delay, full state constraints, and unmodeled dynamics. This serves as an inspiration for this research.

During the operation of practical control systems, the states of systems are inevitably subject to various constraints, including input constraints (Alattas et al., 2022), output constraints (Kong et al., 2023), and state constraints (Wang et al., 2025b). To enhance the applicability of control strategies, researchers have made significant progress in designing controllers for systems with full state constraints (Li et al., 2024b; Liu et al., 2025; Lü and Shen, 2024). To deal with the output constraints in nonlinear systems, a barrier Lyapunov function (BLF) is constructed in Tee et al. (2009). Subsequently, the BLF method was extended to resolve control problems in nonlinear systems with full state constraints, as detailed in Wang et al. (2025a) and Xu et al. (2025). Based on the current research, when it comes to investigating the tracking control of nonlinear systems with full state constraints, there have been few efforts to solve the control problem that integrate input-delay and unmodeled dynamics. Accordingly, the research in this paper addresses a practical importance.

In practice, unmodeled dynamics are ubiquitous in real-world systems, necessitating effective handling to achieve optimal control performance. Consequently, researching the control of nonlinear systems characterized by unmodeled dynamics is of considerable significance (Han et al., 2021; Jiang and Praly, 1998; Ye et al., 2019; Zhai et al., 2023). In Jiang and Praly (1998), a robust adaptive control strategy was proposed, where a dynamic signal was used for the first time to tackle the unmodeled dynamics issue in nonlinear systems. In subsequent studies, although Ye et al. (2019) put forward adaptive control strategy for nonlinear systems with full state constraints and unmodeled dynamics, it did not consider the possible effects of input-delay. The studies referenced in Zhai et al. (2023) and Han et al. (2021) focused solely on the tracking control problem for nonlinear systems constrained by unmodeled dynamics and input-delay, without full state constraints. In conclusion, the adaptive control strategies discussed previously cannot be directly implemented for the nonlinear systems introduced in this paper.

Given prior research, designing an adaptive tracking control strategy for nonlinear systems with input-delay, full state constraints, and unmodeled dynamics remains largely unexplored and challenging. Hence, this paper puts forward a novel adaptive tracking control strategy for such systems. The principal contributions are summarized as follows:

Unlike traditional approximation methods (e.g., radial basis function neural network (RBFNN) (Li et al., 2004)) that rely on Gaussian functions with inherent computational complexity, the adaptive tracking control strategy proposed in this paper adopts an MTN-based framework. This approach not only simplifies system design but also substantially reduces computational burden while maintaining satisfactory control performance.

This paper introduces a framework combining two methods: The Pade approximation simplifies the analysis of input-delay by approximating complex functions with rational polynomials, while the BLF ensures state constraint satisfaction by incorporating barrier terms into the Lyapunov function. Although input-delay and full state constraints were addressed in prior works (Li et al., 2022; Wang et al., 2021), the proposed controller cannot be directly applied to nonlinear systems with unmodeled dynamics. Moreover, integrating the MTN approximation for nonlinear functions into the system architecture significantly simplifies controller development, thereby greatly enhancing implementation efficiency.

The contributions of this paper are twofold: First, it establishes a general framework for nonlinear input-delay systems with full state constraints and unmodeled dynamics. Second, it addresses previously unresolved challenges in such systems, as outlined below. While previous research (Gao et al., 2025; Xia et al., 2025) has investigated nonlinear systems with unmodeled dynamics and input-delay, their strategies are not directly applicable to controlling nonlinear systems with full state constraints. Similarly, research in Li and Wang (2025) and Yin et al. (2022) has tackled nonlinear systems with unmodeled dynamics and full state constraints but overlooked input-delay. In contrast, this paper effectively bridges the existing gap by proposing a unified framework that simultaneously addresses input-delay, full state constraints, and unmodeled dynamics, thereby offering practical solutions.

This paper demonstrates the effectiveness of the proposed strategy in addressing input-delay, full state constraints, and unmodeled dynamics through numerical and practical simulations. The practical simulations consider a class of single-link manipulator systems incorporating input-delay and unmodeled dynamics. Furthermore, comparative experiments clearly demonstrate the system performance when applying the RBFNN. Consequently, it can be conclusively shown that the proposed strategy is effective not only in theoretical analysis but also in practical applications.

This paper is organized as follows: The Problem formulation and preliminaries section formulates the problem with preliminary knowledge. The Main results section develops the MTN-based adaptive controller and provides stability proof. The Simulation results section validates the strategy through numerical and practical simulations. The final section concludes the work with future research directions.

Problem formulation and preliminaries

System statements

Consider the nonlinear system below, which considers input-delay, full state constraints, and unmodeled dynamics

{\begin{matrix} \overset{\cdot}{ζ} = q (ζ, x) \\ ẋ_{i} = x_{i + 1} + φ_{i} ({\bar{x}}_{i}) + Δ_{i} (x, ζ), 1 \leq i \leq n - 1 \\ ẋ_{n} = u (t - τ) + φ_{n} ({\bar{x}}_{n}) + Δ_{n} (x, ζ) \\ y = x_{1} \end{matrix}

(1)

where the state vector of system is denoted by $x = {[x_{1}, \dots, x_{n}]}^{T}$ $\in R^{n}$ with ${\bar{x}}_{i} = {[x_{1}, \dots, x_{i}]}^{T} \in R^{i}$ . $ζ \in R^{n_{0}}$ denotes the unmodeled dynamics of the system. The system output is $y \in R$ . The control input of the system is $u \in R$ , and τ denotes the input-delay of the system, which is taken as a known constant. $Δ_{i} (x, ζ)$ represents the dynamic disturbances within the system. The smooth function $φ_{i} (\cdot)$ is unknown, which satisfies $φ_{i} (0) = 0$ .

To resolve the challenge posed by input-delay, this paper introduces the Pade approximation method (Khanesar et al., 2015). By approximating the delay term with a rational function, the Pade approximation method converts the delay-dependent system into a delay-free form, which simplifies the control design process.

As mentioned in Song et al. (2023) and Yu et al. (2024), for the control input $u (t - τ)$ in system (1), there exists a new variable $x_{n + 1}$ that satisfies $u (t - τ) = x_{n + 1} (t) - u (t)$ , and so we have

ẋ_{n + 1} = - γ x_{n + 1} + 2 γu

(2)

where $γ = \frac{2}{τ}$ is a constant.

Remark 1. As mentioned in Li et al. (2017), the system maintains stability and satisfies the desired performance specifications when τ is small.

Then, nonlinear system (1) can be reformulated as

{\begin{matrix} \overset{\cdot}{ζ} = q (ζ, x) \\ ẋ_{i} = x_{i + 1} + φ_{i} ({\bar{x}}_{i}) + Δ_{i} (x, ζ), 1 \leq i \leq n - 1 \\ ẋ_{n} = x_{n + 1} - u + φ_{n} ({\bar{x}}_{n}) + Δ_{n} (x, ζ) \\ ẋ_{n + 1} = - γ x_{n + 1} + 2 γu \\ y = x_{1} \end{matrix}

(3)

For system (1), three control objectives of this paper are as follows:

The system output y is capable of tracking the reference signal $y_{d}$ with a small tracking error.

All signals in the closed-loop system are bounded.

All system states stay within the pre-established constraints, where $| x_{i} | < k_{ci}$ , and $k_{ci} > 0$ is a constant for $1 \leq i \leq n$ .

Knowledge preliminaries

To facilitate controller design, some useful lemmas are given below.

Lemma 1. (Jiang and Praly, 1998): For $\overset{\cdot}{ζ} = q (ζ, x)$ , an exponentially input-to-state practically stable (exp-ISpS) Lyapunov function $V (ζ)$ is utilized, which satisfies the following conditions

{\bar{α}}_{1} (∥ ζ ∥) \leq V (ζ) \leq {\bar{α}}_{2} (∥ ζ ∥)

(4)

\frac{∂V (ζ)}{∂ζ} q (ζ, x) \leq - cV (ζ) + {\bar{α}}_{3} (∥ x_{1} ∥) + d_{0}

(5)

where ${\bar{α}}_{1} (\cdot)$ , ${\bar{α}}_{2} (\cdot)$ , ${\bar{α}}_{3} (\cdot)$ are functions belonging to class $κ_{\infty}$ , and $c > 0$ and $d_{0} > 0$ are known constants.

