Finite-Time Disturbance Observer-Based Sliding Mode Control for a Vehicle Platoon Subject to Mismatched Disturbance

Abstract

The article focuses on the issue of the sliding mode control of a vehicle platoon with matched and mismatched disturbances. A novel finite-time disturbance observer-based sliding mode control scheme is developed to effectively mitigate the adverse impact of disturbances and achieve the control goals of a platoon. As matched and mismatched disturbances might decrease the control performance or even cause the instability of a vehicle platoon, a finite-time disturbance observer (FTDO) is designed to effectively reduce the effects of both types of disturbances. Unlike previous studies, the proposed FTDO in this article has the capability to directly estimate disturbances without the need to know the precise upper bounds of the disturbances. A feedforward compensation term, derived from disturbance estimation, is incorporated into the FTDO-based sliding mode control scheme to solve the issue of the degradation of control performance. The controlled vehicle platoon’s stability is proven through the Lyapunov approach, which means that the control goals of the platoon can be achieved under the developed FTDO-based sliding mode control scheme. Finally, a numerical example is conducted to confirm the efficacy of the developed control scheme.

1. Introduction

A variety of issues, such as traffic jams, traffic collisions, and air pollution, are becoming more and more serious in modern transport systems [1,2,3,4,5]. Vehicle platoon control refers to coordinating and regulating the movement of a group of vehicles in a transportation network to guarantee that the transport system operates efficiently and safely [6]. Vehicle platoon control has shown significant advantages in increasing the traffic volume of roads, reducing traffic congestion, and improving the efficiency of traveling [7,8,9]. To address traffic problems, the field of vehicle platoon control has attracted growing attention [10,11]. Fundamental issues in the vehicle platoon field which have been studied include homogeneous or heterogeneous dynamics [12], stability analysis [13], spacing policies [14], and others.

Maintaining consistent spacing between adjacent vehicles and achieving velocity synchronization of followers with the leader are the primary control goals of use of a platoon [15]. Issues such as the degradation of control performance and instability of a vehicle platoon in practice are usually caused by unknown external disturbances [16,17]. Disturbances that appear in the control input channel are usually classified as matched disturbances, such as those involving model uncertainty, rolling resistance, and gusts of wind [18]. Disturbances which do not appear in the control input channel are usually classified as mismatched disturbances, such as unmodeled dynamics, dynamic friction, and velocity uncertainty [19]. Numerous studies have been carried out to reduce the impact of unknown external disturbances on vehicle platoons [20,21,22,23]. In [20], the article considers the robust control problem confronted by a vehicle platoon when subjected to matched disturbances, and presents a distributed $H_{\infty }$ control strategy designed to effectively accomplish the control goals of the platoon. Ref. [21] integrates a disturbance observer and a multi-power convergence method into a coupled sliding mode controller, aiming to counteract the adverse effects induced by matched disturbances. An investigation of platoon control issues, particularly those arising from matched disturbances, is presented in [22], which proposes a distributed adaptive sliding mode control approach to effectively handle disturbances and ensure that a vehicle platoon achieves the predetermined tracking performance. The authors of [23] present an adaptive event-triggered control strategy that can effectively deal with matched disturbances and synchronize a vehicle platoon. It is to be noted that only the matched disturbances of a vehicle platoon are considered in the control methods mentioned above. However, in practice, mismatched disturbances may also degrade control performance and even cause instability of the platoon control system [19].

In recent studies, several control schemes have been presented for systems under mismatched disturbances to solve the issues caused by such disturbances [24,25]. In [25], a nonsingular dynamic terminal sliding mode control method is developed, where both matched and mismatched disturbances are effectively compensated, thereby guaranteeing the stability of the vehicle platoon under intermittent communication conditions. Ref. [26] develops a disturbance observer-based nonsingular terminal sliding mode control scheme to effectively mitigate both matched and mismatched disturbances of a vehicle platoon, ensuring system stability while optimizing platoon control performance. Ref. [27] investigates a distributed adaptive singularity-free fixed-time neural network tracking control problem for a vehicle platoon with mismatched disturbance, where a novel backstepping control architecture synergistically integrating a neural network is developed to effectively address mismatched disturbance. It is to be noted that nearly all the aforementioned works assume that the precise upper bounds of mismatched disturbances are known. However, due to the uncertainty of systems in practice, it is difficult or impossible to obtain these precise upper bounds and achieve good control results using the abovementioned methods. Consequently, this presents a challenge and the need to devise a novel control scheme to both achieve the control goals of the platoon and mitigate the influence of disturbances under the condition that the precise upper bounds of disturbances are unknown.

According to the above analysis, we investigate the sliding mode control issue of a vehicle platoon affected by matched and mismatched disturbances in this article. A novel FTDO-based sliding mode control strategy is developed to improve platoon control performance and achieve its control goals. The proposed FTDO, which is designed to solve the issue of degraded control performance and enhance the vehicle platoon’s stability, can directly estimate disturbances without the need to know the precise upper bounds of the disturbances. A feedforward compensation term based on disturbance estimation is incorporated into the FTDO-based sliding mode controller to effectively reduce the influence of disturbances. The primary contributions of this article are as follows:

• Unlike existing studies which focus on vehicle platoons with matched disturbances [20,21,22,23], this work investigates the control issues of a vehicle platoon influenced by matched and mismatched disturbances.
• In contrast to [28,29,30], in which the assumptions regarding the known precise upper bounds of disturbances are made, this work presents a novel FTDO capable of directly estimating matched and mismatched disturbances without the need to know the precise upper bounds of both types of disturbances, which makes this approach more general and extends applicability to a broader range of practical systems.
• For a vehicle platoon affected by matched and mismatched disturbances, a novel FTDO-based sliding mode control approach is developed to achieve the control goals. Moreover, a feedforward compensation term is constructed by utilizing disturbance estimation to effectively reduce the negative effects of disturbances and guarantee the platoon’s stability.

The rest of the article is organized as follows: A dynamics model of the vehicle and the problem formulation are described in Section 2. Section 3 illustrates the development process of the FTDO-based sliding mode control strategy. A numerical example is outlined in Section 4 to confirm the efficacy of the developed FTDO-based sliding mode control approach. The conclusions are outlined in Section 5.

2. Problem Formulation

2.1. Vehicle Dynamics Modeling

A vehicle platoon is considered to contain one leader vehicle, represented by 0, and N follower vehicles, identified by $i\in \{1,\dots ,N\}$. Then, the dynamics of the ith follower vehicle is considered as [31]

$$\begin{array}{cc}\hfill {\dot{x}}_i\left(t\right)& =v_i\left(t\right)+{\varpi }_{i1}\left(t\right)\hfill \end{array}$$

(1)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{cc}\hfill {\dot{v}}_i\left(t\right)& ={\displaystyle \frac{{\eta }_i}{m_i{r}_i}}{T}_i-{\displaystyle \frac{C_{A,i}}{m_i}}{v}_i^2\left(t\right)-f_{i}g+{\varpi }_{i2}\left(t\right)\hfill \end{array}$$

(2)

where $x_i\left(t\right)$ represents the position; $m_i$, $r_i$, and $T_i$ represent the mass, tire radius, and driving torque; $v_i\left(t\right)$ represents the velocity; ${\eta }_i$ and $C_{A,i}$ represent the transmission efficiency and drag coefficient, respectively; $f_i$ and g denote the rolling resistance and gravitational acceleration; ${\varpi }_{i1}$ denotes mismatched disturbances caused by unmodeled dynamics, dynamic friction, and velocity uncertainty [19]; ${\varpi }_{i2}$ represents matched disturbances.

