Adaptive NN Finite-Time Resilient Control for Nonlinear Time-Delay Systems With Unknown False Data Injection and Actuator Faults

This article considers neural network (NN)-based adaptive finite-time resilient control problem for a class of nonlinear time-delay systems with unknown fault data injection attacks and actuator faults. In the procedure of recursive design, a coordinate transformation and a modified fractional-order command-filtered (FOCF) backstepping technique are incorporated to handle the unknown false data injection attacks and overcome the issue of “explosion of complexity” caused by repeatedly taking derivatives for virtual control laws. The theoretical analysis proves that the developed resilient controller can guarantee the finite-time stability of the closed-loop system (CLS) and the stabilization errors converge to an adjustable neighborhood of zero. The foremost contributions of this work include: 1) by means of a modified FOCF technique, the adaptive resilient control problem of more general nonlinear time-delay systems with unknown cyberattacks and actuator faults is first considered; 2) different from most of the existing results, the commonly used assumptions on the sign of attack weight and prior knowledge of actuator faults are fully removed in this article. Finally, two simulation examples are given to demonstrate the effectiveness of the developed control scheme.

SECTION I. Introduction

In recent years, the control issues of time-delay systems have been an attractive research topic due to its significant value in theory and practice (see [1]–​[6] and reference therein). Wu and Wang [2] developed a novel predictor for discrete-time linear systems with input delay and external disturbance. The creative idea in [2] is to make full use of available information to construct an estimation of the future state. Min et al. [6] proposed a relaxed result for stabilizing unknown high-order stochastic nonlinear systems by removing the growth assumption. Furthermore, by combining with the intelligent approximation technique, the adaptive fuzzy or neural network (NN) control approaches have been developed to handle the control problems of unknown nonlinear time-delay systems [7]–​[10]. To mention a few, Wang et al. [8] developed an adaptive output-feedback control scheme for a class of stochastic nonlinear time-delay systems with nonlower-triangular form. Ma et al. [11] addressed the event-triggered adaptive output-feedback control issue of nonstrict-feedback nonlinear systems with time-varying state delays and input delay for the first time, where the ISS assumption was fully removed. Furthermore, the dynamic surface control (DSC) method and the command filtered backstepping technique were employed to avoid the problem of “explosion of complexity” in [12]–​[15], which reduced the computational burden greatly. Still, it is worth noting that the aforementioned results only guaranteed the asymptotic stability of the closed-loop system (CLS), which means that the closed-loop convergence can be ensured as time goes to infinity. However, numerous real systems are required to reach steady state within a finite time under a specified operating condition. Therefore, establishing an adaptive finite-time control scheme for more general nonlinear time-delay systems remains an open and challenging problem.

When a control system suffers the various inferences caused by human-made or natural factors such as cyberattacks and sensor/actuator failures, poor performance and instability may be inevitable. To this end, the reliability of the control system design has been a research topic with important significance and practical value. The existing resilient control schemes were mainly divided into two categories: fault-tolerant control against sensor/actuator faults and secure control against malicious cyberattacks. Regarding the first one, many remarkable achievements have been obtained for different kinds of linear/nonlinear systems [16]–​[21]. Wu et al. [17] designed an observer-based adaptive fault-tolerant controller for nonlinear systems with nonstrict-feedback form, where the bounds of the actuation effectiveness and stuck faults are assumed to be available in advance for facilitating controller design. Song et al. [21] proposed a non-PDC method for nonlinear PDE systems, where a novel event-triggered mechanism was developed to reduce the transmission burden of sensor networks. Correspondingly, considerable attention has been paid to the secure control problem of cyber-physical systems (CPSs) due to the importance of network security (see [22]–​[25] and reference therein). However, these methods proposed in the aforementioned achievements were focused on linear systems and robust control problem. More specifically, only a few studies were conducted for the adaptive secure control problem of nonlinear CPSs [26]–​[28]. By combining with backstepping control framework and linear matrix inequality technique, An and Yang [26] developed an adaptive fuzzy resilient decentralized control method against denial-of-service attacks. In [27], a new coordinate transformation was established to handle false data injection attacks. However, the derivatives of virtual control laws were required for designing controller in [26] and [27]. This also means that the proposed methods may be hard to be implemented when the nonlinear systems with high dimensions were considered. Consequently, the DSC method was adopted in [28] to cope with this issue, where attacks compensator was constructed to eliminate the negative effects caused by deception attacks. Nevertheless, the results obtained in [28] based on the assumption that the attack weight was strictly required to be positive, which increased the conservatism of the proposed method to a certain extent. What’s more, it is necessary to emphasize that all aforementioned results paid attention to the asymptotic secure control problem. To our knowledge, the adaptive finite-time control problem for nonlinear time-delay systems with cyberattacks has not been investigated yet, which is still a challenging problem.

Recently, the command filtered backstepping technique, initially proposed by Farrell et al. [29], provides one effective solution to reduce the computational complexity of the traditional backstepping design. To this end, this technique has been widely employed to fulfill the controller design in the backstepping control framework (see [30]–​[33] and reference therein). However, there is no result reported on command filter-based adaptive secure control problem for nonlinear time-delay systems with cyberattacks. On the other hand, it has been recognized that the FO control provides a new solution for enhancing the degree of freedom of controller design and improving system performance due to the unique advantage of historical memory in fractional calculus [34]–​[36]. Although various FO control strategies have been established by integrating with FO characteristics and IO control methods so far, to the author’s best knowledge, there were a few available results on command filter-based adaptive finite-time resilient control for nonlinear systems with unknown cyberattacks and actuator faults.

Inspired by the above discussion, this work aims to develop a novel adaptive finite-time resilient control scheme for nonlinear time-delay systems with unknown false data injection attacks and actuator faults. The main novelties of this work are summarized as follows.

1. This work makes the first attempt to investigate the command filter-based adaptive finite-time resilient control problem for a class of nonlinear time-delay systems with unknown false data injection attacks and actuator faults. Compared with the existing literature [23], [24], [27], [28], a class of more general nonlinear time-delays systems were investigated in this article, where the unknown false data injection attacks and actuator faults are considered simultaneously. Noted that the system model considered in [23], [24], [27], and [28] can be viewed as a special case of this work. Obviously, the presented results provide a unified framework for addressing the adaptive resilient control problem of nonlinear time-delay systems with strict-feedback form.

2. Different from [28] and [37], the conservative assumption that the sign of attack weight is strictly required to be positive can be fully removed based on the Nussbaum gain technique. Besides, the bounds of the actuation effectiveness and stuck faults are not required in this article. Moreover, by integrating with the command-filtered backstepping technique and fractional calculus, a modified fractional-order command filter (FOCF) is developed such that the computational burden of the existing methods in [26] and [27] is greatly relaxed and filter performance is obviously improved in comparison to [28] and [37].

3. The methods proposed in [23], [24], and [26]–​[28] were concentrated on the asymptotic stability of the CLS. Distinct from these results, the finite-time stability of the controlled system with faster response and higher control precision can be guaranteed by using the proposed method in this work.

SECTION II. Preliminaries and System Description

A. Nonlinear System Model

In this article, we focus on a class of nonlinear time-delay systems with unknown false data injection attacks and actuator faults described by $$\begin{align*} \begin{cases} \displaystyle {\dot {x}_{i}}={x_{i+1}}+f_{i}\left ({\bar {x}_{i}}\right)+g_{i}\left ({\bar {x}_{i,\tau \left ({t}\right)}}\right)+d_{i}\left ({t}\right) \\ \displaystyle {\dot {x}_{n}}=\sum _{j=1}^{p}b_{j}u_{j}+f_{n}\left ({x}\right)+g_{n}\left ({\bar {x}_{n,\tau \left ({t}\right)}}\right)+d_{n}\left ({t}\right)\\ \displaystyle \check {x}_{j}=x_{l}+A_{l,s}\left ({x_{l},t}\right), ~i=1,\ldots,n-1;\quad l=1,\ldots,n \end{cases} \tag{1}\end{align*}$$ where $u_{j}\in \mathbb {R}$ is system input with $j=1,\ldots,p$ , ${\bar {x}}_{l}=[x_{1}, \ldots, x_{l}] ^{T}\in \mathbb {R}^{l}$ are state variables, $\bar {x}_{l,\tau (t)}=[x_{1}(t-\tau _{1}(t)), \ldots, x_{l}(t-\tau _{l}(t))] ^{T}\in \mathbb {R}^{l}$ denote delayed state variables with unknown time-varying delays, and the delays are assumed to be bounded satisfying $0 < \tau _{l}\le \bar {\tau }_{l} < \infty$ and $\dot {\tau }_{l}\le \bar {\tau }_{ld} < 1$ , where $\bar {\tau }_{l}$ and $\bar {\tau }_{ld}$ are unknown positive constants. $\check {x}_{j}$ are the available compromised state variables, and $f_{i}(\bar {x}_{i})$ and $g_{i}(\bar {x}_{i,\tau (t)})$ are unknown but smooth nonlinear functions. $d_{i}(t)$ represent the unknown but bounded external disturbances satisfying $|d_{i}(t)|\le \bar {d}_{i}$ , where $\bar {d}_{i}>0$ is a unknown constant; $A_{l,s}(x_{l},t)$ denote the state-dependent false data injection attacks, which satisfies $A_{l,s}(x_{l},t)=\lambda (t)x_{l}(t)$ with an unknown time-varying signals $\lambda (t)$ . Suppose that a general actuator fault may occur in the system, which can be mathematically described as [38], [39] $$\begin{align*}&u_{j}\left ({t}\right)=l_{jm}v_{j}+\bar {u}_{jd,m}, \quad t\in \left [{t_{jm,s}, t_{jm,e}}\right) \tag{2}\\&l_{jm}\bar {u}_{jd,m}=0, \quad j=1,2,\ldots,p \tag{3}\end{align*}$$ where $0 < l_{jm}\le 1$ , $\bar {u}_{jd,m}$ are all unknown constants, $t_{jm,s}$ and $t_{jm,e}$ stand for the starting time and ending time satisfying $0\ge t_{j1,s} < t_{j1,e} < t_{j2,s} < t_{j2,e} < \cdots < t_{jm,e} < t_{j(m+1),s} < t_{j(m+1),e}$ and so on. Equation (2) means that the $j$ th actuator suffers the fault from starting time $t_{jm,s}$ until ending time $t_{jm,e}$ . This fault model includes the following four cases.

1. $\bar {u}_{jd,m}=0$ with $0 < l_{jm} < 1$ . This case stands for the actuator undergoes the partial loss of effectiveness. For instance, $l_{jm}=0.3$ means that the 30% actuator is normally operating after losing 70% effectiveness.