Lemma 2. (Jiang and Praly, 1998):1 Based on Lemma 1, for $\overset{\cdot}{ζ} = q (ζ, x)$ , there is an exp-ISpS Lyapunov function $V (ζ)$ that satisfies inequalities (4) and (5). There exists a limited time $T_{0} = T_{0} (\bar{r}, s_{0}, ζ_{0})$ for the initial value $ζ_{0} = ζ_{0} (0)$ and a signal expressed by

ṡ = - \bar{r} s + {\tilde{α}}_{3} (x_{1}) + d_{0}, s (0) = s_{0}

(6)

such that

V (ζ (t)) \leq s (t) + D (t)

(7)

where $D (t)$ is a non-negative function that holds $D (t) = 0$ . For any $t \geq T_{0}$ , the function ${\tilde{α}}_{3}$ satisfies ${\tilde{α}}_{3} (x_{1}) \geq {\bar{α}}_{3} (| x_{1} |)$ , $s_{0}$ is an arbitrary positive constant, and constant $r_{0}$ satisfies $\bar{r} \in (0, r_{0})$ .

Remark 2. It should be emphasized that Lemma 1 is the statement about the unmodeled dynamics, and Lemma 2 is the statement about the dynamic signal. These Lemmas provide a theoretical foundation for the subsequent controller design.

Lemma 3. (Young’s inequality) (Deng and Krstić, 1997): Given that $\forall (x, y) \in R^{2}$ , the inequality stated below is true

xy \leq \frac{ε^{p}}{p} {| x |}^{p} + \frac{1}{q ε^{q}} {| y |}^{q}

(8)

where constants $ε > 0$ , $p > 1$ , $q > 1$ , $(p - 1) (q - 1) = 1$ .

Lemma 4. (Jiang and Praly, 1998): For any $x \in R$ and $\tilde{ε} > 0$ , the smooth function $f_{x} (x) : R \to R$ fulfills the condition $| x | \leq x f_{x} (x) + \tilde{ε}$ with $f_{x} (0) = 0$ .

Lemma 5. (Jiang and Praly, 1998): $f (\cdot) : R \to R$ is any given function that maintains continuity with $f (0) = 0$ . Therefore, the inequality stated below is true

| f (x) | \leq \hat{f} (x) + \hat{ε}, \forall x \in R

(9)

where $\hat{f} (\cdot)$ denotes a non-negative smooth function that fulfills the conditions $\hat{f} (0) = 0$ and ${\frac{\partial \hat{f}}{∂x} |}_{x = 0} = 0$ , where $\hat{ε} > 0$ is a constant.

Lemma 6. (Li et al., 2022): For any constant $k_{di} > 0$ and any $z_{i} \in R$ , if $| z_{i} | < k_{di}$ holds, the inequality stated below is true

\log \frac{k_{di}^{2 p}}{k_{di}^{2 p} - z_{i}^{2 p}} < \frac{z_{i}^{2 p}}{k_{di}^{2 p} - z_{i}^{2 p}}

(10)

where p is a positive constant.

Remark 3. Lemma 3 is specifically employed to tackle the cross-terms in the derivative of the Lyapunov function. Lemma 4 and Lemma 5 are dedicated to addressing the unmodeled dynamics within the control system to compensate for the unknown parts of the system. Lemma 6 is utilized in combination with the BLF to effectively manage state constraints. These lemmas are relatively standard and commonly found in the existing literature.

To simplify the controller design process, the assumptions listed below are provided.

Assumption 1. (Han et al., 2021; Li et al., 2019): The given reference signal $y_{d}$ and its n-th derivative in relation to t are both continuously bounded and satisfies $| y_{d} | \leq Y_{0} \leq k_{c 1}$ , $| ẏ_{d} | \leq Y_{1}$ ,…, $| y_{d}^{(n)} | \leq Y_{n}$ , $\forall t \geq 0$ , where $Y_{0}$ , $Y_{1}$ , …, $Y_{n}$ are constants.

Assumption 2. (Wang et al., 2017): The nonlinear dynamic disturbances $Δ_{i} (x, ζ)$ with $1 \leq i \leq n$ fulfill the inequality mentioned below

| Δ_{i} (x, ζ) | \leq ϕ_{i 1} (∥ {\bar{x}}_{i} ∥) + ϕ_{i 2} (∥ ζ ∥)

(11)

where $ϕ_{i 1} (\cdot)$ and $ϕ_{i 2} (\cdot)$ represent non-negative increasing functions that are smooth but unknown, while $ϕ_{i 2} (\cdot)$ satisfies $ϕ_{i 2} (0) = 0$ . $∥ \cdot ∥$ denotes the Euclidean norm of.

Assumption 3. (Han et al., 2021): The smooth function $\bar{α} (\cdot)$ is non-negative, and ${\tilde{α}}_{3} (\cdot)$ satisfies ${\tilde{α}}_{3} (m) = {(m - y_{d})}^{2} \bar{α} (m^{2})$ .

Remark 4. Assumption 1 is an indispensable assumption concerning reference signals within the framework of backstepping control design. Assumption 2 pertains to dynamic disturbances. Assumption 3 defines a function to handle the dynamic signals in Lemma 2. These Assumptions form an integral part of the theoretical foundation for subsequent research.

Multi-dimensional Taylor network

This paper utilizes MTNs for approximating the unknown nonlinear functions encountered in the controller design process. The structure of the MTN, which is a three-layer feedforward NN, is illustrated in Figure 1.

Lemma 7. (Hao et al., 2025b; He et al., 2024): If the continuous function $f (Z) : R^{n} \to R$ is defined on the compact set $Ω_{Z} \subset R^{q}$ , then it will be approximated using the specified MTN as described below

f (Z) = θ^{T} Φ (Z)

(12)

where $Z \in Ω_{Z} \subset R^{q}$ denotes the input vector, the input dimension of MTN is denoted by q, the weight vector is denoted by $θ = {[θ_{1}, θ_{2}, . . ., θ_{l}]}^{T} \in R^{l}$ , $l > 1$ is the quantity of nodes within the network, $Φ (Z) = {[Φ_{1} (Z), Φ_{2} (Z), . . ., Φ_{l} (Z)]}^{T}$ represents the vector of basic functions, and choose $Φ_{i} (Z)$ to be the frequently utilized Gaussian functions. In this case, MTN has the capability to estimate any continuous function $f (Z)$ within the compact set $Ω_{Z} \subset R^{q}$ . For any precision $ς > 0$ , the following condition is met

f (Z) = θ^{T} Φ (Z) + δ (Z), \forall Z \in Ω_{Z} \subset R^{q}

(13)

where the approximation error satisfies $| δ (Z) | \leq ς$ .

Main results

Controller design

To design the adaptive controller, the subsequent transformation of coordinates is performed

{\begin{matrix} z_{1} = y - y_{d} \\ z_{i} = x_{i} - α_{i - 1}, 2 \leq i \leq n - 1 \\ z_{n} = x_{n} - α_{n - 1} + \frac{1}{γ} x_{n + 1} \end{matrix}

(14)

where $α_{i - 1}$ represents an intermediary virtual control signal, which will subsequently be developed through the backstepping method.

Step 1: Since $z_{1} = y - y_{d}$ , its derivative in relation to t can be formulated as $ż_{1} = x_{2} + φ_{1} + Δ_{1} - ẏ_{d}$ . Then, the Lyapunov function $V_{1}$ is selected as follows

V_{1} = \bar{V} + \frac{1}{λ_{0}} s + \frac{1}{2} {\tilde{θ}}_{1}^{T} {\tilde{θ}}_{1}

(15)

with $\bar{V} = \frac{1}{2} \log \frac{k_{d 1}^{2}}{k_{d 1}^{2} - z_{1}^{2}}$ , $k_{d 1} = k_{c 1} - Y_{0}$ , $λ_{0}$ is a positive design parameter, and $z_{1}$ satisfies $| z_{1} | < k_{d 1}$ . It is obvious that when $z_{1}$ satisfies $| z_{1} | = k_{d 1}$ , the barrier Lyapunov candidate function escapes to infinity. Therefore, $\bar{V}$ is positive definite.

Differentiating $V_{1}$ with respect to t yields

{\overset{\cdot}{V}}_{1} = \overset{\cdot}{\bar{V}} + \frac{1}{λ_{0}} ṡ - {\tilde{θ}}_{1}^{T} {\overset{\cdot}{\hat{θ}}}_{1}

(16)

where the parameter estimation error is denoted by ${\tilde{θ}}_{1} = θ_{1} - {\hat{θ}}_{1}$ , and substituting it into (16) and according to Lemma 2 and Assumption 3, one has

\begin{matrix} {\overset{\cdot}{V}}_{1} & = \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} (x_{2} + φ_{1} - ẏ_{d}) + \frac{z_{1} Δ_{1}}{k_{d 1}^{2} - z_{1}^{2}} \\ + \frac{1}{λ_{0}} (z_{1}^{2} \bar{α} (| x_{1} |^{2}) + d_{0}) - \frac{\bar{r}}{λ_{0}} s - {\tilde{θ}}_{1}^{T} {\overset{\cdot}{\hat{θ}}}_{1} \end{matrix}

(17)

Using Assumption 2 as a foundation, the subsequent inequality can be derived

\frac{z_{1} Δ_{1}}{k_{d 1}^{2} - z_{1}^{2}} \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} ϕ_{11} (∥ {\bar{x}}_{1} ∥) + \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} ϕ_{12} (∥ ζ ∥)

(18)

where $ϕ_{11} (\cdot)$ and $ϕ_{12} (\cdot)$ are unknown non-negative increasing smooth functions with $ϕ_{11} (0) = 0$ .

