Multi Time-Scale Engine and Powertrain Control for Autonomous Vehicles Via Lagrange Multipliers

As environmental concerns mount, there is increasing pressure on all industries to reduce their energy usage and carbon footprint. In 2019, the United States transportation industry alone consumed 8.3 trillion kWh, which corresponds to 37% of the nation's total energy usage [1]. Of that energy, 95% comes from fossil fuel sources. Recent developments in Connected and autonomous vehicle (CAV) technology resulted into a significant body of literature focusing on the development of control strategies that utilize global positioning system (GPS) and traffic information available to CAVs to improve energy efficiency. These strategies include vehicle velocity optimization [2–7], eco-routing [6,8], platooning [9,10], gear selection optimization [11–14], optimal traffic signal arrival and departure [8,15–18], hybrid vehicle energy management [19], and integrated powertrain and thermal management [20–23]. In their traditional implementation, these supervisory strategies generate optimal trajectories for the lower level controllers such as the engine control module (ECM) and the transmission control module that are then converted into actuator positions. These supervisory controllers often use a prediction horizon of several seconds, consistently with the V2V and V2I information available, and are operating at a time-step that is consistent with the relevant dynamics of the powertrain (tenth of a second and above) [24]. However, for their in-vehicle implementation, the integration with modules with much faster sampling rates, such as the ECM that operates in the order of milliseconds, is required. This is usually achieved through a zero-order hold (ZOH), where the command from the supervisor is held constant between updates. When the ZOH is updated, the sudden change in the command can lead to several undesirable effects, such as actuator saturation [25], constraint violation, actuator saturation, poor tracking performance, and decreased efficiency [26]. In automotive applications, the ZOH strategy can also result in poor drivability [27] and increased emissions [28–30]. Moreover, because the vehicle supervisory controller operates at a slower time resolution compared to the engine, it cannot account for the fast dynamics related to, for example, air and fuel path dynamics that characterize torque production and engine response [4,9,31].

The problem of coupling a reference signal that has a coarse sampling rate with a control module that operates at a much faster rate while meeting the system constraints has been approached in literature using supplemental controllers. The simplest realization is the input shaper strategy, where the original input is filtered in order to remove sharp changes that would result in actuator saturation and decreased tracking performance [25,27]. While this strategy is real-time implementable, it cannot guarantee constraints satisfaction and the lack of optimality considerations can significantly affect the closed-loop performance [27]. Another approach adopted in literature is the reference governor. As opposed to input shapers, which modify the reference signal at all points in time, reference governors only modify the reference when the reference would lead to constraint violation. The modified reference tracks the original reference as closely as possible, without violating constraints. However, like the input shaper, the reference governor obtains the modified reference without minimizing a performance metric.

A more promising approach involves utilizing model predictive control (MPC), as it provides near optimal performance while guaranteeing constraint satisfaction [32]. In addition, in the context of CAVs, the MPC framework can use prediction and information of future driving conditions within the receding horizon [3,4,18,33]. An MPC that simultaneously considers the fast engine and slow powertrain dynamics with associated constraints has been presented by the authors in Ref. [27]. Results show that the strategy is capable of guaranteeing fast dynamics constraint satisfaction and optimal tracking of the supervisory velocity profile, while improving both ride comfort and fuel economy. However, the computation time of the centralized strategy was not suited for in vehicle implementation as the algorithm was operating about 1000 times slower than real-time when simulated on a 7th generation Intel Core i5 with 16 GB of RAM. A possible solution to retain both slow and fast dynamics in the optimization is to implement a variable rate MPC, where in the first part of the horizon, the centralized problem is solved and, in the latter part, only the slow dynamic is considered [23,34]. Given the computation complexity of the centralized problem and the horizon length required for achieving quasi-optimal performance and stability, it is expected that a variable rate MPC would suffer from the same limitations as the centralized strategy.

