Robust control strategy for networked vehicle system with fake data injection attack

Abstract

This article investigates the safety control problem of intelligent networked vehicle and utilizes the sliding-mode control to solve the system with time delay, system disturbance, and parameter uncertainty. First, the modeling of automobile system is introduced. What’s more, time delay, parameter uncertainty, disturbance, and attack are added to the original model. Then, designed an integral sliding-mode surface and given the sufficient conditions for system stability using linear matrix inequalities. Furthermore, the controller is designed according to the slide approach rate, and it is proved that the designed controller can make the system state reach the sliding surface in finite time. Finally, the simulation is used to verify the effectiveness of the arithmetic.

Introduction

The network control system refers to a real-time network closed-loop control system in which the equipment collects data through on-site data collectors such as sensors, and then transmits data through the network system to share information resources among users, and finally realizes collaborative control through the controller.^1–3 With the development of computer networks, networked control systems have aroused the interest of many researchers and have been widely used in many fields, such as environmental science, ⁴ power industry, ⁵ computer science, ⁶ aerospace ⁷ and weapons industry ⁸ . Intelligent connected vehicle is a typical network control system. In recent years, people’s traffic demand has been increasing. Intelligent connected vehicle has become a hot research topic because it can reduce energy consumption, relieve traffic pressure, and reduce traffic accidents (see Wang et al. ⁹ and Yan et al. ¹⁰ and the references therein). The main target of intelligent networked vehicle control is to make the vehicles drive safely at a set desired distance and speed. ¹¹ In the actual situation, the system will inevitably encounter interference, even malicious attacks by hackers, and so on. The actual system also has time delays due to network transmission, and the uncertainty of vehicle parameters caused by use in different environments. These factors will reduce system performance and even lead to serious accidents, causing damage to human safety and property. Therefore, how to design a control strategy so that the system can still maintain normal working when suffering from the above situation to avoid accidents has become a problem which should be solved instantly.

So as to suppressing system disturbances, many studies have done many investigations in this area.^12–14 Li et al. ¹⁵ studied the external disturbances and network-induced disturbances of vehicles’ follow-up control and proposed a discrete-time sliding-mode control arithmetic which can be adaptive. For the overhead crane systems, Wu et al. ¹⁶ proposed a sliding-mode controller and an observer of nonlinear perturbance for the adjustment and perturbation reckoning. In Li et al., ¹⁷ for the purpose of solving the position interference robust control problem of permanent magnet synchronous motor, a robust control method is proposed for solving the tracking control problem of servo system. Although the above references have achieved good results in suppressing disturbances, they ignore the possible disturbances and even hacking attacks between the controller and the actuator, which makes the designed controller unable to deal with attacks.

Hacker attack is a more important issue. Attackers often maliciously undermine system stability to achieve their illegal purposes. Such interference factors often cause greater damage to the intelligent connected vehicle system. Common network attacks are denial of service attacks, ¹⁸ deception attacks, ¹⁹ and replay attacks. ²⁰ Deception attack is also called false data injection attack. Attackers use the state information or measurement data of the system to attack the communication information between the controller and the actuator. Such attacks can bypass the attack detection mechanism to attack the physical system, so that it is difficult to be detected. In order to solve the damage of hacker attacks to the system, Zhao et al. ²¹ studied the design of false data injection attacks and how to detect it in cyber-physical systems, and a design scheme is proposed using subspace technology for fake data injection attacks in the systems. For solving the distributed attack and recovery in vehicle platoon control systems, Mousavinejad et al. ²² developed a distributed attack discovery algorithm and introduced two restoration mechanisms based on computational prediction ellipsoid sets and estimation ellipsoid sets. In Liu et al., ²³ dynamic event-triggered control of attacked network systems is studied, and a dynamic event-triggered mechanism for hybrid systems is proposed. However, parameter uncertainty and time delay are also problems in practical systems. In the above references, the recovery of systems under attack is discussed in detail, but the effects of system time delay and parameter uncertainty are ignored.

Aiming at the time lag resulted by network transmission and parameter uncertainty engendered by system aging, Xu et al. ²⁴ considered isomeric vehicles on complex road conditions, aerodynamic resistance and network communication lag and proposed an integrated platoon control framework. Han et al. ²⁵ designed a robust forecast controller with tamper-resistance function for the problems of time delay, disturbance, and parameter uncertainty faced by supercavitating vehicles. For linear discrete time-delay systems, Ghrab et al. ²⁶ considered the effects of uncertainties on state and delay state vectors and proposed a controller based on sliding-mode algorithm. The conditions of asymptotic stability are derived, and a new Discrete-Time Sliding Mode Control (DT-SMC) law is obtained. However, the system including parameter uncertainty, time delay, disturbance, and attack has not been entirely studied. It gives inspiration for the research of this article.

In this article, we mainly study the safety control problem of intelligent connected vehicles driving on horizontal roads under parameter uncertainty, time delay, disturbance, and false data injection attacks. The primary work of this article is as follows:

1. An intelligent networked vehicle system model is established, and added time delay, parameter uncertainty, disturbance, and false data injection attack to the original model.

2. For the purpose of making the system state follow the predetermined trajectory, an integral sliding-mode surface is proposed. The stability condition of sliding-mode dynamic is given by combining the linear matrix inequality theory with the Lyapunov theory.

3. In order to make the system state get to the sliding-mode surface and remain stable under the conditions of time delay, parameter uncertainty, attack, and disturbance, a sliding-mode controller is proposed using approach rate.