According to Lemma 4 and Lemma 5, the following inequalities are established

\frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} ϕ_{11} (∥ {\bar{x}}_{1} ∥) \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} {\tilde{ϕ}}_{11} (∥ {\bar{x}}_{1} ∥) + ε_{11}

(19)

\begin{matrix} \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} ϕ_{12} (∥ ζ ∥) & \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} {\tilde{ϕ}}_{12} (z_{1}, s) + 2 ε_{12} \\ + \frac{1}{4} \frac{z_{1}^{2}}{{(k_{d 1}^{2} - z_{1}^{2})}^{2}} + h_{1} (t) \end{matrix}

(20)

where ${\tilde{ϕ}}_{11} (\cdot)$ and ${\tilde{ϕ}}_{12} (z_{1}, s)$ are smooth functions that satisfy ${\tilde{ϕ}}_{11} (0) = 0$ and ${\tilde{ϕ}}_{12} (z_{1}, s) = {\hat{ϕ}}_{14} (z_{1}) + {\hat{ϕ}}_{12} (s) {\hat{ϕ}}_{13} (z_{1}, s)$ , and there are also smooth functions that ${\hat{ϕ}}_{12} (\cdot) = {\hat{ϕ}}_{13} (\cdot) = {\hat{ϕ}}_{14} (\cdot) = 0$ . At the same time, for any $t \geq 0$ , $h_{1} (t) \geq 0$ . If $t \geq T_{0}$ , then $h_{1} (t) = 0$ .

Replacing (17) with (18), (19), and (20) and substituting $x_{2} = z_{2} + α_{1}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} (z_{2} + α_{1} + φ_{1} - ẏ_{d} + {\tilde{ϕ}}_{11} (∥ {\bar{x}}_{1} ∥) \\ + {\tilde{ϕ}}_{12} (z_{1}, s) + \frac{1}{4} \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}}) + h_{1} (t) - {\tilde{θ}}_{1}^{T} {\overset{\cdot}{\hat{θ}}}_{1} \\ + \frac{1}{λ_{0}} (z_{1}^{2} \bar{α} ({| x_{1} |}^{2}) + d_{0}) - \frac{\bar{r}}{λ_{0}} s + ε_{11} + 2 ε_{12} \end{matrix}

(21)

From Lemma 3, we have

\frac{z_{1} z_{2}}{k_{d 1}^{2} - z_{1}^{2}} \leq \frac{z_{1}^{2}}{2 {(k_{d 1}^{2} - z_{1}^{2})}^{2}} + \frac{z_{2}^{2}}{2}

(22)

Substituting (22) into (21), one has

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} (α_{1} + {\bar{φ}}_{1}) - \frac{z_{1}^{2}}{2 {(k_{d 1}^{2} - z_{1}^{2})}^{2}} + \frac{1}{λ_{0}} d_{0} \\ - \frac{\bar{r}}{λ_{0}} s + ε_{11} + 2 ε_{12} + h_{1} (t) + \frac{z_{2}^{2}}{2} - {\tilde{θ}}_{1}^{T} {\overset{\cdot}{\hat{θ}}}_{1} \end{matrix}

(23)

where ${\bar{φ}}_{1} = φ_{1} - ẏ_{d} + {\tilde{ϕ}}_{11} (∥ {\bar{x}}_{1} ∥) + {\tilde{ϕ}}_{12} (z_{1}, s) + \frac{5 z_{1}}{4 (k_{d 1}^{2} - z_{1}^{2})} + \frac{1}{λ_{0}} z_{1} (k_{d 1}^{2} - z_{1}^{2}) \bar{α} ({| x_{1} |}^{2})$ .

Based on Lemma 7, for any given constant $ς_{1} > 0$ , there is an MTN in the form of $θ_{1}^{T} Φ_{1} (z_{1})$ that fulfills the following equality

{\bar{φ}}_{1} = θ_{1}^{T} Φ_{1} (z_{1}) + δ_{1} (z_{1}), | δ_{1} (z_{1}) | \leq ς_{1}

(24)

where $z_{1} = {[z_{1}]}^{T}$ , and $δ_{1} (z_{1})$ denotes the estimate error.

From Lemma 3, we have

\frac{z_{1} δ_{1} (z_{1})}{k_{d 1}^{2} - z_{1}^{2}} \leq \frac{z_{1}^{2}}{2 {(k_{d 1}^{2} - z_{1}^{2})}^{2}} + \frac{ς_{1}^{2}}{2}

(25)

Combining (24) and (25) with (23), one has

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} (α_{1} + θ_{1}^{T} Φ_{1} (z_{1})) + \frac{1}{λ_{0}} d_{0} - \frac{\bar{r}}{λ_{0}} s \\ + ε_{11} + 2 ε_{12} + h_{1} (t) + \frac{z_{2}^{2}}{2} + \frac{ς_{1}^{2}}{2} - {\tilde{θ}}_{1}^{T} {\overset{\cdot}{\hat{θ}}}_{1} \end{matrix}

(26)

The virtual control signal $α_{1}$ and the adaptive law ${\overset{\cdot}{\hat{θ}}}_{1}$ are chosen in the following way

α_{1} = - c_{1} z_{1} - {\hat{θ}}_{1}^{T} Φ_{1} (z_{1})

(27)

{\overset{\cdot}{\hat{θ}}}_{1} = - η_{1} {\hat{θ}}_{1} + \frac{z_{1}}{k_{d 1}^{2} - z_{1}^{2}} Φ_{1} (z_{1})

(28)

where $c_{1} > 0$ and $η_{1} > 0$ are constants.

Based on Lemma 6, by combining (27) and (28) into (26), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - c_{1} \log \frac{k_{d 1}^{2}}{k_{d 1}^{2} - z_{1}^{2}} + η_{1} {\tilde{θ}}_{1}^{T} {\hat{θ}}_{1} + \frac{1}{λ_{0}} d_{0} \\ - \frac{\bar{r}}{λ_{0}} s + h_{1} (t) + \frac{z_{2}^{2}}{2} + d_{1} \end{matrix}

(29)

where $d_{1} = ε_{11} + 2 ε_{12} + \frac{ς_{1}^{2}}{2}$ .

Step 2: Since $z_{2} = x_{2} - α_{1}$ , its derivative in relation to t can be formulated as $ż_{2} = x_{3} + φ_{2} + Δ_{2} - {\overset{\cdot}{α}}_{1}$ , where ${\overset{\cdot}{α}}_{1} = \frac{\partial α_{1}}{\partial y_{d}} ẏ_{d} + \frac{\partial α_{1}}{\partial x_{1}} (x_{2} + φ_{1} + Δ_{1}) + \frac{\partial α_{1}}{\partial {\hat{θ}}_{1}} {\overset{\cdot}{\hat{θ}}}_{1}$ .

Then, the Lyapunov function $V_{2}$ is chosen as presented below

V_{2} = V_{1} + \frac{1}{2} {\tilde{θ}}_{2}^{T} {\tilde{θ}}_{2} + \frac{1}{2} \log \frac{k_{d 2}^{2}}{k_{d 2}^{2} - z_{2}^{2}}

(30)

Taking the time derivative of $V_{2}$ , one has

{\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{1} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} (x_{3} + φ_{2} + Δ_{2} - {\overset{\cdot}{α}}_{1}) - {\tilde{θ}}_{2}^{T} {\overset{\cdot}{\hat{θ}}}_{2}

(31)

where ${\tilde{θ}}_{2} = θ_{2} - {\hat{θ}}_{2}$ .

Supposing ${\bar{Δ}}_{2} = Δ_{2} - \frac{\partial α_{1}}{\partial x_{1}} Δ_{1}$ , then according to Assumption 2, one has

\begin{matrix} | {\bar{Δ}}_{2} | & \leq (ϕ_{21} (∥ {\bar{x}}_{2} ∥) + | \frac{\partial α_{1}}{\partial x_{1}} | ϕ_{11} (∥ {\bar{x}}_{1} ∥)) \\ + (ϕ_{22} (∥ ζ ∥) + | \frac{\partial α_{1}}{\partial x_{1}} | ϕ_{12} (∥ ζ ∥)) \end{matrix}

(32)

where $ϕ_{21} (\cdot)$ and $ϕ_{22} (\cdot)$ are unknown non-negative increasing smooth functions with $ϕ_{21} (0) = 0$ .

