Controller Design for Optimizing Fuel Consumption of Truck Platoon on Hilly Roads

Abstract

Platoons consisting of automated and connected vehicles show great potential in reducing fuel or energy consumption. However, the fuel consumption optimization problem for truck platoons traveling on hilly roads has not been investigated thoroughly. To address that problem, a hierarchical control framework is proposed in this paper as follows: (1) The supervising layer is responsible for generating the fuel-oriented optimal speed profile based on the terrain information; (2) The distributed layer consists of an LQR feedback controller, a DMPC feedforward controller and a tube integration method to integrate the two controllers; it receives the optimal speed profile from the supervising layer and yields the control input to the individual vehicle. In this paper, a novel optimal speed profile generation method is proposed, a novel integration of tube method is applied, and the stability performance is analyzed rigorously. Simulations based on a real hilly road are conducted, and the performance of the proposed controller is evaluated regarding the platoon stability, fuel consumption and computation efficiency. The results of the simulation show that the controller is capable of maintaining the string stability of the truck platoon and reducing fuel consumed on hilly roads while improving computation efficiency.

1. Introduction

Recent years witnessed rapid improvements in autonomous vehicles [1,2,3,4]. Fully automated vehicles, however, are still yet to come into practice. As a transition from human-driven vehicles to fully automated vehicles, vehicle platoons offer the benefit of lower fuel consumption, increased road capacity, fewer road traffic accidents and improved traffic efficiency [5,6]. To exploit the advantage of vehicle platoons, many countries and researchers have conducted relevant research regarding performance exploration and controller design [7,8,9].

For a truck platoon, the most attractive attribute lies in the reduction in fuel consumption, which will benefit such fields as the freight industry if aided by advanced relevant technologies [10,11]. The air drag reduction effect in platoons is investigated in [12,13,14,15], and corresponding controllers are devised to take advantage of this effect, although the fuel reduction performance is doubtful since the optimal operating point for fuel efficiency is not guaranteed for tracking. A method of reducing energy consumption by predicting the preceding vehicle’s movement is presented in [16]; the road slope is also taken into consideration in that method. The terrain information is also considered in controlling electric vehicles in an energy-efficient way [17]. Another method aimed at improving fuel economy is proposed in [18]; a hierarchical architecture of the controller, together with the notion of a fuel-optimal speed profile, is presented to reduce energy consumption on hilly roads. Those methods listed above are all merely concerned with energy optimality and the individual vehicle safety in the platoon; the string stability performance, however, is neglected theoretically and practically. String stability is an essential factor in ensuring the safety and overall stability of the platoon; lack of consideration of string stability can cause a collapse of the platoon.

Many control strategies have been presented for vehicle platoons, e.g., sliding mode control [19], $H_{\infty }$ based control [20,21] and some other control methods [5,22]. Among them, the distributed model predictive control (DMPC) mostly operates in the body of the platoon, and it has the advantage of dealing with multiple constraints—both equality and inequality [23]. Aiming to exploit the merit of fuel economy in vehicle platoons, some researchers have transformed energy consumption into a kind of optimal problem in DPMC, which is also referred to as economic DMPC. For instance, He et al. [24] designed a distributed economic MPC to improve the fuel economy and stability of the platoon subject to nonlinear dynamics and safety constraints. Hu et al. [25] proposed a control strategy by combining the switching feedback control and the economic MPC to ensure asymptotic stability and string stability; the simulation result shows a maximum of 6.84% fuel consumption reduction compared to purely tracking-oriented methods. Although single-vehicle stability and the string stability of the nominal systems can be ensured in these methods, the potential of energy consumption reduction has not been fully exploited for engine-driven vehicles on hilly roads, since the terrain information is not taken into account in these works, and time-domain-based methods tend to fail the spatially varying road topology. When the external disturbances satisfy the PE (persistent excitation) condition [26], DMPC-like methods may fail to meet the state constraints and lead the platoon to an unstable situation.

To avoid the computational burden of DMPC, tube-based methods for vehicle platoons have been applied to handle constraints and disturbances [27,28]. The essence of tube methods lies in the sequence of constraint sets determined by the feedforward control, while the feedback control restricts the states to the interior of the tube (sequence of set). Aiming at maximizing the fuel consumption reduction while maintaining the stability of the platoon, the fuel-oriented speed planning together with the tube-based DMPC control method in the form of a hierarchical framework is proposed in this paper. The hierarchical control framework consists of two layers: the supervising layer is responsible for generating a fuel-optimal reference speed profile, and the distributed layer maps the reference speed profile into actual vehicle commands by taking the vehicle states into account.

The main difference from former research and corresponding contributions are summarized as follows: (1) A novel method of the generation of the platoon reference speed profile is proposed, where a novel fuel consumption minimization method considering the terrain information is proposed with the trade-off between fuel consumption and traffic efficiency. Different from [27,28], this method can minimize the fuel consumption of the platoon as a whole, without a decrease in the traffic flow. (2) A novel integration of the tube method is applied, and the stability performance is analyzed rigorously compared to the DMPC methods [24,25,29]; it can significantly reduce the computational burden and the controller conservation while maintaining the disturbance string stability of the platoon.

The following sections of this paper are organized as follows: firstly, some preliminaries including necessary lemmas and the notation explanation will be given, together with the illustration of the dynamics modeling of the platoon and the control objectives; following that, the hierarchical control framework will be elaborated upon in the section of controller design, followed by some simulations to verify the control performance; finally, some conclusions will be drawn.

2. Preparation

In this section, the lemmas necessary for the proof of controller stability will be listed, and the notations utilized in this paper will also be presented. To better illustrate the lemmas, firstly we model the dynamics of the platoon and present the necessary parameters and variables to be used in the following sections.