Consider the following feedback linearization law [31]

$$T_i={\displaystyle \frac{m_i{r}_i}{{\eta }_i}}\left(u_i\left(t\right)+{\displaystyle \frac{C_{A,i}}{m_i}}{v}_i^2\left(t\right)+f_{i}g\right)$$

(3)

where $u_i\left(t\right)$ represents the control input. Substituting (3) into (2) yields

$$\begin{gathered} \dot{x_i}\left(t\right)=v_i\left(t\right)+{\varpi }_{i1}\left(t\right) \\ {\dot{v}}_i\left(t\right)=u_i\left(t\right)+{\varpi }_{i2}\left(t\right) \end{gathered}$$

(4)

Consider a (virtual) leader [32]

$$\begin{array}{c}{\dot{x}}_0\left(t\right)=v_0\left(t\right)\hfill \\ {\dot{v}}_0\left(t\right)=\psi (x_0\left(t\right),v_0\left(t\right))\hfill \end{array}$$

(5)

where $x_0\left(t\right)$ denotes the position; $v_0\left(t\right)$ and $\psi (x_0\left(t\right),v_0\left(t\right))$ represent the velocity and unknown nonlinear function, respectively.

2.2. Platoon Control Problem Formulation

In this paper, the information flow topology of predecessor following is considered as shown in Figure 1. Each vehicle adjusts its driving status according to its predecessor to accomplish the goals of the platoon. This paper utilizes the CTH (constant time headway) spacing policy, and the configuration of the vehicle platoon system under this policy is depicted in Figure 2.

The spacing error $e_i\left(t\right)$ is defined as

$$\begin{array}{c}\hfill e_i\left(t\right)=d_i\left(t\right)-{d_i}^{*}\left(t\right),i\in \{1,\dots ,N\}\end{array}$$

(6)

where $d_i\left(t\right)$ represents the relative spacing between the ith vehicle and its predecessor. The desired spacing is specified by $d_i^{*}\left(t\right)=h_i{v}_i\left(t\right)+{\delta }_i$, where ${\delta }_i>0$ and $h_i>0$ denote the standstill spacing and time headway, respectively. The velocity error of the ith vehicle is

$$\begin{array}{c}\hfill e_{v_i}\left(t\right)=v_{i-1}\left(t\right)-v_i\left(t\right),i\in \{1,\dots ,N\}\end{array}$$

(7)

To ensure that the vehicle keeps a desired spacing from the preceding vehicle and achieves velocity synchronization with the leader, the control goals can be formally described as follows:

• Each vehicle keeps the predetermined inter-vehicle spacing and achieves velocity synchronization with the leader vehicle;
• The matched and mismatched disturbances of the vehicle platoon are estimated by utilizing the developed FTDO without requiring to know the exact upper bounds of the disturbances;
• An FTDO-based sliding mode control scheme is proposed to compensate for the adverse influences caused by matched and mismatched disturbances and to ensure the vehicle platoon’s stability.

2.3. Definitions and Lemmas

Definition 1

([33]). If for any $\epsilon >0$, there exists $\varsigma >0$ such that $\Vert e_i{\left(0\right)\Vert }_{\infty }<\varsigma \Rightarrow su{p}_i{\Vert e_i(·)\Vert }_{\infty }<\epsilon$, the origins of (6) with vehicle dynamics (4) are said to be string stable.

Definition 2

([34]). Consider

$$\dot{z}\left(t\right)=y\left(z\left(t\right)\right),z\left(0\right)=0,z\left(t\right)\subseteq {\mathbb{R}}^n$$

(8)

where $y(·)$ is a continuous bounded function. The origin is a finite-time convergent equilibrium of (8) if an open neighborhood $U\subseteq D$ of the origin and a function $T_z:U\setminus \left\{0\right\}\to [0,\infty )$ exist, such that for every initial point $z_0\in U\setminus \left\{0\right\}$, the solution trajectory $z(t,z_0)$ is well defined for all $t\in [0,T\left(z_0\right))$ and ${\lim }_{t\to T\left(z_0\right)}z(t,z_0)=0$. The function $T_z\left(z_0\right)$ is known as the convergence-time function. The origin is considered as a finite-time equilibrium if it is both finite-time convergent and Lyapunov stable. If $U=D={\mathbb{R}}^n$, the origin is a globally finite-time stable equilibrium.

Definition 3

([35]). The origins of (6) with vehicle dynamics (4) are regarded as strong string stable, if $|e_N\left(t\right)|\le |e_{N-1}\left(t\right)|\le \cdots \le |e_1\left(t\right)|$ such that the error transfer function $|G_i\left(s\right)|=|E_{i+1}\left(s\right)/E_i\left(s\right)|\le 1$ is satisfied, where $E_i\left(s\right)$ is the Laplace transform of $e_i$.

Lemma 1

([36]). Consider the following differentiator

$$\begin{array}{cc}\hfill {\dot{w}}_0\left(t\right)=& -{\lambda }_0|w_0\left(t\right){|}^{n/(n+1)}\mathrm{sgn}\left(w_0\left(t\right)\right)-{\varrho }_0{w}_0\left(t\right)+w_1\left(t\right)\hfill \\ \hfill {\dot{w}}_1\left(t\right)=& -{\lambda }_1|w_1\left(t\right)-{\dot{w}}_0\left(t\right){|}^{(n-1)/n}\mathrm{sgn}\left(w_1\left(t\right)-{\dot{w}}_0\left(t\right)\right)-{\varrho }_1\left(w_1\left(t\right)-{\dot{w}}_0\left(t\right)\right)+w_2\left(t\right)\hfill \\ \hfill {\dot{w}}_{n-1}\left(t\right)=& -{\lambda }_{n-1}|w_{n-1}\left(t\right)-{\dot{w}}_{n-2}\left(t\right){|}^{1/2}\mathrm{sgn}\left(w_{n-1}\left(t\right)-{\dot{w}}_{n-2}\left(t\right)\right)-\hfill \\ \hfill & {\varrho }_{n-1}\left(w_{n-1}-{\dot{w}}_{n-2}\right)+w_n\hfill \\ \hfill {\dot{w}}_n\left(t\right)=& -{\lambda }_n\mathrm{sgn}\left(w_n\left(t\right)-{\dot{w}}_{n-1}\left(t\right)\right)-{\varrho }_n\left(w_n\left(t\right)-{\dot{w}}_{n-1}\left(t\right)\right)-1/L_i\varphi \left(t\right)\hfill \end{array}$$

(9)

where $w_0,\dots ,w_n$ are the state variables; ${\lambda }_i>0$ and ${\varrho }_i>0$; The disturbance term $\varphi \left(t\right)$ satisfies $|\varphi \left(t\right)|\le L_i$, where $L_i>0$, for $i\in \{1,2,\dots ,n\}$. Then, the differentiator converges to a bounded neighborhood of zero for $t<T_d$.