Alternatively, a hierarchical MPC can be designed by separating the dynamics based on time scale where slow dynamics optimization determines an optimal reference profile for the engine torque, which is used as reference for the fast dynamic problem [35]. The hierarchical approach improves computation efficiency, but has the drawback of neglecting the fast dynamic constraints in the slow dynamics optimization, hence generating an unfeasible profile, which is then saturated in the fast dynamic layer. An iterative approach for addressing the inconsistencies between constraints in the different layers of a hierarchical MPC has been presented in Ref. [36] for the problem of distribution and frequency regulation in the electric grid. The approach is based on the introduction of a binary flag that indicates the ability of the fast dynamic optimization to track the reference profile. If the deviation is too large, a weight in the slow dynamic optimization cost function is modified and the process is repeated until either the tracking performance are acceptable, or a maximum number of iterations is reached. While this method has been proven effective in simulation, the weight tuning process is entirely empirical and requires extensive calibration. In addition, there are no guarantees that the method will lead to the optimal solution in a finite number of iterations [37].

This paper proposes a novel, iterative hierarchical scheme for integrated eco-driving and powertrain setpoint optimization. The proposed solution has the ability of guaranteeing constraint satisfaction, while achieving a near-optimal solution comparable to the centralized solution at a significantly reduced computation time. This is achieved by using the Lagrange multipliers of the fast dynamic as measure for constraints satisfaction in the slow dynamic optimization. This leverages the properties of Lagrange multipliers that are nonzero if and only if the corresponding constraints are active, and their magnitude is a measure of the distance to the constraint. In addition, the sensitivity of the fast dynamic Lagrange multipliers to the slow dynamic cost function is used in the algorithm to reduce the number of required iterations.

The paper is organized as follows: First, the plant model and experimental validation is presented. Then, the general optimization problem is formulated and solved using a centralized scheme and conventional hierarchical method. The novel iterative method based on Lagrange multipliers is derived and demonstrated on a simplified problem next. Finally, the methodology is applied to two case studies related to the autonomous truck operation.

The vehicle considered in this study is a Volvo VNL 300, manufactured by Volvo Trucks headquartered in Gothenburg, Sweden. The VNL 300 is a class 8 semitruck equipped with a 13 L six cylinder turbocharged Diesel engine with exhaust gas recirculation and a twelve speed automated manual transmission. The truck has two driven axles with dual tires in the rear, and one steered axle with single tires in the front, and tows a standard trailer, with a maximum loading such that the gross vehicle weight is 80,000 lbs (35,000 kg). A model of the vehicle that accounts for the engine air and fuel dynamics, the transmission efficiency, rotational inertia, and the longitudinal dynamic has been developed and its structure is shown in Fig. 1.

The engine is modeled using the approach presented in Ref. [38] where the air-handling and fuel dynamics, which determine the instantaneous fuel flow rate in transient conditions, are represented by first-order systems. This model is expected to describe the low-frequency engine dynamics, while higher frequency behavior, such as pressure wave propagation or cycle to cycle combustion variation, is averaged out over several cycles. This simplifying assumption provides enough details for the characterization of the engine torque generation during transient operations and associated constraints without explicitly accounting for gasdynamic effects or modeling the engine actuators. It is worth noting that the objective of the supplemental controller is to modify the engine torque command, but does not have the ability to independently modify engine actuators such as valve timing, or exhaust gas recirculation valve [39,40]. Therefore, a higher order model capable of capturing the effects of engine actuators is unnecessary.

The volumetric efficiency model was calibrated and validated using data collected on an engine dynamometer. A set of 251 steady-state engine operating points was used for the calibration and the model was validated on 79 points. The curve fitting for Eq. (3) is shown in Fig. 2, and the comparison with experimental data is shown in Fig. 3. The model predicts the steady-state air mass flow rate with a mean absolute error of 2.8%. The error distribution shown Fig. 3(b) has a mean of 0.15% and a standard deviation of 3.8%. The outlier corresponds to a condition with low air flow, and therefore low torque output.

and are calibrated to meet the transient responses collected on the engine dynamometer and lump the effect of air and fuel path components.

where FAR_lim represents the fuel-to-air ratio (FAR) limit.

where the indicated mean effective pressure and friction mean effective pressure are mapped as functions of fuel flow and engine speed based on a validated GT-POWER engine model [43].

where $v$ is the vehicle velocity.

where $ρ$ is the density of air, $Cd$ is the drag coefficient, and $Af$ is the frontal area.

