Design and Performance Analysis of a Cascaded Model Predictive Controller and Command Governor for Fuel-Efficient Control of Heavy-Duty Trucks

Heavy-duty trucks are responsible for 23% of transportation-related $CO2$ emissions in the U.S. (see Ref. [1]). There have been many attempts in both government and industry to increase the fuel economy of heavy duty trucks (see Ref. [2]). Current projections forecast freight tonnage increasing 1.4% per year for the next few decades (see Ref. [3]). In order for heavy-duty trucks to achieve better fuel economy while meeting the increasing demand for freight transport, predictive ecological vehicle control technologies offer a promising solution. The overarching objective of the research explored in this paper is to fuse fuel efficiency with safe vehicle-following for heavy-duty vehicles. This requires the consideration of a lengthy horizon for the purpose of fuel efficiency, which requires consideration of a much shorter horizon for safe vehicle following.

For light-duty passenger vehicles, a substantial body of literature has studied the problems of velocity trajectory optimization for fuel efficiency and following constraint satisfaction, both in isolation and in combination. When used for velocity trajectory optimization alone, these control strategies are referred to as ecological cruise control (ECC). In particular, model predictive control (MPC) has been widely used in online ECC strategies (see Refs. [4] and [5]), due to MPC's inherent ability to consider economic performance objectives while explicitly enforcing constraints. Furthermore, multiple papers have incorporated following distance considerations and penalties within MPC formulations, thereby achieving ecological vehicle following. Such control strategies are termed ecological adaptive cruise control (Eco-ACC) strategies. The authors of Ref. [6] designed an Eco-ACC such that the MPC objective function approximated fuel consumption while adding penalties to the objective function to promote driver comfort and safety. Simulation results from Ref. [6] indicated that Eco-ACC improved the fuel economy in comparison with typical ACC. In Refs. [7] and [8], the MPC objective function was designed to improve ecological driving by penalizing nonsmooth acceleration profiles. Additionally, in Refs. [7] and [8], hard constraints on acceptable bounds for various desired vehicle-following qualities (such as vehicle-following distance, maximum jerk, and relative velocity) were incorporated and softened when necessary to ensure feasibility.

The aforementioned Eco-ACCs were designed for passenger vehicles, which benefit substantially through MPC-based Eco-ACC strategies with horizons of hundreds of meters or less. On the other hand, due to their inertial properties, it has been demonstrated in Ref. [9] that heavy-duty trucks only begin to see near-full benefits of ECC under horizon lengths greater than 2000 m. Consequently, MPC-based ECCs designed for heavy-duty trucks in Refs. [10–12] rely on horizon lengths of 2000 m or more. Implementing MPC-based Eco-ACCs on heavy-duty trucks has recently been explored in Refs. [13] and [14]. Typically, MPC-based Eco-ACCs explicitly enforce vehicle-following constraints within the MPC, which is implemented at a time-step of 100–200 ms to account sudden changes in the behavior of the lead vehicle [15]. Due to the limited computational resources on a typical electronic control unit, an MPC-based Eco-ACC that explicitly enforces vehicle-following constraints implemented on a heavy-duty truck would require a shorter horizon (therefore sacrificing ecological performance) to ensure the MPC is real-time feasible (see Ref. [16]). To avoid the above tradeoff between ecological performance and real-time feasibility, the MPC-based Eco-ACCs in Refs. [13] and [14] implemented a proportional plus derivative distance-keeping controller that heuristically maintained safe vehicle following, in parallel to an MPC-based ECC designed for a heavy-duty truck. While solving all issues relating to computational complexity, the plus derivative distance-keeping controller does not provide any guarantees of constraint satisfaction. Indeed, to the best of our knowledge, an MPC-based Eco-ACC designed for a heavy-duty truck that explicitly enforces vehicle-following constraints has yet to be investigated in the literature.

To address the aforementioned challenges, we propose an Eco-ACC that decouples ecological driving and distance keeping through the cascade of an MPC with a command governor (CG). A CG is an add-on scheme that alters the reference to a closed-loop stable system (see Ref. [17]). CGs are predictive control schemes used solely to enforce constraints. Compared to MPC, CGs are less computationally expensive (see Ref. [18]) and can consequently be implemented with much smaller time steps than MPC, at the expense of only being able to enforce constraints (rather than additionally managing competing performance objectives, which is accomplished by MPC).

