Event-based adaptive tracking controller design for nonlinear cyber-physical systems with deception attacks

Abstract

The work studies the tracking control issue for a class of nonlinear strict-feedback cyber-physical systems (CPSs) subject to unknown deception attacks. We propose a coordinate transformation technique with a reference signal, which incorporates the attack gains, and exploit the compromised states to construct robust controllers. The Nussbaum functions are also introduced to resist the inherent uncertainty of the time-varying attack injection signal. Meanwhile, the neural networks are used to handle the nonlinearities of CPSs. In addition, to reduce the network bandwidth of the CPSs, we adopt an event-triggered mechanism that updates the control input only when necessary. Finally, the presented control scheme guarantees that all signals are bounded in the control system and the tracking control of the CPSs can be achieved. Simulation results demonstrate that the developed algorithm is effective.

Keywords

Cyber-physical systems (CPSs)neural networks event-triggered secure control adaptive control deception attacks

Introduction

Cyber-physical systems (CPSs) are commonly used to demonstrate the relationship between the information and physics, aiming at improving the execution efficiency and intelligence of the system (Ding et al., 2020). Meanwhile, they achieve decision-making and scheduling of communication resources through the network. However, the open nature of information transmission in CPSs exposes network components to physical environmental interference and cyber threats (Wang et al., 2022), leaving them vulnerable to miscellaneous attacks (An and Yang, 2018; Cuan et al., 2023; Xu et al., 2024). Among the major security attacks, denial-of-service (DoS) attacks and deception attacks are extensively researched (Li et al., 2017). DoS attacks endeavor to disrupt the transmission of information and control inputs from sensors (Wang et al., 2024; Zhao et al., 2023), whereas deception attacks involve the injection of false information to compromise data integrity (Pan et al., 2024; Zhang et al., 2024). Unlike DoS attacks, deception attacks are highly covert and difficult to detect, allowing them to evade detection while severely compromising the states of CPSs. Therefore, ensuring the security of system under deception attacks has become a critical challenge in practical applications.

In recent years, significant progress has been made in developing defense strategies against deception attacks. From the perspective of system characteristics, for linear CPSs, Wang et al. (2021), Yucelen et al. (2016), Li et al. (2019) had proposed various methods against the attacks. Specifically, Yucelen et al. (2016) designed several adaptive control structures for linear dynamical systems subjected to sensor attacks. By proposing a novel state estimator, the method in the work by Li et al. (2019) solved the estimation problem under stochastic error and unknown attacks. The paper by Liu et al. (2019) employed dual functions to handle different types of network attacks. In contrast, defending nonlinear CPSs against deception attacks presents greater challenges, primarily due to the loss of state availability caused by attacks. To tackle this limitation, a new coordinate transformation was designed in the work by Ren and Yang (2020); the authors designed the safe feedback controller by using compromise states. A novel adaptive fuzzy control scheme for each subsystem of the CPSs was designed in the work by Bi et al. (2022) to mitigate the effects of deception attacks. Xing et al. (2022) discussed how to detect asynchronous attacks on nonlinear CPSs. However, these methods in the works by Liu et al. (2019) and Xing et al. (2022) fail to achieve tracking control since only compromised states are available after the deception attacks. Accordingly, Tian et al. (2024) and Ma et al. (2024) introduced filtering techniques to overcome this shortage. Although the tracking control is realized in the works by Tian et al. (2024) and Ma et al. (2024), the deception attacks destroy the input state of the filter, which in turn lead to a degradation of the system performance. Thus, it is worth considering a coordinate transformation that combines the reference signal and the attack gain.

In addition, CPSs as complex systems with multiple cooperative control loops often encounter resource competition issues due to their shared computing and communication resources. Such competition may lead to inefficient resource utilization. To address these issues, event-triggered schemes (De Persis et al., 2023) have been widely adopted in control applications, significantly improving network resource efficiency. The paper by Xie et al. (2022) developed a novel resilient adaptive event trigger to address resource constraints. A distributed event-triggered mechanism was introduced in the work by Feng et al. (2018) to enhance system flexibility. Moreover, the methods in the works by Ma et al. (2023) and Qi et al. (2021) proposed event-triggered finite-time control algorithms to ensure state convergence within a finite time. The simplified event trigger for fractional-order CPSs was designed in the work by Xiong et al. (2020) to improve adaptability. Qiu et al. (2019) reduced resource waste via a fixed-threshold event-triggered mechanism. Crucially, the aforementioned research breakthroughs have simultaneously given rise to several non-negligible issues. For instance, the event-triggered schemes in the works by Qi et al. (2021) and Xiong et al. (2020) are only applicable to systems with specific structures, lacking generalizability. Meanwhile, the fixed-threshold triggering condition in the work by Qiu et al. (2019) fails to adapt to time-varying system dynamics when reducing communication overhead. Therefore, this study focuses on designing a resource-efficient event-triggered scheme that addresses deception attacks, aiming to enhance both security and communication performance in CPSs.

Building on the above discussion, this paper introduces an event-triggered security program tracking control for CPSs with the attacks. The primary tasks and the symbolic meanings (Table 1) used in this article are as follows.

Different from the robustness problem of the attacked nonlinear CPSs in the works by Tian et al. (2024) and Ma et al. (2024), we design the coordinate transformation with reference signal and the attack gain, constructing the controller by using the available compromise state, which realizes the tracking control of the attacked nonlinear CPSs.

Unlike the works by Li and Zhao (2021) and Yoo (2019), where assumed the sign of attack weight signals are positive. The sign of the time-varying attack weight signal is allowed to be unknown in this article, which eliminates the conservative condition. Moreover, the unknown sign is handled by the Nussbaum-type functions in the iterative process.

To minimize communication load, this paper develops an event-triggering scheme using a relative threshold mechanism, which allows the system to sample under the triggering conditions and reduces the number of communications between sensors and actuators.

Table 1.

Symbol description.

Symbol	Definition
$δ_{M_{1}}$ , $δ (t)$	Deception attacks
$ϖ_{M_{1}}$ , $ϖ (t)$	Time-varying weight
$γ_{a_{i}}, γ_{b_{i}}, i = 1, 2, \dots, n$	Unknown constant
$α_{i}, i = 1, 2, \dots, n$	Virtual controller
$y_{r} (t)$	Reference signal

Guidelines for manuscript preparation

This section proposes a coordinate transformation to achieve adaptive tracking control for nonlinear CPSs (equation (1)). It involves designing the virtual controllers for the parallel subsystem in the initial n steps and proposing an event-triggered scheme to enhance network resource utilization, as depicted in Figure 1.

Figure 1.

Structure of the adaptive secure control design.

System descriptions

Consider the following nonlinear CPSs characterized by

\begin{matrix} ẋ_{i} & = x_{i + 1} + f_{i} ({\bar{x}}_{i}) \\ ẋ_{n} & = v + f_{n} ({\bar{x}}_{n}) \\ y & = x_{1} \end{matrix}

(1)

$y \in R$ and $v \in R$ are the output and input of the system and the unavailable states ${\bar{x}}_{i} = {[x_{1}, x_{2}, \dots, x_{i}]}^{T} \in R^{n}$ for $i = 1, 2 \dots n$ . Meanwhile, the $f_{i} ({\bar{x}}_{i})$ is unknown nonlinear function.

Deception attacks

The impact of deception attacks through sensor networks can be described by

\begin{matrix} {\overset{⌣}{x}}_{1} (t) & = x_{1} (t) + δ_{M_{1}} (t) \\ {\overset{⌣}{x}}_{i} (t) & = x_{i} (t) + δ (t) \end{matrix}

(2)

in which the system state ${\overset{⌣}{x}}_{1} (t)$ and ${\overset{⌣}{x}}_{i} (t)$ are available after attacks. $δ_{M_{1}}$ and $δ (t)$ are the signals of deception attacks.

Assumption 1. The deception attacks are assumed to be parameterized as $δ_{M_{1}} (t) = ϖ_{M_{1}} (t) x_{1} (t)$ , $δ (t) = ϖ (t) x_{i} (t)$ . We further assume $ϖ_{M_{1}} (t)$ is the first known attack with a known time-varying weight, and $ϖ (t)$ is unknown. Besides, $ϖ (t)$ and $ϖ_{M} (t)$ satisfy $1 + ϖ_{M} (t) \neq 0$ and $1 + ϖ (t) \neq 0$ . Here, there exist two unknown positive constants $\bar{ϖ}$ and $\bar{\overset{\cdot}{ϖ}}$ satisfying $| ϖ (t) | \leq \bar{ϖ}$ and $| \overset{\cdot}{ϖ} (t) | \leq \bar{\overset{\cdot}{ϖ}}$ .

Remark 1. Through the above definition, ${\overset{⌣}{x}}_{1} = (1 + ϖ_{M_{1}}) x_{1}$ and ${\overset{⌣}{x}}_{i} = (1 + ϖ) x_{i}$ can be obtained. When $1 + ϖ_{M_{1}} = 0$ and $1 + ϖ = 0$ , this will cause the available compromised states ${\overset{⌣}{x}}_{i}$ and ${\overset{⌣}{x}}_{i} = 0$ . Meanwhile, the controller v cannot be designed to maintain the stability of the system in this case. In addition, we assume that the attack weight is different between the first step and other steps, to realize the tracking control for the attacked system.