The structure of this article is as follows. Section “Problem formulation” describes the intelligent connected vehicle system. In section “Design and analysis of sliding-mode function,” a sliding surface is designed, then the stability of the sliding-mode dynamics is proved, and the stability conditions of the system are given. In section “Design of controller and stability analysis,” a sliding-mode controller is designed and the accessibility of the sliding-mode surface is proved. In section “Illustrative example,” the feasibility of the designed controller is verified by simulation.

Notations

“*” denotes the symmetric term in symmetric entries, E denotes the unit matrix of appropriate dimension, $\mathrm{Ep}[·]$ denotes the expectation. $L_2[0,\infty )$ denotes the energy bounded.

Problem formulation

Consider a vehicle running on a horizontal road. We can obtain the desired acceleration, velocity, and position through the network. The vehicle is equipped with sensors to detect the positional relationship with the previous vehicle. Let $\gamma$ ,v, and a denote the position, velocity, and acceleration of the vehicle, respectively. The dynamic model of the vehicle is modeled as ²⁷

$$\begin{array}{l}\stackrel{·}{\gamma }=v\\ \stackrel{·}{v}=a\\ \stackrel{·}{a}=n(v,a)+l(v)q\end{array}$$

(1)

where q is the engine input for the vehicle, and $n(v,a)$ and $l(v)$ are given by the formula

$$\begin{array}{c}l(v)=\frac{1}{\delta m}\hfill \\ n(v,a)=-\frac{1}{\delta }(\stackrel{·}{v}+\frac{\epsilon S\kappa }{2m}{v}^2+\frac{c}{m})-\frac{\epsilon S\kappa \mathrm{va}}{m}\hfill \end{array}$$

(2)

where q is the engine input of the vehicle; $\delta$ , m, c, $\kappa$ , S, and $\epsilon$ are the engine time constant, mass, mechanical resistance, drag coefficient, cross-sectional area, and the specific mass of the air, respectively. $\epsilon S\kappa /2m$ representing the air resistance.

To facilitate the analysis, the following controller is set by feedback linearization

$$q=\vartheta m+\frac{\epsilon S\kappa }{2}{v}^2+c+\delta \epsilon S\kappa \mathrm{va}$$

(3)

Substituting equation (3) into the third formula in equation (1) gives

$$\stackrel{·}{a}=-\frac{1}{\delta }a+\frac{1}{\delta }\vartheta$$

(4)

Define the state vector as $s(t)=[\gamma ,v,a{]}^T$ , and the output vector as $\lambda (t)=[\gamma ,v,a{]}^T$ . According to equations (1) and (4), the state-space modeling of a single intelligent connected vehicle is as follows

$$\{\begin{array}{c}\stackrel{·}{s}(t)=\mathrm{As}(t)+B\vartheta (t)\hfill \\ \multicolumn{1}{c}{\lambda (t)=\mathrm{Cs}(t)}\end{array}$$

(5)

where

$$\begin{array}{c}A=\left[\begin{array}{ccc}0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1/\delta \hfill \end{array}\right],B={\left[\begin{array}{ccc}0\hfill & 0\hfill & \frac{1}{\delta }\hfill \end{array}\right]}^T\hfill \\ C=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

Define the expected distance of the vehicle as ${\gamma }_d$ , the expected velocity as $v_d$ , the expected acceleration as $a_d$ , then the expected state vector is: $s_d(t)=[{\gamma }_d,v_d,a_d{]}^T$ , the expected output vector is ${\lambda }_d(t)={[{\gamma }_d,v_d,a_d]}^T$ .

Using coordinate transformation to move the expected state of the vehicle to the zero point, we get

$$\{\begin{array}{c}\stackrel{·}{\eta }(t)=A\eta (t)+B\vartheta (t)\hfill \\ \multicolumn{1}{c}{p(t)=C\eta (t)}\end{array}$$

(6)

where $\eta (t)=s(t)-s_d(t)=[\gamma -{\gamma }_d,v-v_d,a-a_d{]}^T=[\mathrm{\Delta }\gamma ,\mathrm{\Delta }v,\mathrm{\Delta }a{]}^T$ and $p(t)=\lambda (t)-{\lambda }_d(t)=[\gamma -{\gamma }_d,v-v_d,a-a_d{]}^T=[\mathrm{\Delta }\gamma ,\mathrm{\Delta }v,\mathrm{\Delta }a{]}^T$ .

Generally speaking, considering the application of the vehicle in the actual system, first, because of the different scenarios when the vehicle system is used, resulting in different temperatures, combined with the service life of the system components aging, system parameters will change, parametric uncertainty should be added to the model. Second, the communication between vehicles has time difference, which is system time delay. Finally, adding system disturbance makes the model closer to the actual system. After adding the above factors, the system is as follows

$$\{\begin{array}{c}\stackrel{·}{\eta }(t)=[A+\mathrm{\Delta }A(t)]\eta (t)+A_d\eta (t-\iota (t))+B\vartheta (t)+\mathrm{Hh}(t)\hfill \\ \multicolumn{1}{c}{p(t)=C\eta (t)}\end{array}$$

(7)

where $\mathrm{\Delta }A(t)$ is the uncertain parameter term, $A_d$ is a known constant real matrix with appropriate dimension, $\iota (t)$ is the time lag, $h(t)$ is the disturbance term, and H is a matrix of appropriate dimension. For convenience, the following assumptions are made.

Assumption 1. $\mathrm{\Delta }A$ satisfies the relationship as

$$\mathrm{\Delta }A(t)=\Lambda g(t)\mathrm{\Xi }$$

(8)

where $\Lambda$ and $\mathrm{\Xi }$ are the known matrices with appropriate dimensions, and the nonlinear term $g(t)$ satisfies

$$g^T(t)g(t)\le E$$

(9)

Assumption 2. $\iota (t)$ satisfies

$$\stackrel{·}{\iota }(t)\le \chi <1,0\le \iota (t)<\infty$$

(10)

where $\chi$ is a constant less than one.