According to Lemma 4 and Lemma 5, we have

\begin{matrix} | \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} | (ϕ_{21} (∥ {\bar{x}}_{2} ∥) + | \frac{\partial α_{1}}{\partial x_{1}} | ϕ_{11} (∥ {\bar{x}}_{1} ∥)) \\ \leq \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} {\tilde{ϕ}}_{21} + ε_{21} \end{matrix}

(33)

\begin{matrix} | \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} | (ϕ_{22} (∥ ζ ∥) + | \frac{\partial α_{1}}{\partial x_{1}} | ϕ_{12} (∥ ζ ∥)) \leq h_{2} (t) \\ + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} {\tilde{ϕ}}_{22} + 4 ε_{22} + \frac{1}{4} \frac{z_{2}^{2}}{{(k_{d 2}^{2} - z_{2}^{2})}^{2}} ι_{1} \end{matrix}

(34)

where $ι_{1} = 1 + \sum_{k = 1}^{1} {(\frac{\partial α_{1}}{\partial x_{k}})}^{2}$ .

Combining (32), (33), and (34) with (31), one has

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq {\overset{\cdot}{V}}_{1} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} (x_{3} + φ_{2}^{*}) + ε_{21} \\ + 4 ε_{22} + h_{2} (t) - {\tilde{θ}}_{2}^{T} {\overset{\cdot}{\hat{θ}}}_{2} \end{matrix}

(35)

where $φ_{2}^{*} = φ_{2} + {\tilde{ϕ}}_{21} + {\tilde{ϕ}}_{22} + \frac{1}{4} \frac{z_{2}}{k_{d 2}^{2}, - z_{2}^{2}} ι_{1} - \frac{\partial α_{1}}{\partial y_{d}} ẏ_{d} - \frac{\partial α_{1}}{\partial x_{1}}$ $(x_{2} + φ_{1}) - \frac{\partial α_{1}}{\partial {\hat{θ}}_{1}} {\overset{\cdot}{\hat{θ}}}_{1}$ .

From Lemma 3, we have

\frac{z_{2} z_{3}}{k_{d 2}^{2} - z_{2}^{2}} \leq \frac{z_{2}^{2}}{2 {(k_{d 2}^{2} - z_{2}^{2})}^{2}} + \frac{z_{3}^{2}}{2}

(36)

Substituting (36) into (35) and considering $x_{3} = z_{3} + α_{2}$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq {\overset{\cdot}{V}}_{1} + ε_{21} + 4 ε_{22} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} (α_{2} + {\bar{φ}}_{2}) \\ - \frac{z_{2}^{2}}{2 {(k_{d 2}^{2} - z_{2}^{2})}^{2}} + \frac{z_{3}^{2}}{2} + h_{2} (t) - {\tilde{θ}}_{2}^{T} {\overset{\cdot}{\hat{θ}}}_{2} \end{matrix}

(37)

where ${\bar{φ}}_{2} = φ_{2}^{*} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}}$ .

Based on Lemma 7, an MTN of the form $θ_{2}^{T} Φ_{2} (z_{2})$ can approximate ${\bar{φ}}_{2}$ . Thus, for any constant $ς_{2} > 0$ , the following equality is true

{\bar{φ}}_{2} = θ_{2}^{T} Φ_{2} (z_{2}) + δ_{2} (z_{2}), | δ_{2} (z_{2}) | \leq ς_{2}

(38)

where $z_{2} = {[z_{1}, z_{2}]}^{T}$ , and $δ_{2} (z_{2})$ denotes the estimate error.

From Lemma 3, we have

\frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} δ_{2} (z_{2}) \leq \frac{z_{2}^{2}}{2 {(k_{d 2}^{2} - z_{2}^{2})}^{2}} + \frac{ς_{2}^{2}}{2}

(39)

Combining (38) and (39) with (37), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq {\overset{\cdot}{V}}_{1} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} (α_{2} + θ_{2}^{T} Φ_{2} (z_{2})) + \frac{z_{3}^{2}}{2} + \frac{ς_{2}^{2}}{2} \\ + ε_{21} + 4 ε_{22} + h_{2} (t) - {\tilde{θ}}_{2}^{T} {\overset{\cdot}{\hat{θ}}}_{2} \end{matrix}

(40)

Selecting the virtual control signal $α_{2}$ and the adaptive law ${\overset{\cdot}{\hat{θ}}}_{2}$ as follows

α_{2} = - c_{2} z_{2} - {\hat{θ}}_{2}^{T} Φ_{2} (z_{2})

(41)

{\overset{\cdot}{\hat{θ}}}_{2} = - η_{2} {\hat{θ}}_{2} + \frac{z_{2}}{k_{d 2}^{2} - z_{2}^{2}} Φ_{2} (z_{2})

(42)

where $c_{2} > 0$ and $η_{2} > 0$ are constants.

Based on Lemma 6 and combining (29), (41), and (42) into (40), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq - \sum_{i = 1}^{2} c_{i} \log \frac{k_{di}^{2}}{k_{di}^{2} - z_{i}^{2}} + \sum_{i = 1}^{2} h_{i} (t) + \frac{1}{λ_{0}} d_{0} \\ + \sum_{i = 1}^{2} η_{i} {\tilde{θ}}_{i}^{T} {\hat{θ}}_{i} + \sum_{i = 1}^{2} d_{i} - \frac{\bar{r}}{λ_{0}} s + z_{2} z_{3} \end{matrix}

(43)

where $d_{2} = ε_{21} + 4 ε_{22} + \frac{ς_{2}^{2}}{2}$ .

Step $3 \leq i \leq n$ : Since $z_{i} = x_{i} - α_{i - 1}$ , its derivative in relation to t can be formulated as $ż_{i} = x_{i + 1} + φ_{i} + Δ_{i} - {\overset{\cdot}{α}}_{i - 1}$ , where ${\overset{\cdot}{α}}_{i - 1} = \sum_{k = 0}^{i - 1} \frac{\partial α_{i - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} + \sum_{k = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial {\hat{θ}}_{k}} {\overset{\cdot}{\hat{θ}}}_{k} + \sum_{k = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial x_{k}}$ $(x_{k + 1} + φ_{k} + Δ_{k})$ .

Choosing the Lyapunov function $V_{i}$ as follows

V_{i} = V_{i - 1} + \frac{1}{2} {\tilde{θ}}_{i}^{T} {\tilde{θ}}_{i} + \frac{1}{2} \log \frac{k_{di}^{2}}{k_{di}^{2} - z_{i}^{2}}

(44)

Differentiating $V_{i}$ with respect to t, we have

{\overset{\cdot}{V}}_{i} = {\overset{\cdot}{V}}_{i - 1} + \frac{z_{i}}{k_{di}^{2} - z_{i}^{2}} (x_{i + 1} + φ_{i} + Δ_{i} - {\overset{\cdot}{α}}_{i - 1}) - {\tilde{θ}}_{i}^{T} {\overset{\cdot}{\hat{θ}}}_{i}

(45)

where ${\tilde{θ}}_{i} = θ_{i} - {\hat{θ}}_{i}$ .

Supposing ${\bar{Δ}}_{i} = Δ_{i} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial x_{k}} Δ_{k}$ , then according to Assumption 2, one has

\begin{matrix} | {\bar{Δ}}_{i} | & \leq (ϕ_{i 1} (∥ {\bar{x}}_{i} ∥) + \sum_{k = 1}^{i - 1} | \frac{\partial α_{i - 1}}{\partial x_{k}} | ϕ_{k 1} (∥ {\bar{x}}_{i} ∥)) \\ + (ϕ_{i 2} (∥ ζ ∥) + \sum_{k = 1}^{i - 1} | \frac{\partial α_{i - 1}}{\partial x_{k}} | ϕ_{k 2} (∥ ζ ∥)) \end{matrix}

(46)

where $ϕ_{i 1} (\cdot)$ and $ϕ_{i 2} (\cdot)$ are unknown non-negative increasing smooth functions with $ϕ_{i 1} (0) = 0$ .

Subsequently, by replicating step 2, we have

\begin{matrix} {\overset{\cdot}{V}}_{i} & \leq {\overset{\cdot}{V}}_{i - 1} + \frac{z_{i}}{k_{di}^{2} - z_{i}^{2}} (α_{i} + θ_{i}^{T} Φ_{i} (z_{i})) + \frac{z_{i + 1}^{2}}{2} \\ + \frac{ς_{i}^{2}}{2} + ε_{i 1} + 2 i ε_{i 2} + h_{i} (t) - {\tilde{θ}}_{i}^{T} {\overset{\cdot}{\hat{θ}}}_{i} \end{matrix}

(47)

The selection of the virtual control signal $α_{i}$ and the adaptive law ${\overset{\cdot}{\hat{θ}}}_{i}$ are detailed below

α_{i} = - c_{i} z_{i} - {\hat{θ}}_{i}^{T} Φ_{i} (z_{i})

(48)

{\overset{\cdot}{\hat{θ}}}_{i} = - η_{i} {\hat{θ}}_{i} + \frac{z_{i}}{k_{di}^{2} - z_{i}^{2}} Φ_{i} (z_{i})

(49)

where $c_{i} > 0$ and $η_{i} > 0$ with $3 \leq i \leq n - 1$ both are constants.

