Data-Based Security Fault Tolerant Iterative Learning Control under Denial-of-Service Attacks

Abstract

This paper mainly studies the data-based security fault tolerant iterative learning control (SFTILC) problem of nonlinear networked control systems (NCSs) under sensor failures and denial-of-service (DoS) attacks. Firstly, the radial basis function neural network (RBFNN) is used to approximate the sensor failure function and a DoS attack compensation mechanism is proposed in the iterative domain to lessen the impact of DoS attacks. Then, using the dynamic linearization technology, the nonlinear system considering failures and network attacks is transformed into a linear data model. Further, based on the designed linearization model, a new data-based SFTILC algorithm is designed to ensure the satisfactory tracking performance of the system. This process only uses the input and output data of the system, and the stability of the system is proved by using the compression mapping principle. Finally, a digital simulation is used to demonstrate the effectiveness of the proposed SFTILC algorithm.

1. Introduction

As the network technology develops, traditional local control systems (LCSs) have been gradually replaced by network control systems (NCSs) [1,2,3,4]. NCSs are integrated by controllers, sensors, actuators, and network transmission channels. NCSs receive sensor signals of equipment through networks and send control signals to control the equipment, so that remote control tasks can be completed accordingly. In NCSs, the separation of controllers and equipment is one of the most important characteristics of NCSs, but it requires a large amount of data transmission, and the existence of data also stimulates the research of data-based control (DDC) [5,6,7].

It is more and more difficult to identify the model of systems, and the control problem becomes more difficult and unrealistic by using the traditional model-based control theory. Therefore, DDC theory began to be researched. There have had many DDC methods in the existing literature, such as PID control, model free adaptive control (MFAC) [8], iterative learning control (ILC) [9,10,11,12], iterative feedback tuning (IFT) [13] and so on. MFAC and ILC are proposed to solve the control problem of nonlinear systems by directly applying the I/O data of the controlled systems. The paper [14] proposed a new MFAC method for discrete-time single input single output nonlinear systems combined with radial basis function neural networks. In [15], the bounded input–output stability and tracking performance of full format dynamic linearized MFAC were strictly proved by using the compression mapping principle. In [16], aiming at the problem of driverless vehicle ground heading control, an MFAC method based on the adaptive forgetting factor was proposed. When the control task is repeated, the system shows the same behavior. For the ILC method of the repeated process, when constructing the current algorithm, the information of the last iteration time stored by storage devices is used to correct the current control input, so as to improve the error accuracy of the system cycle process. In [17], the contraction mapping (CM)-based method was suitable for some robust ILCs that are non-competitive with uncertainties, especially for power system models. In order to ensure the robust convergence of iterative learning control designs, the $H_{\infty }$ norm condition was proposed, and the learning gain matrix was determined by solving norm conditions. In [18], a class of multi-input and multi-output discrete linear systems was studied. Under the assumption that all iterative uncertainties were bounded, conditions to ensure the boundedness of all system trajectories and to ensure the convergence of tracking errors were given in [18]. In the presence of data quantization and DoS attacks, a quantized data-based iterative learning tracking control algorithm was proposed for a class of repetitive nonlinear systems in [19].

It is well-known that NCSs involve the transmission of data that are required by the control system through the network, while many network failures or network attacks exist in networks. Network attacks can be roughly divided into denial-of-service (DoS) [20,21,22] attacks, false data injection (FDI ) [23] attacks, and replay attacks. The basic goal of a DoS attack is to prevent the target network from providing regular service or resource access, as well as to cause the target system to become unresponsive or crash. In this way, NCSs cannot achieve the expected control objectives. Therefore, there has had much research on solving the impacts of DoS attacks in the existing literature. The consensus problem of linear multi-agent systems (MASs) under DoS attacks was investigated in [24]. The paper [25] studied the output tracking problem of distributed model-free adaptive control systems under DoS attacks. If the sensor in an NCS fails, it will also affect the accuracy of data transmission. Inaccurate data measurement leads to that the controller can not well generate control inputs to control the controlled system, resulting in the system can not achieve the expected control objectives. For a class of continuous-time disturbed systems, the paper [26] provided an adaptive compensation technique against state-dependent and partly bounded actuator failures and disturbances. In [27], the author studied the distributed fault tolerance and resilient consensus of heterogeneous multi-agent systems under equipment failures and DoS attacks. Under DoS attacks, the problem of fault-tolerant MFAC for nonlinear networked control systems was investigated in [28]. The above research on fault tolerance is to design fault-tolerant control methods and prove the stability in the time domain. There are few research findings on ILC fault-tolerant control under the repeated operation NCSs. Therefore, we investigate the nonlinear repetitive NCSs under DoS attacks and fault-tolerant control in the iteration domain, which are prompted by the above descriptions. The contributions of this paper can be listed as:

•

The nonlinear system is transformed into a linear data model by using the dynamic linearization technology under failures and DoS attacks. Then, a new data-based security fault tolerant iterative learning control (SFTILC) algorithm is designed to ensure the tracking performance of the system under the DoS attack.

•

The radial basis function neural network (RBFNN) is used to approximate the sensor failure function. A DoS attack compensation mechanism is proposed in the iterative domain to lessen the impacts of DoS attacks.

The structure of this document is as follows. The preparation and explanation of the problem are described in Section 2. In Section 3, the design of a fault tolerant controller is presented. Section 4 provides the stability analysis. The simulations and conclusions are presented in Section 5 and Section 6, respectively.

2. Preliminaries and Problem Formulations

2.1. System Description

The nonlinear discrete-time single-input single-output system that repeatedly runs at finite time intervals is given as follows

$$\begin{array}{c}\hfill y_i(k+1)=f(y_i\left(k\right),\dots ,y_i(k-n_y),u_i\left(k\right),\dots ,u_i(k-n_u)),\end{array}$$

(1)

where $k\in \{0,1,\cdots ,T\}$ is the sampling time, $i\in \{1,2,\cdots \}$ is the number of iterations, $y_i\left(k\right)$ and $u_i\left(k\right)$ are the output and input signals of the system, respectively, $n_y$ and $n_u$ are two unknown positive integers, and $f(\cdots )$ is an unknown nonlinear scalar function.

2.2. Dynamic Linearization

To facilitate the later controller design, the following assumptions are made for the system (1).

Assumption 1.

The partial derivative of $f(\cdots )$ to $u_i\left(k\right)$ is continuous.

Assumption 2.

System (1) satisfies the generalized Lipschitz condition along the iteration axis, namely, $\forall k\in \{1,\cdots T\}$, if $|\Delta u_i\left(k\right)|\ne 0$, the formula below applies

$$|\Delta y_i(k+1)|\le a|\Delta u_i\left(k\right)|,\forall i=1,2,\cdots$$

(2)

where $\Delta y_i(k+1)=y_i(k+1)-y_{i-1}(k+1)$, $\Delta u_i\left(k\right)=u_i\left(k\right)-u_{i-1}\left(k\right)$ and $a>0$ is a constant.

Remark 1.

Assumptions 1 and 2 are reasonable in many actual situations. In the control system design, Assumption 1 is a typical constraint condition for generic nonlinear systems. Assumption 2 limits the system output change rate produced by the change of the control input along the iteration axis. Namely, a bounded input energy change should result in a finite output energy change.

Lemma 1.

For the nonlinear system (1) that meets both Assumptions 1 and 2, when $|\Delta u_i\left(k\right)|\ne 0$, there is a time-varying pseudo partial derivative (PPD) ${\varphi }_i\left(k\right)$ so that system (1) can be transformed into the following linear data model

$$\begin{array}{c}\hfill \Delta y_i(k+1)={\varphi }_i\left(k\right)\Delta u_i\left(k\right),\forall k\in \{0,1,\cdots ,N\},i=1,2,\cdots \end{array}$$

(3)

where $|{\varphi }_i\left(k\right)|\le a$. The proof of Lemma 1 is the same as that in [29].