Lemma 1 (Ren and Yang 2020.). For a multi-variable function $F (\cdot)$ , one has $| F (ι_{1}, ι_{2}) | \leq ψ (ι_{1}) \bar{F} (ι_{2})$ , where $ψ (ι_{1}) \geq 1$ and $\bar{F} (ι_{2}) \geq 1$ are smooth functions.

Neural networks learning algorithm

The unknown nonlinear functions are approximated by the radial basis function neural networks (RBFNNs) with a three-layer architecture, which are described as

\begin{matrix} F (X) = Ξ^{T} K (X) + ξ (X) \end{matrix}

(3)

where $| ξ (X) | \leq \bar{ξ}$ with $\bar{ξ}$ , l is the neuron number, $Ξ = {[ς_{1} \dots ς_{l}]}^{T}$ $\in R^{l}$ is the weight vector, $K (X) = [κ_{1} (X), \dots, κ_{l} (X)] \in R^{l}$ is the basis function vector, and

\begin{matrix} κ_{l} (X) = \exp [- \frac{{(X - c_{l})}^{T} (X - c_{l})}{Λ_{l}^{2}}] \end{matrix}

where $Λ_{l}^{2}$ and $c_{l}$ being the width and center of $κ_{l} (X)$ .

Lemma 2 (Li and Zhao 2021). The basis function vector $K (X) = [κ_{1} (X), \dots, {κ_{l} (X)]}^{T}$ and the input vector $X = {[x_{1}, \dots, x_{n}]}^{T}$ are considered. For an integer $1 \leq p \leq n$ , it holds $∥ K (X) ∥ \leq ∥ K (X_{p}) ∥$ , where $X_{P} = {[x_{1}, \dots, x_{p}]}^{T}$ .

Nussbaum functions

Definition 1 (Lin and Qian 2002). One will be named Nussbaum functions, when the following conditions are satisfied

\begin{matrix} \lim_{d \to \infty} \sup \frac{1}{d} \int_{0}^{d} N (t) dt = + \infty \\ \lim_{d \to \infty} \inf \frac{1}{d} \int_{0}^{d} N (t) dt = - \infty \end{matrix}

Many established functions fall into the category of Nussbaum-type functions and fulfill the aforementioned two conditions such as $e^{t^{2}} \cos (\frac{π}{2} t)$ and $t^{2} \cos (t)$ .

Lemma 3 (Ren and Yang 2020). Define smooth functions $V (\cdot) \in [0, \infty)$ with $V (t) \geq 0$ and $θ (\cdot) \in [0, \infty)$ . The N function N $(\cdot)$ is smooth. When the following term is satisfied

\begin{matrix} \overset{\cdot}{V} (t) \leq - CV + D + \sum_{s = 1}^{n} (λ_{s} (t) N (θ_{s} (t)) + 1) {\overset{\cdot}{θ}}_{s} (t) \end{matrix}

(4)

\begin{matrix} {\overset{\cdot}{θ}}_{s} (t) \geq 0, \forall t > 0 \end{matrix}

(5)

where the $λ_{s} (t) \in U = [λ_{s}^{-}, λ_{s}^{+}]$ is the time-varying function with $0 \notin U$ for constants $λ_{s}^{-}$ and $λ_{s}^{+}$ satisfying $λ_{s}^{-} λ_{s}^{+} > 0$ . $C$ and $D$ are positive constants, then $V (t)$ and $θ_{s} (t)$ are bounded over $[0, \infty)$ .

Main results

Based on the considered deception attacks, one gets that

\begin{matrix} x_{1} (t) = γ_{M_{1}} (t) {\overset{⌣}{x}}_{1} (t), x_{i} (t) = γ (t) {\overset{⌣}{x}}_{1} (t) \end{matrix}

(6)

where $γ_{M_{1}} (t) = {(1 + ϖ_{M_{1}} (t))}^{- 1}$ , $γ (t) = {(1 + ϖ (t))}^{- 1}$ , the real state variables $x_{i}$ for $i = 1 \dots n$ denote unavailable. Besides, we define $\tilde{Δ} = Δ - \hat{Δ}$ , in which $\hat{Δ}$ is the estimation of $Δ$ .

From Assumption 1, as a result of $γ_{m} \leq | γ (t) | \leq γ_{M}$ and $| \overset{\cdot}{γ} (t) | \leq \bar{\overset{\cdot}{γ}}$ , $γ (t)$ and $\overset{\cdot}{γ} (t)$ are bounded, where $γ_{m}$ , $γ_{M}$ , and $\bar{\overset{\cdot}{γ}}$ are unknown positive constants.

To achieve tracking control under deception attacks, a coordinate transformation is proposed as follows

z_{1} = x_{1} - y_{r}, z_{i} = x_{i} - γ (t) α_{i - 1}, i = 2, 3, \dots, n

(7)

Then we obtain

{\overset{⌣}{z}}_{1} = {\overset{⌣}{x}}_{1} - \frac{1}{γ_{M_{1}}} y_{r}, {\overset{⌣}{z}}_{i} = {\overset{⌣}{x}}_{i} - α_{i - 1}, i = 2, 3, \dots, n

(8)

In the following content, event-triggered control (ETC) scheme based on neural network learning is constructed.

Assumption 2. The reference signal $y_{r} (t)$ is continuous and bounded, and so is the $n th$ -order time derivatives of $y_{r} (t)$ .

Step 1. Through the coordinate transformations, the time derivative of $z_{1}$ can be derived

\begin{matrix} ż_{1} & = ẋ_{1} - ẏ_{r} = x_{2} + f_{1} (x_{1}) - ẏ_{r} \\ = z_{2} + γ α_{1} + f_{1} (x_{1}) - ẏ_{r} \end{matrix}

Construct the Lyapunov function candidate as follows

\begin{matrix} V_{1} = \frac{1}{2} z_{1}^{2} + \frac{1}{2} {\tilde{γ}}_{a_{1}}^{2} + \frac{1}{2} {\tilde{γ}}_{b_{1}}^{2} \end{matrix}

(9)

where ${\tilde{γ}}_{a_{1}} = γ_{a_{1}} - {\hat{γ}}_{a_{1}}$ and ${\tilde{γ}}_{b_{1}} = γ_{b_{1}} - {\hat{γ}}_{b_{1}}$ , with $γ_{a_{1}}$ , $γ_{b_{1}}$ being defined later. ${\hat{γ}}_{a_{1}}$ , ${\hat{γ}}_{b_{1}}$ being the estimations of $γ_{a_{1}}$ , $γ_{b_{1}}$ .

Then the derivative of $V_{1}$ is given as

\begin{matrix} {\overset{\cdot}{V}}_{1} & = z_{1} z_{2} + z_{1} γ α_{1} + z_{1} f_{1} (x_{1}) - z_{1} ẏ_{r} - {\tilde{γ}}_{a_{1}} {\overset{\cdot}{\hat{γ}}}_{a_{1}} - {\tilde{γ}}_{b_{1}} {\overset{\cdot}{\hat{γ}}}_{b_{1}} \end{matrix}

For convenience, we denote

F_{1} (γ_{M_{1}}, X_{1}) = k_{1} γ_{M_{1}} {\overset{⌣}{z}}_{1} + \frac{1}{2} γ_{M_{1}} {\overset{⌣}{z}}_{1} + f_{1} (γ_{M_{1}} {\overset{⌣}{x}}_{1})

From Assumption 1, there are positive constants $γ_{0}, γ_{1} < γ_{2}$ , such that $γ_{1} \leq | γ (t) | \leq γ_{2}$ and $| \overset{\cdot}{γ} (t) | \leq γ_{0}$ . Since $γ_{M_{1}}$ is bounded, there is an unknown constant $ω_{1}$ and unknown functions $ψ_{1} (γ)$ and ${\bar{f}}_{1} (X_{1})$ from Lemma 1 such that

F_{1} (γ, X_{1}) \leq ψ_{1} (γ) {\bar{f}}_{1} (X_{1}) \leq ω_{1} {\bar{f}}_{1} (X_{1})

in which $ω_{1} > 0$ satisfying $ω_{1} \geq ψ_{1} (γ)$ . Then, it is deduced that

\begin{matrix} z_{1} F_{1} (γ, X_{1}) & \leq \frac{1}{2 ε_{1}} z_{1}^{2} F_{1}^{2} (γ, X_{1}) + \frac{ε_{1}}{2} \\ \leq \frac{1}{2 ε_{1}} {\overset{⌣}{z}}_{1}^{2} γ_{M_{1}}^{2} ω_{1}^{2} {\bar{f}}_{1}^{2} (X_{1}) + \frac{ε_{1}}{2} \end{matrix}

where $ε_{1}$ is a positive constant.

Since ${\bar{f}}_{1} (X_{1})$ is unknown, the RBFNNs are introduced to approximate ${\bar{f}}_{1} (X_{1})$ such that

\begin{matrix} {\bar{f}}_{1} (X_{1}) = {Ξ_{1}}^{T} K_{1} (X_{1}) + ξ_{1} (X_{1}) \end{matrix}

(10)

with $X_{1} = {\overset{⌣}{x}}_{1}$ , $ξ_{1} (X_{1}) \leq {\bar{ξ}}_{1}$ and ${\bar{ξ}}_{1}$ being any constants. By using equation (10), the following inequality is obtained

\begin{matrix} \frac{1}{2 ε_{1}} {\overset{⌣}{z}}_{1}^{2} γ_{M_{1}}^{2} ω_{1}^{2} {\bar{f}}_{1}^{2} (X_{1}) & \leq \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} γ_{M_{1}}^{2} ω_{1}^{2} ({∥ Ξ_{1} ∥}^{2} {∥ K_{1} (X_{1}) ∥}^{2} + {\bar{ξ}}_{1}^{2}) \\ \leq \frac{1}{ε_{1}} γ_{a_{1}} {\overset{⌣}{z}}_{1}^{2} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} + \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} γ_{b_{1}} \end{matrix}

(11)

where ${\bar{X}}_{1} = {\overset{⌣}{x}}_{1}, γ_{a_{1}} = γ_{M_{1}}^{2} ω_{1}^{2} {∥ Ξ_{1} ∥}^{2}, γ_{b_{1}} = γ_{M_{1}}^{2} ω_{1}^{2} {\bar{ξ}}_{1}^{2}$ .