According to Lemma 6 and substituting (43), (48), and (49) into (47), one has

\begin{matrix} {\overset{\cdot}{V}}_{i} & \leq - \sum_{k = 1}^{i} c_{k} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} + \sum_{k = 1}^{i} h_{k} (t) + \frac{1}{λ_{0}} d_{0} \\ + \sum_{k = 1}^{i} η_{k} {\tilde{θ}}_{k}^{T} {\hat{θ}}_{k} + \sum_{k = 1}^{i} d_{k} - \frac{\bar{r}}{λ_{0}} s + z_{i} z_{i + 1} \end{matrix}

(50)

where $d_{i} = ε_{i 1} + 2 i ε_{i 2} + \frac{ς_{i}^{2}}{2}$ .

Step n: Since $z_{n} = x_{n} - α_{n - 1} + \frac{1}{γ} x_{n + 1}$ and equation (2), the derivative of $z_{n}$ in relation to t can be formulated as $ż_{n} = u + φ_{n} + Δ_{n} - {\overset{\cdot}{α}}_{n - 1}$ , where ${\overset{\cdot}{α}}_{n - 1} = \sum_{k = 0}^{n - 1} \frac{\partial α_{n - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} +$ $\sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial {\hat{θ}}_{k}} {\overset{\cdot}{\hat{θ}}}_{k} + \sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial x_{k}} (x_{k + 1} + φ_{k} + Δ_{k})$ .

The Lyapunov function $V_{n}$ is chosen as presented below

V_{n} = V_{n - 1} + \frac{1}{2} {\tilde{θ}}_{n}^{T} {\tilde{θ}}_{n} + \frac{1}{2} \log \frac{k_{dn}^{2}}{k_{dn}^{2} - z_{n}^{2}}

(51)

Taking the time derivative of $V_{n}$ , one has

{\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} (u + φ_{n} + Δ_{n} - {\overset{\cdot}{α}}_{n - 1}) - {\tilde{θ}}_{n}^{T} {\overset{\cdot}{\hat{θ}}}_{n}

(52)

where the parameter estimation error is denoted by ${\tilde{θ}}_{n} = θ_{n} - {\hat{θ}}_{n}$ .

Supposing ${\bar{Δ}}_{n} = Δ_{n} - \sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial x_{k}} Δ_{k}$ , then according to Assumption 2, one has

\begin{matrix} | {\bar{Δ}}_{n} | & \leq (ϕ_{n 1} (∥ {\bar{x}}_{n} ∥) + \sum_{k = 1}^{n - 1} | \frac{\partial α_{n - 1}}{\partial x_{k}} | ϕ_{k 1} (∥ {\bar{x}}_{n} ∥)) \\ + (ϕ_{n 2} (∥ ζ ∥) + \sum_{k = 1}^{n - 1} | \frac{\partial α_{n - 1}}{\partial x_{k}} | ϕ_{k 2} (∥ ζ ∥)) \end{matrix}

(53)

where the unknown non-negative increasing smooth functions are denoted by $ϕ_{n 1} (\cdot)$ and $ϕ_{n 2} (\cdot)$ with $ϕ_{n 1} (0) = 0$ .

According to Lemma 4 and Lemma 5, the following inequalities are established

\begin{matrix} | \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} | (ϕ_{n 1} (∥ {\bar{x}}_{n} ∥) + \sum_{k = 1}^{n - 1} | \frac{\partial α_{n - 1}}{\partial x_{k}} | ϕ_{k 1} (∥ {\bar{x}}_{n} ∥)) \\ \leq \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} {\tilde{ϕ}}_{n 1} + ε_{n 1} \end{matrix}

(54)

\begin{matrix} | \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} | (ϕ_{n 2} (∥ ζ ∥) + \sum_{k = 1}^{n - 1} | \frac{\partial α_{n - 1}}{\partial x_{k}} | ϕ_{k 2} (∥ ζ ∥)) \\ \leq h_{n} (t) + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} {\tilde{ϕ}}_{n 2} + 2 n ε_{n 2} + \frac{1}{4} \frac{z_{n}^{2}}{{(k_{dn}^{2} - z_{n}^{2})}^{2}} ι_{n - 1} \end{matrix}

(55)

where $ι_{n - 1} = 1 + \sum_{k = 1}^{n - 1} {(\frac{\partial α_{n - 1}}{\partial x_{k}})}^{2}$ .

Combining (53), (54), and (55) with (52), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq {\overset{\cdot}{V}}_{n - 1} + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} (u + φ_{n}^{*}) + ε_{n 1} \\ + 2 n ε_{n 2} + h_{n} (t) - {\tilde{θ}}_{n}^{T} {\overset{\cdot}{\hat{θ}}}_{n} \end{matrix}

(56)

where $φ_{n}^{*} = φ_{n} - \sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial x_{k}} (x_{k + 1} + φ_{k}) - \sum_{k = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial {\hat{θ}}_{k}} {\overset{\cdot}{\hat{θ}}}_{k} - \sum_{k = 0}^{n - 1} \frac{\partial α_{n - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} + \frac{1}{4} \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} ι_{n - 1} + {\tilde{ϕ}}_{n 1} + {\tilde{ϕ}}_{n 2}$

Substituting ${\overset{\cdot}{V}}_{n - 1}$ into (56), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq - \sum_{k = 1}^{n - 1} c_{k} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} + \sum_{k = 1}^{n - 1} h_{k} (t) + ε_{n 1} + 2 n ε_{n 2} \\ + \sum_{k = 1}^{n - 1} η_{k} {\tilde{θ}}_{k}^{T} {\hat{θ}}_{k} + \sum_{k = 1}^{n - 1} d_{k} + \frac{1}{λ_{0}} d_{0} - \frac{\bar{r}}{λ_{0}} s + h_{n} (t) \\ + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} (u + {\bar{φ}}_{n}) - \frac{z_{n}^{2}}{2 {(k_{dn}^{2} - z_{n}^{2})}^{2}} - {\tilde{θ}}_{n}^{T} {\overset{\cdot}{\hat{θ}}}_{n} \end{matrix}

(57)

where ${\bar{φ}}_{n} = φ_{n}^{*} + (k_{dn}^{2} - z_{n}^{2}) z_{n - 1} + \frac{1}{2} \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}}$ .

Utilizing Lemma 7 as a foundation, for any constant $ς_{n} > 0$ , an MTN exists that fulfills the following equality

{\bar{φ}}_{n} = θ_{n}^{T} Φ_{n} (z_{n}) + δ_{n} (z_{n}), | δ_{n} (z_{n}) | \leq ς_{n}

(58)

where $z_{n} = {[z_{1}, . . ., z_{n}]}^{T}$ , and the estimate error is denoted by $δ_{n} (z_{n})$ .

From Lemma 3, we have

\frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} δ_{n} (z_{n}) \leq \frac{z_{n}^{2}}{2 {(k_{dn}^{2} - z_{n}^{2})}^{2}} + \frac{ς_{n}^{2}}{2}

(59)

Combining (58) and (59) with (57), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq - \sum_{k = 1}^{n - 1} c_{i} \log \frac{k_{di}^{2}}{k_{di}^{2} - z_{i}^{2}} + \sum_{k = 1}^{n - 1} h_{k} (t) + ε_{n 1} + 2 n ε_{n 2} \\ + \sum_{k = 1}^{n - 1} η_{k} {\tilde{θ}}_{k}^{T} {\hat{θ}}_{k} + \sum_{k = 1}^{n - 1} d_{k} + \frac{1}{λ_{0}} d_{0} - \frac{\bar{r}}{λ_{0}} s + h_{n} (t) \\ + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} (u + θ_{n}^{T} Φ_{n} (z_{n})) + \frac{ς_{n}^{2}}{2} - {\tilde{θ}}_{n}^{T} {\overset{\cdot}{\hat{θ}}}_{n} \end{matrix}

(60)

The selection of the actual control input u and the adaptive law ${\overset{\cdot}{\hat{θ}}}_{n}$ are detailed below

u = - c_{n} z_{n} - {\hat{θ}}_{n}^{T} Φ_{n} (z_{n})

(61)

{\overset{\cdot}{\hat{θ}}}_{n} = - η_{n} {\hat{θ}}_{n} + \frac{z_{n}}{k_{dn}^{2} - z_{n}^{2}} Φ_{n} (z_{n})

(62)

where $c_{n} > 0$ and $η_{n} > 0$ are constants.