3. Design Fault Tolerant Controller

3.1. Model Free Adaptive Iterative Learning Control

Considering the input cost function as follows

$$\begin{array}{c}\hfill \zeta \left(u_i\left(k\right)\right)={|y_d(k+1)-y_i(k+1)|}^2+\lambda {(\Delta u_i\left(k\right))}^2,\end{array}$$

(4)

where $\lambda$ is a weight coefficient and $y_d(k+1)$ is the time-invariant desired output signal.

The optimal control rate is obtained by minimizing the cost function $\zeta \left(u_i\left(k\right)\right)$. In order to make the algorithm more general, the step factor $\rho \in (0,1]$ is introduced and the following optimal control algorithm is obtained

$$\begin{array}{c}\hfill u_i\left(k\right)=u_{i-1}\left(k\right)+\frac{\rho {\varphi }_i\left(k\right)}{\lambda +{\varphi }_i{\left(k\right)}^2}(y_d(k+1)-y_{i-1}(k+1)).\end{array}$$

(5)

Because ${\varphi }_i\left(k\right)$ are difficult to be measured in real systems, we need to estimate them by introducing the following performance function

$$\begin{array}{c}\hfill \zeta \left({\widehat{\varphi }}_i\left(k\right)\right)=|\Delta y_{i-1}(k+1)-{\widehat{\varphi }}_i\left(k\right)\Delta u_{i-1}{\left(k\right)|}^2+\mu {|{\widehat{\varphi }}_i\left(k\right)-{\widehat{\varphi }}_{i-1}\left(k\right)|}^2\end{array}$$

(6)

where ${\widehat{\varphi }}_i\left(k\right)$ is an estimate of ${\varphi }_i\left(k\right)$, and $\mu$ is a weight factor.

Minimize the performance function $\zeta \left({\widehat{\varphi }}_i\left(k\right)\right)$ to get ${\widehat{\varphi }}_i\left(k\right)$, similarly, in order to make the algorithm more general the step-size factor $\mu \in (0,1]$ is introduced, then one can obtain

$$\begin{array}{c}\hfill {\widehat{\varphi }}_i\left(k\right)={\widehat{\varphi }}_{i-1}\left(k\right)+\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}(\Delta y_{i-1}(k+1)-\Delta u_{i-1}\left(k\right){\widehat{\varphi }}_{i-1}\left(k\right)).\end{array}$$

(7)

The above control algorithm is applied to the nominal system. However, in the practical networked control system, on the one hand, sensor failures will result in some deviations in the measurement data. On the other hand, the emergence of the network may bring cyber attacks. So, the above algorithm may not complete the tracking task well when these problems exist. Therefore, it seems necessary to design the secure fault-tolerant iterative learning control (SFTILC) algorithm under sensor failures and cyber attacks.

Figure 1 depicts the nonlinear NCS architecture under DoS attacks and sensor failures consideration.

3.2. Sensor Failure

According to Figure 1, when a fault occurs, the sensor measurement signal becomes to the following form

$$\begin{array}{c}\hfill y_i^F\left(k\right)=y_i\left(k\right)+F_{s,i}\left(k\right)\end{array}$$

(8)

where $F_{s,i}\left(k\right)$ is a fault function, which can be approximated using the radial basis function neural network (RBFNN), the details are given as

$$\begin{array}{c}\hfill F_{s,i}\left(k\right)=v_i^T\left(k\right)\psi \left(z\left(k\right)\right)\end{array}$$

(9)

where $v_i\left(k\right)\in R^p$ is the ideal weight vector in the output layer, p is the RBFNN’s node number, and $\psi \left(z\left(k\right)\right)={[{\psi }_1\left(z\left(k\right)\right),{\psi }_2\left(z\left(k\right)\right)\cdots ,{\psi }_p\left(z\left(k\right)\right)]}^T\in R^p$ is the hidden layer’s basis function, which is a Gaussian function, and it can be represented as

$$\begin{array}{c}\hfill {\psi }_l\left(z\left(k\right)\right)=exp(-\frac{\parallel z\left(k\right)-{\varrho }_l{\parallel }^2}{{\tau }_l^2})\end{array}$$

(10)

where ${\varrho }_l$ denotes the RBFNN’s center, ${\tau }_l$ denotes the RBFNN’s breadth for $l\in 1,2,\cdots ,p$, $z\left(k\right)={[u_r\left(k\right),F_{s,i}(k-1),y_i\left(k\right)]}^T$ is the RBFNN input.