Therefore, one has

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq & - k_{1} z_{1}^{2} + \frac{1}{2} z_{2}^{2} - {\tilde{γ}}_{a_{1}} {\overset{\cdot}{\hat{γ}}}_{a_{1}} - {\tilde{γ}}_{b_{1}} {\overset{\cdot}{\hat{γ}}}_{b_{1}} + \frac{ε_{1}}{2} - z_{1} ẏ_{r} \\ + {\overset{⌣}{z}}_{1} (γ_{M_{1}} γ α_{1} + \frac{1}{ε_{1}} γ_{a_{1}} {\overset{⌣}{z}}_{1} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} + \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1} γ_{b_{1}}) \end{matrix}

(12)

Design the virtual control $α_{1}$ and the parameter variable $θ_{1}$ as

α_{1} = N (θ_{1}) Φ_{1}, {\overset{\cdot}{θ}}_{1} = {\overset{⌣}{z}}_{1} Φ_{1}

(13)

where $Φ_{1} = \frac{1}{ε_{1}} {\hat{γ}}_{a_{1}} {\overset{⌣}{z}}_{1} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} + \frac{1}{ε_{1}} {\hat{γ}}_{b_{1}} {\overset{⌣}{z}}_{1} - γ_{M_{1}} ẏ_{r}$ , the adaptive law is designed as

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{a_{1}} & = \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} - π_{1, 1} {\hat{γ}}_{a_{1}} \\ {\overset{\cdot}{\hat{γ}}}_{b_{1}} & = \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} - π_{1, 2} {\hat{γ}}_{b_{1}} \end{matrix}

(14)

where $π_{1, 1} > 0$ , $π_{1, 2} > 0$ .

Substituting equations (13) and (14) into equation (12) gives

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq & - k_{1} z_{1}^{2} + \frac{1}{2} z_{2}^{2} + (λ_{1} (t) N (θ_{1}) + 1) {\overset{\cdot}{θ}}_{1} \\ + π_{1, 1} {\tilde{γ}}_{a_{1}} {\hat{γ}}_{a_{1}} + π_{1, 2} {\tilde{γ}}_{b_{1}} {\hat{γ}}_{b_{1}} + \frac{ε_{1}}{2} \end{matrix}

(15)

where $λ_{1} = γ_{M_{1}} γ$ .

Step i ( $2 \leq i \leq n - 1$ ): According to equation (7), one has

\begin{matrix} ż_{i} & = ẋ_{i} - \overset{\cdot}{γ} α_{i - 1} - γ {\overset{\cdot}{α}}_{i - 1} \\ = z_{i + 1} + γ α_{i} + f_{i} ({\bar{x}}_{i}) - \overset{\cdot}{γ} α_{i - 1} - γ {\overset{\cdot}{α}}_{i - 1} \end{matrix}

with ${\overset{\cdot}{α}}_{i - 1} = \sum_{s = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial θ_{s}} {\overset{\cdot}{θ}}_{s} + \frac{\partial α_{i - 1}}{\partial {\overset{⌣}{x}}_{s}} {\overset{\cdot}{\overset{⌣}{x}}}_{s} + \frac{\partial α_{i - 1}}{\partial {\hat{γ}}_{a_{s}}} {\overset{\cdot}{\hat{γ}}}_{a_{s}} + \frac{\partial α_{i - 1}}{\partial {\hat{γ}}_{b_{s}}}$ ${\overset{\cdot}{\hat{γ}}}_{b_{s}} + \sum_{s = 0}^{i - 1} \frac{\partial α_{i - 1}}{\partial {y_{r}}^{s}} {y_{r}}^{s + 1}$ , due to $x_{i} = γ {\overset{⌣}{x}}_{i}$ , a direct calculation gives

\begin{matrix} {\overset{\cdot}{\overset{⌣}{x}}}_{1} = \frac{γ}{γ_{M_{1}}} {\overset{⌣}{x}}_{2} + \frac{f_{1} (x)}{γ_{M_{1}}}, {\overset{\cdot}{\overset{⌣}{x}}}_{s} = {\overset{⌣}{x}}_{s + 1} + \frac{1}{γ} (f_{s} (x) - \overset{\cdot}{γ} {\overset{⌣}{x}}_{i}) \end{matrix}

(16)

with $2 \leq s \leq i$ .

By denoting $α_{h, i - 1} = \sum_{s = 1}^{i - 1} \frac{\partial α_{i - 1}}{\partial θ_{s}} {\overset{\cdot}{θ}}_{s} + \frac{γ}{γ_{M_{1}}} \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} {\overset{⌣}{x}}_{2} + \frac{\partial α_{i - 1}}{\partial {\hat{γ}}_{a_{s}}} {\overset{\cdot}{\hat{γ}}}_{a_{s}} +$ $\frac{\partial α_{i - 1}}{\partial {\hat{γ}}_{b_{s}}} {\overset{\cdot}{\hat{γ}}}_{b_{s}} + \sum_{s = 2}^{i - 1} \frac{\partial α_{i - 1}}{\partial {\overset{⌣}{x}}_{s}} {\overset{⌣}{x}}_{s} + \sum_{s = 0}^{i - 1} \frac{\partial α_{i - 1}}{\partial {y_{r}}^{s}} {y_{r}}^{s + 1}$ , it is known that

\begin{matrix} {\overset{\cdot}{α}}_{i - 1} = α_{h, i - 1} + \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} \frac{f_{1} (x)}{γ_{M_{1}}} + \sum_{s = 2}^{i - 1} \frac{\partial α_{i - 1}}{\partial {\overset{⌣}{x}}_{s}} \frac{f_{s} (x) - \overset{\cdot}{γ} {\overset{⌣}{x}}_{s}}{γ} \end{matrix}

(17)

The Lyapunov function can be chosen as

\begin{matrix} V_{i} = V_{i - 1} + \frac{1}{2} z_{i}^{2} + \frac{1}{2} {\tilde{γ}}_{a_{i}}^{2} + \frac{1}{2} {\tilde{γ}}_{b_{i}}^{2} \end{matrix}

(18)

where ${\tilde{γ}}_{a_{i}} = γ_{a_{i}} - {\hat{γ}}_{a_{i}}$ and ${\tilde{γ}}_{b_{i}} = γ_{b_{i}} - {\hat{γ}}_{b_{i}}$ , with $γ_{a_{i}}$ , $γ_{b_{i}}$ to be defined and ${\hat{γ}}_{a_{i}}$ , ${\hat{γ}}_{b_{i}}$ being the estimations of $γ_{a_{i}}$ , $γ_{b_{i}}$ .

Then, ${\overset{\cdot}{V}}_{i}$ can be given as follows

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq & {\overset{\cdot}{V}}_{i - 1} - k_{i} z_{i}^{2} - \frac{1}{2} z_{i}^{2} + z_{i} z_{i + 1} - {\tilde{γ}}_{a_{i}} {\overset{\cdot}{\hat{γ}}}_{a_{i}} - {\tilde{γ}}_{b_{i}} {\overset{\cdot}{\hat{γ}}}_{b_{i}} \\ + z_{i} (γ α_{i} + F_{i} (γ, \overset{\cdot}{γ}, x_{i}) - γ α_{h, i - 1}) \end{matrix}

(19)

with $X_{i} = {[\overset{⌣}{x}, θ_{1}, \dots, θ_{i - 1}, {\hat{γ}}_{a_{1}}, \dots, {\hat{γ}}_{a_{i - 1}}, {\hat{γ}}_{b_{1}}, \dots, {\hat{γ}}_{b_{i - 1}}]}^{T}$ and

\begin{matrix} F_{i} (γ_{M_{1}}, γ, \overset{\cdot}{γ}, x_{i}) = & k_{i} z_{i} + \frac{1}{2} z_{i} + f_{i} (x) - \overset{\cdot}{γ} α_{i - 1} - \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} \frac{γ}{γ_{M_{1}}} f_{1} (x) \\ - \sum_{s = 2}^{i - 1} \frac{\partial α_{i - 1}}{\partial {\overset{⌣}{x}}_{s}} (f_{s} (x) - \overset{\cdot}{γ} {\overset{⌣}{x}}_{s}) \end{matrix}