2.1. Longitudinal Dynamics Modeling

The vehicle is subject to driving and resistance moments in the single-vehicle driving scenario, while the longitudinal dynamics of a single vehicle are affected by the relative motion of other vehicles in a platoon. Thus, to represent the states of a single vehicle and the vehicles in a platoon, we introduce a two-dynamic-model nonlinear model for a single vehicle and a linear model in a state-space form.

Taking the driving torque of the engine and four kinds of resistance moments into consideration and applying Newton’s second law, we have

$$m{a}_x=\frac{T_e{i}_t}{r}-\frac{1}{2}{A}_w{C}_D\rho v_x^2-mg\sin \theta -mgf\cos \theta$$

(1)

where $m$ is the vehicle mass, $a_x$ is the longitudinal acceleration, $T_e$ is the engine output torque, $i_t$ is the overall gear ratio of the transmission system, $r$ denotes the wheel effective radius, $A_w$ is the characteristic area of the vehicle, $C_D$ is the air drag coefficient, $\theta$ is the slope angle, and $f$ represents the rolling resistance. Note that the air drag is strongly affected by the vehicle gap from the immediate predecessor; in this paper, we adopt the empirical model developed in [18], as follows

$$C_D=C_a\left(1-\frac{C_b}{C_c+d_{i,i-1}}\right)$$

(2)

The expression of (2) implies that a smaller gap from the predecessor leads to less air drag. This observation is quite essential in deriving the fuel-optimal controller of the platoon, as will be detailed in Section 2. To derive the controller, the dynamics model of the platoon is introduced as follows,

$$\begin{array}{l}{\dot{p}}_i=v_i\\ {\dot{v}}_i=a_i\\ {\dot{a}}_i=\frac{1}{\tau }{u}_i-\frac{1}{\tau }{a}_i\end{array}$$

(3)

where $p_i,v_i,a_i$ denote the position, longitudinal velocity and acceleration of vehicle $i \left(i=1,2,3\cdots \right)$, respectively. $\tau$ is the time constant of the driving system, which implies the actuation lag of the vehicle. $u_i$ denotes the control input generated from the controller. Here, we define the set of $\mathcal{N}$ following vehicles in the platoon as $\mathcal{V}$, then $i\in \mathcal{V}, 1\le i\le \mathcal{N}$. Meanwhile, we define the leader of the platoon as vehicle 0 ($i=0$).

Equation (3) is written in a continuous form; to apply the controller, we need to discretize the model, and then, applying a zero-order hold mechanism [30], we have

$$x_i\left(k+1\right)=A{x}_i\left(k\right)+B{u}_i(k)$$

(4)

where $x_i={\left[p_i,v_i,a_i\right]}^T$ denotes the states of the vehicle $i$, $k=0,1,2\cdots$ denotes the time instant; suppose a sampling instant $\iota$, the system matrices $A$ and $B$ are expressed as

$$A=\left[\begin{array}{ccc}1& \iota & 0\\ 0& 1& \iota \\ 0& 0& e^{-\frac{\iota }{\tau }}\end{array}\right], B=\left[\begin{array}{c}0\\ 0\\ 1-e^{-\frac{\iota }{\tau }}\end{array}\right]$$

A nominal system is one without the effect of disturbance; the nominal system form of system (4) can be expressed as

$${\overline{x}}_i\left(k+1\right)=A{\overline{x}}_i\left(k\right)+B{\overline{u}}_i(k)$$

(5)

The difference between the actual values of the states and input is the so-called disturbance; note that system (4) is almost certainly subject to disturbance and dynamic uncertainty. Then, by adding the amplitude-bounded uncertainty term ${\varpi }_i(k)$ to (4), and subtracting the nominal system (5), we have

$${\tilde{x}}_i\left(k+1\right)=A{\tilde{x}}_i\left(k\right)+B{\tilde{u}}_i(k)+{\varpi }_i(k)$$

(6)

where the tilde form denotes the disturbance form of the corresponding term, and ${\varpi }_i(k)={\left[{\varpi }_{pi},{\varpi }_{vi},{\varpi }_{ai}\right]}^T\in {\mathbb{W}}_i$, ${\mathbb{W}}_i$ is an interval with limited upper and lower bounds.

The following relations can be derived,

$${\tilde{x}}_i\left(k\right)=x_i\left(k\right)-{\overline{x}}_i\left(k\right), {\tilde{u}}_i(k)=u_i(k)-{\overline{u}}_i(k)$$

(7)

Here, we define some constraint sets for system (4) that will be referred to in designing the controller as follows.

$$x_i\in {\mathbb{X}}_i=\left\{x_i\in {\mathbb{R}}^{3\times 1}:x_{i,\min }\le x_i\le x_{i,\max }\right\}, u_i\in {\mathbb{U}}_i=\left\{u_i\in \mathbb{R}:\left|u_i\right|\le u_{i,\max }\right\}$$

(8)

Note that sets ${\mathbb{X}}_i$ and ${\mathbb{U}}_i$ are both closed and compact sets with the origins in the interior. Then, the state errors of system (4) are denoted as

$$e_{pi}=p_{i-1}(k)-p_i(k)-d_{i,i-1}(k)-l_i, e_{vi}=v_{s,i}(z_k)-v_s^{\ast }(z_k)$$

(9)

where $z_k$ denotes the spatial variable at time instant $k$, $v_{s,i}(\cdot )$ denotes the velocity of vehicle $i$ defined in spatial domain, and $v_s^{\ast }(\cdot )$ denotes the optimal velocity for the platoon at the specific space. The spatial and time domain variables can be inter-transformed based on the history states information. $d_{i,i-1}(k)$ is the gap between vehicle $i$ and vehicle $i-1$, and $l_i$ is the length of vehicle $i$. For convenience, the errors are written in a compact form as $e_i={\left[e_{pi},e_{vi}\right]}^T$.

