Keywords

1 Introduction

Cooperative Adaptive Cruise Control (CACC) utilizes vehicle-to-vehicle (V2V) communication and on-board sensors like a radar to maintain a desired headway to the preceding vehicle. CACC’s ability to maintain short following distances while attenuating disturbances through the vehicle string (string stability) can enhance traffic flow and safety [5]. To enable adoption of the technology, the controller should be able to deal with heterogeneities in the platoon. Furthermore, actuation delays of the ego vehicle can be detrimental to the performance of controllers that are designed without taking into account this delay [2].

In this paper, we present a prediction based control approach to enable control of heterogeneous platoons where the ego vehicle experiences a delay in the driveline. We use a prediction of the (effects) of the ego vehicle’s acceleration (on the error dynamics) to obtain a controller that is ISS with respect to the preceding vehicle’s acceleration.

The outline of the paper is as follows. The controller design is presented in Sect. 2. An analysis of the string stability of the system, taking into account actuation and communication delays, is presented in Sect. 3. To validate the controller, simulations of a platoon employing the controller are presented in Sect. 4. Finally, the conclusions and recommendation for future research are presented in Sect. 5.

2 Controller Design

Consider a heterogeneous string of n vehicles as depicted in Fig. 1, where each vehicle’s dynamics are described by

$$\begin{aligned} \dot{q}_i (t) &= v_i (t) & \nonumber \\ \dot{v}_i (t) &= a_i (t) & i = 1, 2, ..., n \\ \dot{a}_i (t) &= -\tfrac{1}{\tau _i} a_i(t) + \tfrac{1}{\tau _i} u_i(t - \phi _i). & \nonumber \end{aligned}$$
(1)

Here, \(q_i\), \(v_i\), \(a_i\) denote the position, velocity and acceleration of vehicle i respectively, for the platoon of length \(n \in \mathbb {N}^+\) vehicles. The desired acceleration \(u_i\) is considered the input to the system, and \(\tau _i > 0\) and \(\phi _i \ge 0\) are a time constant and actuation delay associated with the driveline of vehicle i respectively.

The vehicle-following objective, i.e, follow the predecessor at constant headway \(h_i > 0\), can be captured in the error definition

$$\begin{aligned} e_i(t) = q_{i-1}(t) - q_i(t) - h_i v_i(t) - r_i - L_i, \end{aligned}$$
(2)

where \(r_i \ge 0\) is a constant to account for a certain distance at standstill, and \(L_i\) is the length of vehicle i.

Fig. 1.
figure 1

Heterogeneous string of vehicles equipped with CACC.

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2.1 Controller Design

Defining the coordinate transformation along the lines of [3]

$$\begin{aligned} x_1(t) &= e_i(t) = q_{i-1}(t) - q_i(t) - h_i v_i(t) - r_i - L_i\nonumber \\ x_2(t) &= \dot{e}_i(t) = v_{i-1}(t) - v_i(t) - h_i a_i(t) \\ x_3(t) &= v_{i-1}(t) - v_i(t) , \nonumber \end{aligned}$$
(3)

the error (2) and its dynamics are given by

$$\begin{aligned} \dot{x}_1(t) &= x_2(t) \nonumber \\ \dot{x}_2(t) &= a_{i-1}(t) - \tfrac{h_i - \tau _i}{h_i \tau _i} \left[ x_2(t) - x_3(t) \right] -\tfrac{h_i}{\tau _i} u_i(t-\phi _i) \\ \dot{x}_3(t) &= \tfrac{1}{h_i} x_2(t) - \tfrac{1}{h_i} x_3(t) + a_{i-1}(t). \nonumber \end{aligned}$$
(4)

We propose a prediction based control law for vehicle i, given by

$$\begin{aligned} u_i(t) &= \left[ 1 - \tfrac{\tau _i}{h_i} \right] \hat{a}_i^{\phi _i} (t) + \tfrac{\tau _i}{h_i} a_{i-1}(t) - \tfrac{\tau _i}{h_i} \bar{u}_i(t), \end{aligned}$$
(5a)
$$\begin{aligned} \text {with} \ \hat{a}_i^{\phi _i} (t) & = \exp \left( -\tfrac{\phi _i}{\tau _i} \right) a_i(t) + \int _{t - \phi _i}^{t} \tfrac{1}{\tau _i} \exp \left( - \tfrac{t-\sigma }{\tau _i} \right) u_i(\sigma ) \textrm{d} \sigma \end{aligned}$$
(5b)
$$\begin{aligned} \text {and } \ \bar{u}_i (t) &= - \begin{bmatrix} k_p & k_d \end{bmatrix} \left( \begin{bmatrix} 1 & \phi _i \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} + \int _{t - \phi _i}^{t} \begin{bmatrix} t-\sigma \\ 1 \end{bmatrix} \bar{u}_i(\sigma ) \textrm{d} \sigma \right) , \end{aligned}$$
(5c)