Since γ, $γ_{M_{1}}$ , and $\overset{\cdot}{γ}$ are bounded, there is an unknown constant $ω_{i}$ and unknown function $ψ_{i} (γ)$ , $z_{i} F_{i} (X_{i})$ from Lemma 1 such that

\begin{matrix} z_{i} F_{i} (γ_{M_{1}}, γ, \overset{\cdot}{γ}, X_{i}) & \leq \frac{1}{2 ε_{i}} z_{i}^{2} {F_{i}}^{2} (X_{i}) + \frac{ε_{i}}{2} \\ \leq \frac{1}{2 ε_{i}} {\overset{⌣}{z}}_{i}^{2} γ^{2} ω_{i}^{2} {\bar{f}}_{i}^{2} (X_{i}) + \frac{ε_{i}}{2} \end{matrix}

where $ω_{i} > 0$ is a constant satisfying $ω_{i} \geq φ_{i} (γ, \overset{\cdot}{γ})$ , $ε_{i} > 0$ is a constant. Since ${\bar{f}}_{i} (X_{i})$ is unknown, the RBFNNs are used to approximate ${\bar{f}}_{i} (X_{i})$ such that

\begin{matrix} {\bar{f}}_{i} (X_{i}) = Ξ_{i}^{T} K_{i} (X_{i}) + ξ_{i} (X_{i}) \end{matrix}

(20)

where $ξ_{i} (X_{i}) \leq {\bar{ξ}}_{i}$ , with ${\bar{ξ}}_{i} > 0$ being a positive constant. From equation (20), it is deduced that

\begin{matrix} \frac{1}{2 ε_{i}} {\overset{⌣}{z}}_{i}^{2} γ^{2} ω_{i}^{2} {\bar{f}}_{i}^{2} (X_{i}) & \leq \frac{1}{ε_{i}} {\overset{⌣}{z}}_{i}^{2} γ^{2} ω_{i}^{2} ({∥ Ξ_{i} ∥}^{2} {∥ K_{i} (X_{i}) ∥}^{2} + {\bar{ξ}}_{i}^{2}) \\ \leq \frac{1}{ε_{i}} γ_{a_{i}} {\overset{⌣}{z}}_{i}^{2} {∥ K_{i} ({\bar{X}}_{i}) ∥}^{2} + \frac{1}{ε_{i}} {\overset{⌣}{z}}_{i}^{2} γ_{b_{i}} \end{matrix}

(21)

where $γ_{a_{i}} = γ^{2} ω_{i}^{2} {∥ Ξ_{i} ∥}^{2}$ , $γ_{b_{i}} = γ^{2} ω_{i}^{2} {\bar{ξ}}_{1}^{2}$ and ${\bar{X}}_{i} = [{\overset{⌣}{x}}_{1}, \dots, {\overset{⌣}{x}}_{i},$ $\dots, θ_{1}, θ_{i - 1}, {\hat{γ}}_{a_{1}}$ , ${\dots, {\hat{γ}}_{a_{i - 1}}, {\hat{γ}}_{b_{1}}, \dots, {\hat{γ}}_{b_{i - 1}}]}^{T}$ .

Then, one obtains

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq & {\overset{\cdot}{V}}_{i - 1} - k_{i} z_{i}^{2} + \frac{1}{2} z_{i + 1}^{2} + \frac{1}{ε_{i}} γ_{a_{i}} {\overset{⌣}{z}}_{i}^{2} {∥ K_{i} ({\bar{X}}_{i}) ∥}^{2} + \frac{ε_{i}}{2} \\ + \frac{1}{ε_{i}} {\overset{⌣}{z}}_{i}^{2} γ_{b_{i}} - z_{i} γ α_{h, i - 1} + z_{i} γ α_{i} - {\tilde{γ}}_{a_{i}} {\overset{\cdot}{\hat{γ}}}_{a_{i}} - {\tilde{γ}}_{b_{i}} {\overset{\cdot}{\hat{γ}}}_{b_{i}} \end{matrix}

(22)

Design the virtual control $α_{i}$ and the parameter variable $θ_{i}$ as

\begin{matrix} α_{i} & = α_{h, i - 1} + N (θ_{i}) Φ_{i}, {\overset{\cdot}{θ}}_{i} = {\overset{⌣}{z}}_{i} Φ_{i} \end{matrix}

(23)

where $Φ_{i} = \frac{1}{ε_{i}} {\hat{γ}}_{a_{i}} {\overset{⌣}{z}}_{i} {∥ K_{i} ({\bar{X}}_{i}) ∥}^{2} + \frac{1}{ε_{i}} {\hat{γ}}_{b_{i}} {\overset{⌣}{z}}_{i}$ , and the adaptive law is designed as

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{a_{i}} & = \frac{1}{ε_{i}} {\overset{⌣}{z}}_{i}^{2} {∥ K_{i} ({\bar{X}}_{i}) ∥}^{2} - π_{i, 1} {\hat{γ}}_{a_{i}} \\ {\overset{\cdot}{\hat{γ}}}_{b_{i}} & = \frac{1}{ε_{i}} {\overset{⌣}{z}}_{i}^{2} - π_{i, 2} {\hat{γ}}_{b_{i}} \end{matrix}

(24)

where $π_{i, 1} > 0$ , $π_{i, 2} > 0$ .

By substituting equations (23) and (24) into equation (22), one holds

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq & - \sum_{s = 1}^{i} k_{s} z_{s}^{2} + \frac{1}{2} z_{i + 1}^{2} + \sum_{s = 1}^{i} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} \\ + \sum_{s = 1}^{i} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} + \sum_{s = 1}^{i} π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} + \sum_{s = 1}^{i} \frac{ε_{s}}{2} \end{matrix}

with $λ_{i} = γ^{2}$ .

Step n: Consider the coordinate transformation. By equations (1) and (7) one has

\begin{matrix} ż_{n} & = ẋ_{n} - \overset{\cdot}{γ} α_{n - 1} - γ {\overset{\cdot}{α}}_{n - 1} = v + f_{n} ({\bar{x}}_{n}) - \overset{\cdot}{γ} α_{n - 1} - γ {\overset{\cdot}{α}}_{n - 1} \end{matrix}

(25)

Based on the previous step, the derivative of $α_{n - 1}$ can be written as follows

\begin{matrix} {\overset{\cdot}{α}}_{n - 1} = α_{h, n - 1} + \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} \frac{1}{γ_{M_{1}}} f_{1} (x) + \sum_{s = 2}^{n - 1} \frac{\partial α_{n - 1}}{\partial {\overset{⌣}{x}}_{s}} \frac{f_{s} (x) - \overset{\cdot}{γ} {\overset{⌣}{x}}_{s}}{γ} \end{matrix}

(26)

$α_{h, n - 1} = \sum_{s = 1}^{n - 1} \frac{\partial α_{n - 1}}{\partial θ_{s}} {\overset{\cdot}{θ}}_{s} + \frac{γ}{γ_{M_{1}}} \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} {\overset{⌣}{x}}_{2} + \frac{\partial α_{n - 1}}{\partial {\hat{γ}}_{a_{s}}} {\overset{\cdot}{\hat{γ}}}_{a_{s}} +$ $\frac{\partial α_{n - 1}}{\partial {\hat{γ}}_{b_{s}}} {\overset{\cdot}{\hat{γ}}}_{b_{s}} + \sum_{s = 2}^{n - 1} \frac{\partial α_{n - 1}}{\partial {\overset{⌣}{x}}_{s}} {\overset{⌣}{x}}_{s} + \sum_{s = 0}^{n - 1} \frac{\partial α_{n - 1}}{\partial {y_{r}}^{s}} {y_{r}}^{s + 1}$ is defined in the above equation.

In this situation, a relative ETC is defined as $v (t) = l (t_{k})$ when $t \in [t_{k}, t_{k + 1})$ ,

\begin{matrix} t_{k + 1} = \inf {t \in R | o (t) | \geq ρ | v (t) | + q} \end{matrix}

(27)

with $l (t_{k}) = - (1 + ρ) σ \tanh (({\overset{⌣}{z}}_{n} σ) ∕ e)$ and $o (t) = l (t) - v (t)$ , ρ, q being two constants satisfying $0 < ρ < 1$ , $q > 0$ . e is a function chosen to satisfy $e > 0$ . Then, the inequality $| l (t) - v (t) | \leq ρ | v (t) | + q$ hold all the time.

Furthermore, the following situations are considered.

Case 1: $v (t) \geq 0$

\begin{matrix} - ρv (t) - q \leq l (t) - v (t) \leq ρv (t) + q \end{matrix}

(28)

then, it has

\begin{matrix} l (t) - v (t) = ϕ (t) (ρv (t) + q) \end{matrix}

Case 2: When $v (t) < 0$ , one has

\begin{matrix} ρv (t) - q \leq l (t) - v (t) \leq - ρv (t) + q \end{matrix}

(29)

Furthermore

\begin{matrix} l (t) - v (t) = ϕ (t) (ρv (t) - q) \end{matrix}

Considering the above two situations, we can obtain

\begin{matrix} v (t) = \frac{l (t)}{1 + ϕ_{1} (t) ρ} - \frac{ϕ_{2} (t) q}{1 + ϕ_{1} (t) ρ} \end{matrix}

(30)

here $ϕ_{1} (t) = ϕ_{2} (t) = ϕ (t)$ with $v (t) \in [0, + \infty)$ ; $ϕ_{1} (t) = ϕ (t)$ , $ϕ_{2} (t) = - ϕ (t)$ with $v (t) \in (- \infty, 0]$ , in which $ϕ (t)$ is a time-varying parameter satisfying $ϕ (t) \in [- 1, 1]$ .