2. $\bar {u}_{jd,m}=0, l_{jm}=1$ . In this case, the input of the actuator equal to its output, i.e., $u_{j}(t)=v_{j}$ . It also reflects that the actuator works normally.

3. $\bar {u}_{jd,m}\neq 0, l_{jm}=1$ . This situation means that the bias fault occurs.

4. $\bar {u}_{jd,m}\neq 0, l_{jm}=0$ . In this situation, $u_{j}$ is stuck at an unknown constant $\bar {u}_{jd,m}$ such that it cannot be influenced by the control signal $v_{j}$ . This also implies that the actuator suffers from the total loss of effectiveness.

Our control goal is to establish a novel adaptive resilient control framework for nonlinear time-delay systems subject to unknown false data injection attacks and actuator faults as shown in Fig. 1 while ensuring that the finite-time stability of the CLS can be achieved.

Fig. 1.

Adaptive resilient control framework of nonlinear time-delay system.

B. RBF Neural Networks

To fulfill the predefined control goal, RBF NNs shall be adopted to estimate the unknown nonlinearities. It is proved that [40]–​[43] that with enough NNs nodes number, the continuous item $F_{i}(\triangle _{i})$ defined in a compact set $\Omega _{\Delta _{i}}\subset \mathbb {R}^{q}$ could be approximated by RBF NNs satisfying $$\begin{equation*} F_{i}\left ({\triangle _{i}}\right)=\theta _{i}^{\ast T}\varphi _{i}\left ({\Delta _{i}}\right)+o_{i} \end{equation*}$$ where $i=1,\ldots,n, \Delta _{i}\in \Omega _{\Delta _{i}}\subset \mathbb {R}^{q}$ denotes the input vector of RBFNNs, $o_{i}$ is an approximation error, and $\theta _{i}^{\ast }=[\theta _{i1}^{\ast },\ldots,\theta _{im}^{\ast }]^{T}\in \mathbb {R}^{m}$ represents the optimal weight vector satisfying $\theta _{i}^{\ast }=\arg \min _{\hat {\theta }_{i}}\{ \sup _{\triangle _{i}\in \Omega _{\triangle _{i}}}|F_{i}(\triangle _{i})-\hat {\theta }_{i}^{T}\varphi _{i} (\Delta _{i})|\}$ with $m$ being the number of RBFNN nodes. $\varphi _{i}(\Delta _{i})=[\varphi _{i1}(\Delta _{i}),\ldots,\varphi _{im}(\Delta _{i})]^{T}$ is the basis function (BF) vector with the following form: $$\begin{equation*} \varphi _{ik}\left ({\Delta _{ik}}\right)=\frac {1}{\sqrt {2\pi }b_{ik}}\exp \left [{-\frac {\left ({\Delta _{i}-c_{ik}}\right)^{T}\left ({\Delta _{i}-c_{ik}}\right)}{b_{ik}^{2}}}\right]\end{equation*}$$ where $k=1,\ldots,m, c_{ik}$ is the center of the receptive field, and $b_{ik}>0$ denote the width of the neural cell.

Assumption 1 ([44]):

$\theta _{i}^{\ast }$ and $o_{i}$ are bounded satisfying $||\theta _{i}^{\ast }||\le \bar {\theta }_{i}^{\ast }$ and $| o_{i}|\le \bar {o}_{i}$ , where $\bar {\theta }_{i}^{\ast }$ and $\bar {o}_{i}$ are the unknown constants.

Assumption 2 ([28]):

The unknown time-varying signal $\lambda (t)$ satisfies $1+\lambda (t)\neq 0$ . Furthermore, there exist unknown constants $\bar {\lambda }$ , and $\bar {\lambda }_{d}$ such that $|\lambda (t)|\le \bar {\lambda }, |\dot {\lambda }(t)|\le \bar {\lambda }_{d}$ hold.

Assumption 3 ([38]):

For up to $p-1$ total loss of effectiveness of actuator faults, the predefined control goal can be achieved by the remaining actuators.

Remark 1:

Noted that Assumptions 1 has been widely utilized in the existing results about adaptive NN control for nonlinear systems. As similarly discussed in [44], Assumption 1 is not restricted according to the inherent property of Gaussian BFs. What’s more, these unknown upper bounds are required only for analytical purpose, which are not involved in controller design. For Assumptions 2, $1+\lambda (t)=0$ indicates that the unknown false data offset the original state information. However, the actual controller cannot be designed once the comprised state information is not available. Different from [28], the weight $1+\lambda (t)$ is not required to be strictly positive. Hence, the assumption of $1 + \lambda (t)\neq 0$ is mild for fulfilling the task of controller design. Similarly, Assumptions 3 is a standard condition for ensuring the controllability of the investigated system, which does not pose a strong restriction on engineering applications.

C. Nussbaum Function

In this article, the Nussbaum function scheme is adopted to overcome the difficulty in control design caused by the unknown attacks. According to [45], for $N(\xi)\in N$ , the following equations hold: $$\begin{align*}&\lim _{\tau \rightarrow \infty }\sup \frac {1}{\tau }\int _{0}^{\tau }N\left ({\xi }\right)d\xi =+\infty \\&\lim _{\tau \rightarrow \infty }\inf \frac {1}{\tau }\int _{0}^{\tau }N\left ({\xi }\right)d\xi =-\infty.\end{align*}$$

Nowadays, it is recognized that a lot of continuous functions can be regarded as the Nussbaum-type function, such as $e^{\xi ^{2}}\cos (({\pi }/{2})\xi), \xi ^{2}\cos (\xi)$ , and $\xi ^{2}\sin (\xi)$ .

Lemma 1 ([45]):

Assume that $V(t)$ and $\xi (\cdot)$ are smooth functions specified on $[0,t_{s}$ ) and $V(t)\ge 0, \forall t\in [0, t_{s})$ , $N(\xi)$ is a smooth Nussbaum-type function. The following inequality holds: $$\begin{equation*} V\left ({t}\right)\le m_{1}+e^{-m_{2}t}\int _{0}^{t}\left ({g\left ({\varsigma }\right)N\left ({\xi }\right)+1}\right)\dot {\xi }e^{m_{2}\varsigma }d\varsigma\end{equation*}$$ where $\forall t\in [0,t_{s}$ ), $m_{2}$ is a positive constant, $g(\cdot)$ represents a time-varying parameter which takes values in the unknown closed intervals $I=[l^{-}, l^{+}]$ with $0\notin I$ , and $m_{1}$ is a suitable constant, and then, $V(t), \xi (t)$ , and $\int _{0}^{t}g(\varsigma)N(\xi)\dot {\xi }d\varsigma$ must be bounded on $[0,t_{s}$ ).

Definition 1:

[46] The fractional derivative of order $\alpha$ of a function $h(t)\in C^{n}([t_{0},+\infty],R)$ with Caputo’s definition is defined as $$\begin{equation*} D^{\alpha }f\left ({t}\right)=\frac {1}{\Gamma \left ({1-\alpha }\right)}\int _{t_{0}}^{t}\frac {f^{\prime }\left ({\tau }\right)}{\left ({t-\tau }\right)^{\alpha }}d\tau\end{equation*}$$ where $0 < \alpha < 1$ .

Lemma 2 ([30]):

For any real numbers $\kappa _{1}>0 \kappa _{2}>0,0 < q < 1$ , an extended Lyapunov condition of finite-time stability can be given $\dot {V}(x)+\kappa _{1}V(x)+\kappa _{2}V^{q}(x)\le 0$ , where $T_{s}\le T_{0}+[({1}/{\kappa _{1}(1-q)})]\ln [{(\kappa _{1}V^{1-q}(T_{0})+\kappa _{2})}/({\kappa _{2}})]$ .

Lemma 3 ([47]):

For any real variables $z_{1}$ and $z_{2}$ and any positive numbers $\varsigma _{1},\varsigma _{2}$ , and $\iota$ , the following inequality holds: $$\begin{equation*} |z_{1}|^{\varsigma _{1}}|z_{2}|^{\varsigma _{2}}\le \frac {\varsigma _{1}}{\varsigma _{1}+\varsigma _{2}}\iota |z_{1}|^{\varsigma _{1} + \varsigma _{2}}+\frac {\varsigma _{2}}{\varsigma _{1}+\varsigma _{2}}\iota ^{\frac {-\varsigma _{1}}{\varsigma _{2}}}|z_{2}|^{\varsigma _{1}+\varsigma _{2}}.\end{equation*}$$

Lemma 4 ([48]):

Let $\varsigma _{1}\in \mathbb {R}^{n}$ , $\varsigma _{2}\in \mathbb {R}^{m}$ , and $g: \mathbb {R}^{n}\times \mathbb {R}^{m}\longmapsto \mathbb {R}$ be a real-valued continuous function. There exist smooth scalar functions $C(\varsigma _{1})\ge 0$ and $D(\varsigma _{2})\ge 0$ such that the following inequality holds: $$\begin{equation*} |g\left ({\varsigma _{1},\varsigma _{2}}\right)|\le C\left ({\varsigma _{1}}\right)+D\left ({\varsigma _{2}}\right).\end{equation*}$$

Remark 2:

According to Lemma 4, for time-delay terms $g_{j}(\bar {x}_{j,\tau (t)})$ , there exist nonnegative continuous functions $\psi _{j}$ such that the following inequality holds: $$\begin{equation*} |g_{j}\left ({\bar {x}_{j,\tau \left ({t}\right)}}\right)|\le \sum _{m=1}^{j}\psi _{j,m}\left ({x_{m}\left ({t-\tau _{m}\left ({t}\right)}\right)}\right). \tag{4}\end{equation*}$$

SECTION III. Main Results

This section will propose a command-filtered-based adaptive finite-time resilient control scheme for nonlinear time-delay system (1).

A. Adaptive Finite-Time Resilient Control Design

According to the description of the investigated false data injection attacks, we have $$\begin{equation*} x_{i}\left ({t}\right)=\beta \left ({t}\right)\check {x}_{i}\left ({t}\right) \tag{5}\end{equation*}$$ where $\beta (t)=(1+\lambda (t))^{-1}$ satisfying $|\beta (t)|\le \beta _{M}$ and $\beta _{M}>0$ is a unknown constant.

To handle the unknown attacks, a new change of coordinates is defined as $$\begin{align*} \begin{cases} \displaystyle s_{1}=x_{1} \\ \displaystyle s_{i}=x_{i}-\beta \vartheta _{i-1}^{f},\quad \left ({i=2,\ldots,n}\right) \end{cases} \tag{6}\end{align*}$$ where $s_{k}(k=1,\ldots,n)$ denote surface errors and $\vartheta _{i-1}^{f}$ is a filtered signal of the virtual control signal $\eta _{i-1}$ to be designed.