Fake data injection attack is a common attack method. Shown in Figure 1, the attacker uses the system state information or measurement data to attack the communication channel between the controller and the actuator. Such attacks can reduce the system performance or even destroy the normal operation of the entire system. And it is difficult to be detected directly. The fake data injection attack model is modeled as follows ²⁸

$${\vartheta }_f(t)=\theta (t)\varpi (\eta (t),t)$$

(11)

where ${\vartheta }_f(t)$ is the model of fake data injection attack, $\theta (t)$ represents the injection mode of the attack, and $\varpi (\eta (t),t)$ represents the system information exploited by the attacker. Considering the actual situation, the following assumptions are made for $\theta (t)$ and $\varpi (\eta (t),t)$ .

Figure 1.

Control system under false data injection attack.

Assumption 3. $\theta (t)$ is a unknown weighting matrix that satisfies: $\Vert \theta (t)\Vert \le \mu$ .

Assumption 4. $\varpi (\eta (t),t)$ is a nonlinear function and satisfies: $\Vert \varpi (\eta (t),t)\Vert \le \omega (\eta (t),t)$ .

where $\mu$ is a known constant, and $\omega (\eta (t),t)$ is a non-negative function.

Under a fake data injection attack, the control signal accepted by the actuator will be expressed as

$$\tilde{\vartheta }(t)=\vartheta (t)+{\vartheta }_f(t)$$

(12)

Substitute the actual control signal equation (12) into the system state equation to get

$$\{\begin{array}{c}\stackrel{·}{\eta }(t)=[A+\mathrm{\Delta }A(t)]\eta (t)+A_d\eta (t-\iota (t))+B[\vartheta (t)+{\vartheta }_f(t)]+\mathrm{Hh}(t)\hfill \\ \multicolumn{1}{c}{p(t)=C\eta (t)}\end{array}$$

(13)

Design and analysis of sliding-mode function

In this section, first, designed an integral sliding-mode surface in order to make the system state follow the predetermined trajectory. Then, analyzed the stability of the sliding-mode dynamic, and the conditions of system stability are given.

Design of sliding-mode surface

In this section, an integral sliding-mode surface is designed as following

$$\mathrm{{\rm Y}}(t)=\psi \eta (t)-\underset{0}{\overset{t}{\int }}\psi (A+BL)\eta (\tau )d\tau$$

(14)

where L represents the parameter matrix to be designed, which will be solved in the following theorem. Matrix $\psi$ will be designed to make $(\psi B{)}^{-1}$ exist and satisfies $(\psi B{)}^{-1}>0$ . Taking the differential to equation (11), it can be deduced that

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\mathrm{\Upsilon }}(t)=\psi \stackrel{·}{\eta }(t)-\psi (A+\mathrm{BL})\eta (t)}\\ =\psi \{[A+\mathrm{\Delta }A(t)]\eta (t)+A_d\eta (t-\iota (t))+B[\vartheta (t)+{\vartheta }_f(t)]+\mathrm{Hh}(t)\}-\psi (A+\mathrm{BL})\eta (t)\hfill \\ =[\psi \mathrm{\Delta }A(t)-\psi \mathrm{BL}]\eta (t)+\psi A_d\eta (t-\iota (t))+\psi B\vartheta (t)+\psi B{\vartheta }_f(t)+\psi \mathrm{Hh}(t)\hfill \end{array}$$

(15)

Furthermore, let $\stackrel{·}{\mathrm{\Upsilon }}(t)=0$ , the equivalent control input can be given as follows

$$\begin{array}{c}{\vartheta }_{\mathrm{eq}}(t)=(L-{(\psi B)}^{-1}\psi \mathrm{\Delta }A(t))\eta (t)-(\psi B{)}^{-1}\hfill \\ \times \psi A_d\eta (t-\iota (t))-{\vartheta }_f(t)-(\psi B{)}^{-1}\psi \mathrm{Hh}(t)\hfill \end{array}$$

(16)

Substitute equation (16) into the system state equation (13), the following sliding-mode dynamic can be obtained by

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\eta }(t)=\{A+\mathrm{BL}+[E-B{(\psi B)}^{-1}\psi ]\mathrm{\Delta }A(t)\}\eta (t)}\\ \multicolumn{1}{c}{+[E-B{(\psi B)}^{-1}\psi ]A_d\eta (t-\iota (t)+[E-B{(\psi B)}^{-1}\psi ]\mathrm{Hh}(t)}\\ =\tilde{A}\eta (t)+{\tilde{A}}_d\eta (t-\iota (t))+\tilde{H}h(t)\hfill \end{array}$$

(17)

where $\tilde{A}=A+\mathrm{BL}+(E-B(\psi B{)}^{-1}\psi )\mathrm{\Delta }A(t)$ , $\tilde{H}=(E-B(\psi B{)}^{-1}\psi )H$ , and ${\tilde{A}}_d=[E-B(\psi B{)}^{-1}\psi ]A_d$ .

Stability analysis

In this section, we analyze the stability of the sliding-mode dynamic equation (17) and derive the sufficient conditions for the stochastic stability. In order to solve the stability problem, the definition of stability is given as follows.