Lemma 2

([30]). There are $\chi \left(t\right)\in \mathbb{R}$ and $\zeta \left(t\right)\in \mathbb{R}$, if ${ð}_1>0$ and ${ð}_2>0$, then

$$|\chi \left(t\right){|}^{{ð}_1}{\left|\zeta \left(t\right)\right|}^{{ð}_2}\le {\displaystyle \frac{{ð}_1{\left|\chi \left(t\right)\right|}^{{ð}_1+{ð}_2}}{{ð}_1+{ð}_2}}+{\displaystyle \frac{{ð}_2{\left|\zeta \left(t\right)\right|}^{{ð}_1+{ð}_2}}{{ð}_1+{ð}_2}}$$

(10)

Lemma 3

([37]). The super-twisting algorithm (STA) is described as

$$\begin{array}{ccc}\hfill {\dot{p}}_1\left(t\right)& =& -{\kappa }_1{\left|p_1\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(p_1\left(t\right)\right)+p_2\left(t\right)\hfill \\ \hfill {\dot{p}}_2\left(t\right)& =& -{\kappa }_2\mathrm{sgn}\left(p_1\left(t\right)\right)\hfill \end{array}$$

(11)

where ${\kappa }_1>0$, ${\kappa }_2>0$; $p_1\left(t\right)$ and $p_2\left(t\right)$ are the state variables. The trajectories of (11) converge to the origin $p=0$ at $T_s\left(p_0\right)\le {\displaystyle \frac{2}{\vartheta }}{V}^{1/2}\left(p_0\right)$, where $p_0$ is the initial state, $\vartheta ={\displaystyle \frac{{\lambda }_{\min }^{1/2}\left\{P\right\}{\lambda }_{\min }\left\{Q\right\}}{{\lambda }_{\max }\left\{P\right\}}}$, and $V\left(p\right)={\zeta }^{T}P\zeta$ with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\zeta }^T& =\left[\right|p_1{|}^{1/2}\mathrm{sign}\left(p_1\right),p_2]\\ \hfill P& ={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}4{\kappa }_2+{\kappa }_1^2& -{\kappa }_1\\ -{\kappa }_1& 2\end{array}\right]\\ \hfill Q& ={\displaystyle \frac{1}{2}}{\kappa }_1\left[\begin{array}{cc}2{\kappa }_2+{\kappa }_1^2& -{\kappa }_1\\ -{\kappa }_1& 1\end{array}\right]\end{array}\end{array}$$

3. Finite-Time Disturbance Observer and Controller Design

In this part, a novel FTDO-based sliding mode control approach for a vehicle platoon affected by matched and mismatched disturbances is developed to accomplish platoon control goals, with the FTDO designed to directly estimate both types of disturbances.

Assumption 1

([32]). The disturbance ${\varpi }_{ik}$ in (4) is twice differentiable, and there exists $M_i>0$ such that $|{\varpi }_{ik}^{\left(j\right)}\left(t\right)|\le M_i$, for $j=0,1,2$ and $k=1,2$.

Remark 1.

Assumption 1 is employed only for stability analysis and not for controller design, so there is no need to know the exact values of the bounds.

Remark 2.

The bounds involved in Assumption 1 will only be used in stability analysis, and will not be applied in control protocol design, which means that the precise values of these bounds do not need to be known.

3.1. Finite-Time Disturbance Observer Design

Design the novel FTDO for ith vehicle as

$$\begin{array}{cc}\hfill {\dot{\hat{x}}}_i\left(t\right)=& v_i\left(t\right)+{\mu }_{i1}\left(t\right)\hfill \\ \hfill {\mu }_{i1}\left(t\right)=& -{\lambda }_0{L}_i^{1/3}|{\hat{x}}_i\left(t\right)-x_i\left(t\right){|}^{2/3}\mathrm{sgn}\left({\hat{x}}_i\left(t\right)-x_i\left(t\right)\right)\hfill \\ \hfill & -{\varrho }_0\left({\hat{x}}_i\left(t\right)-x_i\left(t\right)\right)+{\hat{\varpi }}_{i1}\left(t\right)\hfill \\ \hfill {\dot{\hat{\varpi }}}_{i_1}\left(t\right)=& -{\lambda }_1{L_i}^{1/2}{\left|{\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right)\right|}^{1/2}\mathrm{sgn}({\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right))\hfill \\ \hfill & -{\varrho }_1({\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right))+{\hat{\dot{\varpi }}}_{i1}\left(t\right)\hfill \\ \hfill {\dot{\hat{\dot{\varpi }}}}_{i1}\left(t\right)=& -{\lambda }_2{L}_i\mathrm{sgn}({\hat{\dot{\varpi }}}_{i1}\left(t\right)-{\dot{\hat{\varpi }}}_{i1}\left(t\right))\hfill \\ \hfill & -{\varrho }_2({\hat{\dot{\varpi }}}_{i1}\left(t\right)-{\dot{\hat{\varpi }}}_{i1}\left(t\right))\hfill \\ \hfill {\dot{\hat{v}}}_i\left(t\right)=& u_i\left(t\right)+{\mu }_{i2}\left(t\right)\hfill \\ \hfill {\mu }_{i2}\left(t\right)=& -{\lambda }_3{L}_i^{1/2}{\left|{\hat{v}}_i\left(t\right)-v_i\left(t\right)\right|}^{1/2}\mathrm{sgn}({\hat{v}}_i\left(t\right)-v_i\left(t\right)\hfill \\ \hfill & -{\varrho }_3({\hat{v}}_i\left(t\right)-v_i\left(t\right))+{\hat{\varpi }}_{i2}\left(t\right))\hfill \\ \hfill {\dot{\hat{\varpi }}}_{i2}\left(t\right)=& -{\lambda }_4{L}_i\mathrm{sgn}({\hat{\varpi }}_{i2}\left(t\right)-{\mu }_{i2}\left(t\right))\hfill \\ \hfill & -{\varrho }_4({\hat{\varpi }}_{i2}\left(t\right)-{\mu }_{i2}\left(t\right))\hfill \end{array}$$

(12)

where ${\lambda }_i>0$, $L_i>0$, and ${\varrho }_i>0$, for $i\in \{1,2,3,4\}$; ${\hat{x}}_i\left(t\right)$ represents the estimate of the position; ${\hat{\varpi }}_{i1}\left(t\right)$ is the estimate of ${\varpi }_{i1}\left(t\right)$; ${\hat{\dot{\varpi }}}_{i1}\left(t\right)$ is the estimate of ${\dot{\varpi }}_{i1}\left(t\right)$; ${\hat{v}}_i\left(t\right)$ and ${\hat{\varpi }}_{i2}\left(t\right)$ represent the estimates of the velocity and ${\varpi }_{i2}\left(t\right)$, respectively. Define the estimation errors of states and disturbances as follows: ${\tilde{x}}_i\left(t\right)={\displaystyle \frac{{\hat{x}}_i\left(t\right)-x_i\left(t\right)}{L_i}}$, ${\tilde{v}}_i\left(t\right)={\displaystyle \frac{{\hat{v}}_i\left(t\right)-v_i\left(t\right)}{L_i}}$, ${\nu }_{i1}\left(t\right)={\displaystyle \frac{{\hat{\dot{\varpi }}}_{i1}\left(t\right)-{\dot{\varpi }}_{i1}\left(t\right)}{L_i}}$, and ${\tilde{\varpi }}_{ik}\left(t\right)={\displaystyle \frac{{\hat{\varpi }}_{ik}\left(t\right)-{\varpi }_{ik}\left(t\right)}{L_i}}$, for $k=1,2$.

To provide a detailed explanation of the FTDO design process, we present its design block diagram as shown in Figure 3.

Theorem 1.

Consider the vehicle dynamics (4) under Assumption 1. The proposed FTDO in (12) is finite-time stable in the sense that there exists a time $T_d$ such that ${\tilde{x}}_i\left(t\right),{\tilde{v}}_i\left(t\right),{\tilde{\varpi }}_{ik}\left(t\right)$, and ${\nu }_{i1}\left(t\right)$ converge to a bounded neighborhood of zero, $\forall t\ge T_d$.