2.2. Notations

Here we define some notations based on the system presented above. The real number set is denoted as $\mathbb{R}$; for a state vector $x\in {\mathbb{R}}^{n\times 1}$, the p-norm is defined as ${‖x‖}_p={\left({\displaystyle \sum x_i^p}\right)}^{1/p}$ with $p\in \left[1,\infty \right)$, the ${\mathcal{L}}_p$ norm is in the form ${‖x‖}_{{\mathcal{L}}_p}={\left({\displaystyle {\int }_{\psi }{‖x‖}_p^{p}dt}\right)}^{1/p}$, and ${‖x‖}_{{\mathcal{L}}_{\infty }}=ess{\sup }_{t\in \psi }\left|x(t)\right|$, where $\psi$ is a Lebesgue measurable set. Suppose a positive semi-definite matrix $R\in {\mathbb{R}}^{n\times n}$, the weighted Euclidean norm is ${‖x‖}_R={\left(x^{T}Rx\right)}^{1/2}$. The Minkowski sum and the Pontryagin difference for sets $\mathbb{Y},\mathbb{Z}$ are stated as, $\mathbb{Y}\oplus \mathbb{Z}=\left\{y+z|y\in \mathbb{Y},z\in \mathbb{Z}\right\}$ and $\mathbb{Y}\ominus \mathbb{Z}=\left\{y|y+z\in \mathbb{Y},z\in \mathbb{Z}\right\}$. Other notations used in the paper will be elaborated upon as they appear, and a list of the main symbols is shown in the

Table A1. Explanation of symbols.

Parameters/ Variables	Explanation	Parameters/ Variables	Explanation
$m$	Vehicle mass	$a_x$	Longitudinal acceleration
$T_e$	Engine torque	$r$	wheel effective radius
$A_w$	Vehicle characteristic area	$C_D$	Air drag coefficient
$\theta$	Slope angle	$f$	Rolling resistance
$p_i,v_i,a_i$	Longitudinal position, velocity and acceleration of vehicle $i$	$\tau$	Time constant of the driving system
$u_i$	Control input	$\mathcal{N}$	Number of vehicles
$\mathcal{V}$	Node set in the platoon	$x_i$	State of vehicle $i$ in actual system
${\overline{\varpi }}_i$	Threshold of triggering mechanism	${\overline{x}}_i$	State of vehicle $i$ in nominal system
$e_i={\left[e_{pi},e_{vi}\right]}^T$	Errors of position and velocity	${\tilde{x}}_i$	State disturbance
${\varpi }_i$	Uncertainty term of vehicle $i$	$\iota$	Sampling time interval
$d_{i,i-1}$	Gap between vehicle $i$$\mathrm{and}i-1$	$l_i$	Length of vehicle $i$
$\Delta l_{DP}$	Space interval length in DP	$N_{DP}$	Number of intervals in DP
$f_{DP}$	Refresh frequency	$F_b$	Braking force
$F_r$	Resistance force	$z_k$	Spatial position at time $k$
$v_{s,i}$	Speed of vehicle $i$ in spatial domain	$v_s^{\ast }$	Spatial reference speed profile
${\zeta }_i(t)$	State for feedback controller	$K_i$	Feedback gain in LQR
${\zeta }_i^a(t_j|t_k)$	Assumed state trajectory	$N_p$	Number of predictive steps
${\mathbb{Z}}_i$	RPI set for vehicle $i$	${\mathbb{U}}_i$	Constraints for control input
${\overline{\mathbb{Z}}}_i$	State constraints for the nominal system	${\mathbb{Z}}_{i,m}$	Minimal RPI set for vehicle $i$
${\mathbb{S}}_i$	String stability constraint for nominal state ${\overline{\zeta }}_i$	${\overline{\mathbb{Z}}}_{\zeta i}$	Terminal constraint for nominal state ${\overline{\zeta }}_i$
${\overline{\mathbb{U}}}_{\zeta i}$	Terminal constraint for nominal control input ${\overline{u}}_i$	${\mathbb{W}}_i$	Bounded disturbance set
${\delta }_{DMPC}$	Discretization time interval of DMPC	$d_{i,i-1}^{des}$	Desired gap between vehicle $i$$\mathrm{and} i-1$

in Appendix A.

2.3. Definitions and Control Objectives

In this part, we will present some necessary definitions to be used in later parts.

Definition 1

[31]. A continuous function $\partial :[0,a)\to [0,\infty )$ is defined as a class $\mathcal{K}$ function if it is strictly increasing and $\partial \left(0\right)=0$.

Definition 2

[32]. For a nominal vehicle platoon system with no leader control input, if the condition $\underset{t\to \infty }{\lim }{\overline{e}}_i(t)=0,i\in \mathcal{V}$ is satisfied, then the nominal system is defined as internally stable.

Definition 3

[31]. Consider the interconnected nominal system (4); if there exists a class  $\mathcal{K}$ function $\partial$ and a constant ${\eta }_{\varpi }>0$ such that, for any bounded initial state $e_{pi}(0)\in {\mathbb{R}}^{\left(\mathcal{N}-1\right)\times 2}<c_p$ and disturbance ${‖{\varpi }_i(k)‖}_{{\mathcal{L}}_{\infty }}<c_{\varpi }$, $c_{\varpi }$, $c_p>0$ are both constant, the condition for the existent solution

$${‖e_{pi}(t)‖}_{{\mathcal{L}}_p}\le \partial \left(e_{pi}(0)\right)+{\eta }_{\varpi }{c}_{\varpi }, t>0$$

(10)

is satisfied; then, we argue that the system meets the criterion of disturbance ${\mathcal{L}}_p$ string stability.