Definition. There is $H\infty$ inhibitory factor $\varsigma >0$ , which makes the sliding-mode dynamic equation (17) stochastically stable. If the conditions follow as:

• 1. When disturb $h(t)=0$ , the sliding-mode dynamic equation (17) is stable for any initial state $x(0)=x_0$ if there is a constant $\mathrm{\Theta }(x_0)>0$ which makes the following equation holds

$$\mathrm{Ep}[\underset{0}{\overset{\infty }{\int }}{\Vert x(t)\Vert }^2\mathrm{dt}|x_0]\le \mathrm{\Theta }(x_0)$$

(18)

• 2. For a non-zero perturbation $h(t)\in L_2[0,\infty )$ , when the initial condition is 0, which is $x_0\equiv 0$ , the following formula holds

$$\underset{0}{\overset{\infty }{\int }}{z}^T(\tau )z(\tau )d\tau \le {\varsigma }^2\underset{0}{\overset{\infty }{\int }}{h}^T(\tau )h(\tau )d\tau$$

(19)

where $z(\tau )$ is the system output vector.

Next, the lemma is given to facilitate the solution of parameter uncertainty $\mathrm{\Delta }A$ .

Lemma. If $b^T(t)b(t)\le E$ , there exists scalar $\sigma >0$ and matrices N and M such that ²⁹

$$\mathrm{Nb}(t)M+M^T{b}^T(t)N^T\le \sigma N{N}^T+\frac{1}{\sigma }{M}^{T}M$$

(20)

The stability of sliding-mode dynamics is proved as follows.

Theorem 1. For the system (17), when disturbance $h(t)=0$ , there are positive definite matrices $R,J>0$ and matrix L satisfying the following conditions

$${\mathrm{\Omega }}_2=\left[\begin{array}{cc}{\tilde{A}}^{T}J+J\tilde{A}+R\hfill & J{\tilde{A}}_d\hfill \\ \multicolumn{1}{c}{*}& -(1-\chi )R\hfill \end{array}\right]<0$$

(21)

Then, the sliding-mode dynamic equation (17) is stable.

Proof. Select the Lyapunov function as

$$N(t)={\eta }^T(t)J\eta (t)+\underset{t-\iota (t)}{\overset{t}{\int }}{\eta }^T(\tau )R\eta (\tau )d\tau$$

(22)

Derivation along the state trajectory of sliding-mode dynamic equation (17)

$$\begin{array}{c}\stackrel{·}{N}(\eta (t),\eta (t-\iota (t)))={\stackrel{·}{\eta }}^T(t)J\eta (t)+{\eta }^T(t)J\stackrel{·}{\eta }(t)\hfill \\ \multicolumn{1}{c}{+{\eta }^T(t)R\eta (t)-(1-\stackrel{·}{\iota }(t)){\eta }^T(t-\iota (t))R\eta (t-\iota (t))}\\ ={[\tilde{A}\eta (t)+{\tilde{A}}_d\eta (t-\iota (t))]}^{T}J\eta (t)\hfill \\ \multicolumn{1}{c}{+{\eta }^T(t)J[\tilde{A}\eta (t)+{\tilde{A}}_d\eta (t-\iota (t))]}\\ +{\eta }^T(t)R\eta (t)-(1-\stackrel{·}{\iota }(t)){\eta }^T(t-\iota (t))R\eta (t-\iota (t))\hfill \\ ={\eta }^T(t){\tilde{A}}^{T}J\eta (t)+{\eta }^T(t)J\tilde{A}\eta (t)+{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^{T}J\eta (t)\hfill \\ +{\eta }^T(t)J{\tilde{A}}_d\eta (t-\iota (t))+{\eta }^T(t)R\eta (t)\hfill \\ \multicolumn{1}{c}{-(1-\stackrel{·}{\iota }(t)){\eta }^T(t-\iota (t))R\eta (t-\iota (t))}\\ ={\eta }^T(t)({\tilde{A}}^{T}J+J\tilde{A}+R)\eta (t)+{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^{T}J\eta (t)\hfill \\ +{\eta }^T(t)J{\tilde{A}}_d\eta (t-\iota (t))-(1-\stackrel{·}{\iota }(t)){\eta }^T(t-\iota (t))R\eta (t-\iota (t))\hfill \end{array}$$

(23)

Using Assumption 2, that is, equation (10), we get

$$\begin{array}{c}\stackrel{·}{N}(\eta (t),\eta (t-\iota (t))))\le {\eta }^T(t)({\tilde{A}}^{T}J+J\tilde{A}+R)\eta (t)\hfill \\ \multicolumn{1}{c}{+{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^{T}J\eta (t)+{\eta }^T(t)J{\tilde{A}}_d\eta (t-\iota (t))}\\ \multicolumn{1}{c}{-(1-\chi ){\eta }^T(t-\iota (t))R\eta (t-\iota (t))}\end{array}$$

(24)

Let $\beta (t)=[\eta (t),\eta (t-\iota (t)){]}^T$ , we can acquire the equation

$$\stackrel{·}{N}(\beta (t))\le {\beta }^T(t)\left[\begin{array}{cc}{\tilde{A}}^{T}J+J\tilde{A}+R\hfill & J{\tilde{A}}_d\hfill \\ \multicolumn{1}{c}{*}& -(1-\chi )R\hfill \end{array}\right]\beta (t)$$

(25)

According to inequality equation (21), it can be further obtained

$${\mathrm{\Omega }}_2\le {\lambda }_m({\mathrm{\Omega }}_2)·E<0$$

(26)

where ${\lambda }_m({\mathrm{\Omega }}_2)$ denotes the maximum eigenvalue of ${\mathrm{\Omega }}_2$ . According to equation (23), it can be derived

$$\stackrel{·}{N}(\beta (t))\le {\beta }^T(t){\lambda }_m({\mathrm{\Omega }}_2)\beta (t)$$