In most cases, $v_i\left(k\right)$ is not available, therefore, the following learning process is introduced to obtain it.

$$\begin{array}{c}\hfill v_i\left(k\right)=\varpi \{\Delta u_i\left(k\right){\psi }^T\left(k\right){\widehat{\varphi }}_i\left(k\right)\}\end{array}$$

(11)

where $\varpi$ stands for the projection operator, $\Delta u_i\left(k\right)$ can be calculated by (18), ${\widehat{\varphi }}_i\left(k\right)$ can be defined in (17). Then $\varpi$ is given as following, which has been employed in [30].

$$\varpi (·)=\left\{\begin{array}{c}\hfill 0,if{v}_i\left(k\right)=\underline{v}and\Delta u_i\left(k\right){\psi }^T\left(k\right){\widehat{\varphi }}_i\left(k\right)<0\\ \hfill 0,if{v}_i\left(k\right)=\overline{v}and\Delta u_i\left(k\right){\psi }^T\left(k\right){\widehat{\varphi }}_i\left(k\right)>0\\ \hfill \Delta u_i\left(k\right){\psi }^T\left(k\right){\widehat{\varphi }}_i\left(k\right),otherwise.\end{array}\right.$$

(12)

where $\underline{v}$ and $\overline{v}$ are the lower and the upper bounds of $v_i\left(k\right)$, respectively.

In the case of sensor failures, the nonlinear repetitive system is linearized dynamically by

$$\begin{array}{c}\hfill y_i^F(k+1)=y_{i-1}(k+1)+{\varphi }_i\left(k\right)\Delta u_i\left(k\right)+F_{s,i}(k+1)\end{array}$$

(13)

where $|F_{s,i}(k+1)|\le \sqrt{p}\overline{v}$ with p being the node number of RBFNN and $\overline{v}$ is the upper bounds of $v_i\left(k\right)$.

3.3. Attack Compensation

When a network attacker launches DoS attacks, the output signal is expressed as follows

$$\begin{array}{c}\hfill y_{ai}^F\left(k\right)={\alpha }_i\left(k\right)y_i^F\left(k\right),\end{array}$$

(14)

where ${\alpha }_i\left(k\right)$ obeys the Bernoulli distribution, ${\alpha }_i\left(k\right)=1$ denotes a failed DoS attack, whereas ${\alpha }_i\left(k\right)=0$ denotes a successful DoS attack. As a result, the probabilities are expressed as

$$Pr\left\{\begin{array}{cc}\hfill Pr[{\alpha }_i\left(k\right)=1]& =\overline{\alpha },\hfill \\ \hfill Pr[{\alpha }_i\left(k\right)=0]& =1-\overline{\alpha }.\hfill \end{array}\right.$$

(15)

To compensate for the impact of DoS attacks, the mechanism is designed as follows

$$\begin{array}{c}\hfill {\overline{y}}_i\left(k\right)={\alpha }_i\left(k\right)y_i^F\left(k\right)+(1-{\alpha }_i\left(k\right))y_{i-1}^F\left(k\right).\end{array}$$

(16)

Consider sensor failure and DoS attacks, the SFTILC algorithm is proposed as follows

$${\widehat{\varphi }}_i\left(k\right)=\left\{\begin{array}{c}{\widehat{\varphi }}_1\left(k\right),\mathrm{if}\left|\right|{\widehat{\varphi }}_i\left(k\right)\left|\right|\le ϵor|\Delta u_i\left(k\right)|\le ϵ\hfill \\ \\ {\widehat{\varphi }}_{i-1}\left(k\right)+\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}({\overline{y}}_{i-1}(k+1)\hfill \\ -y_{i-2}^F(k+1)-\Delta u_{i-1}\left(k\right){\widehat{\varphi }}_{i-1}\left(k\right)),\mathrm{otherwise}.\hfill \end{array}\right.$$