Proof of Theorem 1. 

Taking the derivative of ${\tilde{x}}_i\left(t\right)$ according to (4) and (12), we can obtain

$$\begin{array}{cc}\hfill {\dot{\tilde{x}}}_i\left(t\right)=& {\displaystyle \frac{{\dot{\hat{x}}}_i\left(t\right)-{\dot{x}}_i\left(t\right)}{L_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_0{L}_i^{1/3}{\left|{\hat{x}}_i\left(t\right)-x_i\left(t\right)\right|}^{2/3}\mathrm{sgn}({\hat{x}}_i\left(t\right)-x_i\left(t\right))\hfill \\ \hfill & -{\varrho }_0({\hat{x}}_i\left(t\right)-x_i\left(t\right))+{\hat{\varpi }}_{i1}\left(t\right)-{\varpi }_{i1}\left(t\right))\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_0{L}_i^{1/3}{\left|L_i{\tilde{x}}_i\left(t\right)\right|}^{2/3}\mathrm{sgn}\left(L_i{\tilde{x}}_i\left(t\right)\right)-{\varrho }_0\left(L_i{\tilde{x}}_i\left(t\right)\right)+L_i{\tilde{\varpi }}_{i1}\left(t\right))\hfill \\ \hfill =& -{\lambda }_0{\left|{\tilde{x}}_i\left(t\right)\right|}^{2/3}\mathrm{sgn}\left({\tilde{x}}_i\left(t\right)\right)-{\varrho }_0\left({\tilde{x}}_i\left(t\right)\right)+{\tilde{\varpi }}_{i1}\left(t\right)\hfill \end{array}$$

(13)

By (4) and (12), the derivative of ${\tilde{\varpi }}_{i1}\left(t\right)$ is

$$\begin{array}{cc}\hfill {\dot{\tilde{\varpi }}}_{i1}\left(t\right)=& {\displaystyle \frac{{\dot{\hat{\varpi }}}_{i1}\left(t\right)-{\dot{\varpi }}_{i1}\left(t\right)}{L_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_1{L}_i^{1/2}{\left|{\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right)\right|}^{1/2}\mathrm{sgn}({\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right))\hfill \\ \hfill & -{\varrho }_1({\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right))+{\hat{\dot{\varpi }}}_i\left(t\right)-{\dot{\varpi }}_{i1}\left(t\right))\hfill \end{array}$$

(14)

Using (12), we can obtain

$$\begin{array}{cc}\hfill {\hat{\varpi }}_{i1}\left(t\right)-{\mu }_{i1}\left(t\right)=& {\hat{\varpi }}_{i1}\left(t\right)+{\lambda }_0{L}_i^{1/3}{\left|{\hat{x}}_i\left(t\right)-x_i\left(t\right)\right|}^{2/3}\mathrm{sgn}({\hat{x}}_i\left(t\right)\hfill \\ \hfill & -x_i\left(t\right))+{\varrho }_0({\hat{x}}_i\left(t\right)-x_i\left(t\right))-{\hat{\varpi }}_{i1}\left(t\right)\hfill \\ \hfill =& {\lambda }_0{L}_i^{1/3}{\left|L_i{\tilde{x}}_i\left(t\right)\right|}^{2/3}\mathrm{sgn}\left(L_i{\tilde{x}}_i\left(t\right)\right)+{\varrho }_0\left(L_i{\tilde{x}}_i\left(t\right)\right)\hfill \\ \hfill =& L_i({\lambda }_0|{\tilde{x}}_i\left(t\right){|}^{2/3}\mathrm{sgn}\left({\tilde{x}}_i\left(t\right)\right)+{\varrho }_0\left({\tilde{x}}_i\left(t\right)\right))\hfill \\ \hfill =& L_i({\tilde{\varpi }}_{i1}\left(t\right)-{\dot{\tilde{x}}}_i\left(t\right))\hfill \end{array}$$

(15)

Substituting (15) into (14),

$$\begin{array}{cc}\hfill {\dot{\tilde{\varpi }}}_{i1}(t)=& {\displaystyle \frac{{\dot{\hat{\varpi }}}_{i1}(t)-{\dot{\varpi }}_{i1}(t)}{L_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_1{L}_i^{1/2}|L_i({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t)){|}^{1/2}\mathrm{sgn}(L_i({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t)))\hfill \\ \hfill & -{\varrho }_1(L_i({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t)))+L_i{\nu }_{i1}(t))\hfill \\ \hfill =& -{\lambda }_1{\left|({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))\right|}^{1/2}\mathrm{sgn}({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))\hfill \\ \hfill & -{\varrho }_1({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))+{\nu }_{i1}(t)\hfill \end{array}$$

(16)

Calculating the derivative of ${\tilde{\nu }}_i\left(t\right)$ from (12) yields

$$\begin{array}{cc}\hfill {\dot{\nu }}_{i1}\left(t\right)& ={\displaystyle \frac{{\dot{\hat{\dot{\varpi }}}}_{i1}\left(t\right)-{\ddot{\varpi }}_{i1}\left(t\right)}{L_i}}\hfill \\ \hfill & =-{\lambda }_2\mathrm{sgn}(({\nu }_{i1}(t)-{\dot{\tilde{\varpi }}}_{i1}(t))-{\varrho }_2({\nu }_{i1}(t)-{\dot{\tilde{\varpi }}}_{i1}(t))-{\displaystyle \frac{1}{L_i}}{\ddot{\varpi }}_{i1}(t)\hfill \end{array}$$

(17)

Differentiating ${\tilde{v}}_i\left(t\right)$, one has

$$\begin{array}{cc}\hfill {\dot{\tilde{v}}}_i\left(t\right)=& {\displaystyle \frac{{\hat{v}}_i\left(t\right)-{\dot{v}}_i\left(t\right)}{L_i}}\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_3{L}_i^{1/2}{|{\hat{v}}_i\left(t\right)-v_i\left(t\right)|}^{1/2}\mathrm{sgn}({\hat{v}}_i\left(t\right)-v_i\left(t\right))\hfill \\ \hfill & -{\delta }_3({\hat{v}}_i\left(t\right)-v_i\left(t\right))+{\hat{\varpi }}_{i2}\left(t\right)-{\varpi }_{i2}\left(t\right))\hfill \\ \hfill =& {\displaystyle \frac{1}{L_i}}(-{\lambda }_3{L}_i^{1/2}|L_i\tilde{v}\left(t\right){|}^{1/2}\mathrm{sgn}\left(L_i\tilde{v}\left(t\right)\right)-{\varrho }_3\left(L_i\tilde{v}\left(t\right)\right)+L_i{\tilde{\varpi }}_{i2}\left(t\right))\hfill \\ \hfill =& -{\lambda }_3{\left|\tilde{v}\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(\tilde{v}\left(t\right)\right)-{\varrho }_3\left(\tilde{v}\left(t\right)\right)+{\tilde{\varpi }}_{i2}\left(t\right)\hfill \end{array}$$

(18)

