Cooperative Extended State Observer Based Control of Vehicle Platoons With Arbitrarily Small Time Headway

Vehicle platoon can improve road utilization rate and reduce fuel consumption effectively (Alam, 2011). Therefore, it has attracted worldwide attention (Shladover et al., 1991; Coelingh & Solyom, 2012). From networked control perspective, Li et al. (2015) divided a vehicle platoon system into four basic modules: node dynamics, information flow topology, formation geometry, and distributed controller, among which the formation geometry greatly affects the stability of the vehicle platoon system. Formation geometry is determined by the spacing policy. The constant time headway policy is considered by Klinge & Middleton (2009); Naus et al. (2010); Xiao & Gao (2011); Ploeg et al. (2014); Darbha et al. (2017) and it is well known that a large time headway is conducive to string stabilities of vehicle platoon system (Rajamani & Zhu, 2002; Naus et al., 2010), however, the larger the time headway is, the greater the inter-vehicle distance becomes, which leads to a lower road utilization rate.

It is of interest to guarantee the stability of the vehicle platoon with small time headways. Many researches showed that using the accelerations of the preceding vehicles can reduce the lower bound of the time headway required (Rajamani & Zhu, 2002; Zhou & Peng, 2004; Naus et al., 2010; Darbha et al., 2017; Al-Jhayyish & Schmidt, 2018). All the above works assumed that the accelerations of the preceding vehicles are obtained by the wireless communication network accurately, however, accurate communication doesn’t exist in practical applications. In addition, a control law which relies on communication data runs the risk of failure when the inter-vehicle communication network breaks down under attack. Therefore, a cooperative control law with a small time headway, which can ensure both closed-loop and string stabilities without relying on the inter-vehicle wireless communication network, is of especially significance for practical applications. Ploeg et al. (2015) and Wen & Guo (2019) proposed methods to estimate the acceleration differences between adjacent vehicles, respectively. The control laws in Ploeg et al. (2015) and Wen & Guo (2019) are indeed independent of wireless communication networks. The closed-loop and string stabilities are analyzed by numerical simulations in Ploeg et al. (2015) and Wen & Guo (2019), especially, Ploeg et al. (2015) considered a third-order linear model with input delays, in which the simulation results show that the string stability is not guaranteed if the time headway is small enough.

In this paper, we study platoon control of vehicles with linear third-order longitudinal dynamics under the constant time headway policy. We design a distributed cooperative control law for each follower vehicle only using its own velocity, acceleration, inter-vehicle distance and velocity difference with respect to its immediate predecessor, all of which can be obtained by on-board sensors. Firstly, the models of velocity difference between adjacent vehicles are established, based on which distributed cooperative extended state observers are designed to estimate the acceleration differences between adjacent vehicles. Then based on these estimates, distributed cooperative controllers are designed for follower vehicles. The controller for each follower consists of two parts, where the first part is a feedback term consisting of inter-vehicle distance error and its differential, the other is a feedforward term consisting of the estimate of the acceleration of the preceding vehicle. Thus, the whole cooperative control law only uses the data obtained by on-board sensors without wireless communication networks.

We analyze both closed-loop and $\mathcal{L}_{2}$ string stabilities of the vehicle platoon system. The closed-loop system matrix is decomposed into two matrices, one of which is related to feedback parameters of the distributed controllers, and the other is regarded as the perturbation matrix, which is related to feedforward parameters. Then, we give the range of the control parameters to ensure the closed-loop stability by using the stability theory of perturbed linear systems. In the frequency domain, we analyze the transfer functions which describe the inter-vehicle distance error propagation, and give the range of the control parameters to ensure the $\mathcal{L}_{2}$ string stability. We show that one can design control parameters properly to ensure both closed-loop and $\mathcal{L}_{2}$ string stabilities for any given positive time headway. Our method for estimating the acceleration differences between adjacent vehicles is based on distributed cooperative extended state observers, which is totally different from those in Ploeg et al. (2015) and Wen & Guo (2019). Besides, we give the explicit range of control parameters quantitatively related to the system parameters to ensure both closed-loop and $\mathcal{L}_{2}$ string stabilities for any given positive time headway.

We analyze the robustness of the proposed control law by both theoretical study and numerical simulations. (i) The proposed control law guarantees the exponential closed-loop stability, thus it is naturally robust against bounded sensing noises. (ii) We show that the proposed cooperative control law based on the ideal vehicle model can still ensure the closed-loop and $\mathcal{L}_{2}$ string stabilities provided the parameter uncertainties in the vehicle model are sufficiently small. (iii) In the numerical simulations, we consider the same vehicle model with input delays and the same time headway as Ploeg et al. (2015), with additional sensing noises and uncertain model parameters. Simulation results show that the closed-loop and $\mathcal{L}_{2}$ string stabilities can be guaranteed by the proposed control law based on the ideal vehicle model.