(17)

$$\begin{array}{cc}\hfill u_i\left(k\right)& =u_{i-1}\left(k\right)+\frac{{\varphi }_i\left(k\right)}{\lambda +{\varphi }_i{\left(k\right)}^2}(y_d(k+1)-{\overline{y}}_{i-1}\left(k\right)).\hfill \end{array}$$

(18)

For the convenience, the tracking error $e_i\left(k\right)$ is defined as follows:

$$\begin{array}{c}\hfill e_i\left(k\right)=y_d\left(k\right)-y_i\left(k\right).\end{array}$$

(19)

Problem 1.

This study aims to design the SFTILC algorithm to realize the mean-square bounded output tracking task of system (1) under the influence of sensor failures and DoS attacks, namely,

$$\begin{array}{c}\hfill \{e_i\left(k\right)\mid E\left\{\right|e_i\left(k\right)\left|\right\}\le b\},\end{array}$$

wherebis a constant, where$E\{·\}$denotes the expectation.

4. Stability Analysis

Theorem 1.

If there are parameters $\mu >0$, $\rho \in (0,1]$, $\lambda >\frac{{\rho }^2{a}^2}{4}$, and $\eta \in (0,1]$ such that the SFTILC algorithm constructed of (17) and (18) holds for system (1) subject to sensor failures and DoS attacks, then Problem 1 is solved, where a is the same as described in (2).

Proof. 

The proof of Theorem 1 consists of two parts: the bounded PPD parameter estimation and the bounded result of the output tracking error.

Part I: Boundedness of the PPD parameter estimation

(i) When $|{\widehat{\varphi }}_i\left(k\right)|\le \epsilon$, ${\widehat{\varphi }}_i\left(k\right)={\widehat{\varphi }}_1\left(k\right)$, so ${\widehat{\varphi }}_i\left(k\right)$ is bounded.

(ii) When $|{\widehat{\varphi }}_i\left(k\right)|>\epsilon$, the PPD estimation error is defined as

$$\begin{array}{c}\hfill e_{{\varphi }_i}\left(k\right)={\widehat{\varphi }}_i\left(k\right)-{\varphi }_i\left(k\right).\end{array}$$

(20)

Bring (17) into (20) to obtain the following formula

$$\begin{array}{cc}\hfill e_{{\varphi }_i}\left(k\right)& ={\widehat{\varphi }}_{i-1}\left(k\right)-{\varphi }_i\left(k\right)+\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}\left(y_{a,i-1}(k+1)-y_{i-2}^F(k+1)-\Delta u_{i-1}\left(k\right){\widehat{\varphi }}_{i-1}\left(k\right)\right)\hfill \\ \hfill & =e_{{\varphi }_{i-1}}\left(k\right)-\Delta {\varphi }_i\left(k\right)+\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}({\alpha }_{i-1}(k+1)y_{i-1}^F(k+1)-y_{i-2}^F(k+1)\hfill \\ & +(1-{\alpha }_{i-1}(k+1))y_{i-2}^F(k+1)-\Delta u_{i-1}\left(k\right){\widehat{\varphi }}_{i-1}\left(k\right))\hfill \\ \hfill & =(1-\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)})e_{{\varphi }_{i-1}}\left(k\right)-\Delta {\varphi }_i\left(k\right)\hfill \\ & +\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}\left({\alpha }_{i-1}(k+1)\right)\Delta F_{s,i-1}(k+1)\hfill \\ & +\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}({\alpha }_{i-1}(k+1)-1){\varphi }_{i-1}\left(k\right).\hfill \end{array}$$

(21)