Remark 3:

Influenced by the malicious attacks from attackers, it is worthy to point out that the original state of the system $x_{i}$ and surface error $s_{i}$ are not applicable to controller design, that is, only the compromised state variable $\check {x}_{i}$ is available for achieving feedback control. In this case, the traditional recursive control design methods dependent on error variable $s_{i}$ cannot be directly adopted to fulfill the controller design. To this end, a new coordinate transformation is established for overcoming this problem, where the compromised state variable $\check {x}_{i}$ is used to develop the adaptive resilient controller. Especially, the original error variable $s_{i}$ is only used for stability analysis of the CLS, which will not be involved in the virtual and actual control functions to be designed.

It follows from (6) that: $$\begin{align*} \begin{cases} \displaystyle \check {s}_{1}=\check {x}_{1} \\ \displaystyle \check {s}_{i}=\check {x}_{i}-\vartheta _{i-1}^{f},\quad \left ({i=2,\ldots,n}\right) \end{cases} \tag{7}\end{align*}$$ where filtered signal $\vartheta _{i-1}^{f}$ can be obtained by the following FO filter: $$\begin{align*} \begin{cases} \displaystyle D^{\alpha }{\lambda }_{i-1,1}=\ell _{i-1,1}\\ \displaystyle \ell _{i-1,1}=-\alpha _{i-1,1}\lceil \lambda _{i-1,1}-\eta _{i-1}\rfloor ^{\beta _{1}}\\ \displaystyle \qquad \qquad \quad -\,\alpha _{i-1,2}\lceil \lambda _{i-1,1}-\eta _{i-1}\rfloor ^{\beta _{2}}+\lambda _{i-1,2}\\ \displaystyle D^{\alpha }{\lambda }_{i-1,2}=-\alpha _{i-1,3}\lceil \lambda _{i-1,1}-\eta _{i-1}\rfloor ^{\beta _{3}} \end{cases} \tag{8}\end{align*}$$ where $D^{\alpha }$ denotes the fractional operator with $0 < \alpha < 1$ , which can be referred to Definition 1. The virtual control law $\eta _{i-1}$ is the input, and $\vartheta _{i-1}^{f}=\lambda _{i-1,1}$ and $D^{\alpha }{\vartheta }_{i-1}^{f}=\ell _{i-1,1}$ are the outputs. $0 < \beta _{1} < 1, \beta _{2}>1$ , and $0 < \beta _{3} < 1$ are constants to be selected.

Remark 4:

Due to the unique property of historical memory in fractional calculus, it has been recognized that the FO control provides a possible solution for obtaining higher control accuracy and the degree of freedom for control system design in comparison to the IO counterpart. Therefore, we make the first attempt to develop an FOCF to obtain the filtered signals of virtual control laws in this work. Different from most of the existing results [34], [49], the proposed FOCF not only overcomes the computational complexity of traditional recursive control method but also improves the filter performance. To show the difference between the proposed filter and the existing results, a comparison result is presented by using three filters, including the common integer-order filter (IOF) adopted in [49], the extended fractional-order filter (FOF) adopted in [34] and [50], and the modified FOCF proposed in (8). Defining the signal $2\sin (t)$ as the input signal of three filters, the filter performance is shown in Fig. 2. It is easily observed that filter performance can be improved due to the existence of FO characteristics. What’s more, better performance can be achieved by the proposed filter (8).

Fig. 2.

Filter performance.

Define the filter error $\epsilon _{1}=\vartheta _{1}^{f}-\eta _{1}$ , and then, the following signals are adopted to compensate for the filter errors: $$\begin{align*} \begin{cases} \displaystyle \dot {\gamma }_{i}=-a_{i1}\gamma _{i}-a_{i2}\gamma _{i}^{2q-1}-\Theta \left ({\check {s}_{i},\gamma _{i},\epsilon _{i}}\right)\gamma _{i}+\epsilon _{i} \\ \displaystyle \dot {\gamma }_{n}=-a_{n1}\gamma _{n}-a_{n2}\gamma _{n }^{2q-1},\quad \left ({i=1,\ldots,n-1}\right) \end{cases} \tag{9}\end{align*}$$ where $\Theta (\check {s}_{i},\gamma _{i},\epsilon _{i})=({|\check {s}_{i}\epsilon _{i}|+a_{i3}\epsilon _{i}^{2}})/({||\gamma _{i}||^{2}})$ and $a_{lj}(l=1,\ldots,n;j=1,2,3)$ is a design parameter to be determined.

Design the virtual control function and adaptive laws as $$\begin{align*} \eta _{i}=&-k_{i}\check {s}_{i}+N\left ({\xi _{i}}\right)\bar {\eta }_{i}+\gamma _{i} \tag{10}\\ \bar {\eta }_{i}=&\frac {\check {s}_{i}\hat {\hbar }_{i}\varphi _{i}^{T}\left ({\check {\Delta }_{i}}\right)\varphi _{i}\left ({\check {\Delta }_{i}}\right)}{2\sigma _{i}^{2}}+2c_{i}\check {s}_{i}+\frac {\hat {\Gamma }_{i}\check {s}_{i}}{\sqrt {\check {s}_{i}^{2}+\varpi _{1}^{2}}} \tag{11}\\ \dot {\hat {\hbar }}_{i}=&\frac {1}{2\sigma _{i}^{2}}\check {s}_{i}^{2}\varphi _{i}^{T}\left ({\check {\triangle }_{i}}\right)\varphi _{i}\left ({\check {\triangle }_{i}}\right)-\rho _{i,1}\hat {\hbar }_{i} \tag{12}\\ \dot {\hat {\Gamma }}_{i}=&\frac {\check {s}_{i}^{2}}{\sqrt {\check {s}_{i}^{2}+\varpi _{1}^{2}}}-\rho _{i,2}\hat {\Gamma }_{i}, ~~i=1,\ldots,n \tag{13}\end{align*}$$ and the actual control function and parameter update laws are constructed as $$\begin{align*} v_{j}=&N\left ({\xi _{n+1}}\right)\bar {\eta }_{n+1} \tag{14}\\ \bar {\eta }_{n+1}=&\check {s}_{n}\hat {\chi }_{2}\eta _{n}^{2}+\hat {\chi }_{1}\tanh \left ({\frac {\hat {\chi }_{1}\check {s}_{n}}{\varpi _{2}}}\right) \tag{15}\\ \dot {\hat {\chi }}_{1}=&|\check {s}_{n}|-\rho _{n+1,1}\hat {\chi }_{1} \tag{16}\\ \dot {\hat {\chi }}_{2}=&\check {s}_{n}^{2}\eta _{n}^{2}-\rho _{n+1,2}\hat {\chi }_{2} \tag{17}\end{align*}$$ where $k_{i},c_{i},\rho _{i,1},\rho _{i,2}$ , and $\rho _{n+1,s}(s=1,2)$ are design parameters. $\hat {\chi }_{j}$ denotes the estimation of the unknown parameter $\chi _{j}$ defined in step $n$ .

• Step 1:

  According to (1) and (6), the derivative of $s_{1}$ is calculated as $$\begin{equation*} \dot {s}_{1}=s_{2}+\beta \left ({\eta _{1}+\epsilon _{1}}\right)+f_{1}\left ({\bar {x}_{1}}\right)+g_{1}\left ({\bar {x}_{1,\tau \left ({t}\right)}}\right)+d_{1}\left ({t}\right). \tag{18}\end{equation*}$$

Select the following Lyapunov function: $$\begin{equation*} V_{1}=\frac {1}{2}s_{1}^{2}+\frac {1}{2}\tilde {\hbar }_{1}^{2}+\frac {1}{2}\gamma _{1}^{2}+\frac {1}{2}\tilde {\Gamma }_{1}^{2}+W_{1} \tag{19}\end{equation*}$$ with $W_{1}=({e^{b_{11}\bar {\tau }_{1}}})/({1-\bar {\tau }_{1d})}e^{-b_{11} t}\int _{t-\tau _{1}(t)}^{t}e^{b_{11}s}\psi _{1,1}^{2}(x_{1}(s))ds$ , where $b_{11}$ is a positive constant.

Then, it can be deduced that $$\begin{align*} \dot {V}_{1}\le&-s_{1}^{2q}+ \frac {1}{2}s_{1}^{2}+\frac {1}{2}s_{2}^{2}+\left ({1-2\tanh ^{2}\left ({\frac {s_{1}}{\iota _{1}}}\right)}\right)\Phi _{1} \\&+\,s_{1}\beta \eta _{1}+\check {s}_{1}\left [{F_{1}\left ({x_{1},s_{1},\beta }\right)+D_{1}}\right]+s_{1}\beta \epsilon _{1} \\&+\,s_{1}g_{1}\left ({\bar {x}_{1,\tau \left ({t}\right)}}\right)-\tilde {\hbar }_{1}\dot {\hat {\hbar }}_{1}+\gamma _{1}\dot {\gamma }_{1}-\tilde {\Gamma }_{1}\dot {\hat {\Gamma }}_{1}+\dot {W}_{1} \tag{20}\end{align*}$$ where $\Phi _{1}=[e^{b_{11}\bar {\tau }_{1}}/(1-\bar {\tau }_{1d})]\psi _{1,1}^{2}(x_{1}(t))$ , $F_{1}(x_{1},s_{1},\beta)=\beta (f_{1}(x_{1})+s_{1}^{2q-1})+(2\beta \tanh ^{2}(s_{1}/\iota _{1})\Phi _{1})/s_{1}$ , $D_{1}=\beta d_{1}(t)$ with $|D_{1}|\le \Gamma _{1}$ , and $\Gamma _{1}$ is an unknown constant.

Remark 5:

According to (22), it is easily found that the term $\Phi _{1}/s_{1}$ may not be will defined because $\lim _{s_{1}\rightarrow 0}(\Phi _{1}/s_{1})=\infty$ . In this case, the term in $\Phi _{1}/s_{1}$ cannot be approximated by using the RBFNNs. To this end, the term $2\text {tanh}^{2}(s_{1}/l_{1})\Phi _{1}/s_{1}$ is introduced to handle this problem. Using the fact that $\lim _{s_{1}\rightarrow 0}\text {tanh}^{2}(s_{1}/l_{1})\Phi _{1}/s_{1}=0$ , one gets that the term $F_{1}(x_{1},s_{1},\beta)$ can be handled by means of RBFNNs.