In this work, we present an Eco-ACC, first introduced at a basic level in our conference publication [19], that implements an upper level MPC-based ECC cascaded with a lower level command governor. Such an inner loop command governor, which cascades an economic MPC, is new to the Eco-ACC literature. A conceptual diagram of this Eco-ACC strategy is shown in Fig. 1. The upper level MPC-based ECC, which is fused with an offline dynamic programing (DP) optimization, was originally developed in Ref. [11]. The benefit of fusing an offline DP with online MPC is that the MPC is able to utilize global knowledge of the route that is accounted for in the offline DP solution. This is achieved by using the global and coarsely gridded DP solution to serve as an initial guess for the MPC optimization, and by penalizing large deviations from the DP solution. The DP solution is calculated offline, producing an energy-optimal longitudinal force profile (and corresponding velocity profile) that is optimal with respect to the route topology but does not consider traffic or fine-scale terrain variations. In order for the DP solution to be referenced in real-time by the MPC algorithm, the DP-optimized velocity profile is indexed with respect to position on the route. By indexing the profile with respect to route position, the DP-optimized velocity profile will remain usable by the MPC even in the presence of traffic events that necessitate an altered velocity profile over some portion of the route (thereby leading to the ego vehicle reaching particular points on the route at different times than were anticipated by the DP optimization). The upper level MPC reoptimizes the control sequence to minimize energy expenditure over a receding horizon, introducing a feedback mechanism to account for disturbances, such as when the ego vehicle deviates significantly from the DP optimal solution due to traffic. The MPC in this work minimizes the energy expended as a surrogate for fuel consumption, as is done in Ref. [11], and utilizes an energy-based affine model of the longitudinal dynamics, rendering the optimization convex. The CG maintains distance-keeping constraints by adjusting the output of the upper level MPC, thereby modulating the wheel force command passed to the lower level powertrain controller. By design, the CG only adjusts the output of the upper level MPC the minimum amount necessary to satisfy vehicle-following constraints. The block diagram of this control strategy is shown in Fig. 2. Since the CG enforces distance-keeping constraints, the MPC can be implemented at a much longer time-step, and was shown in Ref. [13] to be real-time feasible when using a horizon length of 2000 m. The CG is formulated such that two vehicle-following constraints are enforced:

1. A desired minimum following distance that can be momentarily infringed upon so long as the ego vehicle ultimately backs off toward this following distance.

2. A minimum safe following distance, which is less than the desired minimum following distance, that must be enforced at all times.

This paper greatly extends our initial conference publication [19], through the presentation of a performance analysis under a wide array of controller parameters. Specific contributions of this paper, relative to those of our earlier conference paper [19], include:

• The utilization of an upper level MPC that is designed for real-time implementation in heavy-duty trucks;

• An analysis of the effect of the CG horizon on fuel economy for various real-world trucking routes and trailer loads; and

• The consideration of drivability as an additional metric against which performance is evaluated.

where $xego$ and $vego$ denote the position and velocity of the ego vehicle, respectively. The simplified control-oriented model is implemented in discrete time, where the time-step is given by t, as

Parameters ρ, C_d, $Aref$⁠, m, g, and $Crr$ are defined as the air density, aerodynamic drag coefficient, reference area, ego vehicle mass, acceleration of gravity, and rolling resistance coefficient. The variable $θ(xego)$ represents road grade as a function of the ego vehicle position along the route.

where $ηdl, τgr$$τax$⁠, and $rwh$ are the driveline efficiency, gear ratio, drive axle gear ratio, and wheel radius. For the initial offline optimization, the gear was chosen using a static shift map, which selected the gear that minimized BSFC for a given u and v.

The Eco-ACC strategy presented in this paper utilizes an initial global, coarsely discretized, offline DP optimization of the ego vehicle's velocity with respect to road terrain. These results of the offline optimization provide a reference velocity profile, indexed by position on the route, for the online MPC. Specifically, this reference is used to set tracking error penalties on deviations from the offline optimized velocity trajectory over the prediction horizon of the MPC, which implements a control sequence that minimizes the energy expended by the ego vehicle. The control signal from the MPC is fed into a CG, which adjusts the control signal, when necessary, to enforce distance-keeping constraints for a nonconnected lead vehicle.

and the vehicle dynamics defined in Eq. (2). $Rfuel$ is the fuel consumption rate, which depends on the applied wheel force. The terms $vmin(xego)$ and $vmax(xego)$ are minimum and maximum velocity limits, respectively, where $vmax$ is determined by the speed limit for the given route at position $xego$⁠, and $vmin$ is 70% of $vmax$⁠. The terms $umin$ and $umax$ are the minimum and maximum limits of applied wheel force, respectively, and are set by the performance capabilities of the simulated heavy-duty truck.