where \(k_p\) and \(k_d\) are controller gains. To compensate for the delay \(\phi _i\), the inputs \(u_i\) and feedback-control actions \(\bar{u}_i\) over the past \(\phi _i\) seconds are used to construct a prediction \(\hat{a}_i^{\phi _i}\) of the acceleration \(a_i\) and evolution of the system (4). Applying the prediction based controller (5) on the system (4), results in closed-loop dynamics

$$\begin{aligned} \dot{x}(t) = \begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \\ \dot{x}_3(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ -k_p & -k_d & 0 \\ 0 & \frac{1}{h_i} & -\frac{1}{h_i} \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 0 \\ w(t) \\ a_{i-1} (t) \end{bmatrix}, \end{aligned}$$
(6)

where

$$\begin{aligned} w(t) = & \ a_{i-1}(t)-a_{i-1}(t-\phi _i) -k_p\int _0^{\phi _i}\sigma [a_{i-1}(t-\sigma )-a_{i-1}(t-\phi _i-\sigma )]{{\,\mathrm{d\!}\,}}\sigma \\ &-k_d\int _0^{\phi _i}[a_{i-1}(t-\sigma )-a_{i-1}(t-\phi _i-\sigma )]{{\,\mathrm{d\!}\,}}\sigma . \end{aligned}$$

The closed loop system (6) is input to state stable (ISS) with respect to the predecessors acceleration \(a_{i-1}(t)\) for \(k_p > 0\), \(k_d > 0\). That is, x(t) remains bounded for bounded \(a_{i-1}(t)\), and x(t) converges to zero when \(a_{i-1}(t)\) converges to zero. It is worth to note that including the preceding vehicle’s future acceleration, \(a_{i-1}(t + \phi _i)\), in the control action (5a) instead of the current acceleration, \(a_{i-1}(t)\), would eliminate the disturbance w(t). However, the future acceleration of the preceding vehicle is generally not available. Instead, a prediction of the preceding vehicle’s future acceleration \(\hat{a}_{i-1}(t + \phi _i)\) could be used to improve performance. Constructing such a prediction is considered outside the scope of this paper and left for future research.

3 String Stability

The attenuation of disturbances through the vehicle string is an important design objective of vehicle platoons. To analyze the string stability of the system employing controller (5) we adopt the notion of \(\mathcal {L}_2\) string stability [4]. To this end, we define the transfer function relating the velocity of (a preceding) vehicle k to the velocity of vehicle \(i > k\), referred to as the String Stability Complementary Sensitivity (SSCS), as \(\varGamma _i^k = \tfrac{v_i(s)}{v_{k}(s)}\), where \(v_i(s)\) is the Laplace transform of \(v_i(t)\). The platoon is \(\mathcal {L}_2\) string stable if [4]

$$\begin{aligned} \Vert \varGamma _i^k(s) \Vert _{\mathcal {H}_\infty } = \sup _{\omega \in \mathbb {R}} |\varGamma _i^k(j\omega )| \le 1, \end{aligned}$$
(7)

where \(\Vert \cdot \Vert _{\mathcal {H}_\infty }\) denotes the \(\mathcal {H}_\infty \) norm. Note that the SSCS for the string of vehicles starting at index k and ending with index i is equal to the product of the SSCS of the individual vehicles with respect to their direct predecessor: \(\varGamma _i^k = \prod _{j=k+1}^i \varGamma _{j}^{j-1}.\) Consequently, if each individual vehicle i satisfies condition (7) with respect to its direct predecessor \(i-1\), it implies (7) is satisfied for the entire platoon. Therefore, we focus the analysis of string stability on an individual vehicle with its direct predecessor, for which condition (7) should hold.

Fig. 2.
figure 2

Minimum required headway \(h_i\) for (9) to satisfy (7) given \(k_p\), \(k_d\) with \(\phi _i = 0.15 \text { s}\), \(\theta _i = 0.02 \text { s}\).