Using the NN approximation technique, one can obtain $F_{1}(x_{1},s_{1},\beta)={\theta }^{\ast T}_{1}\varphi _{1}(\triangle _{1})+o_{1}$ with $\triangle _{1}=s_{1}$ . Using (4), one has $$\begin{align*}&\dot {W}_{1}\le -b_{11}W_{1}+\Phi _{1}-\psi _{1,1}^{2}\left ({x_{1,\tau _{1}\left ({t}\right)}}\right) \tag{21}\\&s_{1}g_{1}\left ({\bar {x}_{1,\tau \left ({t}\right)}}\right)\le \frac {1}{4}s_{1}^{2}+\psi _{1,1}^{2}\left ({x_{1,\tau _{1}\left ({t}\right)}}\right). \tag{22}\end{align*}$$

Invoking (20)–​(22) yields $$\begin{align*} \dot {V}_{1}\le&-s_{1}^{2q}+\frac {3}{4}s_{1}^{2}+\frac {1}{2}s_{2}^{2}+s_{1}\beta \eta _{1}+\check {s}_{1}\left ({\theta _{1}^{\ast T}\varphi _{1}\left ({\check {\Delta }_{1}}\right)+D_{1}}\right) \\[2pt]&+\,\check {s}_{1}\Psi _{1}+s_{1}\beta \epsilon _{1}-\tilde {\hbar }_{1}\dot {\hat {\hbar }}_{1}+\gamma _{1}\dot {\gamma }_{1}-\tilde {\Gamma }_{1}\dot {\hat {\Gamma }}_{1}-b_{11}W_{1} \\[2pt]&+\,\left ({1-2\tanh ^{2}\left ({\frac {s_{1}}{\iota _{1}}}\right)}\right)\Phi _{1} \tag{23}\end{align*}$$ where $\Psi _{1}=\theta _{1}^{\ast T}(\varphi _{1}(\Delta _{1})-\varphi _{1}(\check {\Delta }_{1}))+o_{1}$ .

Based on Assumptions 1 and 2, one gets that $\Psi _{1}$ is bounded satisfying $|\Psi _{1}|\le \bar {\Psi }_{1}$ , where $\bar {\Psi }_{1}>0$ is a constant. Therefore, it holds that $\check {s}_{1}\Psi _{1}\le c_{1}\check {s}_{1}^{2}+(\bar {\Psi }_{1}^{2}/4c_{1})$ .

Using Lemma 7 and Young’s inequality, we have $$\begin{align*}&\hspace {-0.5pc}\check {s}_{1}\left ({\theta _{1}^{\ast T}\varphi _{1}\left ({\check {\Delta }_{1}}\right)+D_{1}}\right)\le \frac {\check {s}_{1}^{2}\hbar _{1}\varphi _{1}^{T}\left ({\check {\Delta }_{1}}\right)\varphi _{1}\left ({\check {\Delta }_{1}}\right)}{2\sigma _{1}^{2}}+\frac {1}{2}\sigma _{1}^{2} \\&+\,\varpi _{1}\Gamma _{1}+\frac {\Gamma _{1}\check {s}_{1}^{2}}{\sqrt {\check {s}_{1}^{2}+\varpi _{1}^{2}}} \tag{24}\end{align*}$$ where $\hbar _{1}=||\theta _{1}^{\ast }||^{2}$ .

Defining $\dot {\xi }_{1}=\check {s}_{1}\bar {\eta }_{1}$ , then substituting (10)–(12) and (24) into (23), one can obtain $$\begin{align*} \dot {V}_{1}\le&-b_{11}W_{1}-\bar {k}_{1}s_{1}^{2}-s_{1}^{2q}+\frac {1}{2}s_{2}^{2}+\left ({g_{1}\left ({t}\right)N\left ({\xi _{1}}\right)+1}\right)\dot {\xi }_{1} \\&+\,\rho _{1,1}\tilde {\hbar }_{1}\hat {\hbar }_{1}+\rho _{1,2}\tilde {\Gamma }_{1}\hat {\Gamma }_{1}-\bar {a}_{11}\gamma _{1}^{2}-a_{12}\gamma _{1}^{2q}-\bar {a}_{13}\epsilon _{1}^{2} \\&+\,\left ({1-2\tanh ^{2}\left ({\frac {s_{1}}{\iota _{1}}}\right)}\right)\Phi _{1}+\zeta _{1} \tag{25}\end{align*}$$ where $\bar {k}_{1}=k_{1}-1,\bar {a}_{11}=a_{11}-(1+\beta _{M}^{2})/2, \bar {a}_{13}=a_{13}-(1/2+\beta _{M}^{2})-(1/4c_{1})$ , and $\zeta _{1}=(\bar {\Psi }_{1}^{2}/4c_{1})+\sigma _{1}^{2}/2+\varpi _{1}\Gamma _{1}$ .

For the terms $\rho _{1,1}\tilde {\hbar }_{1}\hat {\hbar }_{1}$ and $\rho _{1,2}\tilde {\Gamma }_{1}\hat {\Gamma }_{1}$ in (25), the following inequalities hold: $$\begin{align*} \rho _{1,1}\tilde {\hbar }_{1}\hat {\hbar }_{1}\le&-\bar {\rho }_{1,1}\tilde {\hbar }_{1}^{2}+\bar {\rho }_{1,1}\hbar _{1}^{2} \tag{26}\\ \rho _{1,2}\tilde {\Gamma }_{1}\hat {\Gamma }_{1}\le&-\bar {\rho }_{1,2}\tilde {\Gamma }_{1}^{2}+\bar {\rho }_{1,2}\Gamma _{1}^{2} \tag{27}\end{align*}$$ where $\bar {\rho }_{1,l}=\rho _{1,l}/2(l=1,2)$ .

Furthermore, defining $z_{1}=1, z_{2}=({1}/{2})\tilde {\hbar }_{1}^{2}, w_{1}=1-q, w_{2}=q$ , and $\varrho =q^{({q}/{1-q})}$ using Lemma 3 achieves $$\begin{equation*} \left ({\frac {1}{2}\tilde {\hbar }_{1}^{2}}\right)^{q}\le \left ({1-q}\right)\varrho +\frac {1}{2}\tilde {\hbar }_{1}^{2}. \tag{28}\end{equation*}$$

Similarly, one has $$\begin{align*} \left ({\frac {1}{2}\tilde {\Gamma }_{1}^{2}}\right)^{q}\le&\left ({1-q}\right)\varrho +\frac {1}{2}\tilde {\Gamma }_{1}^{2} \tag{29}\\ W_{1}^{q}\le&\left ({1-q}\right)\varrho +W_{1}. \tag{30}\end{align*}$$

It follows from the inequalities (25)–​(30) that: $$\begin{align*} \dot {V}_{1}\le&-\bar {k}_{1}s_{1}^{2}-s_{1}^{2q}+\frac {1}{2}s_{2}^{2}+\left ({g_{1}\left ({t}\right)N\left ({\xi _{1}}\right)+1}\right)\dot {\xi }_{1} \\&-\,\bar {a}_{11}\gamma _{1}^{2}-a_{12}\gamma _{1}^{2q}-\bar {a}_{13}\epsilon _{1}^{2}-\frac {\bar {\rho }_{1,1}}{2}\tilde {\hbar }_{1}^{2}-\frac {\bar {\rho }_{1,2}}{2}\tilde {\Gamma }_{1}^{2} \\&-\,\bar {\rho }_{1,1}\left ({\frac {1}{2}\tilde {\hbar }_{1}^{2}}\right)^{q}-\bar {\rho }_{1,2}\left ({\frac {1}{2}\tilde {\Gamma }_{1}^{2}}\right)^{q}-\bar {b}_{11}W_{1} \\&-\,b_{12}W_{1}^{q}+\left ({1-2\tanh ^{2}\left ({\frac {s_{1}}{\iota _{1}}}\right)}\right)\Phi _{1}+\bar {\zeta }_{1} \tag{31}\end{align*}$$ where $b_{12}$ is a positive constant, $\bar {b}_{11}=b_{11}-b_{12}>0$ , and $\bar {\zeta }_{1}=\zeta _{1}+\bar {\rho }_{1,1}\hbar _{1}^{2}+\bar {\rho }_{1,2}\Gamma _{1}^{2}+(\bar {\rho }_{1,1}+\bar {\rho }_{1,2}+b_{12})(1-q)\varrho$ .

Step $i$ ($i=2,\ldots,n-1$ ): It is easily calculated that $$\begin{align*}&\hspace {-0.5pc}\dot {s}_{i}=s_{i+1}+\beta \eta _{i}+\beta \epsilon _{i}+f_{i}\left ({\bar {x}_{i}}\right)+g_{i}\left ({\bar {x}_{i,\tau \left ({t}\right)}}\right) \\&+\,d_{i}\left ({t}\right)-\dot {\beta }\vartheta _{i-1}^{f}-\beta \dot {\vartheta }_{i-1}^{f} \tag{32}\end{align*}$$ where $\epsilon _{i}=\vartheta _{i}^{f}-\eta _{i}$ .

Select the following Lyapunov function: $$\begin{equation*} V_{i}=V_{i-1}+\frac {1}{2}s_{i}^{2}+\frac {1}{2}\tilde {\hbar }_{i}^{2}+\frac {1}{2}\gamma _{i}^{2}+\frac {1}{2}\tilde {\Gamma }_{i}^{2}+W_{i} \tag{33}\end{equation*}$$ where $W_{i}=\sum _{j=1}^{i}({e^{b_{i1}\bar {\tau }_{j}}}/{(1-\bar {\tau }_{jd})})e^{-b_{i1} t}\int _{t-\tau _{j}(t)}^{t} e^{b_{i1}s}\psi _{i,j}^{2}(x_{j}(s))ds$ , and $b_{i1}$ is a positive constant.

Then, it is calculated that $$\begin{align*} \dot {V}_{i}\le&\dot {V}_{i-1}-s_{i}^{2q}+ \frac {1}{2}s_{i}^{2}+\frac {1}{2}s_{i+1}^{2}-2\tanh ^{2}\left ({\frac {s_{i}}{\iota _{i}}}\right)\Phi _{i} \\&+\,s_{i}\beta \left ({\eta _{i}+\epsilon _{i}}\right)+\check {s}_{i}\left [{F_{i}\left ({\bar {x}_{i},s_{i},\beta,\dot {\beta },\vartheta _{i-1}^{f},\dot {\vartheta }_{i-1}^{f}}\right)+D_{i}}\right] \\&+\,s_{i}g_{i}\left ({\bar {x}_{i,\tau \left ({t}\right)}}\right)-\tilde {\hbar }_{i}\dot {\hat {\hbar }}_{i}+\gamma _{i}\dot {\gamma }_{i}-\tilde {\Gamma }_{i}\dot {\hat {\Gamma }}_{i}+\dot {W}_{i} \tag{34}\end{align*}$$ in which $\Phi _{i}=\sum _{j=1}^{i}[e^{b_{i1}\bar {\tau }_{j}}/(1-\bar {\tau }_{jd})]\psi _{i,j}^{2}(x_{j}(t))$ , $F_{i}(\bar {x}_{i},s_{i},\beta,\dot {\beta },\vartheta _{i-1}^{f},\dot {\vartheta }_{i-1}^{f})=\beta (f_{i}(\bar {x}_{i})-\dot {\beta }\vartheta _{i-1}^{f}-\beta \dot {\vartheta }_{i-1}^{f}+s_{i}^{2q-1})+(2\beta \tanh ^{2}(s_{i}/\iota _{i})\Phi _{i})/s_{i}$ , $D_{i}=\beta d_{i}(t)$ , with $|D_{i}|\le \Gamma _{i}$ , and $\Gamma _{i}$ is an unknown constant. Following the same procedure with Step 1, one has $F_{i}(\bar {x}_{i},s_{i},\beta,\dot {\beta },\vartheta _{i-1}^{f},\dot {\vartheta }_{i-1}^{f})={\theta }^{\ast T}_{i}\varphi _{i}(\triangle _{i})+o_{i}$ with $\triangle _{i}=[\bar {x}_{i},s_{i},\vartheta _{i-1}^{f}]$ .