The rest of this paper is organized as follows. The vehicle model and the control objectives are presented in Section II. In Section III, we first establish dynamic models based on velocity differences between every adjacent vehicles, then distributed cooperative extended state observers are designed to estimate the acceleration differences between adjacent vehicles. Finally, distributed cooperative controllers for follower vehicles are designed. In Section IV, we give the range of control parameters for the closed-loop and $\mathcal{L}_{2}$ string stabilities. In Section V, we analyze the robustness of the proposed control law against model parameter uncertainties. Numerical simulations are carried out in Section VI. In Section VII, we give some conclusions.

The following notation will be used throughout this paper. For a given matrix $A$, its 2-norm and minimum singular value are denoted by $\|A\|$ and $S_{n}(A)$, respectively; $diag(A)$ denotes a block diagonal matrix whose diagonal blocks are all matrix $A$; $\mathbb{C}$ denotes the complex domain; $\mathbb{R}$ denotes the real domain; $O$ and $I$ denote a zero matrix and an identity matrix with an appropriate size, respectively. For a given transfer function $G(s)$, its $\infty$-norm and inverse Laplace transform are denoted by $\|G(s)\|_{\infty}$ and $\mathscr{L}^{-1}[G(s)]$, respectively.

We consider the following third-order linear vehicle model which is commonly used in the vehicle platoon control (Rajamani & Zhu, 2002; Ploeg et al., 2015; Zheng et al., 2016; Wen & Guo, 2019).

where $p_{i}(t)$, $v_{i}(t)$ and $a_{i}(t)$ are the position, velocity and acceleration of the $i$th vehicle, respectively, $u_{0}(t)$ is the control input of the leader vehicle and $u_{i}(t)$ is the control input of the $i$th follower vehicle to be designed, $i=1,2,...,N$. The constant $\tau$ is the inertial delay of vehicle longitudinal dynamics. We consider the constant time headway spacing policy. The expected inter-vehicle distance is denoted by

where the constants $r$ and $h$ are the standstill distance and the time headway, respectively. The inter-vehicle distance error is given by,

which is the difference between the actual inter-vehicle distance and the expected inter-vehicle distance. Denote $G_{ei}(s)=\mathscr{E}_{i}(s)/\mathscr{E}_{i-1}(s)$, where $\mathscr{E}_{i}(s)$ is the Laplace transform of $e_{i}(t)$. The control objectives are to design $u_{i}(t)$, $i=1,2,\ldots,N$, for follower vehicles so that the following two objectives are satisfied.

A. closed-loop stability: all the follower vehicles tend to move at the same velocity as the leader vehicle and the inter-vehicle distance errors converge to zero, i.e. $\lim\limits_{t\to\infty}[v_{i}(t)-v_{0}(t)]=0$, $\lim\limits_{t\to\infty}e_{i}(t)=0,i=1,2,\ldots,N$.

B. $\mathcal{L}_{2}$ string stability: the inter-vehicle distance errors are not amplified during the backward propagation along the platoon, i.e. $\|G_{ei}(s)\|_{\infty}\leq 1,\;i=2,...,N.$

Suppose that there is no wireless communication between vehicles. We consider predecessor following topology, and each follower vehicle in the platoon relies on the on-board sensors to measure its own velocity, acceleration, the inter-vehicle distance and the velocity difference with respect to its immediate predecessor. However, the sensor of each follower vehicle cannot measure the acceleration of its immediate predecessor. Instead, we can design an observer to estimate the acceleration difference between adjacent vehicles. The extended state observer (ESO) was first put forward by Han (1995), whose core idea is to expand the unmodeled dynamics into new state and then according to the new state equation, an extended state observer is designed to estimate all states of the system. However, the extended state observer proposed by Han (1995) is nonlinear, which has difficulty in parameter tuning and stability analysis. Gao (2003) put forward a linear extended state observer, which simplifies parameter tuning and is also beneficial for stability analysis. In this paper, we design a linear extended state observer to estimate the acceleration difference between adjacent vehicles.