Based on the definitions of internal stability and string stability, we now present the control objectives of this research as follows: (1) the internal stability of the nominal system (5) is satisfied; (2) the ${\mathcal{L}}_p$ string stability of the real system (4) is satisfied.

3. Controller Design

Based on the preparation, now we can present the controller design. The controller designed in this paper consists of two layers; the supervising layer receives the system states from the distributed layer and generates the optimal speed profile and the corresponding gap policy. The distributed layer then, based on the optimal speed profile and gap policy, integrates the feedback and feedforward control inputs to compute the final control input for the individual vehicle. The overall control architecture is shown in Figure 1.

3.1. Generation of Optimal Speed Profile

For a single vehicle traveling on a hilly road, there is a so-called fuel-optimal speed profile to make the vehicle travel at a smooth speed profile at a low fuel consumption rate [33]. For a platoon of $\mathcal{N}+1$ vehicles, however, such factors as the vehicle-to-vehicle gap need to be considered in generating the optimal speed profile. The most popular gap policies for platoons include constant spacing (CS) and constant time headway spacing (CTH); these gap policies are not suitable for platoon scenarios due to the uphill and downhill processes, as illustrated in Figure 2.

As can be seen in Figure 2, there are vehicles traveling uphill and downhill at the same time; if the leader starts to decelerate while climbing uphill, followers equipped with CD or CTH gap policies need to decelerate as well to maintain the consensus state, which obviously will lead to excessive waste of energy. In summary, the CD and CTH policies will cause the inconsonance of vehicle velocity within the platoon in the spatial sense. Based on this analysis, we will derive a spatial consonant speed profile and the corresponding gap policy.

Firstly, we develop a fuel consumption model based on the engine data collected from the commercial software TruckSim (Trucksim 2016), as shown in Figure 3.

The fuel consumption rate model is then fitted with a third-order polynomial as

$$\begin{array}{l}{f}_r(T_e,n_e)=C_{00}+C_{10}{n}_e+C_{01}{T}_e+C_{11}{T}_e{n}_e+C_{02}{n}_e^2+C_{20}{T}_e^2+C_{21}{n}_e^2{T}_e\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+C_{12}{T}_e^2{n}_e+C_{03}{T}_e^3\end{array}$$

(11)

where $C_{ii}$ is the constant coefficient of the polynomial. Assuming a linear transmission system, we have

$$n_e=\frac{i_t}{r}{v}_x\times \frac{60}{2\pi }$$

(12)

Combining (11) and (12), the fuel consumption rate with respect to the longitudinal velocity and the engine output torque can be obtained.

Secondly, we derive a spatial domain dynamic programming strategy to handle such constraints as the state and input constraint. Defining the programming space interval length as $\Delta l_{DP}$, the number of intervals as $N_{DP}$, and the refresh frequency $f_{DP}$, then the total space length is $L_{DP}=\Delta l_{DP}\cdot N_{DP}$. Then, the discretized model can be designed based on (1)–(3) as

$$m\frac{v_{s,i}(z_k)-v_{s,i}(z_{k-1})}{\Delta l_{DP}}=\frac{T_{ei}(z_k)i_t}{r}+F_{bi}(z_k)-F_r$$

(13)

where $F_r$ is the resistance force including air drag and rolling resistance.

Then, we define the control input as ${\xi }_k=T_e(z_k)$ and the output $y(k)=f_r(T_{ei}(z_k),n_{ei}(z_k))$, and formulate the fuel consumption-oriented cost function as,

$$J_{DP}={\displaystyle \sum _{i=1}^{\mathcal{N}+1}{\displaystyle \sum _{j=k}^{k+N_{DP}}y\left(j\right)}}$$

(14)

Cost function (14) is only concerned about fuel consumption, which is somewhat unreasonable considering the traffic flow; for instance, if the platoon travels as slowly as possible, the fuel consumed may drop to the bottom, but this situation is not desirable. Thus, we propose a speed-relevant constraint to facilitate the controller design

$$fd\ge f{d}_{\max }-0.01\lambda \left(f{d}_{\max }-f{d}_{\min }\right)$$

(15)

where $fd$ is the traveled distance by the leader in the horizon of prediction; note that the distance is defined in a Frenet frame to precisely record the fuel consumed. $f{d}_{\max }$ and $f{d}_{\min }$ are the maximum and minimum possible distance, respectively; $\lambda$ is a tuning parameter for the tradeoff between the minimum fuel consumed and the optimal speed. With a smaller $\lambda$, the platoon will consume more fuel and travel faster, and vice versa.

As analyzed above, the vehicle speed needs to be consistent with the spatial position, expressed as

$$v_{s,i}(z)=v_{s,i-1}(z)=v_s^{\ast }(z)$$

(16)

One problem arises given the spatial domain constraint presented in (16), as the distance defined in (15) is in the sense of the time domain. Then, an approximation function of the distance traveled is applied here,

$$f{d}_{s,i}\left(z_k\right)=v_{s,i}\left(z_k\right)T_i-l_i$$

(17)

where $T_i$ is the time gap between vehicle $i$ and vehicle $i-1$.