Based on Lemma 6, by combining (61) and (21) into (60), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} & \leq - \sum_{k = 1}^{n} c_{k} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} + \sum_{k = 1}^{n} h_{k} (t) + \sum_{k = 1}^{n} d_{k} \\ + \sum_{k = 1}^{n} η_{k} {\tilde{θ}}_{k}^{T} {\hat{θ}}_{k} + \frac{1}{λ_{0}} d_{0} - \frac{\bar{r}}{λ_{0}} s \end{matrix}

(63)

where $d_{n} = ε_{n 1} + 2 n ε_{n 2} + \frac{ς_{n}^{2}}{2}$ .

Until now, an adaptive tracking controller design employing MTN and backstepping method has been accomplished, which is shown in Figure 2.

Stability analysis

Theorem 1: Considering the nonlinear system (1), the intermediary virtual control signals (27), (41), and (48), the actual control input (61), and the adaptive laws (28), (42), (49), and (62), the system satisfies the following aims for any initial conditions:

The system output y is capable of tracking the reference signal $y_{d}$ with a small tracking error.

All signals in the closed-loop system are bounded.

All system states stay within the pre-established constraints, where $| x_{i} | < k_{ci}$ , and $k_{ci} > 0$ is a constant for $1 \leq i \leq n$ .

Proof: For the closed-loop system, the Lyapunov function $V$ can be defined as

V = V_{n} = V_{n - 1} + \frac{1}{2} {\tilde{θ}}_{n}^{T} {\tilde{θ}}_{n} + \frac{1}{2} \log \frac{k_{dn}^{2}}{k_{dn}^{2} - z_{n}^{2}}

(64)

According to (63), and taking ${\tilde{θ}}_{k}^{T} {\hat{θ}}_{k} \leq - \frac{1}{2} {\tilde{θ}}_{k}^{T} {\tilde{θ}}_{k} + \frac{1}{2} θ_{k}^{T} θ_{k}$ into account, one has

\begin{matrix} \overset{\cdot}{V} & \leq - \sum_{k = 1}^{n} c_{k} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} - \frac{1}{2} \sum_{k = 1}^{n} η_{k} {\tilde{θ}}_{k}^{T} {\tilde{θ}}_{k} + \sum_{k = 1}^{n} d_{k} \\ + \frac{1}{2} \sum_{k = 1}^{n} η_{k} \frac{1}{2} θ_{k}^{T} θ_{k} + \sum_{k = 1}^{n} h_{k} (t) + \frac{1}{λ_{0}} d_{0} - \frac{\bar{r}}{λ_{0}} s \end{matrix}

(65)

Lets define $c = \min {2 c_{k}, η_{k}, \bar{r}, k = 1, 2, . . ., n}$ and $λ =$ $\frac{1}{2} \sum_{k = 1}^{n} η_{k} θ_{k}^{T} θ_{k} + \sum_{k = 1}^{n} d_{k} + \frac{1}{λ_{0}} d_{0}$ .

\overset{\cdot}{V} \leq - \frac{c}{2} \sum_{k = 1}^{n} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} - \frac{c}{2} \sum_{k = 1}^{n} {\tilde{θ}}_{k}^{T} {\tilde{θ}}_{k} + λ + \sum_{k = 1}^{n} h_{k} (t) - \frac{\bar{r}}{λ_{0}} s

(66)

Since $V \geq \frac{1}{2} \sum_{k = 1}^{n} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} + \frac{1}{2} \sum_{k = 1}^{n} {\tilde{θ}}_{k}^{T} {\tilde{θ}}_{k}$ , we have

\begin{matrix} \overset{\cdot}{V} \leq - c (\sum_{k = 1}^{n} \frac{1}{2} \log \frac{k_{dk}^{2}}{k_{dk}^{2} - z_{k}^{2}} + \frac{1}{2} \sum_{k = 1}^{n} {\tilde{θ}}_{k}^{T} {\tilde{θ}}_{k}) + λ + \sum_{k = 1}^{n} h_{k} (t) \\ - \frac{\bar{r}}{λ_{0}} s \end{matrix}

(67)

where $\bar{r}$ and $λ_{0}$ are both positive parameters.

Subsequently, the inequality (67) can be expressed in the following way

\overset{\cdot}{V} \leq - cV + λ + \sum_{k = 1}^{n} h_{k} (t)

(68)

Because when $t \geq 0$ , $\sum_{k = 1}^{n} h_{k} (t) \geq 0$ holds. And when $t \geq T_{0}$ , $\sum_{k = 1}^{n} h_{k} (t) = 0$ holds, that is

\int_{0}^{\infty} \sum_{k = 1}^{n} h_{k} (t) dt < + \infty

(69)

Hence, a positive number H exists such that $\sum_{k = 1}^{n} h_{k} (t) dt \leq H$ . It follows that

\overset{\cdot}{V} \leq - cV + H^{'}

(70)

where $H^{'} = H + λ$ .

Then solving the above differential inequality, we have

V (t) \leq V (0) e^{- ct} + \frac{H^{'}}{c} (1 - e^{- ct}) \leq V (0) + \frac{H^{'}}{c}

(71)

At this point, in accordance with similar arguments in Jiang and Praly (1998), it follows that the system output y can successfully track the reference signal $y_{d}$ , and all states $x_{i}$ of the system will not exceed the pre-established constraints.

Simulation results

This section shows simulation results from both a numerical example and a practical application to validate the effectiveness of the proposed control strategy. Additionally, a comparative experiment is included to further validate the superiority of the method presented in this paper.

Example 1: Consider the input-delay nonlinear system with unmodeled dynamics below:

{\begin{matrix} \overset{\cdot}{ζ} = - ζ + 0.5 x_{1}^{2} \\ ẋ_{1} = x_{2} + \sin x_{1} + 0.2 ζ \sin x_{1} \\ ẋ_{2} = x_{3} + \sin x_{2} + 0.1 ζ \cos x_{2} \\ ẋ_{3} = u (t - τ) + \sin x_{3} + 0.5 ζ \sin x_{3} \\ y = x_{1} \end{matrix}

(72)

where the system state variables are denoted by $x_{1}$ , $x_{2}$ , $x_{3}$ and satisfy $| x_{1} | \leq 1$ , $| x_{2} | \leq 1.5$ , $| x_{3} | \leq 10$ . The control input of the system is denoted by u. The reference signal is selected as $y_{d} = 0.5 (\sin (t) + \sin (0.5 t))$ .

Based on Theorem 1, the virtual control signals $α_{i}$ , the actual control input u, and adaptive parameters ${\overset{\cdot}{\hat{θ}}}_{i}$ are designed as follows:

\begin{matrix} α_{i} = - c_{i} z_{i} - {\hat{θ}}_{i}^{T} Φ_{i} (Z_{i}) \\ u = - c_{n} z_{n} - {\hat{θ}}_{n}^{T} Φ_{n} (Z_{n}) \end{matrix}

{\overset{\cdot}{\hat{θ}}}_{i} = - η_{i} {\hat{θ}}_{i} + \frac{z_{i}}{k_{di}^{2} - z_{i}^{2}} Φ_{i} (Z_{i})

where $z_{1} = x_{1} - y_{d}$ , $z_{2} = x_{2} - α_{1}$ , $z_{3} = x_{3} - α_{2}$ , $Z_{1} = {[z_{1}]}^{T}$ , $Z_{2} = {[z_{1}, z_{2}]}^{T}$ , $Z_{3} = {[z_{1}, z_{2}, z_{3}]}^{T}$ .

In the simulation, the design parameters are specified as follows: $c_{1} = 12$ , $c_{2} = 15$ , $c_{3} = 50$ , $η_{1} = η_{2} = η_{3} = 15$ , $k_{d 1} = 0.4$ , $k_{d 2} = 0.5$ , $k_{d 3} = 0.5$ . The simulation was initiated with $ζ (0) = 0$ , $τ = 0.005$ , and $x_{1} (0) = x_{2} (0) = x_{3} (0) = 0$ as the initial states. Then, the boundaries of the states are chosen, that is, $k_{c 1} = \pm 1.2$ , $k_{c 2} = \pm 1.5$ , $k_{c 3} = \pm 6.5$ . Figures 3 –6 display the simulation results.