And, the time derivative of ${\tilde{\varpi }}_{i2}\left(t\right)$ is

$$\begin{array}{cc}\hfill {\dot{\tilde{\varpi }}}_{i2}\left(t\right)& ={\displaystyle \frac{{\dot{\hat{\varpi }}}_{i2}\left(t\right)-{\dot{\varpi }}_{i2}\left(t\right)}{L_i}}\hfill \\ \hfill & ={\displaystyle \frac{1}{L_i}}(-{\lambda }_4{L}_i\mathrm{sgn}({\hat{\varpi }}_{i2}\left(t\right)-{\mu }_{i2}\left(t\right))-{\varrho }_4({\hat{\varpi }}_{i2}\left(t\right)-{\mu }_{i2}\left(t\right))-{\dot{\varpi }}_{i2}\left(t\right))\hfill \end{array}$$

(19)

According to (12), one can obtain

$$\begin{array}{cc}\hfill {\hat{\varpi }}_{i2}(t)-{\mu }_{i2}(t)=& {\hat{\varpi }}_{i2}(t)+{\lambda }_3{L}_i^{{\displaystyle \frac{1}{2}}}{\left|\begin{array}{c}{\hat{v}}_i(t)-v_i(t)\end{array}\right|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}({\hat{v}}_i(t)-v_i(t))\hfill \\ \hfill & +{\varrho }_3({\hat{v}}_i(t)-v_i(t))-{\hat{\varpi }}_{i2}(t)\hfill \\ \hfill =& {\lambda }_3{L}_i^{1/2}{\left|L_i{\tilde{v}}_i(t)\right|}^{1/2}\mathrm{sgn}(L_i{\tilde{v}}_i(t))+{\varrho }_3(L_i{\tilde{v}}_i(t))\hfill \\ \hfill =& L_i({\lambda }_3|{\tilde{v}}_i(t){|}^{1/2}\mathrm{sgn}({\tilde{v}}_i(t))+{\varrho }_3({\tilde{v}}_i(t)))\hfill \\ \hfill =& L_i({\tilde{\varpi }}_{i2}(t)-\dot{\tilde{v}}(t))\hfill \end{array}$$

(20)

Substituting (20) into (19), we have

$$\begin{array}{cc}\hfill {\dot{\tilde{\varpi }}}_{i2}(t)& ={\displaystyle \frac{{\dot{\hat{\varpi }}}_{i2}(t)-{\dot{\varpi }}_{i2}(t)}{L_i}}\hfill \\ \hfill & ={\displaystyle \frac{1}{L_i}}(-{\lambda }_4{L}_i\mathrm{sgn}({\hat{\varpi }}_{i2}(t)-{\mu }_{i2}(t))-{\varrho }_4({\hat{\varpi }}_{i2}(t)-{\mu }_{i2}(t))-{\dot{\varpi }}_{i2}(t))\hfill \\ \hfill & ={\displaystyle \frac{1}{L_i}}(-{\lambda }_4{L}_i\mathrm{sgn}(L_i({\tilde{\varpi }}_{i2}(t)-\dot{\tilde{v}}(t)))-{\varrho }_4(L_i({\tilde{\varpi }}_{i2}(t)-\dot{\tilde{v}}(t)))-{\dot{\varpi }}_{i2}(t))\hfill \\ \hfill & =-{\lambda }_4\mathrm{sgn}({\tilde{\varpi }}_{i2}(t)-\dot{\tilde{v}}(t))-{\varrho }_4({\tilde{\varpi }}_{i2}(t)-\dot{\tilde{v}}(t))-{\displaystyle \frac{1}{L_i}}{\dot{\varpi }}_{i2}(t)\hfill \end{array}$$

(21)

Then, the observation error can be expressed as

$$\begin{array}{cc}\hfill {\dot{\tilde{x}}}_i(t)& =-{\lambda }_0{\left|{\tilde{x}}_i(t)\right|}^{2/3}\mathrm{sgn}({\tilde{x}}_i(t))-{\varrho }_0({\tilde{x}}_i(t))+{\tilde{\varpi }}_{i1}(t)\hfill \\ \hfill {\dot{\tilde{\varpi }}}_{i1}(t)& =-{\lambda }_1{\left|({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))\right|}^{1/2}\mathrm{sgn}({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))-{\varrho }_1({\tilde{\varpi }}_{i1}(t)-{\dot{\tilde{x}}}_i(t))+{\nu }_{i1}(t)\hfill \\ \hfill {\dot{\nu }}_{i1}(t)& =-{\lambda }_2\mathrm{sgn}({\nu }_{i1}(t)-{\dot{\tilde{\varpi }}}_{i1}(t))-{\varrho }_2({\nu }_{i1}(t)-{\dot{\tilde{\varpi }}}_{i1}(t))-{\displaystyle \frac{1}{L_i}}{\ddot{\varpi }}_{i1}(t)\hfill \\ \hfill {\dot{\tilde{v}}}_i(t)& =-{\lambda }_3{\left|\tilde{v}(t)\right|}^{1/2}\mathrm{sgn}(\tilde{v}(t))-{\varrho }_3(\tilde{v}(t))+{\tilde{\varpi }}_{i2}(t)\hfill \\ \hfill {\dot{\tilde{\varpi }}}_{i2}(t)& =-{\lambda }_4\mathrm{sgn}({\varpi }_{i2}(t)-\dot{\tilde{v}}(t))-{\varrho }_4({\varpi }_{i2}(t)-\dot{\tilde{v}}(t))-{\displaystyle \frac{1}{L_i}}{\dot{\varpi }}_{i2}(t)\hfill \end{array}$$

(22)

According to Lemma 1, by designing observer gains ${\lambda }_i$, the observation error (22) is finite-time stable, which means that ${\tilde{x}}_i\left(t\right),{\tilde{\varpi }}_{i1}\left(t\right),{\tilde{\nu }}_{i1}\left(t\right),{\tilde{v}}_i\left(t\right)$, and ${\tilde{\varpi }}_{i2}\left(t\right)$ converge to a bounded neighborhood of zero for $t<T_d$ [30]. The proof of Theorem 1 is complete. □

3.2. FTDO-Based Sliding Mode Controller Design and Stability Analysis

A novel FTDO-based sliding mode controller for a vehicle platoon affected by matched and mismatched disturbances is developed in this section to accomplish the platoon control goals. Define the sliding mode surface as

$$\begin{array}{c}\hfill s_i\left(t\right)=e_i\left(t\right)+{\int }_0^t{\mathcal{l}}_i{e}_i\left(\tau \right)d\tau \end{array}$$

(23)

where ${\mathcal{l}}_i>0$.

To ensure string stability, a coupled sliding mode surface is designed as [16]

$$\begin{array}{c}\hfill S_i\left(t\right)=\left\{\begin{array}{cc}\gamma s_i\left(t\right)-s_{i+1}\left(t\right),\hfill & i=1,\dots ,N-1\hfill \\ \gamma s_i\left(t\right),\hfill & i=N\hfill \end{array}\right.\end{array}$$

(24)

where $\gamma >0$.

Then, from (24), we have

$$\begin{array}{c}\hfill S\left(t\right)={\rm Y}s\left(t\right)\end{array}$$

(25)

where $S\left(t\right)={\left[S_1\left(t\right)S_2\left(t\right)\cdots S_n\left(t\right)\right]}^T,s\left(t\right)={\left[s_1\left(t\right)s_2\left(t\right)\cdots s_n\left(t\right)\right]}^T$, and

$$\begin{array}{c}\hfill {\rm Y}=\left[\begin{array}{ccccccccc}\gamma & & -1& & \dots & & 0& & 0\\ 0& & \gamma & & -1& & \dots & & 0\\ & & & \ddots & & & & \\ 0& & 0& & \dots & & \gamma & & -1\\ 0& & 0& & \dots & & 0& & \gamma \end{array}\right]\end{array}$$

As $\gamma \ne 0$, ${\rm Y}$ is an invertible matrix. According to (24), $S_i\left(t\right)$ converges to a bounded neighborhood of zero if, and only if, $s_i\left(t\right)$ converges to a bounded neighborhood of zero at the same time, and vice versa [35].