(27)

According to Dynkin’s formula, it is obtained that

$$\begin{array}{c}\mathrm{Ep}\left[N(\beta (t))\right]-N({\beta }_0)=\mathrm{Ep}[\underset{0}{\overset{t}{\int }}\stackrel{·}{N}(\beta (t))d\tau |{\beta }_0]\hfill \\ \le {\lambda }_m({\mathrm{\Omega }}_2)\mathrm{Ep}[\underset{0}{\overset{t}{\int }}{\Vert \beta (\tau )\Vert }^{2}d\tau |{\beta }_0]<0\hfill \end{array}$$

(28)

Using $N(\beta (t))>0$ , we can deduce that

$$\mathrm{Ep}\left[N(\beta (t))\right]-N({\beta }_0)\le {\lambda }_m({\mathrm{\Omega }}_2)\mathrm{Ep}[\underset{0}{\overset{t}{\int }}{\Vert \beta (\tau )\Vert }^{2}d\tau |{\beta }_0]$$

(29)

Let $\mathrm{\Theta }({\beta }_0)=-N({\beta }_0)/{\lambda }_m({\mathrm{\Omega }}_2)$ , when $t\to \infty$ , we can obtain that

$$\mathrm{Ep}[\underset{0}{\overset{\infty }{\int }}{\Vert \beta (\tau )\Vert }^{2}d\tau |{\beta }_0]\le \mathrm{\Theta }({\beta }_0)$$

(30)

According to the definition, when the disturbance $h(t)=0$ , the sliding-mode dynamic is stable. Next, the stability of sliding-mode dynamic will be discussed in the case of $h(t)\ne 0$ .

Theorem 2. For automatic driving system, the sliding-mode surface is determined by equation (14), if there is a positive definite matrix J, scalar $\varsigma >0$ , and $H\infty$ inhibition factor $\sigma >0$ , the following linear matrix inequality is satisfied

$$\left[\begin{array}{cccccc}{\mathrm{\Omega }}_1\hfill & \tilde{H}\hfill & J^{-1}{C}^T\hfill & (E-B{(\psi B)}^{-1}\psi )\Lambda \hfill & J^{-1}{\mathrm{\Xi }}^T\hfill & J^{-1}\hfill \\ \multicolumn{1}{c}{*}& -{\varsigma }^{2}E\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -E\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -\sigma E\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\frac{1}{\sigma }E\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -R^{-1}\hfill \end{array}\right]<0$$

(31)

where ${\mathrm{\Omega }}_1=J^{-1}{A}^T+A{J}^{-1}+Z^T{B}^T+ZB+{\tilde{A}}_d{[(1-\chi )R]}^{-1}{\tilde{A}}_d^T$ . Then, the sliding-mode dynamic equation (17) is asymptotically stable and the sliding-mode parameter matrix L in equation (11) can be solved by the following equation

$$L=\mathrm{ZJ}$$

(32)

Proof. Select the function as following

$$\mathrm{\Gamma }=\underset{0}{\overset{\infty }{\int }}[p^T(t)p(t)-{\varsigma }^2{h}^T(t)h(t)]\mathrm{dt}$$

(33)

According to Theorem 1, it can be derived that

$$\begin{array}{c}\mathrm{\Gamma }\le \underset{0}{\overset{\infty }{\int }}[p^T(t)p(t)-{\varsigma }^2{h}^T(t)h(t)]\mathrm{dt}+\underset{0}{\overset{\infty }{\int }}\stackrel{·}{N}(\eta (t))\mathrm{dt}\hfill \\ =\underset{0}{\overset{\infty }{\int }}[p^T(t)p(t)-{\varsigma }^2{h}^T(t)h(t)+\stackrel{·}{N}(\eta (t))]\mathrm{dt}\hfill \\ =\underset{0}{\overset{\infty }{\int }}\{{\eta }^T(t)C^{T}C\eta (t)-{\varsigma }^2{h}^T(t)h(t)+[{\eta }^T(t){\tilde{A}}^T+{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^T+h^T(t){\tilde{H}}^T]J\eta (t)\hfill \\ +{\eta }^T(t)J[\tilde{A}\eta (t)+{\tilde{A}}_d\eta (t-\iota (t))+\tilde{H}h(t)]+{\eta }^T(t)R\eta (t)-(1-\chi ){\eta }^T(t-\iota (t))R\eta (t-\iota (t))\}\mathrm{dt}\hfill \\ =\underset{0}{\overset{\infty }{\int }}\{{\eta }^T(t)C^{T}C\eta (t)+{\eta }^T(t)R\eta (t)+{\eta }^T(t){\tilde{A}}^{T}J\eta (t)+{\eta }^T(t)J\tilde{A}\eta (t)+{\eta }^T(t)J\tilde{H}h(t)+h^T(t){\tilde{H}}^{T}J\eta (t)\hfill \\ +{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^{T}J\eta (t)+{\eta }^T(t)J{\tilde{A}}_d\eta (t-\iota (t))-(1-\chi ){\eta }^T(t-\iota (t))R\eta (t-\iota (t))-{\varsigma }^2{h}^T(t)h(t)\}\mathrm{dt}\hfill \\ =\underset{0}{\overset{\infty }{\int }}\{{\eta }^T(t)({\tilde{A}}^{T}J+J\tilde{A}+C^{T}C+R)\eta (t)+h^T(t){\tilde{H}}^{T}J\eta (t)+{\eta }^T(t)J\tilde{H}h(t)\hfill \\ +{\eta }^T(t-\iota (t)){{\tilde{A}}_d}^{T}J\eta (t)+{\eta }^T(t)J{\tilde{A}}_d\eta (t-\iota (t))-(1-\chi ){\eta }^T(t-\iota (t))R\eta (t-\iota (t))-{\varsigma }^2{h}^T(t)h(t)\}\mathrm{dt}\hfill \end{array}$$