Taking the absolute value and expectation on both sides of (21), we can get

$$\begin{array}{cc}\hfill E\left\{\right|e_{{\varphi }_i}\left(k\right)\left|\right\}& \le |1-\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}\left|E\right\{|e_{{\varphi }_{i-1}}\left(k\right)\left|\right\}+|\Delta {\varphi }_i\left(k\right)|+|\frac{\overline{\alpha }\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}\left(k\right)}\Delta F_{s,i-1}(k+1)|\hfill \\ & +|\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}(\overline{\alpha }-1){\varphi }_{i-1}\left(k\right)|\hfill \end{array}$$

(22)

where $|{\varphi }_i\left(k\right)|\le a$, $0<\overline{\alpha }<1$ and $\mu +\Delta u_{i-1}^2\left(k\right)>\Delta u_{i-1}^2\left(k\right)>0$, so

$$\begin{array}{c}\hfill 0<\frac{\eta \Delta u_{i-1}\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}<\frac{\eta }{2\sqrt{\mu }}\end{array}$$

(23)

$$\begin{array}{c}\hfill 0<|\frac{\eta (\overline{\alpha }-1)\Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}|<\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}\le m_1<1\end{array}$$

(24)

$$\begin{array}{c}\hfill 0<1-\frac{\eta \Delta u_{i-1}^2\left(k\right)}{\mu +\Delta u_{i-1}^2\left(k\right)}\le m_2<1\end{array}$$

(25)

where $m_1$ and $m_2$ are constants.

From (23)–(25), we can reduce (22) to the following inequality

$$\begin{array}{c}\hfill E\left\{\right|e_{{\varphi }_i}\left(k\right)\left|\right\}\le m_{2}E\left\{\right|e_{{\varphi }_{i-1}}\left(k\right)\left|\right\}+m_3,\end{array}$$

(26)

where $m_3=\frac{\overline{\alpha }\eta \sqrt{p}\overline{v}}{\sqrt{\mu }}+3a$.

By using the compressed mapping principle, we can get

$$\begin{array}{cc}\hfill E\left\{\right|e_{{\varphi }_i}\left(k\right)\left|\right\}& \le m_2^{2}E\left\{\right|e_{{\varphi }_{i-2}}\left(k\right)\left|\right\}+m_2{m}_3+m_3\hfill \\ \hfill & \le \cdots \hfill \\ \hfill & \le m_2^{i-1}E\left\{\right|e_{{\varphi }_1}\left(k\right)\left|\right\}+m_2^{i-2}{m}_3+\cdots +m_3\hfill \\ \hfill & =m_2^{i-1}E\left\{\right|e_{{\varphi }_1}\left(k\right)\left|\right\}+\frac{m_3(1-m_2^{i-1})}{1-m_2},\hfill \end{array}$$

(27)

so, $e_{{\varphi }_i}\left(k\right)$ converges to the following set

$$\begin{array}{c}\hfill \left\{e_{{\varphi }_i}\left(k\right)\left|E\left\{\right|e_{{\varphi }_i}\left(k\right)\left|\right\}\le \frac{m_3}{1-m_2}\right.\right\}.\end{array}$$

(28)

So far, the proof of the boundedness of $e_{{\varphi }_i}\left(k\right)$ is completed because ${\varphi }_i\left(k\right)$ is bounded, leading to ${\widehat{\varphi }}_i\left(k\right)$ also being bounded.

Part II: Tracking performance analysis

The boundedness of the PPD parameter estimation proved above can be used as a condition to prove the boundedness of the system tracking error.