Differentiating $S_i$, one can obtain

$${\dot{S}}_i\left(t\right)=\left\{\begin{array}{cc}\hfill & \gamma \{v_0\left(t\right)-v_i\left(t\right)-{\varpi }_{i1}\left(t\right)-h_i[u_i\left(t\right)+{\varpi }_{i2}\left(t\right)]+{\mathcal{l}}_i{e}_i\left(t\right)\hfill \\ & -A_i\left(t\right)\},i=1\hfill \\ \hfill & \gamma \{v_{i-1}\left(t\right)+{\varpi }_{i-1,1}\left(t\right)-v_i\left(t\right)-{\varpi }_{i1}\left(t\right)-h_i[u_i\left(t\right)+{\varpi }_{i2}\left(t\right)]\hfill \\ & +{\mathcal{l}}_i{e}_i\left(t\right)-A_i\left(t\right)\},i\in \{2,\dots ,N-1\}\hfill \\ \hfill & \gamma \{v_{i-1}\left(t\right)+{\varpi }_{i-1,1}\left(t\right)-v_i\left(t\right)-{\varpi }_{i1}\left(t\right)-h_i[u_i\left(t\right)+{\varpi }_{i2}\left(t\right)]\hfill \\ \hfill & +{\mathcal{l}}_i{e}_i\left(t\right)\},i=N\hfill \end{array}\right.$$

(26)

where $A_i\left(t\right)={\displaystyle \frac{1}{\gamma }}({\dot{e}}_{i+1}+{\mathcal{l}}_{i+1}{e}_{i+1})$, $i\in \{1,2,\cdots ,N-1\}$.

The novel FTDO-based sliding mode controller is designed as

$$\begin{array}{c}\hfill u_i\left(t\right)=\left\{\begin{array}{c}-{\hat{\varpi }}_{i2}\left(t\right)-{\hat{\dot{\varpi }}}_{i1}\left(t\right)+{\displaystyle \frac{1}{h_i}}[{\mathcal{l}}_i{e}_i\left(t\right)+{\displaystyle \frac{1}{\gamma }}(k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)+{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau )\hfill \\ +v_0\left(t\right)-\left(v_i\left(t\right)+{\hat{\varpi }}_{i1}\left(t\right)\right)-A_i\left(t\right)],i=1\hfill \\ -{\hat{\varpi }}_{i2}\left(t\right)-{\hat{\dot{\varpi }}}_{i1}\left(t\right)+{\displaystyle \frac{1}{h_i}}[\mathcal{l}{e}_i\left(t\right)+{\displaystyle \frac{1}{\gamma }}(k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)+{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau )\hfill \\ +v_{i-1}\left(t\right)+{\hat{\varpi }}_{i-1,1}\left(t\right)-\left(v_i\left(t\right)+{\hat{\varpi }}_{i1}\left(t\right)\right)-A_i\left(t\right)],i=2,\dots ,N\hfill \end{array}\right.\end{array}$$

(27)

where $k_1>0$ and $k_2>0$; ${\hat{\varpi }}_{i2}\left(t\right)$ and ${\hat{\dot{\varpi }}}_{i1}\left(t\right)$ are the feedforward compensation terms, constructed by utilizing the estimation of mismatched and matched disturbances.

To provide a clearer explanation of the structure of the controller (27), we present its design block diagram as shown in Figure 3.

Theorem 2.

Consider the vehicle dynamics (4), the leader (5), and the FTDO (12) under Assumption 1. The FTDO-based sliding mode controller (27) can guarantee that $e_i\left(t\right)$ for $i\in \{1,2,\dots ,N\}$ converges to a bounded neighborhood of zero, while the string stability of the entire vehicle platoon can also be ensured when $0<\gamma \le 1$.

Proof of Theorem 2. 

The proof procedure first analyzes the state boundedness of $e_i\left(t\right)$ for $t\in \{0,T_d\}$, and then addresses the global stabilization of $e_i\left(t\right)$ in the second part.

Part 1 (State boundedness for $t\in \left[0,T_d\right)$): According to (4) and (6), the differential of $e_i\left(t\right)$ is

$$\begin{array}{c}\hfill {\dot{e}}_i\left(t\right)=\left\{\begin{array}{c}{v}_0\left(t\right)-v_i\left(t\right)-{\varpi }_{i1}\left(t\right)-h_i(u_i\left(t\right)+{\varpi }_{i2}\left(t\right)\hfill \\ +{\dot{\varpi }}_{i1}\left(t\right)),i=1\hfill \\ v_{i-1}\left(t\right)+{\varpi }_{i-1,1}\left(t\right)-v_i\left(t\right)-{\varpi }_{i1}\left(t\right)-h_i(u_i\left(t\right)\hfill \\ +{\varpi }_{i2}\left(t\right)+{\dot{\varpi }}_{i1}\left(t\right)),i=2,\dots ,N\hfill \end{array}\right.\end{array}$$

(28)

Substituting (27) into (28), one can obtain

$$\begin{array}{c}\hfill {\dot{e}}_i\left(t\right)=\left\{\begin{array}{c}-{\mathcal{l}}_i{e}_i\left(t\right)-{\displaystyle \frac{1}{\gamma }}\left({\dot{e}}_{i+1}\left(t\right)+{\mathcal{l}}_{i+1}{e}_{i+1}\left(t\right)\right)-{\displaystyle \frac{1}{\gamma }}(k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)\hfill \\ +E_{{\varpi }_i}\left(t\right)+{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau ),i=1,\dots ,N-1\hfill \\ -{\mathcal{l}}_i{e}_i\left(t\right)-{\displaystyle \frac{1}{\gamma }}\left(k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)+{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau \right)\hfill \\ +E_{{\varpi }_i}\left(t\right),i=N\hfill \end{array}\right.\end{array}$$

(29)

where

$$\begin{array}{c}\hfill E_{{\varpi }_i}\left(t\right)=\left\{\begin{array}{c}({\hat{\varpi }}_{i1}\left(t\right)-{\varpi }_{i1}\left(t\right))+h_i({\hat{\varpi }}_{i2}\left(t\right)-{\varpi }_{i2}\left(t\right)+{\hat{\dot{\varpi }}}_{i1}\left(t\right)\hfill \\ -{\dot{\varpi }}_{i1}\left(t\right)),i=1\hfill \\ ({\varpi }_{i-1,1}\left(t\right)-{\hat{\varpi }}_{i-1,1}\left(t\right))+({\hat{\varpi }}_{i1}\left(t\right)-{\varpi }_{i1}\left(t\right))+h_i({\hat{\varpi }}_{i2}\left(t\right)\hfill \\ -{\varpi }_{i2}\left(t\right)+{\hat{\dot{\varpi }}}_{i1}\left(t\right)-{\dot{\varpi }}_{i1}\left(t\right)),i=2,\dots ,N\hfill \end{array}\right.\end{array}$$

According to Theorem 1, $E_{{\varpi }_i}\left(t\right)$ is bounded for $t\in [0,T_d)$. Based on (27), (26) becomes

$$\begin{array}{c}\hfill \dot{S_i}\left(t\right)=-k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)-{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)d\tau \right)+\gamma E_{{\varpi }_i}\left(t\right)\end{array}$$