(34)

Let $\xi =[\eta (t),\eta (t-\iota (t)),h(t){]}^T$ , we get

$$\mathrm{\Gamma }\le \underset{0}{\overset{\infty }{\int }}{\xi }^T\left[\begin{array}{ccc}{\mathrm{\Omega }}_3& J{\tilde{A}}_d& J\tilde{H}\\ *& -(1-\chi )R& 0\\ *& *& -{\varsigma }^{2}E\end{array}\right]\xi dt$$

(35)

where ${\mathrm{\Omega }}_3={\tilde{A}}^{T}J+J\tilde{A}+C^{T}C+R$ .

Let $\tilde{J}=J(E-B(\psi B{)}^{-1}\psi )$ . From the lemma, it can be deduced that

$$\begin{array}{c}{\tilde{A}}^{T}J+J\tilde{A}=J(A+\mathrm{BL})+(A+\mathrm{BL}{)}^{T}J+\mathrm{\Delta }{A}^T(t)(E\hfill \\ {-B{(\psi B)}^{-1}\psi )}^{T}J+J(E-B{(\psi B)}^{-1}\psi )\mathrm{\Delta }A(t)\hfill \\ =J(A+\mathrm{BL})+(A+\mathrm{BL}{)}^{T}J+\mathrm{\Delta }{A}^T(t){\tilde{J}}^T+\tilde{J}\mathrm{\Delta }A(t)\hfill \\ =J(A+\mathrm{BL})+(A+\mathrm{BL}{)}^{T}J+{\mathrm{\Xi }}^T{g}^T(t){\Lambda }^T{\tilde{J}}^T+\tilde{J}\Lambda g(t)\mathrm{\Xi }\hfill \\ \le J(A+\mathrm{BL})+(A+\mathrm{BL}{)}^{T}J+\sigma \tilde{J}\Lambda (\tilde{J}\Lambda {)}^T+\frac{1}{\sigma }{\mathrm{\Xi }}^T\mathrm{\Xi }\hfill \end{array}$$

(36)

Combining equations (35) and (36) and using Shur’s complement theorem, we get

$$\mathrm{\Gamma }\le \underset{0}{\overset{\infty }{\int }}{\xi }^T{\mathrm{\Omega }}_4\xi \mathrm{dt}$$

(37)

where ${\mathrm{\Omega }}_4$ is the following matrix

$${\mathrm{\Omega }}_4=\left[\begin{array}{ccccccc}{\mathrm{\Omega }}_5\hfill & J{\tilde{A}}_d\hfill & J\tilde{H}\hfill & C^T\hfill & {\mathrm{\Omega }}_6\hfill & {\mathrm{\Xi }}^T\hfill & E\hfill \\ \multicolumn{1}{c}{*}& -(1-\chi )R\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\varsigma }^{2}E\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -E\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\sigma E\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -\frac{1}{\sigma }E\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & -R^{-1}\hfill \end{array}\right]$$

(38)

where ${\mathrm{\Omega }}_5=J(A+BL)+{(A+BL)}^{T}J$ and ${\mathrm{\Omega }}_6=J(E-B(\psi B{)}^{-1}\psi )\Lambda$ .

Further simplification yields

$${\mathrm{\Omega }}_4=\left[\begin{array}{cccccc}{\mathrm{\Omega }}_7\hfill & J\tilde{H}\hfill & C^T\hfill & {\mathrm{\Omega }}_8\hfill & {\mathrm{\Xi }}^T\hfill & E\hfill \\ \multicolumn{1}{c}{*}& -{\varsigma }^{2}E\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -E\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -\sigma E\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\frac{1}{\sigma }E\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -R^{-1}\hfill \end{array}\right]$$

(39)

where ${\mathrm{\Omega }}_7=J(A+BL)+{(A+BL)}^{T}J+J{\tilde{A}}_d{[(1-\chi )R]}^{-1}{\tilde{A}}_d^{T}J$ and ${\mathrm{\Omega }}_8=J(E-B(\psi B{)}^{-1}\psi )\Lambda$ .

Then pre- and post-multiplying ${\mathrm{\Omega }}_4$ by $\mathrm{diag}\{J^{-1},E,E,E,E,E\}$ , and substituting $L=Z{J}^{-1}$ into the matrix, it is deduced that

$$\left[\begin{array}{cccccc}{\mathrm{\Omega }}_1\hfill & \tilde{H}\hfill & J^{-1}{C}^T\hfill & (E-B{(\psi B)}^{-1}\psi )\Lambda \hfill & J^{-1}{\mathrm{\Xi }}^T\hfill & J^{-1}\hfill \\ \multicolumn{1}{c}{*}& -{\varsigma }^{2}E\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -E\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -\sigma E\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -\frac{1}{\sigma }E\hfill & ⋮\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -R^{-1}\hfill \end{array}\right]$$

(40)

From equation (31), it can be known that equation (40) is less than 0. According to Theorem 2, it can be deduced that the system is stable.

This completes the proof.

Design of controller and stability analysis

In this section, a sliding-mode controller is designed and then it is proved that the system state can close to the sliding-mode surface in a certain time by the designed controller. The controller is designed as follows.

In the actual vehicle system, the state value is limited. For the purpose of facilitating the processing of the time lag term, the assumptions are made as following.