Similarly, one can obtain $$\begin{align*} \dot {W}_{i}\le&-b_{i1}W_{i}+\Phi _{i}-\sum _{j=1}^{i}\psi _{i,j}^{2}\left ({x_{j,\tau _{j}\left ({t}\right)}}\right) \tag{35}\\ s_{i}g_{i}\left ({\bar {x}_{i,\tau \left ({t}\right)}}\right)\le&\frac {1}{4}s_{i}^{2}+\sum _{j=1}^{i}\psi _{i,j}^{2}\left ({x_{j,\tau _{j}\left ({t}\right)}}\right). \tag{36}\end{align*}$$

Using (35) and (36), (34) can be written as $$\begin{align*} \dot {V}_{i}\le&\dot {V}_{i-1}\!-\!s_{i}^{2q}\!+\!\frac {3}{4}s_{i}^{2}+\frac {1}{2}s_{i+1}^{2}+s_{i}\beta \eta _{i}+\check {s}_{i}\left ({\theta _{i}^{\ast T}\varphi _{i}\left ({\check {\Delta }_{i}}\right)\!+\!D_{i}}\right) \\&+\check {s}_{i}\Psi _{i}+s_{i}\beta \epsilon _{i}-\tilde {\hbar }_{i}\dot {\hat {\hbar }}_{i}+\gamma _{i}\dot {\gamma }_{i}-\tilde {\Gamma }_{i}\dot {\hat {\Gamma }}_{i} \\&-b_{i1}W_{i}+\left ({1-2\tanh ^{2}\left ({\frac {s_{i}}{\iota _{i}}}\right)}\right)\Phi _{i} \tag{37}\end{align*}$$ where $\Psi _{i}=\theta _{i}^{\ast T}(\varphi _{i}(\Delta _{i})-\varphi _{i}(\check {\Delta }_{i}))+o_{i}$ .

Furthermore, the following equality holds: $$\begin{align*}&\hspace {-0.5pc}\check {s}_{i}\left ({\theta _{i}^{\ast T}\varphi _{i}\left ({\check {\Delta }_{i}}\right)+D_{i}}\right)\le \frac {\check {s}_{i}^{2}\hbar _{i}\varphi _{i}^{T}\left ({\check {\Delta }_{i}}\right)\varphi _{i}\left ({\check {\Delta }_{i}}\right)}{2\sigma _{i}^{2}}+\frac {1}{2}\sigma _{i}^{2} \\&+\,\varpi _{1}\Gamma _{i}+\frac {\Gamma _{i}\check {s}_{i}^{2}}{\sqrt {\check {s}_{i}^{2}+\varpi _{1}^{2}}}. \tag{38}\end{align*}$$

Using Assumptions 1 and 2 again, one gets that $\Psi _{i}$ is bounded satisfying $|\Psi _{i}|\le \bar {\Psi }_{i}$ , where $\bar {\Psi }_{i}>0$ is a constant. Therefore, it holds that $\check {s}_{i}\Psi _{i}\le c_{i}\check {s}_{i}^{2}+(\bar {\Psi }_{i}^{2})/(4c_{i})$ .

Defining $\dot {\xi }_{i}=\check {s}_{i}\bar {\eta }_{i}$ , then invoking (10)–(13) and (38) gets $$\begin{align*} \dot {V}_{i}\le&-\sum _{j=1}^{i}\left ({\bar {k}_{j}s_{j}^{2}+s_{j}^{2q}}\right)+\sum _{j=1}^{i}\left ({g_{j}\left ({t}\right)N\left ({\xi _{j}}\right)+1}\right)\dot {\xi }_{j}+\frac {1}{2}s_{i+1}^{2} \\&-\sum _{j=1}^{i}\left [{\bar {a}_{j1}\gamma _{j}^{2}+a_{j2}\gamma _{j}^{2q}+\bar {a}_{j3}\epsilon _{j}^{2}+\frac {\bar {\rho }_{j,1}}{2}\tilde {\hbar }_{j}^{2}}\right. \\&\qquad \qquad \left.{+\frac {\bar {\rho }_{j,2}}{2}\tilde {\Gamma }_{j}^{2}+\bar {\rho }_{j,1}\left ({\frac {1}{2}\tilde {\hbar }_{j}^{2}}\right)^{q}+\bar {\rho }_{j,2}\left ({\frac {1}{2}\tilde {\Gamma }_{j}^{2}}\right)^{q}}\right] \\&+\sum _{j=1}^{i}\left [{\bar {\zeta }_{j}-\bar {b}_{j1}W_{j}-b_{j2}W_{j}^{q}+\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}}\right] \\\tag{39}\end{align*}$$ where $\bar {a}_{i1}=a_{i1}-(1+\beta _{M}^{2})/2, \bar {a}_{i3}=a_{i3}-(1+\beta _{M}^{2})/2-1/(4c_{i}), \bar {k}_{i}=k_{i}-(7/4)$ , and $\bar {\zeta }_{i}=\bar {\Psi }_{i}^{2}/(4c_{i})+\sigma _{i}^{2}/2+\varpi _{1}\Gamma _{i}+\bar {\rho }_{i1}\hbar _{i}^{2}+\bar {\rho }_{i2}\Gamma _{i}^{2}+(\bar {\rho }_{i1}+\bar {\rho }_{i2}+b_{i2})(1-q)\varrho$ .

Step $n$ : In this step, the control input will be designed. Along with the previous steps, we have $$\begin{align*}&\hspace {-0.5pc}\dot {s}_{n}=\sum _{j=1}^{p}b_{j}l_{jm}v_{j}+w^{T}\phi +f_{n}\left ({\bar {x}_{n}}\right)+g_{n}\left ({\bar {x}_{n,\tau \left ({t}\right)}}\right) \\&+\,d_{n}\left ({t}\right)-\dot {\beta }\vartheta _{n-1}^{f}-\beta \dot {\vartheta }_{n-1}^{f} \tag{40}\end{align*}$$ where $$\begin{align*} w=\left [{ \begin{matrix} b_{1}\bar {u}_{1d,m}\left ({t}\right) \\ b_{2}\bar {u}_{2d,m}\left ({t}\right) \\ \vdots \\ b_{p}\bar {u}_{pd,m}\left ({t}\right) \end{matrix} }\right], \phi =\left [{ \begin{matrix} 1 \\ 1 \\ \vdots \\ 1 \end{matrix} }\right].\end{align*}$$

To handle the unknown actuator faults and facilitate the design of the actual control signal, we define $\chi _{1}=|\beta |\sup _{t\ge 0}||w||$ and $\chi _{2}=c_{n}\beta _{M}^{4}$ . Then, choose the Lyapunov function as $$\begin{align*} V_{n}=V_{n-1}+\frac {1}{2}s_{n}^{2}+\frac {1}{2}\tilde {\hbar }_{n}^{2}+\frac {1}{2}\gamma _{n}^{2}+\frac {1}{2}\tilde {\Gamma }_{n}^{2} +\tilde {\chi }_{1}^{2}+\tilde {\chi }_{2}^{2}+W_{n} \\ \tag{41}\end{align*}$$ where $W_{n}=\sum _{j=1}^{n}({e^{b_{n1}\bar {\tau }_{j}}}/{(1-\bar {\tau }_{jd})})e^{-b_{n1} t}\int _{t-\tau _{j}(t)}^{t}e^{b_{n1}s}\psi _{n,j}^{2} (x_{j}(s))ds$ , $b_{n1}$ is a positive constant, and $\tilde {\chi }_{j}=\chi _{j}-\hat {\chi }_{j}(j=1,2)$ .

Furthermore, introducing a new variable $\eta _{n}$ yields $$\begin{align*} \dot {V}_{n}\le&\dot {V}_{n-1}-s_{n}^{2q}+s_{n}\left ({\sum _{j=1}^{p}b_{j}l_{jm}v_{j}+\beta \eta _{n}-\beta \eta _{n}}\right) \\&+\,s_{n}w^{T}\phi -2\tanh ^{2}\left ({\frac {s_{n}}{\iota _{n}}}\right)\Phi _{n}+\check {s}_{n} \\&\left [{D_{n}+F_{n}\left ({\bar {x}_{n},s_{n},\beta,\dot {\beta },\vartheta _{n-1}^{f},\dot {\vartheta }_{n-1}^{f}}\right)}\right]+s_{n}g_{n}\left ({\bar {x}_{n,\tau \left ({t}\right)}}\right) \\&-\,\tilde {\hbar }_{n}\dot {\hat {\hbar }}_{n}+\gamma _{n}\dot {\gamma }_{n}-\tilde {\Gamma }_{n}\dot {\hat {\Gamma }}_{n}-\sum _{j=1}^{2}\tilde {\chi }_{j}\dot {\hat {\chi }}_{j}+\dot {W}_{n} \tag{42}\end{align*}$$ where $\Phi _{n}=\sum _{j=1}^{n}[e^{b_{n1}\bar {\tau }_{j}}/(1-\bar {\tau }_{jd})]\psi _{n,j}^{2}(x_{j}(t))$ , $F_{n}(\bar {x}_{n},s_{n},\beta,\dot {\beta },\vartheta _{n-1}^{f},\dot {\vartheta }_{n-1}^{f})=(2\beta \tanh ^{2}(s_{n}/\iota _{n})\Phi _{n})/s_{n}+\beta (f_{n}(\bar {x}_{n})-\dot {\beta }\vartheta _{n-1}^{f}-\beta \dot {\vartheta }_{n-1}^{f}+s_{n}^{2q-1})$ , $D_{n}=\beta d_{n}(t)$ , with $|D_{n}|\le \Gamma _{n}$ , and $\Gamma _{n}$ is an unknown constant.