Cost function (14) is also subject to such constraints as the velocity limit, the engine torque limit and the braking system limitation as $v_x\in \left[v_{x,\min },v_{x,\max }\right],$ $T_e\in \left[T_{e,\min },T_{e,\max }\right]$, $F_b\in \left[F_b{}_{,\min },0\right]$. $v_{x,\min }$, $v_{x,\max }$, $T_{e,\min }$, $T_{e,\max }$, $F_b{}_{,\min }$ denote the minimum and maximum of longitudinal speed, minimum and maximum of engine torque, and minimum brake force, respectively.

Then, the final dynamic programming (DP) problem can be established as

$$\begin{array}{l}\underset{u_s(z_j)}{\min }{J}_{DP}={\displaystyle \sum _{i=1}^{\mathcal{N}+1}{\displaystyle \sum _{j=k}^{k+N_{DP}}y\left(j\right)}}\\ s.t.\hspace{1em}model\left(13\right), constraints\left(15\right),\left(16\right),\left(18\right)\\ \hspace{1em}\hspace{1em}{z}_k=f{d}_1(t)\\ \hspace{1em}\hspace{1em}{v}_s(z_k)=v_1(t)\end{array}$$

(18)

In the DP problem (19), $u_s(z_j)=\left[T_e(z_j),F_b(z_j)\right]$ represent the control input; the last two constraints stand for the initial conditions. The spatial domain optimal speed profile and the corresponding time gap can be generated upon obtaining the control input from solving the DP problem.

3.2. Feedback Controller Design

The feedback controller is designed to diminish the error between real-time vehicle states and the generated reference states obtained in Section 2.1; thus, the easy-to-implement, computationally efficient LQR controller is applied here. Based on the error definition and the control objectives described in (9) and (10), a new controller-oriented system state is defined as

$${\zeta }_i(t)={\left[f{d}_i(t),v_i(t)\right]}^T=\Xi \cdot x_i(t)$$

(19)

with $\Xi =\left[\begin{array}{c}1 0 0\\ 0 1 0\end{array}\right]$.

The controller is formulated as ${\tilde{u}}_i(k)=K_i{\tilde{\zeta }}_i$; then, based on the disturbance system (6) the following relation can be achieved:

$${\tilde{\zeta }}_i(k+1)=\underset{A_{BKi}}{\underbrace{\left(A_i+B_i\cdot K_i\right)}}{\tilde{\zeta }}_i(k)+{\varpi }_i(k)$$

(20)

To obtain the coefficient matrix $K_i$, we formulate a discrete linear quadratic regulator problem as follows,

$$\begin{array}{l}\min J_{fb}={\displaystyle \sum _{k=0}^{\infty }\left({‖R_{fb}{\tilde{\zeta }}_{si}(k)‖}_2^2+{‖Q_{fb}{\tilde{\zeta }}_{vi}(k)‖}_2^2+{‖M_{fb}{\tilde{u}}_i(k)‖}_2^2\right)}\\ s.t. equation \left(6\right)\end{array}$$

(21)

where the positive definite and symmetric matrices $R_{fb}$, $Q_{fb}$ and $M_{fb}$ add weight to the corresponding variables.

Assume a robust positively invariant (RPI) set [34] ${\mathbb{Z}}_i$ exists about the system (20), i.e.,

$$A_{BKi}{\tilde{\zeta }}_i(k)+{\varpi }_i(k)\in {\mathbb{Z}}_i, \forall {\tilde{\zeta }}_i(k)\in {\mathbb{Z}}_i \forall u_i\in {\mathbb{U}}_i$$

(22)

The condition in (22) implies that properly tuned parameters involved in the controller can drive the system to a stable state.

3.3. DPMC Based Feedforward Controller Design

The feedforward controller is designed to achieve the ${\mathcal{L}}_p$ string stability for the nominal system ${\overline{\zeta }}_i(t)=\Xi \cdot {\overline{x}}_i(t)$. Firstly, we define the optimal state trajectory as ${\zeta }_i^{\ast }(t_j|t_k)$ and the assumed state trajectory as ${\zeta }_i^a(t_j|t_k)$ over the predictive horizon $\left[t_k,t_{k+N_p}\right]$, $j=k,\cdots ,k+N_p$. The assumed state denotes the most precise state trajectories of the real system, and will be transmitted to the adjacent connected vehicles, which can be precomputed as

$${\zeta }_i^a(t_j|t_k)=\left\{\begin{array}{l}{\zeta }_i(t_j),\hspace{1em}\hspace{1em}j<k\\ {\zeta }_i^{\ast }(t_j|t_{k-1}),k\le j<k+N_p\end{array}\right.$$

(23)

Based on the optimal speed profile generated in Section 2.1, the assumed state and control input can be obtained.

Then, the DMPC cost function can be formulated as

$$J_{DMPC,i}={\displaystyle \sum _{j=k}^{k+N_p}\left\{\begin{array}{l}{‖{\overline{\zeta }}_i(t_j|t_k)-{\zeta }_i^{\ast }(t_j|t_k)‖}_{P_{DMPC}}+{‖{\overline{a}}_i(t_j|t_k)-a_i^{\ast }(t_j|t_k)‖}_{R_{DMPC}}\\ +{‖{\overline{\zeta }}_i(t_j|t_k)-{\overline{\zeta }}_{i-1}(t_j|t_k)+d_{i,i-1}^{des}‖}_{Q_{DMPC}}\end{array}\right\}}$$

(24)

where $P_{DMPC}$, $R_{DMPC}$, $Q_{DMPC}$ are the positive semi-definite matrices; the reference control input is defined based on the discretization time of DMPC ${\delta }_{DMPC}$ as

$$a_i^{\ast }(t_j|t_k)=\left(v_s^{\ast }\left(f{\overline{d}}_i(t_{j+1}|t_k)\right)-v_s^{\ast }\left(f{\overline{d}}_i(t_j|t_k)\right)\right)/{\delta }_{DMPC}$$