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3.1 String Stability with Communication Delays

The control action (5) requires the preceding vehicle’s acceleration \(a_{i-1}\), which cannot be directly measured with sensors on-board of vehicle i. This means that in practice V2V communication is used to obtain this information. Including a communication delay \(\theta _i \ge 0\) in the control action (5a) gives

$$\begin{aligned} u_i(t) = \left[ 1 - \tfrac{\tau _i}{h_i} \right] \hat{a}_i^{\phi _i} (t) + \tfrac{\tau _i}{h_i} a_{i-1}(t - \theta _i) - \tfrac{\tau _i}{h_i} \bar{u}_i(t). \end{aligned}$$
(8)

The closed-loop system (6) employing (8) results in the SSCS as

$$\begin{aligned} \varGamma _i^{i-1}(s) = \frac{(s^2 + k_d s + k_p) e^{-\theta _i s} + \left[ (k_d + k_p \phi _i)s + k_p \right] \left[ 1 - e^{-(\phi _i + \theta _i)s}\right] }{(h_i s + 1)(s^2 + k_d s + k_p)} e^{-\phi _i s}. \end{aligned}$$
(9)

The SSCS (9) can be used to determine appropriate controller gains \(k_p\) and \(k_d\) to satisfy (7), given the actuation and communication delays, \(\phi _i\) and \(\theta _i\) respectively.

4 Tuning and Simulation

We experimentally validated the discrete time equivalent of the CACC controller with a platoon of two full-scale vehicles in [1]. There, we show that the closed-loop response of the vehicle employing controller (5) indeed behaves according to (6). In this section, we consider the tuning of the controller with respect to string stability and illustrate the results with time-domain simulations.

4.1 Tuning for String Stability

To determine an appropriate tuning for the controller that achieves string stability, given the characteristics of the experimental vehicle from [1] (time constant \(\tau _i = 0.067\) seconds, actuation delay \(\phi _i = 0.15\) seconds and a communication delay \(\theta _i = 0.02\) seconds), we use a bi-section algorithm to numerically determine the minimum required headway \(h_i\) for which condition (7) is satisfied for (9) for a grid of \(k_p\) and \(k_d\). The resulting Fig. 2 shows the minimum headway that can be achieved, greatly depends on the choice of controller gains. Especially \(k_d\) should be chosen sufficiently large to be able to employ small headways.

Table 1. Vehicle parameters of the platoon used in the simulation.
Full size table
Fig. 3.
figure 3

Magnitude of SSCS (9) and time domain velocity response of the platoon of six vehicles (black - light gray: \(i = 1, 2, ..., 6\)) with distinct \(\tau _i\) and \(\phi _i\) as listed in Table 1, employing (5).

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4.2 Simulation

To illustrate the effectiveness of the prediction based controller, we consider a heterogeneous platoon of \(n = 6\) vehicles, with the parameters as listed in Table 1. Here, vehicle 2 and 3 have the characteristics of the experimental vehicles from [1] and [4] respectively. The leader vehicle is modeled without actuation delay, to illustrate the effectiveness of the controller in a setting where the preceding vehicle does not experience a delay. The communication delay \(\theta _i = 0.02 \text { seconds}\) is assumed to be identical for all vehicles. Furthermore, all vehicles use the same tuning of the feedback controller with \(k_p = 1\) and \(k_d = 4\). At the adopted headway \(h_i = 0.5 \text { seconds}\), the string stability condition (7) is satisfied for each vehicle in the platoon, as can be seen in the magnitude plot of the SSCS in Fig. 3a.

Figure 3b shows a time domain simulation of the platoon employing the discrete time equivalent of (5), obtained by discretizing system with ZOH at a sample time \(T_s = 0.01 \text { seconds}\) as described in [1]. The leader vehicle is prescribed an acceleration step of \(u_1 = 1 \text { m/s}^2\) for \(t \in [5, 10]\), followed by a negative step \(u_1 = -1 \text { m/s}^2\) for \(t \in [15, 20]\). No velocity overshoots are observed, indicating the platoon indeed exhibits string stable behavior.

5 Conclusion

In this paper, we present a prediction based control approach for platoons with actuation delay. The controller does not require any information of the driveline dynamics of the preceding vehicle, making it suitable for platoons that are heterogeneous with respect to both the driveline time constant and actuation delays. An analysis of the string stability properties considering both actuation and communication delays is presented, which shows the controller is able to achieve string stability at short headways (smaller than 0.5 seconds), given the properties of the experimental setup. A simulation confirms the effectiveness of the controller.

The presented prediction based controller only considers the predicted states of the ego vehicle and uses the current (communicated) state of the preceding vehicle. Consequently, a disturbance acts on the system which is a function of the preceding vehicle’s acceleration. Although the system is ISS with respect to this disturbance regardless of the size of the actuation delay, a certain minimum headway time is required for string stability. Future research entails including the preceding vehicle’s acceleration in a different manner in the control action to decrease or eliminate the minimum headway required for string stability. For instance, by predicting the preceding vehicle’s future acceleration. Finally, in this paper only simulations are used to demonstrate the efficacy of the controller. For future research, we plan to conduct experiments with full-scale vehicles to validate the performance with respect to string stability.