The model was validated using engine-in-the-loop (EIL) data. For the test, the Volvo engine was instrumented and, together with the ECM, coupled with a dynamometer. The drivetrain and vehicle were emulated using high-fidelity proprietary models developed by Volvo. Two driving cycles representing city and highway were tested, each driving cycle was run for three different trailer load configurations (empty, half, and fully loaded) and repeated twice [31].

A comparison of the predicted and experimental engine torque is shown in Fig. 4(a) and has an average root-mean-square (RMS) error of 2.7% over all tests. The comparison between predicted and measured instantaneous fuel consumption is shown in Fig. 4(b), and has an average RMS error of 3.7% and an average cumulative error of 2.05% over all tests. The velocity validation is shown in Fig. 4(c) and has an average RMS error of 4.2% over all tests and test conditions. Figures 5(a)–5(c) show a zoomed in view of these figures, making the deviation more clear, and highlighting the high-frequency oscillation inherent in the experimental data.

where N_p is the length of the horizon, $Δt$ is the discretization time, γ₁, γ₂, and γ₃ are weights penalizing fuel flow rate, change in torque command, and the distance traveled at the final time, respectively. The function G includes the state and input constraints.

The nonlinear MPC problem defined previously can be solved in its original centralized form using numerical solution tools, such as dynamic programming [45]. Alternatively, to improve computation time, the plant model and constraints can be convexified, which allows to convert the problem into a standard quadratic programming (QP) problem [44]. In this paper, the latter approach is adopted and the centralized solution is used as optimality benchmark for the novel hierarchical approach.

While the centralized approach is able to minimize cost and satisfy constraints from both time-scales, it is generally computationally intensive and, depending on the applications, might not be real-time implementable [27,46]. In addition, when simultaneously considering different time-scales in the plant, the optimization problem may result to be numerically ill-conditioned [47].

which are the value of the fast states obtained under the assumption that the changes are instantaneous. The fuel flow rate is determined from the static fueling map, while the air flow rate is given by Eq. (1).

where N_f is the length of horizon for the fast dynamic problem, and $x¯s$ are the nominal trajectories of the slow states obtained from the slow dynamic optimization problem and are included as external inputs. The function g^f includes the state, input and the FAR constraints. The combined hierarchical problem is then solved sequentially [48]. To provide a consistent comparison against the centralized MPC solution, the objective functions and state dynamics have been convexified and the problems are solved using the same batch approach algorithm and QP solver.

While it is clear that the hierarchical approach will result in a significant computation time improvement and possibly lead to a real-time executable algorithm [48], the sequential formulation cannot guarantee that the optimal trajectories for $m˙f,kref$ are feasible due to the constraints on the fast dynamics. To address this issue, a novel iterative approach based on the Lagrange multipliers is presented in this paper.

where $u*$ is the optimal control sequence, G is the function of the constraints, and J is the objective function. The Lagrange multipliers λ can be interpreted as the change of the objective function due to a perturbation of the constraints [45,49]. This concept is leveraged in this work for determining the minimum value of $γ2s$ in Eq. (26) that does not result in a fast dynamic constraint violation. That is, find the minimum value of $γ2s$⁠, such that $∑λ*$ is as small as possible. The associated solution of the slow time-scale optimization problem would then result in a trajectory for the fast dynamic problem that is feasible and obtained with the smallest possible penalty on actuation.

Two illustrative examples are discussed here. First the case of a convex optimization problem with equality constraints is presented and the general analytical expression for the Lagrange multipliers is determined. Then, the process is extended to include inequality constraints, as this is crucial for the practical implementation of the method.

which can be solved under the conditions that Q_s and W_s are positive semidefinite, R_s is positive definite, and the rows of S_u are linearly independent, hence if the pair $(As,Bs)$ is controllable.

The augmented system matrix $A¯kf$ contains the term $ykf,ref(Us*)$⁠, ultimately coupling the fast and slow dynamic optimization problems. Note that because of the structure of the augmented state input matrix $B¯f$⁠, only $Sf,x$ is a function of $Us*$⁠. Hence, only the vector f_f contains the coupling term.