(25)

To drive the nominal system to reach the stable terminal state, the terminal constraints can be established as

$${\overline{\zeta }}_i\in {\overline{\mathbb{Z}}}_{\zeta i}=\left\{{\overline{\zeta }}_i\in {\mathbb{R}}^{2\times 1}:{\overline{e}}_i\left(N_p\right)=0\right\}, {\overline{u}}_i\in {\overline{\mathbb{U}}}_{\zeta i}=\left\{{\overline{u}}_i\in \mathbb{R}:{\overline{u}}_i\left(N_p\right)=0\right\}$$

(26)

Then, the constraint for meeting the criteria of ${\mathcal{L}}_p$ string stability is constructed as

$${\overline{\zeta }}_i\in {\mathbb{S}}_i=\left\{{\overline{\zeta }}_i\in {\mathbb{R}}^{2\times 1}:{‖{\overline{e}}_{pi}‖}_{{\mathcal{L}}_p}\le \partial \left(\left|{\overline{e}}_{pi}(0)\right|\right)\right\}$$

(27)

Finally, the DMPC-based feedforward control input can be obtained by solving the problem as follows:

$$\begin{array}{l}\underset{{\overline{u}}_i(t_k)}{\mathrm{minimize}} J_{DMPC,i}={\displaystyle \sum _{j=k}^{k+N_p}\left\{\begin{array}{l}{‖{\overline{\zeta }}_i(t_j|t_k)-{\zeta }_i^{\ast }(t_j|t_k)‖}_{P_{DMPC}}\\ +{‖{\overline{a}}_i(t_j|t_k)-a_i^{\ast }(t_j|t_k)‖}_{R_{DMPC}}\\ +{‖{\overline{\zeta }}_i(t_j|t_k)-{\overline{\zeta }}_{i-1}(t_j|t_k)+d_{i,i-1}^{des}‖}_{Q_{DMPC}}\end{array}\right\}}\\ s.t.\hspace{1em}{\overline{\zeta }}_i(t_k)\in {\overline{\mathbb{Z}}}_i\cap {\overline{\mathbb{Z}}}_{\zeta i}\cap {\mathbb{S}}_i,{\overline{u}}_i\in {\overline{\mathbb{U}}}_i\cap {\overline{\mathbb{U}}}_{\zeta i}\end{array}$$

(28)

where ${\overline{\mathbb{Z}}}_i$ and ${\overline{\mathbb{U}}}_i$ denote the state and control input constraints for the nominal system, respectively, the calculation of which is presented as follows:

$${\overline{\zeta }}_i(t_k)\in {\overline{\mathbb{Z}}}_i={\mathbb{X}}_i\ominus {\mathbb{Z}}_i,{\overline{u}}_i(t_k)\in {\mathbb{U}}_i\ominus K_i{\mathbb{Z}}_i\ominus {\mathbb{W}}_i$$

(29)

The number of prediction steps $N_p$ can be tuned to achieve the feasibility of the MPC problem (28).

3.4. Integration of Controllers with Tube Methods

The tight sets calculated in (29) will not exist if the RPI set ${\mathbb{Z}}_i$ is not sufficiently small; to avoid that condition, we then develop a minimal RPI (mRPI) set with the following method [35]:

$${\mathbb{Z}}_{i,m}=\underset{m\to \infty }{\lim }{F}_m=\underset{j=0}{\overset{m-1}{\oplus }}{A}_{BKi}^j{\mathbb{W}}_i,\mathrm{with} F_0=\left\{0\right\}$$

(30)

Then, we replace the RPI set in (29) with mRPI set calculated in (30) as

$${\overline{\zeta }}_i(t_k)\in {\overline{\mathbb{Z}}}_i={\mathbb{X}}_i\ominus {\mathbb{Z}}_{i,m},{\overline{u}}_i(t_k)\in {\overline{\mathbb{U}}}_i\ominus K_i{\mathbb{Z}}_{i,m}\ominus {\mathbb{W}}_i$$

(31)

This replacement adds the probability of the existence of the tight sets. Let us further assume that the mRPI set is small enough so that all tight sets exist. Note that this kind of assumption makes sense in practical circumstances, as proven in [27,36].

3.5. Triggering Mechanism and Stability Analysis

The feedforward controller is designed based on the MPC algorithm, which is quite computationally intensive. To avoid this situation, a triggering mechanism is devised based on the amplitude of the disturbance. If the amplitude exceeds the bounds of the set ${\mathbb{W}}_i$, then the trigger starts functioning and the feedforward controller generates a corresponding control input to drive the system back to stability. The tuning process of the bounds of the set ${\mathbb{W}}_i$ is conducted by trading off between the state tracking errors and the fuel economy performance; the ultimate goal is to achieve the overall stability of the system and only trigger the feedforward controller when necessary.

The stability performance, as stated by the control objectives in Section 2.3, is analyzed with the following theorems.

Theorem 1.

Suppose that the mRPI set ${\mathbb{Z}}_{i,m}$ exists, then system (5) can achieve internal stability defined in Definition 2.

Proof of Theorem 1.

Under the circumstance of excessive disturbance located outside the tight set ${\mathbb{W}}_i$, the constraints defined in (26) guarantee that the solution of the feedforward MPC problem will reach the terminal states, i.e., the equilibrium point of zero error. If the disturbance falls inside the set ${\mathbb{W}}_i$, then the state of the system will reach the state sets defined in (26) before the terminal prediction step. To sum up, system (5) is internally stable under the reasonable assumption. □

Theorem 2.