By taking (3) and (18) into (19), we can get

$$\begin{array}{cc}\hfill e_i(k+1)& =y_d(k+1)-y_{i-1}(k+1)-{\varphi }_i\left(k\right)\Delta u_i\left(k\right)\hfill \\ \hfill & =e_{i-1}(k+1)-\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}\hfill \\ & \times (y_d(k+1)+y_{i-1}^F(k+1)-y_{i-1}^F(k+1)-{\alpha }_{i-1}(k+1)y_{i-1}^F(k+1)\hfill \\ & -(1-{\alpha }_{i-1}(k+1))y_{i-2}^F(k+1))\hfill \\ \hfill & =(1-\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)})e_{i-1}(k+1)-\frac{\rho {\varphi }_i\left(k\right){\varphi }_{i-1}\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}(1-{\alpha }_{i-1}(k+1))\Delta u_{i-1}\left(k\right)\hfill \\ \hfill & -\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}((1-{\alpha }_{i-1}(k+1))\Delta F_{s,i-1}(k+1)+F_{s,i-1}(k+1)).\hfill \end{array}$$

(29)

Taking the absolute value and expectation on both sides of (29), we can get

$$\begin{array}{cc}\hfill E\left\{\right|e_i(k+1)\left|\right\}& \le |1-\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}\left|E\right\{|e_{i-1}(k+1)\left|\right\}\hfill \\ & +|\frac{\rho {\varphi }_i\left(k\right){\varphi }_{i-1}\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}(1-\overline{\alpha })\Delta u_{i-1}\left(k\right)|\hfill \\ \hfill & +|\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}((1-\overline{\alpha })\Delta F_{s,i-1}(k+1)+F_{s,i-1}(k+1))|.\hfill \end{array}$$

(30)

According to Theorem 1, (3) and (28), we can get

$$\begin{array}{c}\hfill 0<m_4\le \frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}\le \frac{\rho a}{2\sqrt{\lambda }}=m_5<1,\end{array}$$

(31)

$$\begin{array}{c}\hfill 0<1-\frac{\rho {\varphi }_i\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}\le m_6<1,\end{array}$$

(32)

$$\begin{array}{c}\hfill 0<\frac{\rho {\varphi }_i\left(k\right){\varphi }_{i-1}\left(k\right){\widehat{\varphi }}_i\left(k\right)}{\lambda +{\widehat{\varphi }}_i^2\left(k\right)}\le a{m}_5<a,\end{array}$$

(33)

where $m_4$, $m_5$, and $m_6$ are constants.

In addition, generally because the energy of the equipment cannot be unlimited, the input increment $\Delta u_{i-1}\left(k\right)$ is similarly bounded with $|\Delta u_{i-1}\left(k\right)|\le \sigma$. From (31)–(33) and (13), we can reduce (30) to the following inequality

$$\begin{array}{c}\hfill E\left\{\right|e_i(k+1)\left|\right\}\le m_6\left|E\right\{|e_{i-1}(k+1)\left|\right\}+|a(1-\overline{\alpha })\Delta u_{i-1}\left(k\right)|+(2-\overline{\alpha })\sqrt{p}\overline{v},\end{array}$$

(34)

$$\begin{array}{c}\hfill E\left\{\right|e_i(k+1)\left|\right\}\le m_6\left|E\right\{|e_{i-1}(k+1)\left|\right\}+m_7,\end{array}$$

(35)

where $m_7=a(1-\overline{\alpha })\sigma +(2-\overline{\alpha })\sqrt{p}\overline{v}$. Further, by using the principle of the compressed mapping, we can get

$$\begin{array}{cc}\hfill E\left\{\right|e_i(k+1)\left|\right\}& \le m_6^2\left|E\right\{|e_{i-2}(k+1)\left|\right\}+m_6{m}_7+m_7\hfill \\ \hfill & \le \cdots \hfill \\ \hfill & \le m_6^{i-1}\left|E\right\{|e_1(k+1)\left|\right\}+m_6^{i-2}{m}_7+\cdots +m_7\hfill \\ \hfill & =m_6^{i-1}\left|E\right\{|e_1(k+1)\left|\right\}+\frac{m_7(1-m_6^{i-1})}{1-m_6}\hfill \end{array}$$

(36)

so, $e_i(k+1)$ converges to the following set

$$\begin{array}{c}\hfill \left\{e_i(k+1)\left|E\left\{\right|e_i(k+1)\left|\right\}\le \frac{m_7}{1-m_6}\right.\right\}.\end{array}$$

(37)