and the optimization is repeated over the receding time horizon at the next position step, k + 1. The offline optimal control sequence, $uDP*$⁠, implicitly defines optimal position and velocity sequences, denoted by $xDP*$ and $vDP*$⁠, respectively. These optimal position and velocity sequences are utilized by the MPC in two ways:

1. The position and velocity sequences are used to formulate tracking error penalties for the MPC optimization.

2. The DP solution, $uDP*$⁠, is used as an initial guess for the MPC optimization.

where λ_b, and λ_d are tuning weights. By penalizing $Emech$ and $Ebrake$⁠, the MPC minimizes the energy put into the system in order to overcome the energy expended due to aerodynamic drag, rolling resistance and road grade, as dictated by constraint. By penalizing $Ebrake$ (in addition to $Emech$⁠), the MPC strategy avoids energy loss due to unnecessary braking events that must be recovered afterward through mechanical work in order to maintain the desired kinetic energy. Increasing λ_d prioritizes how strictly the MPC tracks the offline DP trajectory and completes the trip by the desired trip time, $Tdes,trip$⁠. Increasing λ_b prioritizes avoiding unnecessary braking, thereby further improving fuel economy. Values for λ_b and λ_d are given Sec. 4.1.

The aforementioned DP and MPC formulations work together to minimize fuel consumption without sacrificing trip time. However, they do not address vehicle-following constraints or the drivability implications associated with these constraints (e.g., high-jerk events associated with vehicle cut-ins). The lower level CG is designed specifically for addressing these concerns. Traditionally, reference and command governors are used to enforce constraints by modifying the reference input to an already stable system. In particular, reference and command governors modify the reference to a value that, if held constant, would satisfy all constraints over all future time. The set of all originating states and constant inputs that satisfy constraints is commonly referred to as the maximal output admissible set. Typically, reference and command governors can always adjust the reference such that the input and current state are within the maximal admissible set, a property referred to as recursive feasibility. In realistic vehicle-following scenarios, however, there is no constant input that can be guaranteed with certainty to satisfy distance-keeping constraints over all time. This is the case for two reasons:

1. The future velocity profile of the lead vehicle is unknown; thus, the future constraints are unknown.

2. If the velocity profiles of the lead vehicles vary significantly (i.e., the disturbances are large), no constant input could be applied for the duration of the trip that upholds distance-keeping constraints and satisfies upper and lower speed bounds.

A recovery reference governor, developed in Ref. [20], can recover a closed-loop system when an unmodeled disturbance causes a constraint violation; however it is applied to linear systems. Additionally, conventional reference and command governor theory places the governor outside of the feedback loop (see Ref. [21]), adjusting only the reference signal of the closed-loop system. However, the upper level MPC in this work has an economic objective function, i.e., the MPC does not track a reference. For us to enforce vehicle-following constraints, we have chosen to place our command governor inside the feedback loop, cascaded from the upper level MPC. Though reference and command governor theory for placing the governor inside the control loop has only thus far resulted in recursive feasibility results for a set of specific controller and plant configurations (see Refs. [22] and [23]), experiments performed in Refs. [24] and [25] were successful in using an inner loop reference governor to enforce compressor surge constraints in turbocharged gasoline engines.

Due to the aforementioned limitations arising from uncertain future lead vehicle velocity, the CG strategy in our work uses a truncated analogy of the maximal output admissible set, denoted as O_T, where O_T is the set of all originating states, $x(t)$⁠, and inputs, $uCG$⁠, such that if the input is held constant over some finite prediction horizon, constraints are satisfied (also referred to as the i-step output admissible set in Ref. [26]). The finite horizon is chosen to reasonably reflect the future horizon over which the lead vehicle's velocity profile can be predicted. In the block diagram of Fig. 2, the CG lies directly downstream of the MPC, receiving a prescribed wheel force, $uMPC$⁠, as an input and outputting an adjusted wheel force $uCG$⁠, whenever not adjusting $uMPC$ would result in constraint violation. The term $uCG$ is the input to the lower level powertrain controller. The CG strategy is formulated as follows:

The CG adjusts $uCG(t)$ only when it is necessary to satisfy constraints. If no constraint violation occurs over the CG horizon, $uCG(t)=uMPC(t)$⁠. If a constraint violation occurs over the CG horizon, the minimization in Eq. (19) is performed via bisection search. The maximum correction the CG can apply is $uCG=umax,brk$⁠, where $umax,brk$ is the maximum negative force that can be applied to the road through the wheel via braking.