According to (1), the models of the velocity difference between adjacent vehicles are given by

where

Here, $q_{i}(t)$ is the unmodeled dynamics, which contains the control input and the acceleration of $i-1$th vehicle that cannot be measured directly by the $i$th vehicle. We design a linear extended state observer

where $z_{2,i}(t)$ is the estimate of the acceleration difference $a_{d,i}(t)$ between the $i-1$th vehicle and the $i$th vehicle. The constants $\beta_{1}>0$, $\beta_{2}>0$ and $\beta_{3}>0$ are the observer gains to be designed. Then by the acceleration of $i$th vehicle, the estimate of the acceleration of $i-1$th vehicle is given by $z_{2,i}(t)+a_{i}(t)$. The controller of $i$th follower vehicle is designed as

where $k_{p}>0$, $k_{v}>0$, $k_{a}>0$ are the control parameters to be designed. The controller (3) consists of two parts. The first part $k_{p}e_{i}(t)+k_{v}(v_{d,i}(t)-ha_{i}(t))$ is the feedback item, which consists of the inter-vehicle distance error between the adjacent vehicles and its differential; while the second part $k_{a}(z_{2,i}(t)+a_{i}(t))$ is the feedforward item, which consists of the estimate of the acceleration of the $i-1$th vehicle. It can be seen that the extended state observer (9) and the controller (3) only use the information obtained by on-board sensors of followers.

In reality, the velocity of the leader vehicle will reach at a steady state within a finite time $t_{f}$, that is, there exists $t_{f}$, such that $u_{0}(t)=0$, $t\geq t_{f}$. We make the following assumption.

Denote $\Psi(k_{p},k_{v})={\left[\begin{array}[]{cc}\Psi_{11}&O\\ \Psi_{21}&\Psi_{22}\end{array}\right]}\in\mathbb{R}^{6N\times 6N}$, where $\Psi_{11}$, $\Psi_{21}$ and $\Psi_{22}$ are $3N$ dimensional matrices with $\Psi_{22}=diag(\mathcal{H})$,

For the closed-loop and $\mathcal{L}_{2}$ string stabilities, we have the following theorems.

The proof of Theorem 1 is given in Appendix B.

The proof of Theorem 2 is given in Appendix C.

The control law (9) and (3) is designed based on the ideal vehicle model (1) with completely known parameters. In practice, the vehicle model may be inaccurate. In this section, we will give the range of control parameters to ensure the closed-loop and $\mathcal{L}_{2}$ string stabilities when there are parameter uncertainties in the vehicle model. It is shown that the control law (9) and (3) based on the ideal vehicle model can ensure the closed-loop and $\mathcal{L}_{2}$ string stabilities with small model parameter uncertainties.

Suppose that the real vehicles have the following dynamics

where the definitions of $u_{i}(t),i=0,1,...,N$ are the same as in (1). The constant $\tau$ is the known nominal inertial delay of vehicle longitudinal dynamics. The constant $\epsilon_{i}$ is the parameter uncertainty. We assume that $|\epsilon_{i}|\leq\overline{\epsilon}$, $i=0,1,...,N$, where $\overline{\epsilon}\in[0,1/\tau)$. We have the following theorems.

The proof of Theorem 3 is given in Appendix D.

The proof of Theorem 4 is given in Appendix E.

Suppose there are 1 leader vehicle and 5 follower vehicles with the following dynamics with parameter uncertainties

where $\tau=0.1$. The parameter uncertainties are given by $\epsilon_{0}=-0.8$, $\epsilon_{1}=0.1$, $\epsilon_{2}=0.5$, $\epsilon_{3}=-0.2$, $\epsilon_{4}=0.65$, $\epsilon_{5}=-0.3$.

To make the behavior of the leader vehicle close to a real vehicle, we construct a virtual vehicle in CarSim and record its acceleration. Then we use the recorded acceleration as the expected acceleration of the leader vehicle in the following simulations. The initial velocities are given by $v_{i}(0)=10\,m/s$, $i=0,1,...,5$. The initial accelerations are given by $a_{i}(0)=0\,m/s^{2}$, $i=0,1,...,5$. The initial positions are taken as $p_{0}(0)=30\,m$, $p_{1}(0)=24\,m$, $p_{2}(0)=18\,m$, $p_{3}(0)=12\,m$, $p_{4}(0)=6\,m$, $p_{5}(0)=0\,m$. The standstill distance is given by $r=3\,m$. Let $h=0.3s$. We choose $k_{p}=8$, $k_{v}=40$, $k_{a}=1.2$ and $\omega_{o}=15$, $\beta_{1}=45$, $\beta_{2}=675$ and $\beta_{3}=3375$. The evolution of $|G_{vi}(j\omega)|$ and $\mathscr{L}^{-1}[G_{vi}(s)]$ are shown in Fig.1.