(30)

Define the Lyapunov function as $V_i\left(t\right)={\displaystyle \frac{1}{2}}{S}_i^2\left(t\right)$, and its time derivative is

$$\begin{array}{cc}\hfill {\dot{V}}_i\left(t\right)& ={\dot{S}}_i\left(t\right)S_i\left(t\right)\hfill \\ & =S_i\left(t\right)(-k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)-{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau +\gamma E_{{\varpi }_i}\left(t\right))\hfill \\ \hfill & =-k_1{\left|S_i\left(t\right)\right|}^{3/2}-k_2{S}_i\left(t\right){\int }_0^t\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau +\gamma E_{{\varpi }_i}\left(t\right)S_i\left(t\right)\hfill \\ \hfill & \le -k_2{S}_i\left(t\right){\int }_0^t\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau +\gamma E_{{\varpi }_i}\left(t\right)S_i\left(t\right)\hfill \\ \hfill & \le k_2\left|S_i\left(t\right)\right|{\int }_0^t\left|\mathrm{sgn}\left(S_i\left(\tau \right)\right)\right|d\tau +\left|\gamma E_{{\varpi }_i}\left(t\right)\right|\left|S_i\left(t\right)\right|\hfill \end{array}$$

(31)

By Lemma 2, we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_i\left(t\right)& \le k_2{T}_d({\displaystyle \frac{1}{2}}{S}_i^2\left(t\right)+{\displaystyle \frac{1}{2}})+\left|\gamma E_{{\varpi }_i}\left(t\right)\right|({\displaystyle \frac{1}{2}}{S}_i^2\left(t\right)+{\displaystyle \frac{1}{2}})\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}(k_4{T}_d+|\gamma E_{{\varpi }_i}\left(t\right)\left|\right)S_i^2\left(t\right)+{\displaystyle \frac{1}{2}}(k_2{T}_d+|\gamma E_{{\varpi }_i}\left(t\right)\left|\right)\hfill \\ \hfill & \le {\Gamma }_{\max }{V}_i\left(t\right)+{\Lambda }_{\max }\hfill \end{array}$$

(32)

where ${\Gamma }_{\max }=$ max$\left\{k_2{T}_d+\left|\gamma E_{{\varpi }_i}\left(t\right)\right|\right\}$ and ${\Lambda }_{max}=$${\displaystyle \frac{1}{2}}$max$\left\{k_2{T}_d+\left|\gamma E_{{\varpi }_i}\left(t\right)\right|\right\}$. As $E_{{\varpi }_i}\left(t\right)$ is bounded for $t\in [0,T_d)$, both ${\Gamma }_{\max }$ and ${\Lambda }_{max}$ are bounded. From (32), we can get that $V_i\left(t\right)$ and $S_i\left(t\right)$ are also bounded for $t\in [0,T_d)$ [30].

Define ${\Xi }_N\left(t\right)=-{\displaystyle \frac{1}{\gamma }}(k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)+{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau )+E_{{\varpi }_i}\left(t\right)$. Then, (29) can be rewritten as

$$\begin{array}{c}\hfill {\dot{e}}_i\left(t\right)=\left\{\begin{array}{c}-{\mathcal{l}}_i{e}_i\left(t\right)+{\Xi }_N\left(t\right)-A_i\left(t\right),i=1,\dots N-1\hfill \\ -{\mathcal{l}}_i{e}_i\left(t\right)+{\Xi }_N\left(t\right),i=N\hfill \end{array}\right.\end{array}$$

(33)

For $i=N$, define $V_{e_N}\left(t\right)={\displaystyle \frac{1}{2}}{e}_N^2\left(t\right)$, and its differential is

$$\begin{array}{cc}\hfill {\dot{V}}_{e_N}\left(t\right)& ={\dot{e}}_N\left(t\right)e_N\left(t\right)\hfill \\ \hfill & =-{\mathcal{l}}_i{e_N}^2\left(t\right)+{\Xi }_N\left(t\right)e_N\left(t\right)\hfill \\ \hfill & \le \left|{\Xi }_N\left(t\right)\right|\left|e_N\left(t\right)\right|\le {\displaystyle \frac{e_N^2\left(t\right)}{2}}+{\displaystyle \frac{{\Xi }_N^2\left(t\right)}{2}}\le V_{e_N}\left(t\right)+{\displaystyle \frac{{\Xi }_N^2\left(t\right)}{2}}\hfill \end{array}$$

(34)

As ${\Xi }_N$ is bounded for $t\in \left[0,T_d\right)$, $V_{e_N}$ and $e_N$ are also bounded for $t\in \left[0,T_d\right)$ [30].

For $i=N-1$, let ${\overline{\Xi }}_{N-1}\left(t\right)=-{\displaystyle \frac{1}{\gamma }}(e_N\left(t\right)+{\mathcal{l}}_N{e}_N\left(t\right))+{\Xi }_{N-1}\left(t\right)$, which is bounded for $t\in \left[0,T_d\right)$. Then, we have ${\dot{e}}_{N-1}\left(t\right)=-{\mathcal{l}}_{N-1}{e}_{N-1}\left(t\right)+{\overline{\Xi }}_{N-1}\left(t\right)$. Similar to the derivation of $e_N$ which is bounded, we can obtain that $V_{e_{N-1}}\left(t\right)$ and $e_{N-1}\left(t\right)$ are bounded for $t\in \left[0,T_d\right)$, and $e_i\left(t\right)$ is also bounded for $t<T_d$.

Part 2 (Global stabilization): As $E_{{\varpi }_i}\left(t\right)$ converges to a bounded neighborhood of zero according to Theorem 1 for $t>T_d$, (30) is rewritten as

$$\begin{array}{c}\hfill \dot{S_i}\left(t\right)=-k_1{\left|S_i\left(t\right)\right|}^{1/2}\mathrm{sgn}\left(S_i\left(t\right)\right)-{\int }_0^t{k}_2\mathrm{sgn}\left(S_i\left(\tau \right)\right)d\tau \end{array}$$

(35)

According to Lemma 3, (35) is finite-time stable, which means that $S_i\left(t\right)$ converges to a bounded neighborhood of zero for $t<T_s$. From (24), as $s_i\left(t\right)$ and $S_i\left(t\right)$ exhibit the same convergence, $s_i\left(t\right)$ also converges to a bounded neighborhood of zero for $t<T_s$. Then, (33) is rewritten as

$$\begin{array}{c}\hfill {\dot{e}}_i\left(t\right)=-{\mathcal{l}}_i{e}_i\left(t\right)\end{array}$$

(36)

Choose the Lyapunov function $V_{e_i}\left(t\right)={\displaystyle \frac{1}{2}}{e}_i^2\left(t\right)$, and its derivative is

$$\begin{array}{c}\hfill {\dot{V}}_{e_i}\left(t\right)={\dot{e}}_i\left(t\right)e_i\left(t\right)=-{\mathcal{l}}_i{e}_i^2\left(t\right)=-2{\mathcal{l}}_i{V}_{e_i}\left(t\right)\end{array}$$

(37)

The above analysis indicates that $e_i$ converges to a bounded neighborhood of zero, which means that the string stability of the vehicle platoon is guaranteed.