Assumption 5. Suppose system state is bounded and satisfies

$$\Vert \eta (t)\Vert <\alpha$$

(41)

Remark 1. For the automotive system, the system state $\eta (t)$ will always remain within a certain range due to the power of the car engine, the strength of the auto spare parts, and so on, that is, the state of the system is bounded, so Assumption 5 is reasonable.

Select the constant velocity approach rate $\stackrel{·}{\mathrm{\Upsilon }}(t)=-\zeta \mathrm{sign}(\mathrm{\Upsilon }(t))$ , and substitute it into equation (15) to get the controller

$$\begin{array}{c}\vartheta (t)=L\eta (t)-\mu \omega (\eta (t),t)\mathrm{sign}(\mathrm{\Upsilon }(t))-\zeta (\psi B{)}^{-1}\mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \\ -(\psi B{)}^{-1}\psi \mathrm{Hh}(t)-\alpha \Vert \psi A_d\Vert (\psi B{)}^{-1}\mathrm{sign}(\mathrm{\Upsilon }(t))-\rho \mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \end{array}$$

(42)

where $\zeta$ is a constant to be designed, $\rho \ge \Vert {(\psi B)}^{-1}\Vert ·\Vert \psi \Lambda \mathrm{\Xi }\Vert ·\alpha$ .

It is demonstrated below that the designed controller can satisfy the accessibility of the sliding-mode surface.

Theorem 3. For the automobile system, when the sliding-mode function is determined by equation (11) and ${(\psi B)}^{-1}>0$ (where $J>0$ is the solution of equation (31)), the sliding-mode controller equation (42) can make the state of the system converges to the sliding-mode surface $S(t)=0$ within a certain time.

Proof. Choose the Lyapunov function as

$$N(t)=\frac{1}{2}{\mathrm{\Upsilon }}^T(t)\mathrm{\Upsilon }(t)$$

(43)

Taking the derivative of $N(t)$ and combined with equations (11), (15), and (42), we get

$$\begin{array}{c}\stackrel{·}{N}(t)={\mathrm{\Upsilon }}^T(t)\stackrel{·}{\mathrm{\Upsilon }}(t)\hfill \\ ={\mathrm{\Upsilon }}^T(t)\{-\zeta \mathrm{sign}(\mathrm{\Upsilon }(t))+\psi \mathrm{\Delta }A(t)\eta (t)+\psi A_d\eta (t-\iota (t))-\alpha \Vert \psi A_d\Vert \mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \\ -\psi B\rho \mathrm{sign}(\mathrm{\Upsilon }(t))+\psi B\theta (t)\varpi (\eta (t),t)-\psi B\mu \omega (\eta (t),t)\mathrm{sign}(\mathrm{\Upsilon }(t))\}\hfill \\ \le -\zeta {\mathrm{\Upsilon }}^T(t)\mathrm{sign}(\mathrm{\Upsilon }(t))+{\mathrm{\Upsilon }}^T(t)\psi B\Vert \theta (t)\varpi (\eta (t),t)\Vert -{\mathrm{\Upsilon }}^T(t)\psi B\mu \omega (\eta (t),t)\mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \\ +{\mathrm{\Upsilon }}^T(t)\Vert \psi \Lambda \mathrm{\Xi }\Vert \Vert \eta (t)\Vert -\rho {\mathrm{\Upsilon }}^T(t)\psi \mathrm{Bsign}(\mathrm{\Upsilon }(t))+{\mathrm{\Upsilon }}^T(t)\Vert \psi A_d\Vert \Vert \eta (t-\iota (t))\Vert -\alpha \Vert \psi A_d\Vert {\mathrm{\Upsilon }}^T(t)\mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \end{array}$$

(44)

According to Assumptions 1 and 2, we can get

$$\begin{array}{c}{\mathrm{\Upsilon }}^T(t)\psi B\Vert \theta (t)\varpi (\eta (t),t)\Vert -{\mathrm{\Upsilon }}^T(t)\psi B\mu \omega (\eta (t),t)\mathrm{sign}(\mathrm{\Upsilon }(t))\hfill \\ \le \Vert {\mathrm{\Upsilon }}^T(t)\Vert \Vert \psi B\Vert \Vert \theta (t)\varpi (\eta (t),t)\Vert -\Vert {\mathrm{\Upsilon }}^T(t)\Vert \Vert \psi B\Vert \mu \omega (\eta (t),t)\le 0\hfill \end{array}$$

(45)

Combined Assumption 5 and substituting $\rho$ into the formula, we can get

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Upsilon }}^T(t)\Vert \psi \Lambda \mathrm{\Xi }\Vert \Vert \eta (t)\Vert -\rho {\mathrm{\Upsilon }}^T(t)\psi \mathrm{Bsign}(\mathrm{\Upsilon }(t))\le \Vert \eta (t)\Vert }\\ \multicolumn{1}{c}{\Vert {\mathrm{\Upsilon }}^T(t)\Vert \Vert \psi \Lambda \mathrm{\Xi }\Vert -\alpha \Vert {\mathrm{\Upsilon }}^T(t)\Vert \Vert \psi \Lambda \mathrm{\Xi }\Vert \le 0}\end{array}$$

(46)

Similarly, according to Assumption 5, we can get

$${\mathrm{\Upsilon }}^T(t)\Vert \psi A_d\Vert \Vert \eta (t-\iota (t))\Vert -\alpha \Vert \psi A_d\Vert {\mathrm{\Upsilon }}^T(t)\mathrm{sign}(\mathrm{\Upsilon }(t))\le 0$$

(47)

Furthermore, it can be deduced that

$$\stackrel{·}{N}(t)\le -\zeta {\mathrm{\Upsilon }}^T(t)\psi B(\psi B{)}^{-1}\mathrm{sign}(\mathrm{\Upsilon }(t))\le -\zeta \Vert {\mathrm{\Upsilon }}^T(t)\Vert \le 0$$

(48)

According to the Lyapunov stability condition, it can be obtained from $\stackrel{·}{N}(t)\le 0$ that the sliding-mode controller equation (42) can ensure that the system state trajectory converges to the sliding surface $\mathrm{\Upsilon }(t)=0$ in finite time.