Specifically, it is possible to a priori determine what is the minimum value for the slow dynamic actuator penalty that would not result in a fast dynamic constraints violation.

which includes both equality and inequality constraints. The same procedure shown for the case of equality constraint can be followed here. However, in the presence of inequality constraints, Eqs. (65) and (66) cannot be solved analytically because it is unknown which inequality constraints are active. To overcome this issue, numerical methods such as interior point or method of multipliers are adopted [45]. However, these approaches are not suited for this application because they would not provide any information regarding the sensitivity of $λ*$ to the penalty R_s. For this reason, this paper proposes an iterative approximation method for estimating the behavior of $λ*$ as function of R_s. The process is summarized in the flowchart in Fig. 6 and Algorithm 1.

While the iterative algorithm is presented under the assumption that R_s is scalar, it can be extended to nonscalar R_s by fitting a function for each diagonal element in R_s.

The weights used in the cost functions are summarized in Table 1. In this example, in the first iteration, R_s is selected as 0.1, and the corresponding $∑λ$ is 4. A nonzero value of $∑λ$ indicates that the fast dynamics constraints are active, which can be seen in Fig. 7(c) and results in poor tracking performance, as shown in Fig. 7(a). The corresponding slow and fast dynamic control inputs are shown in Figs. 7(b) and 7(d). Following the estimation scheme proposed before and switching from linear to exponential fitting at q_exp = 4, the corresponding values for R_s are shown in Fig. 8. By the fifth iteration, $∑λ$ is near zero, which results in the constraint satisfaction and improved tracking as shown in Figs. 7(c) and 7(a). Hence, the numerical estimation strategy is able to find the minimum value of R_s large to guarantee constraint satisfaction and reference tracking.

To test the performance of the proposed algorithm, two case studies have been selected where significant acceleration events occur; hence, they are most likely to lead to fast dynamic constraint violation. The controller is verified in simulation using the nonlinear vehicle model as plant model, while for the controller, the system is linearized at each time steps and the matrices updated. It is worth noting that while only fast dynamics associated with the engine torque generation are considered in this paper, the methodology can account for additional fast dynamics, such as gear changes or backlash events, under the same mathematical framework.

The simulation parameters and controller settings are summarized in Table 2 and are the same for both case studies. For consistency, the value of the weights penalizing fuel flow rate, change in torque command, and the distance traveled at the final time in the centralized problem and hierarchical problem are the same. The iterative hierarchical algorithm modifies $γ2s$ starting from the nominal value shown in Table 2.

All simulations were performed on a desktop PC running an Intel^® Xeon^® CPU E5-1650 v2 at 3.5 GHz with 64 GB.

The first case study examines the scenario in which the CAV is approaching a traffic light and driving below the speed limit due to a slow lead vehicle. At some point before the traffic light, the slow vehicle changes lane allowing the CAV to increase its speed and catch an earlier green phase. This scenario is simulated by providing a reference distance d_ref as the input to the supplemental controller. During the transient, the gear is constant. The simulation results for the centralized, hierarchical, and iterative algorithms are shown in Fig. 9. Figure 9(a) shows a sharp increase in engine torque at about 3 s, which corresponds to a change in the optimal distance provided by the supervisor in response to the absence of the lead vehicle. The centralized MPC is able to increase the engine torque faster than the two the hierarchical approaches and takes advantage of a pulse and glide strategy [51]. The smaller average engine torque commanded by the centralized controller results in a longer acceleration time compared to the two hierarchical strategies as shown in Fig. 9(c). All strategies are guaranteeing the constraint satisfaction, as shown in Fig. 9(d). It is worth noting that, if the slow dynamic optimizer generates feasible trajectories, the hierarchical and the iterative schemes will provide the same solution with the same computation time, as the iteration would only take one step. Finally, the effect of considering the Lagrange multipliers in the cost function is shown in Fig. 9(e), where the value of the smoothing penalty during the maneuver is shown. When the slow dynamic optimizer generates unfeasible trajectories due to fast dynamic constraint violation, the weight on the rate of change of the fuel flow rate $γs,2$ is increased from its nominal value. Once the nominal trajectory is feasible, the value of the weight returns to its nominal value. It is worth noting that, it is possible to empirically find a value for $γs,2$ that always results in a feasible reference trajectory for the fast dynamic controller. This, however, would require an extensive tuning process and result in a conservative control design that might not incentivize fuel savings.