Following the same procedure with Step 1, one has $F_{n}(\bar {x}_{n},s_{n},\beta,\dot {\beta },\vartheta _{n-1}^{f},\dot {\vartheta }_{n-1}^{f})={\theta }^{\ast T}_{n}\varphi _{n}(\triangle _{n})+o_{n}$ with $\triangle _{n}=[\bar {x}_{n},s_{n},\vartheta _{n-1}^{f}]$ .

Similarly, we have $$\begin{align*}&\dot {W}_{n}\le -b_{n1}W_{n}+\Phi _{n}-\sum _{j=1}^{n}\psi _{n,j}^{2}\left ({x_{j,\tau _{j}\left ({t}\right)}}\right) \tag{43}\\&s_{n}g_{n}\left ({\bar {x}_{n,\tau \left ({t}\right)}}\right)\le \frac {1}{4}s_{n}^{2}+\sum _{j=1}^{n}\psi _{n,j}^{2}\left ({x_{j,\tau _{j}\left ({t}\right)}}\right). \tag{44}\end{align*}$$

Invoking (42)–​(44) gets $$\begin{align*} \dot {V}_{n}\le&\dot {V}_{n-1}-s_{n}^{2q}+s_{n}\left ({\sum _{j=1}^{p}b_{j}l_{jm}v_{j}+\beta \eta _{n}-\beta \eta _{n}}\right) \\&+\,\check {s}_{n}\left ({\theta _{n}^{\ast T}\varphi _{n}\left ({\check {\Delta }_{n}}\right)+D_{n}}\right)+s_{n}w^{T}\phi +\check {s}_{n}\Psi _{n} \\&-\,\tilde {\hbar }_{n}\dot {\hat {\hbar }}_{n}+\gamma _{n}\dot {\gamma }_{n}-\tilde {\Gamma }_{n}\dot {\hat {\Gamma }}_{n}-\sum _{j=1}^{2}\tilde {\chi }_{j}\dot {\hat {\chi }}_{j} \\&-\,b_{n1}W_{n}+\left ({1-2\tanh ^{2}\left ({\frac {s_{n}}{\iota _{n}}}\right)}\right)\Phi _{n} \tag{45}\end{align*}$$ where $\Psi _{n}=\theta _{n}^{\ast T}(\varphi _{n}(\Delta _{n})-\varphi _{n}(\check {\Delta }_{n}))+o_{n}$ .

Similarly, one gets that $\Psi _{n}$ is bounded satisfying $|\Psi _{n}|\le \bar {\Psi }_{n}$ , where $\bar {\Psi }_{n}>0$ is a constant. Therefore, it holds that $\check {s}_{n}\Psi _{n}\le c_{n}\check {s}_{n}^{2}+(\bar {\Psi }_{n}^{2}/4c_{n})$ .

Furthermore, it is easily derived that $$\begin{align*}&\hspace {-2pc}\check {s}_{n}\left ({\theta _{n}^{\ast T}\varphi _{n}\left ({\check {\Delta }_{n}}\right)+D_{n}}\right) \\\le&\frac {\check {s}_{n}^{2}\hbar _{n}\varphi _{n}^{T}\left ({\check {\Delta }_{n}}\right)\varphi _{n}\left ({\check {\Delta }_{n}}\right)}{2\sigma _{n}^{2}}+\frac {1}{2}\sigma _{n}^{2} \\&+\,\varpi _{1}\Gamma _{n}+\frac {\Gamma _{n}\check {s}_{n}^{2}}{\sqrt {\check {s}_{n}^{2}+\varpi _{1}^{2}}} \tag{46}\\ s_{n}w^{T}\phi\le&\tilde {\chi }_{1}|\check {s}_{n}|\!+\!\hat {\chi }_{1}\check {s}_{n}\tanh \left ({\frac {\hat {\chi }_{1}\check {s}_{n}}{\varpi _{2}}}\right)\!+\!0.2785\varpi _{2}. \tag{47}\end{align*}$$

Defining $\dot {\xi }_{n}=\check {s}_{n}\bar {\eta }_{n}$ and invoking (10)–(13) and (46) and (47), one can obtain $$\begin{align*} \dot {V}_{n}\le&\dot {V}_{n-1}-\left ({k_{n}-\frac {1}{2}}\right)s_{n}^{2}-s_{n}^{2q}+\left ({g_{n}\left ({t}\right)N\left ({\xi _{n}}\right)+1}\right)\dot {\xi }_{n} \\&+\,\rho _{n,1}\tilde {\hbar }_{n}\hat {\hbar }_{n}+\rho _{n,2}\tilde {\Gamma }_{n}\hat {\Gamma }_{n}-\bar {a}_{n1}\gamma _{n}^{2}-a_{n2}\gamma _{n}^{2q} \\&-\,\sum _{j=1}^{2}\tilde {\chi }_{j}\dot {\hat {\chi }}_{j}+\tilde {\chi }_{1}|\check {s}_{n}|+\hat {\chi }_{1}\check {s}_{n}\tanh \left ({\frac {\hat {\chi }_{1}\check {s}_{n}}{\varpi _{2}}}\right) \\&+\,s_{n}\left ({\sum _{j=1}^{p}b_{j}l_{jm}v_{j}-\beta \eta _{n}}\right)+\zeta _{n}-b_{n1}W_{n} \\&+\,\left ({1-2\tanh ^{2}\left ({\frac {s_{n}}{\iota _{n}}}\right)}\right)\Phi _{n} \tag{48}\end{align*}$$ where $g_{n}(t)=\beta ^{2},\bar {a}_{n1}=a_{n1}-\beta _{M}^{2}$ , and $\zeta _{n}=\bar {\Psi }_{n}^{2}/(4c_{n})+\sigma _{n}^{2}/2+\varpi _{1}\Gamma _{n}+0.2785\varpi _{2}$ .

Furthermore, the inequality (48) can be rewritten as $$\begin{align*} \dot {V}_{n}\le&-\sum _{j=1}^{n}\left ({\bar {k}_{j}s_{j}^{2}+s_{j}^{2q}}\right)+\sum _{j=1}^{n}\left ({g_{j}\left ({t}\right)N\left ({\xi _{j}}\right)+1}\right)\dot {\xi }_{j} \\[-4pt]&-\,\sum _{j=1}^{n}\left [{\bar {a}_{j1}\gamma _{j}^{2}+a_{j2}\gamma _{j}^{2q}+\frac {\bar {\rho }_{j,1}}{2}\tilde {\hbar }_{j}^{2}+\bar {\rho }_{j,1}\left ({\frac {1}{2}\tilde {\hbar }_{j}^{2}}\right)^{q}}\right. \\[-4pt]&\qquad \qquad \left.{+\frac {\bar {\rho }_{j,2}}{2}\tilde {\Gamma }_{j}^{2}+\bar {\rho }_{j,2}\left ({\frac {1}{2}\tilde {\Gamma }_{j}^{2}}\right)^{q}}\right]-\sum _{j=1}^{n-1}\bar {a}_{j3}\epsilon _{j}^{2}+\tilde {\chi }_{1}|\check {s}_{n}| \\[-4pt]&-\,\sum _{j=1}^{2}\tilde {\chi }_{j}\dot {\hat {\chi }}_{j}+\hat {\chi }_{1}\check {s}_{n}\tanh \left ({\frac {\hat {\chi }_{1}\check {s}_{n}}{\varpi _{2}}}\right)+\sum _{j=1}^{n-1}\bar {\zeta }_{j} +\underline {\zeta }_{n}\\[-4pt]&+\sum _{j=1}^{n}\left [{\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}-\bar {b}_{j1}W_{j}-b_{j2}W_{j}^{q}}\right] \\[-4pt]&+\,s_{n}\left ({\sum _{j=1}^{p}b_{j}l_{jm}v_{j}-\beta \eta _{n}}\right) \tag{49}\end{align*}$$ where $\bar {k}_{n}=k_{n}-1$ and $\underline {\zeta }_{n}=\zeta _{n}+\bar {\rho }_{n1}\hbar _{n}^{2}+\bar {\rho }_{n2}\Gamma _{n}^{2}+(\bar {\rho }_{n1}+\bar {\rho }_{n2}+b_{22})(1-q)\varrho$ .

Defining $\dot {\xi }_{n+1}=\check {s}_{n}\bar {\eta }_{n+1}$ and substituting (14) and (15) into (49), one has $$\begin{align*} \dot {V}_{n}\le&-\sum _{j=1}^{n}\left ({\bar {k}_{j}s_{j}^{2}+s_{j}^{2q}}\right)+\sum _{j=1}^{n+1}\left ({g_{j}\left ({t}\right)N\left ({\xi _{j}}\right)+1}\right)\dot {\xi }_{j} \\[-4pt]&-\,\sum _{j=1}^{n}\left [{\bar {a}_{j1}\gamma _{j}^{2}+a_{j2}\gamma _{j}^{2q}+\frac {\bar {\rho }_{j,1}}{2}\tilde {\hbar }_{j}^{2}+\bar {\rho }_{j,1}\left ({\frac {1}{2}\tilde {\hbar }_{j}^{2}}\right)^{q}}\right. \\[-4pt]&\qquad \qquad \left.{+\frac {\bar {\rho }_{j,2}}{2}\tilde {\Gamma }_{j}^{2}+\bar {\rho }_{j,2}\left ({\frac {1}{2}\tilde {\Gamma }_{j}^{2}}\right)^{q}}\right] \\[-4pt]&-\,\sum _{j=1}^{n-1}\bar {a}_{j3}\epsilon _{j}^{2}-\frac {\bar {\rho }_{n+1,1}}{2}\tilde {\chi }_{1}^{2} \\[-4pt]&-\frac {\bar {\rho }_{n+1,2}}{2}\tilde {\chi }_{2}^{2}-\bar {\rho }_{n+1,1}\left ({\frac {1}{2}\tilde {\chi }_{1}^{2}}\right)^{q}-\bar {\rho }_{n+1,2}\left ({\frac {1}{2}\tilde {\chi }_{2}^{2}}\right)^{q} \\[-4pt]&+\sum _{j=1}^{n}\left [{\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}-\bar {b}_{j1}W_{j}}\right. \\[-4pt]&\qquad \qquad \qquad \left.{-b_{j2}W_{j}^{q}+\bar {\zeta }_{j}\vphantom {\left [{\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}-\bar {b}_{j1}W_{j}}\right.}}\right] \tag{50}\end{align*}$$ where $g_{n+1}(t)=\beta \sum _{j=1}^{p}b_{j}l_{jm}$ and $\bar {\zeta }_{n}=\underline {\zeta }_{n}+(\bar {\rho }_{n+1,1}+\bar {\rho }_{n+1,2})(1-q)\varrho +\bar {\rho }_{n+1,1}\chi _{1}^{2}+\bar {\rho }_{n+1,2}\chi _{2}^{2}+1/(4c_{n})$ .