Suppose that the mRPI set ${\mathbb{Z}}_{i,m}$ exists, then the real system (4) can achieve disturbance ${\mathcal{L}}_p$ string stability defined in Definition 3.

Proof of Theorem 2.

The feedback controller designed in Section 2.2 guarantees the asymptotic stability of $A_{BKi}$; then, with a disturbance within ${\mathbb{W}}_i$, the RPI set ${\mathbb{Z}}_i$ exists for system (21) [37]. Then, given the assumption that the mRPI set ${\mathbb{Z}}_{i,m}$ exists, if the nominal states and input locate inside the sets ${\overline{\mathbb{Z}}}_{\zeta i}$ and ${\overline{\mathbb{U}}}_i$, then, together with the generation of $K_i$, the states and control input conform to the scope of the tight sets as ${\zeta }_i(t_k)\in {\mathbb{Z}}_i$, $u_i(t_k)\in {\mathbb{U}}_i$. Then, we can conclude that there exists a constant $\gamma$ that makes the inequality ${‖{\tilde{e}}_{pi}‖}_{{\mathcal{L}}_p}\le \gamma$ hold. Based on the triangular inequality principle and constraint (28), the following derivation can be obtained:

$${‖e_{pi}‖}_{{\mathcal{L}}_p}={‖{\overline{e}}_{pi}+{\tilde{e}}_{pi}‖}_{{\mathcal{L}}_p}\le {‖{\overline{e}}_{pi}‖}_{{\mathcal{L}}_p}+{‖{\tilde{e}}_{pi}‖}_{{\mathcal{L}}_p}\le \partial \left(\left|{\overline{e}}_{pi}(0)\right|\right)+\gamma$$

(32)

We can tell that the right-hand side of (33) is of no relevance to the number of vehicles in the platoon, both implicitly and explicitly. Then, referring to Definition 3, we conclude that platoon system (4) can achieve disturbance ${\mathcal{L}}_p$ string stability. □

Remark 1.

As the controller was designed based on the construction of the RPI set, some parametric errors or variations shall not cause performance deterioration as long as the affected maximal acceleration/deceleration of the vehicle located in the mRPI set defined in the paper, the string stability, and corresponding optimal fuel economy can still be obtained.

4. Simulation and Results Discussion

To validate the effectiveness and performance of the proposed control strategy, numerical simulations regarding truck platoons traveling on hilly roads are conducted. The testing road and the corresponding elevation profile are shown in Figure 4, and the total length of the road is about 2.5 km. The truck platoon applied in the simulation consists of six homogeneous trucks in total; the leader is numbered as vehicle $v_0$, and the following vehicles are numbered as vehicles $v_i,i\in \left\{1,2,3,4,5\right\}$.

4.1. Simulation Settings

Simulations will be carried out with two kinds of disturbances added to the vehicles to validate the performance of the controller regarding string stability, fuel consumption and computation efficiency. In the simulations, the vehicle model in (1)–(3) is adopted and built in Matlab/Simulink. The vehicle’s optimal speed profile is pre-computed or calculated intermittently with the DP problem. The first simulation is conducted to simulate the scenario of an external vehicle cutting in and cutting out with a position-error-sudden-change disturbance added to the second following vehicle. This maneuver serves as a low-frequency large-amplitude disturbance to the platoon.

In the second simulation, a random disturbance with high frequency, is added to the vehicles to simulate the dynamic uncertainty and external disturbances such as the wind. Thus, the disturbance is simulated with Gaussian white noise due to the unpredictability and stochastic property. Then, the vehicle’s optimal speed profile with added disturbance, together with the speed profile in the spatial domain, is illustrated in Figure 5.

In this simulation, to further verify the high-frequency disturbance attenuation capability of the controller, a random disturbance with an amplitude up to $0.2 \mathrm{m}\cdot {\mathrm{s}}^{-2}$ is directly added to the acceleration of the vehicle. The parameters of the homogeneous truck platoon and some of the variables related to the controller are listed in

Table 1. Vehicle parameters and controller variables.

Parameters/Variables	Value	Unit
$m$	4455	$\mathrm{kg}$
$l_i$	5	$\mathrm{m}$
$A_w$	6.8	${\mathrm{m}}^2$
$r$	0.51	$\mathrm{m}$
$C=\left[C_a,C_b,C_c\right]$	$\left[0.836,4.318,7.588\right]$	-
$f$	0.02	-
$N_p$	40	-
$\tau$	20	$\mathrm{ms}$
$T_{e,\max }$	704.6	$\mathrm{N}\cdot \mathrm{m}$
$x\left(t=0\right)$	$\left[75,60,45,30,15,0\right]$	$\mathrm{m}$

. In both simulations, the threshold for triggering the feedforward control is set as ${\overline{\varpi }}_i\left(k\right)={\left[0.25,1,5\right]}^T$, where the position error is most rigorously constrained because maintaining the string stability regarding the position is deemed as the fundamental and primary goal of controlling the platoon.

4.2. Simulation Results and Discussion

The first simulation scenario is carried out to simulate the cut-in and cut-out maneuvers of an external vehicle, usually a human-driven vehicle. The results are illustrated in Figure 6. As can be seen from Figure 6a, the gap error of the second following vehicle $v_2$ compared to $v_1$ changes abruptly at 10 s and 70 s, respectively.

At 10 s of the simulation, the gap error is positive, indicating a cut-in maneuver of an external vehicle, and at 70 s, the negative gap error implies a cut-out maneuver of that same external vehicle; the cut-in and cut-out actions can also be seen from Figure 6b. The disturbance attenuation effect can be clearly seen in Figure 6a during and after the cut-in and cut-out actions occurring, which demonstrates the capability of string stability of the proposed controller directly.