It is calculated that

\begin{matrix} ż_{n} & = \frac{l (t)}{1 + ϕ_{1} (t) ρ} - \frac{ϕ_{2} (t) q}{1 + ϕ_{1} (t) ρ} + f_{n} ({\bar{x}}_{n}) - \overset{\cdot}{γ} α_{n - 1} - γ {\overset{\cdot}{α}}_{n - 1} \end{matrix}

(31)

Select the Lyapunov function as

\begin{matrix} V_{n} = V_{n - 1} + \frac{1}{2} z_{n}^{2} + \frac{1}{2} {\tilde{γ}}_{a_{n}}^{2} + \frac{1}{2} {\tilde{γ}}_{b_{n}}^{2} + \frac{1}{2} {\tilde{γ}}_{β}^{2} + \frac{1}{2} {\tilde{γ}}_{g}^{2} \end{matrix}

(32)

where ${\tilde{γ}}_{a_{n}}$ , ${\tilde{γ}}_{b_{n}}$ , ${\tilde{γ}}_{β}$ , and ${\tilde{γ}}_{g}$ are defined as the previous step.

Then, similar to step i, one obtains

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \sum_{s = 1}^{n} k_{s} z_{s}^{2} + \sum_{s = 1}^{n - 1} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + \sum_{s = 1}^{n - 1} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} \\ + \sum_{s = 1}^{n - 1} π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} + \sum_{s = 1}^{n - 1} \frac{ε_{s}}{2} + z_{n} (v - γ α_{n}) \\ + {\overset{⌣}{z}}_{n} (γ^{2} α_{n} - γ^{2} α_{h, n - 1} + \frac{1}{ε_{n}} {\tilde{γ}}_{a_{n}} {\overset{⌣}{z}}_{n} {∥ K_{n} ({\bar{X}}_{n}) ∥}^{2} \\ + \frac{1}{ε_{n}} {\overset{⌣}{z}}_{n} {\tilde{γ}}_{b_{n}}) - {\tilde{γ}}_{a_{n}} {\overset{\cdot}{\hat{γ}}}_{a_{n}} - {\tilde{γ}}_{b_{n}} {\overset{\cdot}{\hat{γ}}}_{b_{n}} - {\tilde{γ}}_{β} {\overset{\cdot}{\hat{γ}}}_{β} - {\tilde{γ}}_{g} {\overset{\cdot}{\hat{γ}}}_{g} \end{matrix}

(33)

where $γ_{a_{n}} = γ^{2} ω_{n}^{2} {∥ Ξ_{n} ∥}^{2}$ , $γ_{b_{n}} = γ^{2} ω_{n}^{2} {\bar{ξ}}_{n}^{2}$ , ${\bar{X}}_{n} = [{\overset{⌣}{x}}_{1}, \dots, {\overset{⌣}{x}}_{n}, θ_{1},$ $\dots, θ_{n - 1}, {\hat{γ}}_{a_{1}}, \dots,$ ${{\hat{γ}}_{a_{n - 1}}, {\hat{γ}}_{b_{1}}, \dots, {\hat{γ}}_{b_{n - 1}}]}^{T}$ , $ω_{n} \geq ψ_{n} (γ)$ is a constant.

Design the $α_{n}$ , $θ_{n}$ , and adaptive laws as

\begin{matrix} α_{n} & = α_{h, n - 1} + N (θ_{n}) Φ_{n}, {\overset{\cdot}{θ}}_{n} = {\overset{⌣}{z}}_{n} Φ_{n} \end{matrix}

(34)

where $Φ_{n} = \frac{1}{ε_{n}} {\hat{γ}}_{a_{n}} {\overset{⌣}{z}}_{n} {∥ K_{n} ({\bar{X}}_{n}) ∥}^{2} + \frac{1}{ε_{n}} {\hat{γ}}_{b_{n}} {\overset{⌣}{z}}_{n}$ , and design the adaptive law

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{a_{n}} & = \frac{1}{ε_{n}} {\overset{⌣}{z}}_{n}^{2} {∥ K_{n} ({\bar{X}}_{n}) ∥}^{2} - π_{n, 1} {\hat{γ}}_{a_{n}} \\ {\overset{\cdot}{\hat{γ}}}_{b_{n}} & = \frac{1}{ε_{n}} {\overset{⌣}{z}}_{n}^{2} - π_{n, 2} {\hat{γ}}_{b_{n}} \end{matrix}

(35)

Thus, one has

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \sum_{s = 1}^{n} k_{s} z_{s}^{2} + \sum_{s = 1}^{n} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + \sum_{s = 1}^{n} \frac{ε_{s}}{2} \\ + z_{n} (\frac{l (t)}{1 + ϕ_{1} (t) ρ} + g (t) - γ α_{n}) - {\tilde{γ}}_{β} {\overset{\cdot}{\hat{γ}}}_{β} - {\tilde{γ}}_{g} {\overset{\cdot}{\hat{γ}}}_{g} \\ + \sum_{s = 1}^{n} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} + \sum_{s = 1}^{n} π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} \end{matrix}

(36)

with $λ_{n} = γ^{2}$ , $g (t) = - (ϕ_{2} (t) q) ∕ (1 + ϕ_{1} (t) ρ)$ . It is known that

\begin{matrix} | \frac{ϕ_{2} (t) q}{1 + ϕ_{1} (t) ρ} | \leq \frac{q}{1 - ρ}, | \frac{l (t)}{1 + ϕ_{1} (t) ρ} | \leq \frac{l (t)}{1 + ρ} \end{matrix}

(37)

Then, one can obtain that

\begin{matrix} - z_{n} g (t) \leq \frac{1}{2 ε_{g}} {\overset{⌣}{z}}_{n}^{2} \frac{γ^{2} q^{2}}{{(1 - ρ)}^{2}} + \frac{ε_{g}}{2} \leq \frac{1}{2 ε_{g}} {\overset{⌣}{z}}_{n}^{2} γ_{g} + \frac{ε_{g}}{2} \end{matrix}

(38)

\begin{matrix} - z_{n} γ α_{n} \leq \frac{1}{2 ε_{β}} {\overset{⌣}{z}}_{n}^{2} γ^{4} α_{n}^{2} + \frac{ε_{β}}{2} \leq \frac{1}{2 ε_{β}} {\overset{⌣}{z}}_{n}^{2} α_{n}^{2} γ_{β} + \frac{ε_{β}}{2} \end{matrix}

(39)

where $γ_{g} (t) = (γ^{2} (t) q^{2}) ∕ {(1 - ρ)}^{2}$ and $γ_{β} (t) = γ^{4} (t)$ .

It is noted that the function tanh () has the following property as in the work by Wen et al. (2011).

\begin{matrix} 0 < | Γ | - Γ \tanh (\frac{Γ}{e}) \leq 0.2785 e, \forall Γ \in R \end{matrix}

(40)

where e is a function satisfying $e > 0$ .

According to equation (40), one can obtain

\begin{matrix} - z_{n} σ \tanh (\frac{{\overset{⌣}{z}}_{n} σ}{e}) & \leq γ (- {\overset{⌣}{z}}_{n} σ \tanh (\frac{{\overset{⌣}{z}}_{n} σ}{e}) + {\overset{⌣}{z}}_{n} σ + | {\overset{⌣}{z}}_{n} σ |) \\ \leq 0.2785 e γ_{M} + γ {\overset{⌣}{z}}_{n} σ \end{matrix}

(41)

Substituting equations (38), (39), and (41) into equation (36), one obtains

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \sum_{s = 1}^{n - 1} k_{s} z_{s}^{2} + \sum_{s = 1}^{n} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + 0.2785 e γ_{M} \\ + γ {\overset{⌣}{z}}_{n} σ + \frac{1}{2 ε_{g}} {\overset{⌣}{z}}_{n}^{2} γ_{g} + \sum_{s = 1}^{n} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} + π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} \\ + \frac{1}{2 ε_{β}} {\overset{⌣}{z}}_{n}^{2} α_{n}^{2} γ_{β} - {\tilde{γ}}_{β} {\overset{\cdot}{\hat{γ}}}_{β} - {\tilde{γ}}_{g} {\overset{\cdot}{\hat{γ}}}_{g} + \frac{ε_{g}}{2} + \frac{ε_{β}}{2} + \frac{ε_{s}}{2} \end{matrix}

(42)

Then choose σ and $θ_{n}$ as

\begin{matrix} σ & = N (θ_{n + 1}) Φ_{n + 1}, {\overset{\cdot}{θ}}_{n + 1} = {\overset{⌣}{z}}_{n} Φ_{n + 1} \end{matrix}

(43)

where $Φ_{n + 1} = \frac{1}{ε_{β}} {\overset{⌣}{z}}_{n} α_{n}^{2} {\hat{γ}}_{β} + \frac{1}{ε_{g}} {\overset{⌣}{z}}_{n} {\hat{γ}}_{g}$ , and the adaptive law is designed as

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{g} = \frac{1}{ε_{g}} {\overset{⌣}{z}}_{n}^{2} - π_{g} {\hat{γ}}_{g} \\ {\overset{\cdot}{\hat{γ}}}_{β} = \frac{1}{ε_{β}} {\overset{⌣}{z}}_{n}^{2} α_{n}^{2} - π_{β} {\hat{γ}}_{β} \end{matrix}

(44)

By equations (43) and (44), equation (42) will be converted to

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \sum_{s = 1}^{n} k_{s} z_{s}^{2} + \sum_{s = 1}^{n + 1} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + \sum_{s = 1}^{n} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} \\ + \sum_{s = 1}^{n} π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} + π_{β} {\tilde{γ}}_{β} {\hat{γ}}_{β} + π_{g} {\tilde{γ}}_{g} {\hat{γ}}_{g} + \frac{ε_{β}}{2} + \frac{ε_{g}}{2} \\ + \sum_{s = 1}^{n} \frac{ε_{s}}{2} + 0.2785 e γ_{M} \end{matrix}

(45)

with $λ_{n + 1} = γ$ .