By choosing the appropriate design parameters, the inequality (50) can be rewritten as $$\begin{align*}&\hspace {-0.5pc}\dot {V}_{n} \le -\kappa _{1}V_{n}-\kappa _{2}V_{n}^{q}+\sum _{j=1}^{n+1}\left ({g_{j}\left ({t}\right)N\left ({\xi _{j}}\right)+1}\right)\dot {\xi }_{j} \\[-2pt]&+\,\sum _{j=1}^{n}\left [{\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}+\bar {\zeta }_{j}}\right] \tag{51}\end{align*}$$ with $\kappa _{1}=\min \{2\bar {k}_{j}, 2\bar {a}_{j1},\bar {\rho }_{m,1},\bar {\rho }_{m,2},\bar {b}_{j1}\}, \kappa _{2}=\min \{2^{q}, 2^{q}a_{j2},\bar {\rho }_{m,1},\bar {\rho }_{m,2},b_{j2}\}, j=1,\ldots,n; m=1,\ldots,n+1$ .

According to the aforementioned analysis, we can derive the main result in the following theorem.

Theorem 1:

Consider nonlinear time-delay system (1) under unknown false data injection attacks and actuator faults under Assumption 1–​3, the proposed adaptive resilient control scheme, including the FO filter (8), compensation function (9), the intermediate control law (10)–(11), the actual control law (14) and (15), and the adaptive laws (12), (13), (16), and (17), can ensure that the following properties hold.

1. All the resulting closed-loop signals are semiglobal practical finite-time stable (SGPFT).

2. The stabilization error will converge to an arbitrary small neighborhood of the origin within a finite time $$\begin{align*} T_{f}\le&\max \left \{{ \frac {1}{\nu \kappa _{1}(1-q)}\ln \frac {\nu \kappa _{1}V_{n}^{1-q}(0)+\kappa _{2}}{\kappa _{2}} }\right. \\[-2pt]&\qquad \qquad \left.{\frac {1}{\kappa _{1}(1-q)}\ln \frac {\kappa _{1}V_{n}^{1-q}(0)+\nu \kappa _{2}}{\nu \kappa _{2}}}\right \}.\end{align*}$$

Proof:

The term $\sum _{j=1}^{n}(1-2\tanh ^{2}({s_{j}}/{\iota _{j}}))\Phi _{j}$ in (51) can be rewritten as $$\begin{align*} \sum _{j=1}^{n}\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}=&\sum _{s_{i}\in \Pi _{s_{i}}}^{}\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j} \\[-2pt]&+\,\sum _{s_{i}\bar {\in }\Pi _{s_{i}}}^{}\left ({1-2\tanh ^{2}\left ({\frac {s_{j}}{\iota _{j}}}\right)}\right)\Phi _{j}. \\[-2pt]\tag{52}\end{align*}$$

According to (51) and (52), using [51, Lemma 3] yields $$\begin{equation*} \dot {V}_{n} \le -\kappa _{1}V_{n}-\kappa _{2}V_{n}^{q}+\sum _{j=1}^{n+1}\left ({g_{j}\left ({t}\right)N\left ({\xi _{j}}\right)+1}\right)\dot {\xi }_{j}+\Lambda \tag{53}\end{equation*}$$ where $\Lambda$ is a positive parameter satisfying $|\sum _{j=1}^{n}\bar {\zeta }_{j}+\pi _{1}+\pi _{2}| < \Lambda$ with $\pi _{1}=\sum _{s_{i}\in \Pi _{s_{i}}}^{}(1-2\tanh ^{2}({s_{j}}/{\iota _{j}}))\Phi _{j}$ and $\pi _{2}=\sum _{s_{i}\bar {\in }\Pi _{s_{i}}}^{}(1-2\tanh ^{2}({s_{j}}/{\iota _{j}}))\Phi _{j}$ .

Furthermore, applying Lemma 1, it can be obtained that $\sum _{j=1}^{n+1}(g_{j}(t)N(\xi _{j})+1)\dot {\xi }_{j}$ is bounded over the interval $[0,t_{s}$ ), and $\Lambda _{max}=\max _{t\in [0,t_{s})}\sum _{j=1}^{n+1}(g_{j}(t)N(\xi _{j})+1)\dot {\xi }_{j}$ . Furthermore, by defining $\bar {\Lambda }=\Lambda +\Lambda _{max}$ , the inequality (53) can be written as $$\begin{equation*} \dot {V}_{n} \le -\kappa _{1}V_{n}-\kappa _{2}V_{n}^{q}+\bar {\Lambda }. \tag{54}\end{equation*}$$

Following the similar procedure with [30, Corollary 1], it can be concluded that all signals in the CLS are SGPFS. According to (53), it is easily obtained that $$\begin{equation*} \left \{{\lim _{t\rightarrow T_{f}}|V_{n}\le \min \left \{{\frac {\bar {\Lambda }}{\left ({1-\nu }\right)\kappa _{1}}, \left [{\frac {\bar {\Lambda }}{\left ({1-\nu }\right)\kappa _{2}}}\right]^{\frac {1}{q}} }\right \} }\right \} \tag{55}\end{equation*}$$ where $\nu$ is a positive constant satisfying $0 < \nu < 1$ . Based on the definition of $V_{n}$ , one can obtain $$\begin{equation*} |s_{i}|\le \min \left \{{\sqrt {\frac {2\bar {\Lambda }}{\left ({1-\nu }\right)\kappa _{1}}}, \sqrt {2\left [{\frac {\bar {\Lambda }}{\left ({1-\nu }\right)\kappa _{2}}}\right]^{\frac {1}{q}}} }\right \} \tag{56}\end{equation*}$$ under a finite time $$\begin{align*}&\hspace {-0.5pc}T_{f}\le \max \left \{{ \frac {1}{\nu \kappa _{1}\left ({1-q}\right)}\ln \frac {\nu \kappa _{1}V_{n}^{1-q}(0)+\kappa _{2}}{\kappa _{2}} }\right.\\[-2pt]&\left.{\frac {1}{\kappa _{1}\left ({1-q}\right)}\ln \frac {\kappa _{1}V_{n}^{1-q}(0)+\nu \kappa _{2}}{\nu \kappa _{2}}}\right \}\end{align*}$$ which implies that the stabilization errors converge to an arbitrary small region of the zero with a finite time. This completes the proof.

Remark 6:

Similar studies about the adaptive resilient control problems for nonlinear systems have been investigated in [18], [19], [23], [24], [27], and [28]. In [23], [24], [27], and [28], the obtained results were mainly focused on the asymptotic stability of the CLS. However, there is no result reported on the finite-time stability of nonlinear time-delay systems (1). Although some adaptive NN finite-time resilient control methods have been developed in [18] and [19], it worth noting that these methods are just valid for actuator faults, not suitable for handling unknown cyberattacks. What is more, the bounds of the actuation effectiveness and stuck faults are not required in this work different from [18] and [19].

Remark 7:

It worth pointing out that the investigated resilient control problem reduces to fault-tolerant control problem of nonlinear time-delay systems with sensor faults when $0 < \beta (t) < 1$ . From the perspective of system theory, as stated in [23] and [27], it can be recognized that the faults and attacks are fundamentally the same. However, due to the existence of the false data injection attacks through the sensor networks, the sign of $\beta (t)$ is unavailable and may change before and after the occurrence of the attacks. In this case, the effectiveness of the ordinary fault-tolerant control methods is hard to be guaranteed. To this end, it is significant for developing a resilient control method for both unknown attacks and faults.

SECTION IV. Simulation Studies

In this section, two simulation examples are provided to illustrate the effectiveness of the developed method in theory and application.

A. Numerical Example

Example 1:

We consider the following nonlinear system: $$\begin{align*} \begin{cases} \displaystyle \dot {x}_{1}=x_{2}+f_{1}\left ({\bar {x}_{1}}\right)+g_{1}\left ({\bar {x}_{1,\tau \left ({t}\right)}}\right)+d_{1}\left ({t}\right) \\ \displaystyle \dot {x}_{2}=\sum _{j=1}^{2}b_{j}u_{j}+f_{2}\left ({\bar {x}_{2}}\right)+g_{2}\left ({\bar {x}_{2,\tau \left ({t}\right)}}\right)+d_{2}\left ({t}\right)\\ \displaystyle \check {x}_{i}=x_{i}+A_{i,s}\left ({x_{i}\left ({t}\right),t}\right), \quad i=1,2 \end{cases} \tag{57}\end{align*}$$ where $f_{1}(\bar {x}_{1})=0.1x_{1}, f_{2}(\bar {x}_{2})=-2.5x_{2}-0.5x_{1}x_{2}$ , $g_{1}(\bar {x}_{1,\tau (t)})=x_{1}^{3}(t-\tau _{1}(t))/(1+x_{1}^{2}(t-\tau _{1}(t)))$ , and $g_{2}(\bar {x}_{2,\tau (t)})=x_{1}^{4}(t-\tau _{1}(t))\sin (x_{2})/(1+x_{2}^{2}(t-\tau _{2}(t))), d_{1}(t)=0.01\sin (2t), d_{2}(t)=0.01\cos (t)$ . The time-varying state delays are selected as $\tau _{1}=0.5(3+\cos (2t))$ and $\tau _{2}=0.5(5+0.5\sin (t))$ . The unknown actuator faults are modeled as $$\begin{align*} u_{1}=&\begin{cases} \displaystyle v_{1}, & t\in \left [{8\,\text {m}, 8\,\text {m}+4}\right)\\ \displaystyle \bar {u}_{1d,m}, & t\in \left [{8\,\text {m}+4, 8\,\text {m}+8}\right) \end{cases} \\ u_{2}=&\begin{cases} \displaystyle v_{2}, & t\in \left [{4\,\text {m}, 4\,\text {m}+2}\right)\\ \displaystyle l_{jm}v_{2}, & t\in \left [{4\,\text {m}+2, 4\,\text {m}+4}\right) \end{cases}\end{align*}$$ and the unknown sensor attacks are chosen as $A_{1,s}=(-0.3-0.5\cos (t))x_{1}$ and $A_{2,s}=(-0.3-0.5\cos (t))x_{2}$ , where $\bar {u}_{1d,m}=0.001, l_{jm}=0.7$ . The above fault model implies that the first actuator normally works every 4 s and undergoes the total loss of effectiveness for the next 4 s. For the second actuator, it works normally every 2 s and undergoes the partial loss of effectiveness for the next 2 s.