Figure 3 depicts the trajectories of the system output y and the reference signal $y_{d}$ with $c_{1} = 12$ , $c_{2} = 15$ , $c_{3} = 50$ , $η_{1} = η_{2} = η_{3} = 15$ . Figure 4 depicts the tracking performance when varying parameters $c_{i}$ and $η_{i}$ : The desired trajectory (1) represents the case where the parameters are set to $c_{1} = 5$ , $c_{2} = 20$ , $c_{3} = 100$ , $η_{1} = η_{2} = η_{3} = 15$ , while the desired trajectory (2) represents the case where the parameters are set to $c_{1} = 12$ , $c_{2} = 15$ , $c_{3} = 50$ , $η_{1} = 50$ , $η_{2} = 80$ , $η_{3} = 10$ . By comparing the two figures, it can be seen that the desired trajectory of the system varies with different values. The parameters selected in this paper better achieve tracking performance. Figure 5 illustrates that the system’s tracking error $y - y_{d}$ approaches a small domain around the origin and shows the trajectory of the system control input u. Figure 6 depicts the trajectories of the system’s unmodeled dynamics ζ and the system states $x_{1}$ , $x_{2}$ , $x_{3}$ , indicating that all states $x_{i}$ are confined to stay within the pre-established constraints. From the simulation results presented earlier, it is evident that the system output y effectively tracks the reference signal $y_{d}$ .

Remark 5. Figure 4 shows that the desired trajectory of the system varies with different values. In Example 1, the sensitivity of parameter $c_{i}$ is higher, so adjusting the parameter $c_{i}$ has a greater impact on the tracking performance, while adjusting the parameter $η_{i}$ has a smaller impact on tracking performance.

Example 2: To further confirm the efficacy of the control strategy presented, a class of single-link manipulator system that incorporates input-delay and unmodeled dynamics is considered. As detailed in Li et al. (2024a), its system can be given as:

{\begin{matrix} \overset{\cdot}{ζ} = - ζ + 0.5 x_{1}^{2} \\ ẋ_{1} = x_{2} \\ ẋ_{2} = x_{3} - x_{2} - 10 \sin x_{1} + x_{2}^{2} \cos (x_{2} x_{3}) + ζ x_{2} \cos x_{1} \\ ẋ_{3} = u (t - τ) + 0.2 x_{2} - x_{3} + 0.05 ζ x_{1} x_{2} \sin x_{3} \\ y = x_{1} \end{matrix}

(73)

where the system state variables are denoted by $x_{1}$ , $x_{2}$ , $x_{3}$ , and u is the system control input. The reference signal is selected as $y_{d} = \sin (0.5 t)$ . The initial conditions selected for the simulation system are identical to those used in Example 1. $ζ (0) = 0$ , $τ = 0.005$ and $x_{1} (0) = x_{2} (0) = x_{3} (0) = 0$ are set.

The parameters necessary for the controller design are listed as follows: $c_{1} = 20$ , $c_{2} = 15$ , $c_{3} = 50$ , $η_{1} = η_{2} = η_{3} = 15$ , $k_{d 1} = 0.4$ , $k_{d 2} = 0.5$ , $k_{d 3} = 0.5$ . The boundaries of the states are chosen such that $k_{c 1} = \pm 1$ , $k_{c 2} = \pm 0.7$ , $k_{c 3} = \pm 8.5$ . The simulation results are presented in Figures 7 –9.

Figure 7 illustrates the trajectories of the system output y and the reference signal $y_{d}$ , clearly showing that the system has achieved satisfactory tracking performance. Figure 8 illustrates that the system’s tracking error $y - y_{d}$ approaches a small area near the origin and shows the trajrctory of the system control input u. Figures 9 depicts the trajectories of the system’s unmodelled dynamics ζ and the system states $x_{1}$ , $x_{2}$ , $x_{3}$ , indicating all signals $x_{i}$ within the closed-loop system are bounded. The simulation outcomes demonstrate that the system output y accurately tracks the reference signal $y_{d}$ . Additionally, all states $x_{i}$ of this system cannot violate the specified constraints.

Example 3 (Comparative example): To further demonstrate the effectiveness of the method put forward in Example 1, a comparative example is carried out between the MTN method and the RBFNN method. Specifically, within the control strategy presented in Theorem 1, MTNs are replaced with RBFNNs, respectively. The comparison results are displayed in Figure 10. As can be clearly seen in Figure 10, although both the MTN-based controller and the RBFNN-based controller can achieve tracking control, the former performs slightly better than the latter. The results of this comparative experiment further strongly demonstrate the effectiveness of the proposed control strategy.

Remark 6. Although both the MTN-based controller and the RBFNN-based controller can achieve tracking control, as can be seen from Figure 1, the former has a simpler structure than the latter. Therefore, the MTN method chosen in this paper has the advantage of low computational complexity.

Conclusion

This paper has developed an adaptive tracking control strategy based on MTN technology for a specific class of nonlinear systems with input-delay, full state constraints, and unmodeled dynamics simultaneously. The Pade approximation technique is employed as an effective tool to address the issue of input-delay. Meanwhile, BLFs are utilized to ensure that the system states remain within the pre-established constraints, and a dynamic signal is designed to handle the unmodeled dynamics. The design of an adaptive tracking controller has successfully guaranteed that all the system’s signals remain bounded, while the tracking error approaches a small region near the origin. Ultimately, the simulation results have effectively illustrated the effectiveness of the designed controller. Future research will focus on incorporating further complexities inherent to real-world nonlinear systems, including input saturation, time-varying delays, and other such factors.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Shandong Provincial Natural Science Foundation, China (No. ZR2020QF055).

ORCID iDs

Xin-Ru Mu

Tian-Tian Wang

Yu-Qun Han

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

Alattas

Mofid

, et al. (2022) Barrier function-based nonsingular finite-time tracker for quadrotor UAVs subject to uncertainties and input constraints. Mathematics 10(10): 1659.

Aly

El-Sousy

FFM

, et al. (2022) Adaptive neural network-based fixed-time tracking controller for disabilities exoskeleton wheelchair robotic system. Mathematics 10(20): 3853.

Deng

Krstić

(1997) Stochastic nonlinear stabilization — I: A backstepping design. Systems and Control Letters 32(3): 143–150.

Feng

Zhang

, et al. (2024) Neuroadaptive output-feedback tracking control for stochastic nonlower triangular nonlinear systems with dead-zone input. IEEE Transactions on Cybernetics 54(12): 7839–7850.

Gao

Liu

Qian

(2025) Decentralized adaptive tracking control for interconnected nonlinear systems with unmodeled dynamics and input delays. International Journal of Control 98(4): 887–901.

Han

, et al. (2021) Adaptive tracking control of a class of nonlinear systems with input delay and dynamic uncertainties using multi-dimensional taylor network. International Journal of Control, Automation and Systems 19: 4078–4089.

Hao

Sun

Zong

, et al. (2025a) Adaptive predefined time tracking control for nonlinear multi-agent systems subject to input saturation: An improved command filtering approach. Nonlinear Dynamics 113: 19697–19716.

Hao

Zong

Wei

, et al. (2025b) Adaptive tracking control of nonlinear multi-agent systems subject to multiple constraints via multi-dimensional taylor network. IEEE Transactions on Automation Science and Engineering 22: 16913–16924.

Zhu

, et al. (2024) Adaptive multi-switching-based global tracking control for switched nonlinear systems with prescribed performance. IEEE Transactions on Automation Science and Engineering 21(3): 3243–3252.

10.

Jain

Katiyar

(2023) Robust adaptive control design of nonlinear systems with input delay. International Journal of Adaptive Control and Signal Processing 37(5): 1193–1202.

11.

Jiang

Praly

(1998) Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties. Automatica 34(7): 825–840.

12.

Khanesar

Kaynak

Yin

, et al. (2015) Adaptive indirect fuzzy sliding mode controller for networked control systems subject to time-varying network-induced time delay. IEEE Transactions on Fuzzy Systems 23(1): 205–214.

13.

Kharrat

(2024) Adaptive fault-tolerant control for a class of nonstrict-feedback nonlinear systems with unmodeled dynamics and dead-zone output using multi-dimensional Taylor networks. Nonlinear Dynamics 112: 13289–13306.

14.

Kong

Liu

, et al. (2023) Adaptive tracking control with global performance for output-constrained MIMO nonlinear systems. IEEE Transactions on Automatic Control 68(6): 3760–3767.

15.

Kong

Zhang

, et al. (2025) Adaptive tracking control of constrained nonlinear systems and its application to circuit systems. IEEE Transactions on Circuits and Systems I: Regular Papers 72(4): 1731–1740.

16.

Krstić

Kanellakopoulos

Kokotović

(1992) Adaptive nonlinear control without overparametrization. Systems and Control Letters 19(3): 177–185.

17.

Liu

Tong

, et al. (2019) Neural networks-based adaptive control for nonlinear state constrained systems with input delay. IEEE Transactions on Cybernetics 49(4): 1249–1258.

18.

Yang

Liu

(2024a) Command filter-based adaptive predefined-time control for non-strict feedback nonlinear systems with unmodeled dynamics. Journal of the Franklin Institute 361(12): 106936.

19.