The CG is formulated to enforce two minimum following distance constraints:

1. A desired minimum following distance, $slarge$⁠, at the end of the CG horizon, $Tlarge$⁠.

2. A minimum safe following distance, $ssmall$⁠, that must be enforced at every time step along some beginning portion of the CG horizon.

The reason for the incorporation of two following distance constraints in the CG formulation is driven by the fact that the lead vehicle's velocity profile will often differ from what the CG assumes. Specifically, while the actual velocity profile of the lead vehicle is unknown, its velocity is assumed by the CG to be constant over the prediction horizon. If only the minimum following distance, $ssmall$⁠, were enforced over the prediction horizon, then the CG would allow the ego vehicle to travel at a following distance near that minimum, under the assumption of a constant-speed lead vehicle. If the lead vehicle slowed down unexpectedly instead of remaining at the assumed constant speed, then the CG would have no choice but to apply a large braking force at the subsequent time-step. To mitigate this issue, our proposed algorithm introduces an additional terminal constraint at time $Tlarge$⁠, which requires a deceleration profile that forces the ego vehicle ultimately to back off to a larger following distance, $slarge$⁠. As illustrated in Fig. 3, this results in a buffer between the ego vehicle's actual following distance, s(t), and the absolute minimum allowable following distance (⁠$ssmall$⁠), thereby avoiding large braking events upon every unexpected deceleration of the lead vehicle.

Based on the constraints of Fig. 3, the truncated output admissible set, O_T, is defined as follows:

Simulations of the cascaded MPC and CG-based Eco-ACC strategy were performed on a heavy-duty truck over a highway route for various CG horizon lengths, $Tlarge$⁠. In this section, we present these results and assess the impact of CG parameters on fuel economy and drivability.

Distance and elevation data for three representative trucking routes furnished by Volvo Group North America, denoted by HWY-1, HWY-2, and HWY-3, with lengths of 180 km, 360 km, and 240 km, respectively, were used in simulations. Additionally, simulations were performed for a fully loaded and a half-loaded trailer, resulting in total truck weights of 35,000 kg and 20,000 kg, respectively. A stochastic lead vehicle traffic model, developed and calibrated using highway traffic data in Ref. [13], was used to provide multiple lead vehicle velocity profiles over the entire route. Over the representative trucking routes HWY-1, HWY-2, and HWY-3, a total of 29, 32, and 40 lead vehicles were encountered, respectively. For the baseline strategy, a Gipps' car-following model was used to determine the velocity setpoint, where the desired following distance was set to $slarge$⁠. In all the simulations performed, $tlarge=3$ s and $tsmall=1$ s, where $slarge=tlargevlead$ and $ssmall=tsmallvlead$⁠. The time-step of the CG, $ΔtCG$⁠, was set to 200 ms for all simulations. $Tsmall$ was held at a relatively short time horizon of 2 s to prevent unnecessary adjustments of applied wheel force, as the approximation of the lead vehicle's future behavior had less accuracy at longer horizons. The MPC used a horizon length of 2000 m and a step size, $ΔxMPC$⁠, of 40 m. Weight λ_b in Eq. (18) was set 100 to place a greater penalty on the energy loss due to braking than the mechanical energy expended, as it was shown in Ref. [11] that the majority of fuel savings realized by ecological driving stem from avoiding unnecessary braking events. Weight λ_d in Eq. (18) was tuned heuristically via preliminary highway driving simulations for the Eco-ACC presented in this work where $Tlarge$ was set to 30 s. Based on the preliminary highway driving simulation results shown in Fig. 4, λ_d was set to 0.1 in order to achieve the most fuel savings without compromising trip time. The MPC optimization was implemented every 1 s, around every 20–30 m at the simulated highway speeds.

where $uwh$ is the realized propulsive force applied through the wheel to the road. Two critical components of total energy expenditure are:

• The total energy dissipated by braking, $Ebraketot$⁠.

• The total energy lost due to aerodynamic drag, $Eaerotot$⁠.