In the light of the NN design, the BF of NNs $\varphi _{i}(\check {\Delta }_{i})=[\varphi _{i1}(\check {\Delta }_{i}),\ldots,\varphi _{im_{i}}(\check {\Delta }_{i})]^{T}$ is achieved by calculating the Gaussian function $\varphi _{ik}(\check {\Delta }_{i})=[1/(({2\pi })^{1/2}b_{ik})]\exp [-(\check {\Delta }_{i}-c_{ik})^{T}(\check {\Delta }_{i}-c_{ik})/b_{ik}^{2}]$ , where $c_{1k}=k-5$ and $c_{2k}=(k-5)*[{1,1,1,1,1}]^{T}$ . The number of nodes and the spread of BF are chosen as $m_{1}=m_{2}=9$ and $b_{ik}=1$ with $\check {\Delta }_{1}=\check {s}_{1}\,\,\text {and}\,\,\check {\Delta }_{2}=[\check {x}_{1},\check {x}_{2}, \check {s}_{1},\check {s}_{2}, \vartheta _{1}^{f}]^{T}$ . According to (31), (39), and (49), choose the control parameters as: $k_{1}=8, k_{2}=5, c_{1}=0.4,c_{2}=0.1, \sigma _{1}=\sigma _{2}=0.2$ , and $\rho _{i,1}=\rho _{i,2}=2$ with $j=1,2,3$ , $\alpha =0.9, \alpha _{1,1}=\alpha _{1,2}=3, \alpha _{1,3}=2, \beta _{1}=\beta _{3}=0.5$ , and $\beta _{2}=1.5$ . It can be easily verified that the conditions $\bar {k}_{j}>0, \bar {a}_{j1}>0,\bar {\rho }_{m,1}>0,\bar {\rho }_{m,2}>0$ , and $\bar {b}_{j1}>0$ hold. Then, the simulation results with initial condition $[x_{1}(0), x_{2}(0)] =[{0.8,-2}]$ are presented in Figs. 3–​6. Fig. 3 shows the trajectory of the system state $x_{i}$ . From Fig. 3, we can see that good stabilization performance can be achieved even if the attacks and actuator failures occur. Figs. 4 and 5 plot the dynamic curves of adaptive parameters $\hat {\hbar }_{i},\hat {\Gamma }_{i}, \hat {\chi }_{i}$ , and $\xi _{i}$ and Nussbaum parameter $N(\xi _{i})$ . The time response of the control signal $u_{i}$ is shown in Fig. 6. It can be easily observed from Figs. 3–​6 that all signals in the CLS are bounded, which also reflects the validity of the proposed control algorithm.

Fig. 3.

Trajectory of system state $x_{i}$ .

Fig. 4.

Trajectory of adaptive parameters $\hat {\hbar }_{i}, \hat {\Gamma }_{i}$ , and $\hat {\chi }_{i}$ .

Fig. 5.

Trajectory of parameters $\xi _{i}$ and $N(\xi _{i})$ .

Fig. 6.

Trajectory of control signal $u_{i}$ .

B. Application to chemical reactor system

Example 2:

Consider a two-stage chemical reactor system with delayed recycle streams, as shown in Fig. 7. Assume that two reactors are continuous stirred tanks reactors with a constant temperature and the feeder encounters the unknown sensor attacks and actuator faults. Then, the mass balance equation that governs the reactors is given by $$\begin{align*} \begin{cases} \displaystyle \dot {x}_{1}=\frac {1-F_{R2}}{V_{1}}x_{2}-\frac {1}{T_{1}}x_{1}-\frac {1}{T_{1}}x_{1}\left ({t-\tau _{1}\left ({t}\right)}\right)\\ \displaystyle \qquad \,\,-Q_{1}x_{1}+d_{1}\left ({t}\right) \\ \displaystyle \dot {x}_{2}= \sum _{j=1}^{2}\frac {F_{RF}^{j}}{V_{2}}u_{j}-Q_{2}x_{2}-\frac {1}{T_{2}}x_{2}^{2}-\frac {2E}{T_{2}}x_{2}\\ \displaystyle \qquad \,\,+\frac {F_{R1}}{V_{2}}x_{1}\left ({t-\tau _{1}\left ({t}\right)}\right)+\frac {F_{R2}}{V_{2}}x_{2}\left ({t-\tau _{2}\left ({t}\right)}\right)+d_{2}\left ({t}\right)\\ \displaystyle \check {x}_{i}=x_{i}+A_{i,s}\left ({x_{i}\left ({t}\right),t}\right),\quad i=1,2 \end{cases} \tag{58}\end{align*}$$ where $x_{1}$ and $x_{2}$ are compositions to be controlled of the produce streams from two reactors. The representation and values of the involved system parameters are given in Table I.

TABLE I Description of Model Parameters

Fig. 7.

Chemical reactor with delayed recycle streams.

The state delays $\tau (t)$ are selected as $\tau _{1}(t)=0.5 (2.5 + \sin (t))$ and $\tau _{2}(t)=0.5(3 + 0.4\cos (2t))$ . The unknown actuator faults are described by $$\begin{align*} u_{1}=&\begin{cases} \displaystyle \bar {u}_{1d,m}, & t\in \left [{6\,\text {m}, 6\,\text {m}+3}\right)\\ \displaystyle v_{1}, & t\in \left [{6\,\text {m}+3, 6\,\text {m}+6}\right) \end{cases} \\ u_{2}=&\begin{cases} \displaystyle l_{jm}v_{2}, & t\in \left [{10\,\text {m}, 10\,\text {m}+5}\right)\\ \displaystyle v_{2}, & t\in \left [{10\,\text {m}+5, 10\,\text {m}+10}\right) \end{cases}\end{align*}$$ where $\bar {u}_{1d,m}=0.01$ and $l_{jm}=0.2$ . It follows from the given fault model that the first actuator suffers the total loss of effectiveness every 3 s and normally works for the next 3 s. Accordingly, the second actuator undergoes the partial loss of effectiveness every 5 s and works normally for the next 5 s. The configuration of NNs is the same as that in Example 1. To show the validity and applicability of the proposed control method in comparison to the existing results, the simulation results are divided into the following two cases.

• Case 1:

  The sign of $\beta (t)=(1+\lambda (t))^{-1}$ is positive. Define the unknown sensor attacks as $A_{1,s}=(-0.5-0.25\cos (t))x_{1}$ and $A_{2,s}=(-0.5-0.25\cos (t))x_{2}$ . The control parameters and initial conditions are selected as: $k_{1}=5, k_{2}=5, c_{1}=c_{2}=0.1, and [x_{1}(0), x_{2}(0)] =[{1,-1.4}]$ , and other design parameters can be referred to Example 1. The simulation results are shown in Figs. 8–​11. Fig. 8 shows the state trajectory of the system (58). The boundedness of the adaptive parameters $\hat {\hbar }_{i},\hat {\Gamma }_{i}, \hat {\chi }_{i}$ , and $\xi _{j}$ and Nussbaum parameter $N(\xi _{j}) (i=1,2;j=1,2,3)$ is shown in Figs. 9 and 10. Fig. 11 shows the curve of control signal. From Fig. 8, it can be seen that the proposed resilient controller against unknown attacks and actuator faults can guarantee that the system states reach to zero with a faster convergence rate.

• Case 2:

  The sign of $\beta (t)=(1+\lambda (t))^{-1}$ is negative. Define the unknown sensor attacks as $A_{1,s}=(-1.2-0.1\cos (t))x_{1}$ and $A_{2,s}=(-1.2-0.1\cos (t))x_{2}$ . Figs. 12–​15 show the control performance of the proposed control scheme. The trajectory of the system states $x_{1}$ and $x_{2}$ is shown in Fig. 12. It is obvious that the system states can be effectively stabilized within a finite time. The time responses of the adaptive parameters $\hat {\hbar }_{i},\hat {\Gamma }_{i}, \hat {\chi }_{i}$ , and $\xi _{j}$ and Nussbaum parameter $N(\xi _{j}) (i=1,2;j=1,2,3)$ are shown in Figs. 13 and 14, respectively. Fig. 15 plots the curve of control signals $u_{1}$ and $u_{2}$ . From Figs. 12–​15, it can be concluded that the boundedness of all closed-loop signals can be ensured by using the proposed adaptive resilient controller. What is more, Figs. 8 and 12 show that the predefined control objective can be achieved whether the sign of weight $\beta$ is positive or negative.

Fig. 8.

Trajectory of system state $x_{i}$ in Case 1.

Fig. 9.

Trajectory of parameters $\hat {\hbar }_{i},\hat {\Gamma }_{i},$ and $\hat {\chi }_{i}$ in Case 1.

Fig. 10.

Trajectory of parameters $\xi _{i} and N(\xi _{i})$ in Case 1.

Fig. 11.

Trajectory of control signal $u_{i}$ in Case 1.

Fig. 12.

Trajectory of system state $x_{i}$ in Case 2.

Fig. 13.

Trajectory of parameters $\hat {\hbar }_{i},\hat {\Gamma }_{i}$ , and $\hat {\chi }_{i}$ in Case 2.

Fig. 14.

Trajectory of parameters $\xi _{i}$ and $N(\xi _{i})$ in Case 2.

Fig. 15.

Trajectory of control signal $u_{i}$ in Case 2.

Remark 8:

Some similar studies have been carried out for a class of cyber-physics systems in the existing literature [28], [37]. In [28] and [37], the weight $\beta (t)$ is required to be strictly positive in this article. This implies that the stability of the system may not be guaranteed when the sign of $\beta (t)$ changes. In this article, this restriction has been fully removed. By using the proposed method, it has proved that the desired control performance can be achieved whether the sign of the weight $\beta (t)$ is positive or negative as shown in the simulation results of Example 2. However, the Nussbaum function-based control method may lead to an unexpected high-gain signal. Noted that a switching-type adaptive control scheme has been proposed in [52] and [53] for handling the unknown control directions. As an improvement, this method provides a potential solution for avoiding the high gain caused by the Nussbaum-type control method. Consequently, developing a novel switching-type adaptive resilient control scheme for system (1) is still open and interesting in the recursive control field of nonlinear time-delay systems.

SECTION V. Conclusion

This work develops an adaptive finite-time resilient control method for nonlinear time-delay systems with unknown false data injection attacks and actuator failures. A new coordinate transformation is established to eliminate the negative effects caused by the false data injection attacks. Furthermore, the Nussbaum gain technique and a modified FOCF backstepping approach are adopted to handle the unknown time-varying weight and overcome the issue of “explosion of complexity” existing in the traditional recursive design procedure. Finally, the simulation results show the effectiveness of the presented method. However, it worthy to note that the proposed method cannot provide an optimal solution for control design distinct from [54]–​[58]. Inspired by these results, developing an adaptive optimal resilient control scheme for nonlinear systems with cyberattacks will be one of our future research work.

ACKNOWLEDGMENT

The authors would like to sincerely thank the editor and anonymous reviewers for their valuable comments and suggestions that are helpful for improving this work.