Illustrative example

For the purpose of proving the sliding mode in this article when there are parameter uncertainties, time delays, system disturbances, and false data injection attacks in the automotive system, a simulation example is designed to proving the feasibility of controller in this section.

The parameter settings are ${\delta }_i=0.2$ , $\varsigma =0.3$ , $\sigma =0.5$ , $\zeta =0.2$ , $\alpha =0.7$ , $\chi =0.5$ , time delay is 0.03 + 0.03 s, interference gain matrix $H=[0.01,0.05,0.1{]}^T$ , external interference $h(t)=0.2\cos (t)$ , parameter uncertainty is $\mathrm{\Delta }A(t)=\Lambda 0.3\sin (t)\mathrm{\Xi }$ , where $\Lambda =[0.01,0.02,0.1{]}^T$ , $\mathrm{\Xi }=[0.01,0.02,0.1]$ , matrix $\psi$ is designed as $\psi =[0.4,0.3,0.1]$ to make $(\psi B{)}^{-1}$ exist and $(\psi B{)}^{-1}>0$ , false data injection attack ${\vartheta }_f(t)=\theta (t)$ $\omega (\eta (t),t)$ , where $\theta (t)=[0.1\cos (t),0.2\sin (t),0.5\cos (t)]$ , $\omega (\eta (t),t)=0.2\eta (t)\cos (t)$ , initial state $\eta (0)=[10,10,10{]}^T$ , simulation time is 30 s, and sampling period is 0.01 s.

Through Theorem 2, matrix J and sliding-mode parameter matrix L can be solved as

$$\begin{array}{c}J=\left[\begin{array}{ccc}6.0175\hfill & 6.1356\hfill & 1.8718\hfill \\ \multicolumn{1}{c}{6.1356}& 9.6829\hfill & 3.0610\hfill \\ \multicolumn{1}{c}{1.87178}& 3.0610\hfill & 1.6945\hfill \end{array}\right]\hfill \\ L=\left[\begin{array}{ccc}-1.5090\hfill & -2.4999\hfill & -0.3231\hfill \end{array}\right]\hfill \end{array}$$

Figure 2 displays the trajectory of the sliding surface. It can be known that the sliding surface converges to 0 in about 2.6 s and the convergence of the sliding surface is narrow, which proves that the dynamic performance of the sliding surface is good. Figure 3 displays the trajectory of the acceleration error. It can be known that the acceleration error is stabilized at 0 in about 3 s, and the small chattering is the normal phenomenon of the sliding-mode controller, which is within the acceptable range. Figure 4 displays the trajectory of the position error, which shows that the position error converges to 0 in about 3 s. Figure 5 shows the trajectory of velocity error, which tends to be stable in about 3 s. Figure 6 displays the comparison between the actual acceleration and the expected acceleration. It can be known that in addition to a certain error at the beginning, the tracking trajectory almost coincides with the expected trajectory. Figure 7 shows the comparison between the actual position and the expected position. It can be known that the position tracking effect is good. Figure 8 displays the comparison between the velocity trajectory and the expected velocity trajectory, and it can be known that the tracking effect is also good. Figure 9 displays the trajectory of the control input, which is stable in about 3 s, and then fluctuated around 0. The simulation results show that under the influence of initial error and parameter uncertainty, time delay, false data injection attack, and disturbance in the vehicle system, the designed controller can still guarantee the convergence of the system state in finite time and the error is small.

Figure 2.

Sliding-mode surface.

Figure 3.

Acceleration error.

Figure 4.

Position error.

Figure 5.

Velocity error.

Figure 6.

System output acceleration.

Figure 7.

System output position.

Figure 8.

System output velocity.

Figure 9.

Controller input.

Furthermore, in order to demonstrate the effectiveness of the designed controller, the larger parameter uncertainty is used in the simulation. The specific parameters are $\mathrm{\Delta }A(t)=\Lambda 0.9\sin (t)\mathrm{\Xi }$ , where $\Lambda =[[0.13,0.13,0.3{]]}^T$ , $\mathrm{\Xi }=[0.13,0.13,0.3]$ , and other parameters are set as same as above. The results are shown in Figures 10–12. Figure 10 gives the acceleration error, Figure 11 shows the velocity error, and Figure 11 describes the position error. Compared to the above results, the control effect has slightly deteriorated under greater parameter uncertainty, but the errors can still converge to 0 at the end.

Figure 10.

Acceleration error.

Figure 11.

Velocity error.

Figure 12.

Position error.

Conclusion

In this article, we studied the safety control problem of intelligent connected vehicles. In which, designed an integral sliding surface and proposed a sliding-mode control strategy. First, considering a vehicle running on a horizontal road, a vehicle model with disturbance, attack, time delay, and parameter uncertainty is established. Second, for the purpose to make the system state slide according to the predetermined trajectory, we proposed an integral sliding surface. Then, we proved the stability of the sliding-mode dynamics, and the stability conditions of the system are given using the linear matrix inequality and Lyapunov theorem. Third, we derived a sliding-mode controller, and then it is proved that the system state can reach the sliding surface by the proposed sliding-mode controller within the certain time limit. Finally, we used a simulation to verify the correctness of the sliding-mode controller.