Model Predictive Control for Automotive Applications

Due to the computational complexity of the OCP, the MPC has been applied mainly for slow dynamic systems, such as the chemical and process industry. During the last decades, however, several developments including modern computing hardware have allowed using these methods also for fast dynamic systems with a growing interest for automotive applications (Re et al., 2010). One of the earliest MPC design to improve fuel economy of theCCwas introduced by Gilbert (1976), while one of the earliestMPC

designs for the ACC was presented by Goodrich et al. (1998) and Seki et al. (1999). Spacing-control laws were computed by formulating the objective of a transitional ma- neuvers as anOCPby Bageshwar et al. (2004). A linearMPCfor theACC system was designed by Corona et al. (2007), and implemented on a SMART vehicle. An explicit

MPC was designed and implemented for the ACC with Stop&Go function by Gerrit Naus et al. (2008) and G.J.L. Naus et al. (2010). Vehicular followingMPC considering fuel economy and tracking capability was introduced by Shengbo Li et al. (2008).

SMPCfor improving the performance of powertrain control algorithms such as theACC

was introduced by Bichi et al. (2010). Challenges of tracking capability, fuel economy and driver desired response for theACCwas addressed utilising a linearMPCapproach by S. Li et al. (2011). A single NMPC was designed by Shakouri, Ordys, and Askari (2012) for the ACC with the objective of the distance tracking. This approach was demonstrated to be more effective in tracking the speed and distance by eliminating the necessity of switching between the two controllers. An ACC was introduced on a HEV platform by K. Li et al. (2012). An OCP framework for the ACC with the assumption of stationary conditions for the dynamics of other vehicles was introduced by M. Wang, W. Daamen, et al. (2012). A linear MPC approach for the CACC that directly minimizing the fuel consumption rather than the acceleration of the vehicle presented by Stanger et al. (2013). A real-time novel control system to drive a vehicle efficiently on roads containing varying traffic and signals at intersections for improved fuel economy was presented by Md Abdus Samad Kamal et al. (2013). In this system, the relevant information of the current road and traffic, simplified prediction of the future states of the preceding vehicle were taken into account. An NMPC was introduced by Shakouri and Ordys (2014) for theACCand theCCsystems which carry out automatic switching between ACC and CC, depending on the situation in front of the vehicle. A novel rolling horizon control framework for non/cooperative driver assistance systems was introduced by Meng Wang, S. Hoogendoorn, et al. (2014) and W. Wang et al. (2014). A terrain-information, actuator-efficiency-incorporated energy management and driving strategy for maximizing the travel distance of in-wheel motor BEV was introduced by Chen et al. (2014). A control methodology that unifies control barrier functions and control Lyapunov functions through quadratic programs was developed by Ames et al. (2014).

A CACC system using stochastic, linear MPCstrategies with the goal of minimisation of fuel consumption in a car-following scenario was presented by Moser, Waschl, et al. (2015). A real-time algorithm to reduce the online computational burden of the MPC

by combining an move blocking strategy with a constraint-set compression strategy was introduced by S. E. Li, Z. Jia, et al. (2015). ANMPCscheme with the target of emission and fuel-efficient driving for the CACC system introduced by Schmied et al. (2015). A novel energy-efficientMPCwas designed for theBEVs Eco-CCsystem by T. Schwickart et al. (2015) and Tim Schwickart, Voos, Hadji-Minaglou, et al. (2016). AnotherNMPC

was introduced by Vajedi et al. (2016) to optimally control the velocity profile for the

with main focus on communications, driver characteristics, and controls was presented by Dey et al. (2016). A novel strategy to enhance string stability of autonomous vehicles with sensor delay and actuator lag based on a MPC framework was proposed by M. Wang, S. P. Hoogendoorn, et al. (2016). A self-tuning control algorithm for an ACC

system that can adapt its behaviour to variations of vehicle dynamics and uncertain road grade was proposed by Marzbanrad et al. (2016). A comprehensive review of power management strategy inHEVs with an emphasis onMPCbased strategies and the factors that affect the performance of theMPCwas presented by Y. Huang et al. (2017). A RMPC approach that regulates a minimum safety distance between vehicles taking into account the overall system delays and braking capacity of each vehicle proposed by Filho et al. (2017). The introduced RMPC was developed to guarantee the minimum safety distance should not be violated due to uncertainties in the lead vehicle behaviour. An enhanced MPC for the ACC system considering road elevation information was presented by S. E. Li, Guo, et al. (2017). A real-time NMPC for the ecological CC of a HEVwas presented by Tajeddin et al. (2017). Another real-timeRMPC for the Eco-

CACC system with the consideration of gear shift was presented by Shao et al. (2017). A SMPC for the ACC and CACC systems under uncertainty based on the constant time gap policy was presented by Y. Zhou et al. (2017). A four-component analysis framework for platoon CACC systems from a networked control perspective, including a literature review by network awareness, unified models of key components, and two application cases for controller synthesis was reviews by S. E. Li, Zheng, et al. (2017). A safe- and eco-driving control system that enables the connected and automated EV

to accelerate or to decelerate optimally while preventing both collision with preceding vehicle and violation of speed limitations was proposed by Ojeda et al. (2017). Last but not least, flexible spacingACCsystem usingSMPCwas introduced by Moser, Schmied, et al. (2018). In this paper, a conditional linear Gauss model is developed and trained with real measurements to estimate the probability distribution of the future velocity of the preceding vehicle (Moser, Schmied, et al.,2018).

To conclude, the optimal control methods have made a considerable impact on ADAS

and especially the ACC systems. The next sections review more details of the general

MPCdesign methods for a class of dynamic system. Throughout this study, Rn_denotes

the n-dimensional Euclidean space. R+ := [0, ∞). N = {1, 2, . . .} is set of natural

numbers. N+ := N ∪ {0}. Z[a,b] := {a, a + 1, . . . , b} is set of integers from a to b.

E denotes expectation and Ex[·] := E[·|x(0) = x] is the conditional expectation. Pr

denotes probability, and Prx[·|x(0) = x] is the conditional probability distribution of