where m_f is the total fuel over the maneuver, KE is the kinetic energy at the end of the maneuver, the subscript c refers to the centralized controller, and LHV is the lower heating value of the fuel. The results based on adjusted fuel consumption show a similar trend. Finally, the two hierarchical approaches are reducing computation time by a factor of one hundred and are both faster than real-time. The additional computation time required by the iterative approach compared to the hierarchical is negligible when evaluated over the maneuver since the algorithm for the selection of R_s converges quickly.

The second case study examines the scenario in which the truck is cruising at highway speed, when a vehicle enters the lane directly in front without maintaining a safety distance. Once in front of the truck, the lead vehicle travels at a constant speed. The CAV immediately reduces the torque command to zero in order to reestablish an adequate safety buffer with the preceding vehicle. Once sufficient buffer distance has been established, the CAV accelerates back to highway speed and tracks the reference distance provided by the supervisor. During the transient, the gear is constant. The simulation results for the centralized, conventional hierarchical, and iterative hierarchical MPC are shown in Fig. 10. Figure 10(a) shows that engine torque command drops to zero in response to the vehicle cut-in. By the time an adequate safety buffer has been established, vehicle velocity has dropped from 25 m/s to 22 m/s, as shown in Fig. 10(c). The engine torque command then increases to accelerate back to about 25 m/s. Similarly to the previous case, the centralized torque command has a chattering profile and is on average lower than the two hierarchical approaches resulting in a longer acceleration time. All controllers meet the FAR constraints as shown in Figs. 10(d) and 10(e) shows that the value of $γs,2$ increases dramatically when large changes in torque are requested. The same metrics defined for the previous case study are calculated here a summarized in Table 4. The conventional hierarchical MPC algorithm sacrifices fuel economy for computational efficiency when compared to the centralized approach. The proposed iterative hierarchical MPC with Lagrange multiplier feedback achieves a negligible impact on fuel when compared to the centralized approach, while still achieving significant improvement in computational efficiency. Contrary to the first case study, the iterative MPC, while more than ten faster than the centralized, is more than four times slower than real-time. The increased computation time is a result of the number of iterations required to find the minimum R_s. One simple approach to improve the real-time factor is to either increase the iteration threshold λ_max or decrease the maximum number of iteration. Furthermore, the algorithms are currently implemented in Matlab as interpreted code. It is expected that by compiling the algorithms in C will improve computation time and enable real-time implementation [53].

In this paper, a method for satisfying inequality constraints in multitime-scale optimal control problems has been developed. The algorithm uses a feedback on the Lagrange multipliers to find the optimal weigh in the slow dynamic cost function that results in a feasible trajectory for the fast dynamics problem. The Lagrange multiplier feedback mechanism eliminates the need for calibration of an appropriate weight in case constraints are not met, while systematically avoiding over-penalizing changes in control action which would result in worsened control performance. This is achieved by leveraging the sensitivity of the Lagrange multiplier to the penalty on control action. The method is demonstrated in this paper for the coordinated ecodriving and powertrain control in a connected and autonomous heavy duty vehicle. Results on two different maneuvers show fuel consumption performance similar to a centralized approach, with less than 1% difference. This is achieved with a computation time that about 10 times faster than centralized MPC and comparable to a traditional sequential hierarchical scheme. While in this paper only fast dynamics associated with the engine are considered, the same mathematical framework can be used to account for additional fast dynamics in the system, and can be extended to more than two time-scales.

This research was funded by the ARPA-E NEXTCAR (Next-Generation Energy Technologies for Connected and Automated On-Road Vehicles) Program, part of the Maximizing Vehicle Fuel Economy through the RealTime Collaborative, and Predictive Co-Optimization of Routing, Speed, and Powertrain Control project (DEAR0000801). The authors would like to acknowledge the assistance and support from Volvo Group North America.

• Advanced Research Projects Agency – Energy (Grant No. 10.13039/100006133; Funder ID: 10.13039/100006133).