Stability analysis

Theorem 1. For the nonlinear CPSs (equation (1)) under the attacks (equation (2)), the coordinate transformation is used to realize the tracking control under the attacks. The communication load is reduced by the designed control scheme (equation (27)) with the trigger mechanism, and the boundedness of the obtained closed-loop CPSs signals is ensured.

Proof. By applying equations (1) and (7), one can get $ż_{n}$ as

\begin{matrix} ż_{n} & = ẋ_{n} - \overset{\cdot}{γ} α_{n - 1} - γ {\overset{\cdot}{α}}_{n - 1} = v + f_{n} ({\bar{x}}_{n}) - \overset{\cdot}{γ} α_{n - 1} - γ {\overset{\cdot}{α}}_{n - 1} \end{matrix}

Choosing the final Lyapunov candidate as follows

\begin{matrix} V_{n} = V_{n - 1} + \frac{1}{2} z_{n}^{2} + \frac{1}{2} {\tilde{γ}}_{a_{n}}^{2} + \frac{1}{2} {\tilde{γ}}_{b_{n}}^{2} + \frac{1}{2} {\tilde{γ}}_{β}^{2} + \frac{1}{2} {\tilde{γ}}_{g}^{2} \end{matrix}

one gets

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \sum_{s = 1}^{n} k_{s} z_{s}^{2} + \sum_{s = 1}^{n} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + \sum_{s = 1}^{n} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} \\ + \sum_{s = 1}^{n} π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} + \sum_{s = 1}^{n} \frac{ε_{s}}{2} + z_{n} (v - γ α_{n}) - {\tilde{γ}}_{β} {\overset{\cdot}{\hat{γ}}}_{β} - {\tilde{γ}}_{g} {\overset{\cdot}{\hat{γ}}}_{g} \end{matrix}

(46)

Based on

\begin{matrix} v (t) & = l (t_{k}), \forall t \in [t_{k}, t_{k + 1}) \end{matrix}

(47)

\begin{matrix} t_{k + 1} & = \inf {t \in R | o (t) | \geq ρ | v (t) | + q} \end{matrix}

(48)

where $l (t) = - (1 + ρ) σ \tanh (\frac{{\overset{⌣}{z}}_{n} σ}{e})$ , $t_{k}, k \in z^{+}$ , e, $0 < ρ < 1$ , and $\bar{q} > q ∕ (1 - ρ)$ are all positive design parameters. From equations (28) and (29), we have $v (t) = \frac{l (t)}{1 + ϕ_{1} (t) ρ} - \frac{ϕ_{2} (t) q}{1 + ϕ_{1} (t) ρ}$ , where $ϕ_{1}$ and $ϕ_{2}$ are time-varying parameters satisfying $| ϕ_{1} | \leq 1$ and $| ϕ_{2} | \leq 1$ . Besides, based on the adaptive laws (35) and (44), one can obtain that

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - \underset{s = 1}{\sum^{n}} k_{s} z_{s}^{2} + \underset{s = 1}{\sum^{n + 1}} (λ_{s} (t) N (θ_{s}) + 1) {\overset{\cdot}{θ}}_{s} + \sum_{s = 1}^{n} π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} \\ + π_{s, 2} {\tilde{γ}}_{b_{s}} {\hat{γ}}_{b_{s}} + \sum_{s = 1}^{n} \frac{ε_{s}}{2} + \frac{ε_{g}}{2} + \frac{ε_{β}}{2} \\ + π_{β} {\tilde{γ}}_{β} {\hat{γ}}_{β} + π_{g} {\tilde{γ}}_{g} {\hat{γ}}_{g} + 0.2785 e γ_{M} \end{matrix}

By using $π_{s, 1} {\tilde{γ}}_{a_{s}} {\hat{γ}}_{a_{s}} \leq \frac{1}{2} π_{s, 1} γ_{a_{s}}^{2} - \frac{1}{2} π_{s, 1} {\tilde{γ}}_{a_{s}}^{2}$ , $γ_{b_{s}}$ , $γ_{β}$ , $γ_{g}$ have also the same relationship, one has

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq & - C V_{n} + D + \underset{s = 1}{\sum^{n + 1}} (λ_{s} (t) N (θ_{s}) + 1) \overset{\cdot}{θ} \end{matrix}

(49)

where $C = \min {2 k_{s}, π_{s, 1}, π_{s, 2}, π_{β}, π_{g}}$ , $D = \sum_{s = 1}^{n} (\frac{1}{2} π_{s, 1} γ_{a_{s}}^{2} + \frac{1}{2} π_{s, 2} γ_{b_{s}}^{2} + \frac{1}{2} ε_{s}) + \frac{1}{2} π_{β} γ_{β}^{2} + \frac{1}{2} π_{g} γ_{g}^{2} + \frac{1}{2} ε_{β} + \frac{1}{2} ε_{g} + 0.2785 e γ_{M}$ .

On the other hand, based on the $Φ_{s}$ , $s = 1, \dots, n, n + 1$ , below equations (13), (23), (34), one knows that ${\overset{\cdot}{θ}}_{s} (t) \geq 0$ , $s = 1, \dots, n, n + 1$ . Therefore, this leads to the deduction that $V (t)$ and $θ_{s} (t)$ are bounded by Lemma 3 and equation (49). Through the coordinate transformation (equation (7)) and the definitions of $γ_{a_{s}}$ , $γ_{b_{s}}$ , $γ_{β}$ , $γ_{g}$ , it can be concluded that all signals of the resulting closed-loop CPSs are bounded by using the developed controller. Furthermore, one gets that the tracking error $z_{1}$ is also bounded because of $z_{1} = x_{1} - y_{r}$ .

Next, we prove the avoidance of the Zeno phenomenon by examining the minimum interval times of events. First, $t_{k + 1} = \inf {t \in R | o (t) | \geq ρ | v (t) | + q}$ , $o (t) = v (t) - l (t)$ , $\forall t \in [t_{k}, t_{k + 1})$ , then we have

\begin{matrix} \frac{d}{dt} | o (t) | \leq | \overset{\cdot}{l} (t) | \end{matrix}

(50)

\begin{matrix} \overset{\cdot}{l} (t) = \frac{d}{dt} (- (1 + ρ) σ \tanh (\frac{{\overset{⌣}{z}}_{n} σ}{e})) \end{matrix}

(51)

Because $\frac{d}{dt} (σ \tanh ({\overset{⌣}{z}}_{n} σ ∕ e)) = (1 ∕ \cosh^{2} ({\overset{⌣}{z}}_{n} σ ∕ e)) \frac{d}{dt} ({\overset{⌣}{z}}_{n} σ)$ is bounded, we have $\overset{\cdot}{l} (t)$ as a bounded function. Recalling equation (50), it is reasonable to assume that $\frac{d}{dt} | o (t) |$ is bounded; there exists a constant $d > 0$ such that $| \overset{\cdot}{l} (t) | \leq d$ . By noting that $o (t_{k}) = v (t) - l (t)$ and $\lim_{t \to t_{k + 1}} o (t) = ρ | v (t) | + q$ , we obtain that the lower bound of inter-execution intervals $t^{*}$ satisfies $t^{*} \geq \frac{ρ | v (t) | + q}{d}$ . Then, the Zeno behavior in the work by Johansson et al. (1999) is overcome. □

Remark 2. For the attacked system (1), we aimed to study the tracking control in this work. Based on the above analysis, it is clearly observed that $k_{s}$ , $π_{s, 1}$ , $π_{s, 2}$ , $π_{β}$ , and $π_{g}$ affect the tracking error $z_{1}$ . It is worth noting that the tracking performance of the system will be improved by reducing $π_{s, 1}$ , $π_{s, 2}$ , $π_{β}$ , and $π_{g}$ , but increasing $k_{s}$ .

Remark 3. Unlike the stabilization problem of the attacked nonlinear CPSs in the works by Bi et al. (2022) and Ma et al. (2019), the required Lyapunov inequality is designed by the coordinate transformation in equation (10), and through the coordinate transformation in equation (11) to construct the controller, so as to realize the tracking control under the attacks.

Remark 4. In contrast to the works by Liu et al. (2019) and Ren and Yang (2020), the isolated uncertain nonlinearity is approximated by the RBFNNs, and the event-triggering mechanism is designed on this basis. This allows for the safety goal of a nonlinear structure that is more general than a system.

Simulation results

The following numerical examples and practical models are given to illustrate the validity and feasibility of the methodology.

Example 1. Consider the uncertain nonlinear CPSs with deception attacks as

\begin{matrix} ẋ_{1} = x_{2} + f_{1} (x_{1}) \\ ẋ_{2} = v + f_{2} ({\bar{x}}_{2}) \end{matrix}

(52)

where $f_{1} (x_{1}) = 0$ and $f_{2} ({\bar{x}}_{2}) = x_{1}^{2} x_{2}$ .