Ecological performance and drivability are reported in Figs. 5 and 6 for a fully loaded and half-loaded trailer, respectively. Figures 5(a) and 6(a) show the total energy expended and fuel consumed for various values of $Tlarge$⁠. Fuel economy improves as $Tlarge$ increases, and then plateaus for $Tlarge≥25 s$⁠. In Figs. 5(b) and 6(b), the energy savings due to the reduction of braking and aerodynamic drag are compared to the baseline. The energy expended due to aerodynamic drag is largely dependent on trip time. As a result, negative energy savings due to drag indicate that the Eco-ACC strategy completed the trip faster than the baseline. The results in Figs. 5(b) and 6(b) indicate that a reduction in braking was responsible for the vast majority of total energy savings.

where $jego$ is the jerk of the ego vehicle and $Ttrip$ is the total trip time. Drivability results in Figs. 5(c) and 6(c) indicate comparable drivability to the baseline strategy. The best drivability results occurred when $10 s≤Tlarge≤30 s$⁠. In Sec. 4.3, we analyze how $Tlarge$ affected vehicle-following behavior and how these behaviors impacted performance.

In order to gain an understanding of how the Eco-ACC strategy modulates the velocity and following distance of the ego vehicle under various CG horizon lengths, $Tlarge$⁠, a stretch of simulated highway driving where the heavy-duty truck engaged in vehicle-following is examined. The vehicle-following scenario in Fig. 7 occurred on HWY-3 for a fully loaded trailer, during which a lead vehicle entered and exited the lane at around 146 km and 151 km, respectively. The lead vehicle entered the lane at an initial following distance near $ssmall$⁠, traveling around 1 m s⁻¹ slower than the Eco-ACC strategy. Consequently, upon the lead vehicle entering the lane, every Eco-ACC simulation applied a braking force.

Before the lead vehicle entered the lane, the Eco-ACC maintained a velocity near the offline optimized velocity trajectory, $vDP*$⁠, which is expected due to the penalty term in Eq. (18). When the lead vehicle entered the lane, the CG began modulating the prescribed wheel torque by the MPC to enforce the predicted vehicle-following constraints. Additionally, the desired velocity, $vDP*$⁠, for this vehicle-following scenario was greater than the lead vehicle's velocity. Consequently, the realized velocity of the Eco-ACC simulations deviated from $vDP*$ while the lead vehicle was present. However, the MPC was still able to engage in ecological driving whenever the prescribed MPC wheel torque satisfied the vehicle-following constraints enforced by the CG. For instance, as the Eco-ACC simulations climbed a relatively steep hill, near 150 km, the MPC prescribed a reduced velocity for the truck, and the Eco-ACC simulations began to follow a similar velocity trend as the offline optimized velocity trajectory. After the lead vehicle exited the lane, the Eco-ACC maintained a slower velocity than the offline optimized solution, but followed a similar velocity trend, until utilizing a downhill portion of the route, starting at 152 km, to accelerate back to a velocity near $vDP*$⁠.

To investigate reasons for the plateau in ecological performance for $Tlarge≥25 s$⁠, the cumulative energy loss due to braking is shown in Fig. 7(b). The greatest energy losses due to braking occurred when the lead vehicle entered the lane. Eco-ACC simulations with smaller $Tlarge$ values expended more braking energy upon the initial entry of the lead vehicle, specifically when $Tlarge<20 s$⁠, due to applying a larger braking effort to quickly achieve $s≥slarge$⁠. However, Eco-ACC simulations where $Tlarge$ was longer had the tendency to stay near $ssmall$ after the lead vehicle entered the lane. This required the CG to apply a large braking effort if the lead vehicle suddenly slowed down, an example instance of which is seen for $Tlarge=150 s$ around 147.5 km in Fig. 7.

Drivability for the vehicle-following scenario of Fig. 7 was evaluated using the cumulative sum of the vehicle jerk squared, shown in Fig. 7(d). Drivability was affected by $Tlarge$ similarly to the way ecological performance was affected by $Tlarge$⁠. The large braking events that occurred for $Tlarge<20 s$ and $Tlarge>52 s$ had a negative effect on drivability.

This paper presented algorithmic details and simulation results for an Eco-ACC strategy that combines MPC for ecological driving with a command governor for safe vehicle following. An analysis on the effect controller parameters have on performance, in terms of both efficiency and drivability, was performed. Results from our analysis demonstrated a consistent reduction in fuel consumption when the CG horizon was increased up to a point. Simulations performed with a longer $Tlarge$ had the tendency to result in a following distance that approached $ssmall$ more often. In these cases, any small braking action on the part of the lead vehicle resulted in a sudden corrective ego vehicle braking effort that had a negative effect on fuel economy and drivability. Specifically, for the heavy-duty truck considered in this work, the best ecological performance was realized for $20 s≤Tlarge≤40 s$⁠.

This research was funded by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award No. DE-AR-0000801. The authors would also like to acknowledge Volvo Group North America for their assistance and support.

• Advanced Research Projects Agency–Energy (Funder ID: 10.13039/100006133).