To further validate the string stability of the controller against high-frequency disturbance, and to verify the performance of fuel consumption and computation efficiency comprehensively, the second simulation is conducted with the settings presented above. The simulation results regarding distance, vehicle gap and spatial domain velocity are displayed in Figure 7. The spacing profile of each truck relative to the predecessor, shown in Figure 7b, shows that trucks in the platoon can maintain a safe distance on the hilly roads. As can be seen from Figure 7c, the vehicle gap errors caused by such factors as external disturbance and parameter uncertainties are attenuated downstream of the platoon as the error diminishes with the increase in the vehicle index, which demonstrates the disturbance string stability of the controller. Moreover, the velocity of each truck in the spatial domain, as illustrated in Figure 7d, is consistent with the reference speed profile with a maximum velocity error of less than 0.1 $\mathrm{m}/\mathrm{s}$, which conforms to the concept of the fuel-optimal speed profile of a single vehicle.

The fuel consumption performance of the designed controller, as the major target of this research, is compared with the controllers proposed in the previous literature, i.e., the controller constructed with constant spacing [38] and constant time headway [39] policy. The comparison results regarding the fuel consumed are illustrated in Figure 8. As shown in the figure, the controller proposed in this paper can reduce the fuel consumed by as much as 6.48% compared to the CS controller and 5.07% compared to the CTH controller. One major cause for the increased fuel consumption of CS and CTH is the inconsonant vehicle profile; we take the spatial domain speed profile of the platoon under the CS policy shown in Figure 9 as an example. The vehicles travel at different velocities from the optimal spatial speed profile marked as ‘$v_0$’, which will cause more fuel consumption due to unnecessary acceleration and deceleration.

An interesting point worth noting is that fuel consumption decreases more significantly downstream of the truck platoon; in other words, trucks in the latter part of the platoon consume less fuel than the ones in the front part. An important reason for this phenomenon is the reduced vehicle gap and the attenuated effect of disturbance due to the string stability performance of the proposed controller. As the disturbance effect is diminished, the truck can maintain the state close to the reference profile without excessive acceleration or deceleration; the fuel consumption is therefore reduced to a large extent.

One important property of the control algorithm when applied in a practical scenario is the computation resource occupation, closely related to which is the computation time. To verify the performance regarding the computation aspect, the computation time of the proposed controller is compared to a controller designed based on DMPC [9]. As shown in Figure 10, the computation time of the proposed controller is much lower than the DMPC-like controller, and the communication triggering times count as about 1/20 of the DMPC controller, which implies the scarcity of the participation of the feedforward controller. The result illustrated in Figure 10a demonstrates the superiority in computation efficiency and potential for practical application.

To further investigate the relation of threshold setting and the computation efficiency of the proposed controller, we conduct simulations with both the position-sudden-change disturbance and the high-frequency random disturbance, and set the threshold of the position error to different values, considering that the position error is the most rigorously constrained variable. The comparison results are gathered in Figure 10b, and the computation time of DMPC is also shown as a comparison. As can be seen from the figure, the computation time of the proposed controller, when the threshold of position error is less than 0.06 m, is almost equal to that of the DMPC, so we mark this zone as ‘DMPC zone’ with a green rectangle; as the threshold increases with an interval of $\left[0.06,0.5\right]$ m, the computation time decreases dramatically, indicating the rapid decrease in triggering frequency of the feedforward controller. The string stability can be maintained as a whole, although some minor instability can be observed in conditions of relatively large amplitude disturbances.

The instability here denotes the chaotic distribution of position error in the platoon. Such instability expands as the threshold reaches nearly 0.5 m; when the threshold exceeds 0.5 m, the possibility of string instability exists when some high-frequency disturbance is added to the system. This kind of disturbance can result in oscillation of position error of the vehicles, as depicted in Figure 11a. In the figure, the threshold is set as 5 m to better illustrate the property of instability. As can be seen from the figure, when the position error exceeds the threshold, the controller can attenuate the error and maintain string stability, but within the threshold, the error amplifies downstream of the platoon, which indicates a certain extent of instability. The oscillation of the position error or turbulence (marked in grey in Figure 10b) can cause a dramatic increase in fuel consumption as the vehicles will operate farther from the optimal fuel economy speed profile, as can be observed in Figure 11b.

Thus, for the system presented in this paper, the threshold of position error is best located in the interval of $\left[0.06,0.5\right]$ m; the final value in practical application is left to the tradeoff between real-time performance (computation efficiency) and fuel consumption performance.

5. Conclusions

In this paper, a controller for optimizing fuel consumption in a truck platoon traveling on hilly roads is proposed based on the generation of a spatial domain optimal speed profile. The controller consists of an LQR feedback controller and a DMPC feedforward controller, and the two controllers are integrated with a tube method. The proposed controller is able to attenuate the disturbance downstream of the platoon, and it outperforms the CS and CTH-based controllers in terms of fuel consumption on hilly roads; meanwhile, due to the event-triggering mechanism involved in the feedforward controller, the proposed controller is more computation efficient than the DMPC kind of controllers. In sum, the controller designed in this paper is promising for application in practical scenarios.

As the simulation results demonstrate the excellent performance of the proposed controller, a set of real truck tests is necessary for the application of the controller. However, the heterogeneity of trucks is not taken into consideration in the controller design, and heterogeneous trucks forming a platoon may be of interest to the freight industry. Thus, designing a controller for heterogeneous truck platoons and on-road tests will be the future topic of our research.