Through the adaptive control scheme, the control scheme is as follows

\begin{matrix} α_{1} = N (θ_{1}) Φ_{1} \end{matrix}

(53)

\begin{matrix} α_{2} = \frac{\partial α_{1}}{\partial θ_{1}} {\overset{\cdot}{θ}}_{1} + \frac{γ}{γ_{M_{1}}} \frac{\partial α_{1}}{\partial {\overset{⌣}{x}}_{1}} {\overset{⌣}{x}}_{2} + \frac{\partial α_{1}}{\partial y_{r}} ẏ_{r} + \frac{\partial α_{1}}{\partial ẏ_{r}} ÿ_{r} + N (θ_{2}) Φ_{2} \end{matrix}

(54)

\begin{matrix} v = N (θ_{3}) Φ_{3} \end{matrix}

(55)

with

\begin{matrix} Φ_{1} = \frac{1}{ε_{1}} {\hat{γ}}_{a_{1}} {\overset{⌣}{z}}_{1} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} + \frac{1}{ε_{1}} {\hat{γ}}_{b_{1}} {\overset{⌣}{z}}_{1} - γ_{M_{1}} ẏ_{r} \\ Φ_{2} = \frac{1}{ε_{2}} {\hat{γ}}_{a_{2}} {\overset{⌣}{z}}_{2} {∥ K_{2} ({\bar{X}}_{2}) ∥}^{2} + \frac{1}{ε_{2}} {\hat{γ}}_{b_{2}} {\overset{⌣}{z}}_{2} \\ Φ_{3} = \frac{1}{ε_{β}} {\overset{⌣}{z}}_{2} α_{2}^{2} {\hat{γ}}_{β} + \frac{1}{ε_{g}} {\overset{⌣}{z}}_{2} {\hat{γ}}_{g} \\ {\overset{\cdot}{θ}}_{1} = {\overset{⌣}{z}}_{1} Φ_{1}, {\overset{\cdot}{θ}}_{2} = {\overset{⌣}{z}}_{2} Φ_{2}, {\overset{\cdot}{θ}}_{3} = {\overset{⌣}{z}}_{3} Φ_{3} \\ {\overset{\cdot}{\hat{γ}}}_{a_{1}} = \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} {∥ K_{1} ({\bar{X}}_{1}) ∥}^{2} - π_{1, 1} {\hat{γ}}_{a_{1}}, {\overset{\cdot}{\hat{γ}}}_{b_{1}} = \frac{1}{ε_{1}} {\overset{⌣}{z}}_{1}^{2} - π_{1, 2} {\hat{γ}}_{b_{1}} \end{matrix}

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{a_{2}} = \frac{1}{ε_{2}} {\overset{⌣}{z}}_{2}^{2} {∥ K_{2} ({\bar{X}}_{2}) ∥}^{2} - π_{2, 1} {\hat{γ}}_{a_{2}}, {\overset{\cdot}{\hat{γ}}}_{b_{2}} = \frac{1}{ε_{2}} {\overset{⌣}{z}}_{2}^{2} - π_{2, 2} {\hat{γ}}_{b_{2}} \end{matrix}

\begin{matrix} {\overset{\cdot}{\hat{γ}}}_{β} = \frac{1}{ε_{β}} {\overset{⌣}{z}}_{n}^{2} α_{n}^{2} - π_{β} {\hat{γ}}_{β}, {\overset{\cdot}{\hat{γ}}}_{g} = \frac{1}{ε_{g}} {\overset{⌣}{z}}_{n}^{2} - π_{g} {\hat{γ}}_{g} \end{matrix}

The attack weight is given with $ϖ (t) = 6 + 0.2 \cos (t)$ from $t = 30 s$ . The parameters are chosen as $ε_{1} = 0.5$ , $ε_{2} = 0.5$ , $ε_{β} = 1$ , $ε_{g} = 1$ . $π_{1, 1} = 0.1$ , $π_{1, 2} = 0.1$ , $π_{2, 1} = 0.1$ , $π_{2, 2} = 0.1$ , $π_{β} = 0.8$ , $π_{g} = 0.1$ . $q = 0.01$ , $ρ = 0.01$ , $e = 0.01$ . We choose the primary condition as $x (0) = {[0.5, 1]}^{T}$ , ${\hat{γ}}_{a_{1}} = 0.2$ , ${\overset{\cdot}{\hat{γ}}}_{β} = 0.5$ , ${\overset{\cdot}{\hat{γ}}}_{b_{1}} = 0.2$ , and the remaining values are all assigned as zero.

Figure 2 shows the state trajectories of $x_{1}$ and $y_{r}$ . The attack occurs at 30 s, resulting in a deviation of the tracking trajectory. Then, the tracking performance is resumed after 1.39 s. Figure 3 shows the trajectory of $x_{2}$ . Figures 4 and 5 represent the changes of adaptive laws. The triggers are illustrated in Figure 6. In Figure 7, the control signal v is exhibited. The input signal is standardized at 3.16 s under normal operation. After the attack, the input signal is stabilized again within 1.39 s. Figures 8 and 9 present comparative results between the proposed method and the approach in the work by Qiu et al. (2019). The performance data are presented in Table 2 (only the number of triggers is full-time data, the rest of the data are counted after the attack appears). From Table 2, we can see that the proposed method reduces the number of triggers from 1346 to 510, cutting 62% of unnecessary activation. Besides, the tracking error, control signal amplitude, and recovery time after the attack are lower than the work by Qiu et al. (2019), validating the advantages of the designed approach.

Figure 2.

The tracking trajectories of $x_{1}$ and $y_{r}$ .

Figure 3.

The trajectories of state $x_{2}$ .

Figure 4.

The trajectories of adaptive laws ${\hat{γ}}_{a_{1}}$ , ${\hat{γ}}_{a_{2}}$ , and ${\hat{γ}}_{β}$ .

Figure 5.

The trajectories of adaptive laws ${\hat{γ}}_{b_{1}}$ , ${\hat{γ}}_{b_{2}}$ , and ${\hat{γ}}_{g}$ .

Figure 6.

The time interval of the event-triggered mechanism.

Figure 7.

The trajectories of control inputs v.

Figure 8.

The comparison trajectories of tracking error.

Figure 9.

The comparison trajectories of control signals v.

Table 2.

Quantitative analysis of Example 1.

	Qiu et al. (2019)	The proposed method
Number of triggers	1346	510
Control input peak	−378 to 631	−221 to 441
Peak tracking error	−0.08 to 0.03	−0.05 to 0.01
Recovery time of control input	1.5 s	1.39 s

Example 2. Consider the electromechanical system considered in the work by Shi et al. (2018):

\begin{matrix} M \overset{\cdot\cdot}{q} + B \overset{\cdot}{q} + N \sin (q) = I \\ Lİ = V_{ε} - RI - K_{B} \overset{\cdot}{q} \end{matrix}

(56)

then, the state space of its dynamic model is described as

\begin{matrix} ẋ_{1} = x_{2} \\ ẋ_{2} = \frac{1}{M} x_{3} - \frac{N}{M} \sin (x_{1}) - \frac{B}{M} x_{2} + \frac{B}{M} \cos (x_{2}) \sin (x_{3}) \\ ẋ_{3} = \frac{1}{L} v - \frac{K_{B}}{L} - \frac{R}{L} x_{3} + \cos (x_{2}) \sin (x_{3}) \\ y = x_{1} \end{matrix}

(57)

with

\begin{matrix} M = \frac{J}{K_{τ}} + \frac{m L_{0}^{2}}{3 K_{τ}} + \frac{M_{0} L_{0}^{2}}{K_{τ}} + \frac{2 M_{0} R_{0}^{2}}{5 K_{τ}} \\ N = \frac{m L_{0} G}{2 K_{τ}} + \frac{M_{0} L_{0} G}{K_{τ}}, B = \frac{B_{0}}{K_{τ}} \end{matrix}

where $x_{1} = q$ , $x_{2} = \overset{\cdot}{q}$ , $x_{3} = I ∕ M$ , $v = V_{ε}$ . The definition of the electromechanical system parameters is defined in the work by Ren and Yang (2020). Furthermore, these parameters are chosen as $J = 1.625 \times 10^{- 3} kg \cdot m^{2}$ , $m = 0.506 kg$ , $M_{0} = 0.434 kg$ , $L_{0} = 0.305 m$ , $R_{0} = 0.023 m$ , $G = 9.8 N/kg$ , $B_{0} = 16.25 \times 10^{- 3} N \cdot m ∕ rad$ , $K_{τ} = 0.9 N \cdot m ∕ A$ , $K_{B} = 0.9 N \cdot m ∕ A$ , $L = 25 \times 10^{- 3} H$ , $R = 5 Ω$ .

In this simulation, the attack weight is given with $ϖ (t) = - 0.24 + 0.02 \sin (t)$ from $t = 30$ . The parameters are chosen as $ε_{1} = 6$ , $ε_{2} = 2$ , $ε_{3} = 2$ , $ε_{β} = 2$ , $ε_{g} = 2$ . $π_{1, 1} = 5$ , $π_{1, 2} = 0.2$ , $π_{2, 1} = 5$ , $π_{2, 2} = 10$ , $π_{3, 1} = 5$ , $π_{3, 2} = 5$ , $π_{β} = 0.01$ , $π_{g} = 0.25$ , $q = 0.05$ , $ρ = 0.03$ , $e = 0.2$ . We choose the primary condition as $x (0) = {[- 1, 2]}^{T}$ , ${\hat{γ}}_{a_{1}} = 0.2$ , and the remaining values are all assigned as zero.