An approach similar to [35] is utilized to demonstrate strong string stability of the entire platoon. As $S_i\left(t\right)$ converges to a bounded neighborhood of zero, we have

$$\begin{array}{c}\hfill \gamma s_i\left(t\right)-s_{i+1}\left(t\right)=0\end{array}$$

(38)

Substituting (23) into (38), we have

$$\begin{array}{c}\hfill \gamma \left((e_i\left(t\right)+{\int }_0^t{\mathcal{l}}_i{e}_i\left(\tau \right)d\tau )\right)=e_{i+1}\left(t\right)+{\int }_0^t{\mathcal{l}}_{i+1}{e}_{i+1}\left(\tau \right)d\tau \end{array}$$

(39)

The Laplace transform of (39) yields

$$\gamma \left(E_i\left(s\right)+{\displaystyle \frac{{\mathcal{l}}_i}{s}}{E}_i\left(s\right)\right)=(E_{i+1}\left(s\right)+{\displaystyle \frac{{\mathcal{l}}_{i+1}}{s}}{E}_{i+1}\left(s\right))$$

(40)

where ${\mathcal{l}}_i$ is the design parameter. Assume ${\mathcal{l}}_i={\mathcal{l}}_{i+1}$ such that $G_i\left(s\right)=E_{i+1}\left(s\right)/E_i\left(s\right)=\gamma$. Then, if $0<\gamma \le 1$ holds, $|G_i\left(s\right)|\le 1$ is satisfied, which indicates that strong string stability is ensured for the vehicle platoon. The proof of Theorem 2 is complete. □

4. Performance Results

The accuracy of the developed FTDO (12) and controller (27) is validated for a vehicle platoon subject to matched and mismatched disturbances through a numerical example in this section. The proposed FTDO-based sliding mode controller is applied to a vehicle platoon that includes a leader vehicle (5) and six follower vehicles (4). The disturbances are given by ${\varpi }_{i1}\left(t\right)=0.5sin\left(t\right)Q$ and ${\varpi }_{i2}\left(t\right)=1.5sin\left(3t\right)Q$, with $Q=e^{-{(t-5-0.2i)}^2}$. The acceleration profile of the leader vehicle is given by [16]

$$\begin{array}{c}\hfill a_0\left(t\right)=\left\{\begin{array}{cc}0.5t\mathrm{m}/{\mathrm{s}}^2,\hfill & 2s<t\le 5\mathrm{s}\hfill \\ -1\mathrm{m}/{\mathrm{s}}^2,\hfill & 10\mathrm{s}<t<12\mathrm{s}\hfill \\ 0\mathrm{m}/{\mathrm{s}}^2,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

The parameters of the FTDO and controller are ${\lambda }_0=8,{\lambda }_1=6,{\lambda }_2=5,{\lambda }_3=4,{\lambda }_4=3,{\varrho }_0=8,{\varrho }_1=6,{\varrho }_2=5,{\varrho }_3=4,{\varrho }_4=1$, $L_i=0.07$, $k_1=1,k_2=0.05,\gamma =0.9,{\mathcal{l}}_i=50$, $h_i=1$ s, ${\delta }_i=10$ m, for $i\in \{1,2,\dots ,6\}$ [16,36].

By utilizing the proposed FTDO (12) and FTDO-based sliding mode controller (27), a numerical example is executed on the vehicle platoon (4) and (5) under the topology shown in Figure 1. The results are illustrated in Figure 4 and Figure 5. As shown in Figure 4a, the mismatched disturbance error exhibits significant fluctuations around the 5 s mark, but rapidly converges to zero after 21 s, indicating that the method proposed in this paper can effectively estimate mismatched disturbances. From Figure 4b, we observe that ${\tilde{\varpi }}_{i2}$ exhibits minor fluctuations at 5 s but rapidly converges to zero, which implies that the matched disturbance is accurately estimated within finite time. The profiles of the position are given in Figure 5a. As there are no crossed and overlapped profiles in Figure 5b, collisions between neighboring vehicles are avoided. The curves of the velocity and velocity error are given in Figure 5b and Figure 5c, respectively. It is evident that velocity synchronization is realized among all follower vehicles and the leader. As shown in Figure 5d, the amplitude of $e_i$ does not increase along the platoon, indicating that strong string stability is achieved. The numerical example results demonstrate that the vehicle platoon can achieve the control goals by the designed FTDO-based sliding mode controller. The efficacy of the proposed FTDO and controller is verified.

To demonstrate the superior performance of the proposed FTDO-based sliding mode controller, the results of the disturbance observer (DO)-based sliding mode controller in [38] are presented in Figure 6 and Figure 7 for comparison. From Figure 6a, we can see that ${\tilde{\varpi }}_{i1}$ fluctuates slightly at 5 s, but quickly converges to zero after 24 s, indicating that mismatched disturbances are effectively estimated. As shown in Figure 6b, ${\tilde{\varpi }}_{i2}$ fluctuates slightly but converges to zero quickly, which illustrates matched disturbances are effectively estimated by DO. From Figure 7a, we can obtain that the collisions between neighboring vehicles are avoided. The velocity and velocity error curves in Figure 7b,c show that the vehicle platoon achieves velocity synchronization. Figure 7d shows that $e_i$ decreases along the platoon, which indicates that strong string stability is achieved.

To achieve a quantitative comparison of the performance between the proposed method and the DO-based sliding mode control strategy, the overshoot and integral of the time-weighted absolute error (ITAE) index of $e_i$ are incorporated as shown in

Table 1. Comparative results under different controllers.

	FTDO-Based Sliding Mode Controller	DO-Based Sliding Mode Controller
Overshoot of $e_i$	0.0732	0.1717
$ITAE={\sum }_{i=1}^6{\int }_0^{t}t\left|e_i\left(\tau \right)\right|d\tau$	0.0047	0.0076

. According to Table 1, we can observe that compared to the DO-based sliding mode controller in [38], the FTDO-based controller can reduce the overshoot of $e_i$ by $57.3\%$ and improve the tracking accuracy by $38.1\%$. A quantitative comparison of the performance between FTDO and DO is presented in

Table 2. Comparative results under different DOs.

	FTDO	DO
$ITAE={\sum }_{i=1}^6{\int }_0^{t}t\left|{\tilde{\varpi }}_{i1}\left(\tau \right)\right|d\tau$	0.0176	0.0260
$ITAE={\sum }_{i=1}^6{\int }_0^{t}t\left|{\tilde{\varpi }}_{i2}\left(\tau \right)\right|d\tau$	0.1156	0.1398

. According to Table 2, compared to the DO in [38], FTDO can improve the estimation accuracy of the mismatched disturbance by $31.2\%$ and that of the matched disturbance by $17.3\%$.

By analyzing the results, we can conclude that the proposed control strategy enhances the tracking accuracy and reduces the overshoot of $e_i$, thereby enabling the vehicle platoon to achieve better control performance. Moreover, the proposed FTDO can effectively improve the estimation accuracy of the disturbances.

5. Conclusions

This study proposes a novel FTDO-based sliding mode control scheme for a vehicle platoon affected by matched and mismatched disturbances to accomplish control goals. The novel FTDO is designed to estimate the two types of disturbances directly without the need to know the precise upper bounds of the disturbances. To mitigate the adverse impact of disturbances, a feedforward compensation term derived from the disturbance estimation is incorporated into the FTDO-based sliding mode control scheme to address the control performance degradation caused by disturbances and to ensure the vehicle platoon’s stability. The example results validate the efficacy of the proposed FTDO-based sliding mode control scheme. Note that we study the control problem of vehicle platoons with mismatched disturbances under the assumption of ideal communication conditions, while neglecting the impact of time-varying communication delays. In future work, it will be of value to investigate control strategies for vehicle platoons with communication delays.