Wang

(2025) Adaptive RL control based on projection operator for stochastic nonlinear systems with full state constraints and unmodeled dynamics. Asian Journal of Control. Epub ahead of print 11 July. DOI: 10.1002/asjc.3778.

20.

Wang

, et al. (2017) Adaptive fuzzy backstepping tracking control for strict-feedback systems with input delay. IEEE Transactions on Fuzzy Systems 25(3): 642–652.

21.

Han

, et al. (2022) Control design for stochastic nonlinear systems with full-state constraints and input delay: A new adaptive approximation method. International Journal of Control, Automation and Systems 20(8): 2768–2778.

22.

Shi

Liu

, et al. (2024b) Improved command-filtering-based fixed-time fuzzy adaptive control for uncertain nonlinear systems with full state constraints. IEEE Transactions on Fuzzy Systems 32(5): 3286–3292.

23.

Qiang

Zhuang

, et al. (2004) Robust and adaptive backstepping control for nonlinear systems using RBF neural networks. IEEE Transactions on Neural Networks 15(3): 693–701.

24.

Liu

, et al. (2025) Event-triggered adaptive fault estimation and fault-tolerant control for strict-feedback nonlinear systems with full state constraints. Nonlinear Dynamics 113: 2505–2520.

25.

Zhu

Wang

, et al. (2024) Distributed adaptive fault-tolerant control with prescribed performance for nonlinear multiagent systems. Communications in Nonlinear Science and Numerical Simulation 138: 108222.

26.

Wang

(2025) Neural network prescribed-time observer-based output-feedback control for uncertain pure-feedback nonlinear systems. Expert Systems with Applications 264(10): 125813.

27.

Lü

Shen

(2024) Adaptive fuzzy asymptotic tracking control of uncertain nonlinear systems with full state constraints. IEEE Transactions on Fuzzy Systems 32(5): 2750–2761.

28.

Mao

Wang

, et al. (2025) Adaptive neural fixed-time sampled-data output-feedback stabilization for a class of nonlinear systems. Applied Mathematics and Computation 507: 129550.

29.

Mao

Xiang

Huang

, et al. (2022) Practical finite-time sampled-data output feedback stabilization for a class of upper-triangular nonlinear systems with input delay. IEEE Transactions on Systems Man and Cybernetics Systems 32(12): 6939–6961.

30.

Mao

Zou

Guo

, et al. (2024) Observer-based finite-time sampled-data control for a class of nonlinear time-delay systems. IEEE Transactions on Systems Man and Cybernetics Systems 54(3): 1645–1657.

31.

Mokhtare

Mobayen

, et al. (2022) An adaptive barrier function terminal sliding mode controller for partial seizure disease based on the pinsky–rinzel mathematical model. Mathematics 10(16): 2940.

32.

Wang

Huang

, et al. (2024) Intermittent sampled-data control for local stabilization of neural networks subject to actuator saturation: A work-interval-dependent functional approach. IEEE Transactions on Neural Networks and Learning Systems 35(1): 1087–1097.

33.

Peng

, et al. (2025) Robust time-varying formation control of one-sided Lipschitz nonlinear multiagent system with delays via optimization algorithm. IEEE Transactions on Cybernetics 55(4): 1801–1813.

34.

Qiu

Lam

(2023) A new design of fuzzy affine model-based output feedback control for discrete-time nonlinear systems. IEEE Transactions on Fuzzy Systems 31(5): 1434–1444.

35.

Song

Chen

Liu

, et al. (2023) Fixed-time fuzzy adaptive fault-tolerant control for strict-feedback nonlinear systems with input delay. IEEE Transactions on Systems, Man, and Cybernetics: Systems 53(11): 6999–7010.

36.

Tee

Tay

(2009) Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica 45(4): 918–927.

37.

Tsai

Lee

Tanaka

, et al. (2024) Polynomial fuzzy observer-based feedback control for nonlinear hyperbolic PDEs systems. IEEE Transactions on Cybernetics 54(9): 5257–5269.

38.

Wang

Gao

Zhang

, et al. (2024a) Output feedback control of stochastic nonlinear systems with external disturbances and inverse dynamics. International Journal of Adaptive Control and Signal Processing 38(11): 3615–3635.

39.

Wang

Chen

Zhen

, et al. (2024b) Prescribed performance adaptive robust control for robotic manipulators with fuzzy uncertainty. IEEE Transactions on Fuzzy Systems 32(3): 1318–1330.

40.

Wang

Yan

Zheng

(2023) Contraction-based stochastic model predictive control for nonlinear systems with input delay using multidimensional taylor network. IEEE Transactions on Automatic Control 68(12): 7498–7513.

41.

Wang

Shi

, et al. (2017) Adaptive neural tracking control for a class of nonlinear systems with dynamic uncertainties. IEEE Transactions on Cybernetics 47(10): 3075–3087.

42.

Wang

Chen

, et al. (2024c) Adaptive PI event-triggered control for MIMO nonlinear systems with input delay. Information Sciences 677: 120817.

43.

Wang

Yan

Qiu

, et al. (2024d) Fuzzy-model-based finite frequency fault detection filtering design for two-dimensional nonlinear systems. IEEE/CAA Journal of Automatica Sinica 11(10): 2099–2110.

44.

Wang

Tao

, et al. (2021) A simplified adaptive tracking control for nonlinear pure-feedback systems with input delay and full-state constraints. International Journal of Adaptive Control and Signal Processing 35(12): 2521–2536.

45.

Wang

Tao

Fan

, et al. (2025a) A simplified adaptive fixed time control of pure-feedback stochastic nonlinear systems subject to full state constraints. International Journal of Adaptive Control and Signal Processing 39(4): 829–840.

46.

Wang

Sun

Chen

, et al. (2025b) Composite learning fixed-time control for nonlinear servo systems with state constraints and unknown dynamics. IEEE Transactions on Systems, Man, and Cybernetics: Systems 55(3): 2332–2342.

47.

Wang

(2024e) Adaptive neural optimised control for stochastic nonlinear systems with time-varying input delay via self-triggered mechanism. International Journal of Systems Science 55(2): 176–190.

48.

Wang

Niu

Shang

, et al. (2024f) Distributed resilient adaptive consensus tracking control of nonlinear multi-agent systems dealing with deception attacks via K-filters approach. Automatica 169: 111871.

49.

Wang

Yan

Fan

, et al. (2025c) Switching event-triggered-based gain-scheduled control for bipartite synchronization of coupled coopetitive memristive neural networks. IEEE Transactions on Systems, Man, and Cybernetics: Systems 55(10): 6951–6963.

50.

Xia

Zhang

, et al. (2025) Adaptive tracking control of high-order nonlinear systems with unknown time-varying input delay and unmodeled dynamics. Nonlinear Dynamics 113: 3543–3561.

51.

Xia

Zhang

Kang

, et al. (2022) Adaptive control of uncertain nonlinear systems with discontinuous input and time-varying input delay. IEEE Transactions on Systems, Man, and Cybernetics: Systems 52(11): 7248–7258.

52.

Xiong

Chen

(2025) RBFNN-based parameter adaptive sliding mode control for an uncertain TQUAV with time-varying mass. International Journal of Robust and Nonlinear Control 35(11): 4658–4668.

53.

(2025) Event-triggered adaptive asymptotic tracking control for uncertain nonlinear system with time-varying parameters and full state constraints. Nonlinear Dynamics 113: 13297–13311.

54.

Yan

Wang

(2023) Adaptive tracking control for stochastic nonlinear systems with time-varying delays using multi-dimensional Taylor network. ISA Transactions 132: 246–257.

55.

Yang

Sun

Zong

, et al. (2025) Dynamic event-triggered adaptive tracking control for switched nonlinear system with vanishing control gains. IEEE Transactions on Automation Science and Engineering 22: 11693–11705.

56.

Cai

Yang

, et al. (2019) Adaptive neural-based control for non-strict feedback systems with full-state constraints and unmodeled dynamics. Nonlinear Dynamics 97: 715–732.

57.

Yin

Niu

Jiang

, et al. (2022) Event-triggered adaptive decentralised control for switched interconnected nonlinear systems with unmodeled dynamics and full state constraints. International Journal of Systems Science 58(3): 1639–1658.

58.

Wang

Jia

, et al. (2024) Event-triggered practical fixed-time containment control for stochastic multiagent systems with input delay. IEEE Transactions on Fuzzy Systems 32(5): 2762–2774.

59.

Zhai

Wang

Tao

, et al. (2023) Observer-based adaptive fuzzy finite time control for non-strict feedback nonlinear systems with unmodeled dynamics and input delay. Nonlinear Dynamics 111: 1417–1440.

60.

Zhang

Wang

Zhang

, et al. (2025) Adaptive event-triggered control combined with high-order backstepping for pure feedback nonlinear systems. IEEE Transactions on Cybernetics 55(4): 1884–1892.