Figure 10 demonstrates that the motor angular position achieves high-precision tracking of the desired reference signal. After the attack, the reference signal is tracked on again within 0.63 s. The bounded trajectories of the three system states are depicted in Figure 11. Figures 12 and 13 present the convergence process of the adaptive laws. Notably, Figure 14 illustrates the trajectories of control signal. The event-triggering statistics in Figure 15 highlight the communication efficiency, and the proposed trigger mechanism cuts 31% of unnecessary activation compared with Qiu et al. Figures 16 and 17 exhibit the performance compared to the work by Qiu et al. (2019) and form the core of Table 3. It is easy to see that the proposed method is lower than the work by Qiu et al. (2019) both in terms of tracking error and control signal amplitude, as well as the recovery time after the attack, thus emphasizing the superiority of the proposed method.

Figure 10.

The tracking trajectories of $x_{1}$ and $y_{r}$ .

Figure 11.

The trajectories of state $x_{1}$ , $x_{2}$ , $x_{3}$ .

Figure 12.

The trajectories of adaptive laws ${\hat{γ}}_{a_{1}}$ , ${\hat{γ}}_{a_{2}}$ , ${\hat{γ}}_{a_{3}}$ , and ${\hat{γ}}_{β}$ .

Figure 13.

The trajectories of adaptive laws ${\hat{γ}}_{b_{1}}$ , ${\hat{γ}}_{b_{2}}$ , ${\hat{γ}}_{b_{3}}$ , and ${\hat{γ}}_{g}$ .

Figure 14.

The trajectories of control inputs v.

Figure 15.

The time interval of the event-triggered mechanism.

Figure 16.

The comparison trajectories of tracking error.

Figure 17.

The comparison trajectories of control signals v.

Table 3.

Quantitative analysis of Example 2.

	Qiu et al. (2019)	The proposed method
Number of triggers	673	464
Control input peak	−398 to 182	−370 to 160
Peak tracking error	−0.17 to 0.17	−0.14 to 0.11
Tracking recovery time of $x_{1}$	0.66 s	0.63 s

Only the number of triggers is full-time data; the rest of the data are counted after the attack appears.

Conclusion

This work focuses on the tracking control problem of nonlinear CPSs under deception attacks. Within the framework of backstepping recursive control, a coordinate transformation incorporating reference signals is designed to achieve tracking control while addressing the challenge of unavailable system states. Then, the neural network techniques are adopted to approximate the system’s nonlinearities. Additionally, the relative threshold-based event-triggered mechanism is proposed. Its superiority is demonstrated through theoretical analysis and comparative experiments. Finally, numerical and practical simulations are conducted to validate the effectiveness and broad applicability of the proposed approach. Future research will extend this method to more complex systems, such as multi-input multi-output CPSs or distributed multi-agent CPSs, incorporating advanced attack detection mechanisms to achieve cooperative tracking control under cyberattacks, thereby significantly enhancing the robustness and security of CPSs against emerging attack strategies.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported in part by the National Natural Science Foundation of China under grants 62025303, 62173173, and 62403225; the Doctoral Research Initiation Foundation of Liaoning Province under grant 2024-BS-237; and the Project of Education Department of Liaoning Province under grant JYTQN2023222.

ORCID iD

Qiang Zeng

Data availability statement

The data sets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

Yang

(2018) Decentralized adaptive fuzzy secure control for nonlinear uncertain interconnected systems against intermittent DoS attacks. IEEE Transactions on Cybernetics 49(3): 827–838.

Wang

Qiu

, et al. (2022) Adaptive decentralized finite-time fuzzy secure control for uncertain nonlinear CPSs under deception attacks. Journal of the Franklin Institute 356(17): 10260–10276.

Cuan

Ding

Liu

, et al. (2023) Adaptive fuzzy control for state-constrained nonlinear cyber-physical systems with unmodeled dynamics against malicious attacks. IEEE Transactions on Industrial Cyber-Physical Systems 1: 56–65.

De Persis

Postoyan

, et al. (2023) Event-triggered control from data. IEEE Transactions on Automatic Control 69(6): 3780–3795.

Ding

Han

, et al. (2020) Secure state estimation and control of cyber-physical systems: A survey. IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(1): 176–190.

Feng

Hao

, et al. (2018) Design of distributed cyber–physical systems for connected and automated vehicles with implementing methodologies. IEEE Transactions on Industrial Informatics 14(9): 4200–4211.

Johansson

Egerstedt

Lygeros

, et al. (1999) On the regularization of Zeno hybrid automata. Systems & Control Letters 38(3): 141–150.

Yang

Xia

, et al. (2019) State estimation for linear systems with unknown input and random false data injection attack. IET Control Theory and Applications 13(6): 823–831.

Shi

Chen

(2017) Detection against linear deception attacks on multi-sensor remote state estimation. IEEE Transactions on Control of Network Systems 5(3): 846–856.

10.

Zhao

(2021) Resilient adaptive control of switched nonlinear cyber-physical systems under uncertain deception attacks. Information Sciences 543: 398–409.

11.

Lin

Qian

(2002) Adaptive control of nonlinearly parameterized systems: The smooth feedback case. IEEE Transactions on Automatic Control 47(8): 1249–1266.

12.

Liu

Tian

Xie

, et al. (2019) Distributed event-triggered control for networked control systems with stochastic cyber-attacks. Journal of the Franklin Institute 356(17): 10260–10276.

13.

Park

(2019) Global adaptive control for uncertain nonlinear systems with sensor and actuator faults. IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(9): 5503–5510.

14.

Wang

, et al. (2023) Event-triggered adaptive finite-time secure control for nonlinear cyber-physical systems against unknown deception attacks. International Journal of Robust and Nonlinear Control 33(3): 1593–1606.

15.

Wang

, et al. (2024) Event-triggered finite-time command-filtered tracking control for nonlinear time-delay cyber physical systems against cyber attacks. Frontiers of Information Technology & Electronic Engineering 25(2): 225–236.

16.

Pan

Ding

Zhang

(2024) Adaptive consensus control and attack detection of nonlinear multiagent systems under false data injection attacks and unknown control directions. Transactions of the Institute of Measurement and Control 47: 1736–1747.

17.

Hou

Zong

, et al. (2021) Finite-time event-triggered control for semi-Markovian switching cyber-physical systems with FDI attacks and applications. IEEE Transactions on Circuits and Systems I: Regular Papers 68(6): 2665–2674.

18.

Qiu

Sun

Wang

, et al. (2019) Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance. IEEE Transactions on Fuzzy Systems 27(11): 2152–2162.

19.

Ren

Yang

(2020) Adaptive control for nonlinear cyber-physical systems under false data injection attacks through sensor networks. International Journal of Robust and Nonlinear Control 30(1): 65–79.

20.

Shi

Lim

Shi

, et al. (2018) Adaptive neural dynamic surface control for nonstrict-feedback systems with output dead zone. IEEE Transactions on Neural Networks and Learning Systems 29(11): 5200–5213.

21.

Tian

Zhang

Liu

, et al. (2024) Event-triggered adaptive secure tracking control for nonlinear cyber–physical systems against unknown deception attacks. Mathematics and Computers in Simulation 221: 79–93.

22.

Wang

Huang

Wang

, et al. (2021) A secure strategy for a cyber physical system with multi-sensor under linear deception attack. Journal of the Franklin Institute 358(13): 6666–6683.

23.

Wang

Park

(2024) Resilient prediction-based and event-triggered control for CPSs under DoS attacks. IEEE Transactions on Industrial Informatics 20(8): 10419–10427.

24.

Wang

Zhang

, et al. (2022) Research on cyber-physical system control strategy under false data injection attack perception. Transactions of the Institute of Measurement and Control. Epub ahead of print January 12. DOI: 10.1177/01423312211069371.

25.

Wen

Zhou

Liu

, et al. (2011) Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance. IEEE Transactions on Automatic Control 56(7): 1672–1678.

26.

Xie

Liu

, et al. (2022) Resilient adaptive event-triggered h_∞ fuzzy filtering for cyber-physical systems under stochastic-sampling and denial-of-service attacks. IEEE Transactions on Fuzzy Systems 31(1): 278–292.

27.

Xing

Wang

Pang

, et al. (2022) Dynamic-memory event-based asynchronous attack detection filtering for a class of nonlinear cyber-physical systems. IEEE Transactions on Cybernetics 53(1): 653–667.

28.

Xiong

Tan

, et al. (2020) Observer-based event-triggered output feedback control for fractional-order cyber–physical systems subject to stochastic network attacks. ISA Transactions 104: 15–25.

29.

Wang

Sun

, et al. (2024) Model-based adaptive security control strategy against false data injection attacks in cyber–physical systems. Transactions of the Institute of Measurement and Control 46(15): 2909–2920.

30.

Yoo

(2019) Neural-network-based adaptive resilient dynamic surface control against unknown deception attacks of uncertain nonlinear time-delay cyber physical systems. IEEE Transactions on Neural Networks and Learning Systems 31(10): 4341–4353.

31.

Yucelen

Haddad

Feron

(2016) Adaptive control architectures for mitigating sensor attacks in cyber-physical systems. Cyber-Physical Systems 2(1–4): 24–52.

32.

Zhang

Huang

(2024) Secure iterative interval estimation method for cyber-physical systems subject to stealthy deception attacks. Transactions of the Institute of Measurement and Control 46(6): 1084–1092.

33.

Zhao

Zhu

Zhang

, et al. (2023) Security switching control of cyber-physical system with incremental quadratic constraints under denial-of-service attack. Transactions of the Institute of Measurement and Control 45(1): 157–167.