Now, the proof of the boundedness of the tracking error is completed. □

5. Simulation Results

The nonlinear repetitive operation system described below is used

$$\begin{array}{c}\hfill y_i(k+1)=\frac{y_i\left(k\right)}{1+y_i^2\left(k\right)}+{(u_i\left(k\right)+0.01round\left(\frac{k}{500}\right)sin\left(y_i\left(k\right)\right))}^3+F_{s,i}\left(k\right).\end{array}$$

The desired output is given as

$$y_d\left(k\right)=\left\{\begin{array}{cc}\hfill & 0.5{(-1)}^{round\left(\frac{t}{100}\right)},0\le t\le 500,\hfill \\ \hfill & 0.5sin\left(\frac{t\pi }{100}\right)+0.3cos\left(\frac{t\pi }{50}\right),501\le t\le 1000.\hfill \end{array}\right.$$

The control input and control output have their initial values with $u_1\left(1\right)=u_1\left(2\right)=0.5$ and $y_1\left(1\right)=y_1\left(2\right)=y_1\left(3\right)=0.5$, respectively, and the estimated values of the PPD parameters are set to ${\widehat{\varphi }}_1\left(1\right)={\widehat{\varphi }}_1\left(2\right)=0.5$. The controller parameters are selected as $\eta =1,\mu =1,\rho =0.6,\lambda =20$ and $\epsilon ={10}^{-5}$. The successful transmission rate is selected to be $\overline{\alpha }=0.5$.

Consider the actual sensor fault function as

$$\begin{array}{c}\hfill F_{s,i}\left(k\right)=2u_r^2\left(k\right)+\frac{F_{s,i}(k-1)}{1+F_{s,i}^2(k-1)},\end{array}$$

where $u_r\left(k\right)=sin\left(\frac{k}{1000}\right)$. In addition, the initial value of the hazard function is set to $F_{s,i}\left(1\right)=0$.

In this paper, the learning rate of the RBFNN is set to $\chi =0.05$, the RBFNN input defined as $z\left(k\right)={[u_r\left(k\right),F_{s,i}(k-1),y_i\left(k\right)]}^T$. The initial value of the Gaussian function is set to ${\varrho }_0=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$ and ${\tau }_0={\left[\begin{array}{ccccc}5& 5& 5& 5& 5\end{array}\right]}^T.$

The input layer weight vector is set to ${\overline{v}}_i={\left[\begin{array}{c}0.62940.8116-0.74600.82680.4574\end{array}\right]}^T.$

The Bernoulli distribution is shown in Figure 2 for a random sequence of DoS attacks. Figure 3 shows the simulation diagram approaching the sensor fault function by RBFNN. From Figure 3, we can clearly see the superiority of the approximation performance of the RBFNN. In Figure 4a–c, it shows the performance of for different iterations.

Figure 5 shows the comparison of the maximum tracking error with the number of times of iterations by using the SFTILC algorithm and the traditional MFAILC [28] under DoS attacks. In Figure 5, the SFTILC can make the tracking error converge to a smaller bound. Comparative results of tracking the performance of the system are depicted in Figure 6. From Figure 6, we can see that the tracking of the desired trajectory with the SFTILC algorithm is obviously better than that of the traditional MFAILC.

According to simulation results, the SFTILC algorithm can handle the output tracking tasks well when taking sensor failure and DoS attacks into account.

6. Conclusions

The data-based SFTILC problem of nonlinear networked control systems has been studied in this paper with sensor failures and DoS attacks. Firstly, the radial basis function neural network has been used to approximate the sensor failure function and a DoS attack compensation mechanism has been proposed in iterative domain to lessen the impact of the DoS attack. Then, using the dynamic linearization technology, the nonlinear system considering fault and network attack has been transformed into a linear data model. Further, a new data-based SFTILC algorithm has been designed to ensure the tracking performance of the system. Finally, a digital simulation has been used to demonstrate the effectiveness of the proposed SFTILC algorithm. In the future, we will consider the partial form dynamic linearization-based model-free adaptive algorithm that uses more input information to improve the control performance of the system.