Comfort Oriented Robust Adaptive Cruise Control in Multi-Lane Traffic Conditions

Preprints, 8th IFAC International Symposium on Preprints, 8th IFAC International on Advances in Automotive Control Symposium Preprints, 8th 8th IFAC International International Symposium on on Preprints, IFAC Symposium Advances Automotive Control June 19-23,in 2016. Norrköping, Sweden Available online at www.sciencedirect.com Advances in Automotive Control Advances Automotive Control June 19-23,in2016. Norrköping, Sweden June June 19-23, 19-23, 2016. 2016. Norrköping, Norrköping, Sweden Sweden

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Comfort Comfort Comfort Control Control Control

Oriented Robust Adaptive Cruise Oriented Robust Adaptive Cruise Oriented Robust Adaptive Cruise in Multi-Lane Traffic Conditions in Multi-Lane Traffic Conditions in Multi-Lane Traffic Conditions ∗ ∗ ∗

Roman Schmied ∗ Harald Waschl ∗ Luigi del Re ∗ Roman Schmied ∗∗ Harald Waschl ∗∗ Luigi del Re ∗ Roman Roman Schmied Schmied Harald Harald Waschl Waschl Luigi Luigi del del Re Re ∗ ∗ Institute for Design and Control of Mechatronical Systems, Johannes ∗ for Design and Control of Mechatronical Johannes ∗ ∗ Institute Kepler Linz, Austria. Systems, Institute and Control Systems, Institute for for Design Design and University Control of of Mechatronical Mechatronical Systems, Johannes Johannes Kepler University Linz, Austria. Kepler Kepler University University Linz, Linz, Austria. Austria. Abstract: The increasing complexity of nowadays traffic situations requires further developAbstract: The increasing complexity of situations further development of already Advanced Driver Assistant traffic Systems (ADAS).requires In the case of Adaptive Abstract: The existing increasing complexity of nowadays nowadays traffic situations requires further developAbstract: The increasing complexity of nowadays traffic situations requires further development of already existing Advanced Driver Assistant Systems (ADAS). In the case of Adaptive Cruise Control (ACC) such enhancements concern the consideration of lane change and/or ment of already existing Advanced Driver Assistant Systems (ADAS). In the case of Adaptive ment of already existing Advanced Driver Assistant Systems (ADAS). In the case of Adaptive Cruise Control (ACC) such enhancements concern the consideration of lane change and/or multi-vehicle scenarios. Such lane change maneuvers may lead to switching processes for the Cruise Control (ACC) such enhancements concern the consideration of lane change Cruise Control (ACC) such enhancements concern the consideration of laneprocesses change and/or and/or multi-vehicle scenarios. Such lane change maneuvers may lead to switching for the controller which often lead to undesired and intense braking with detrimental effects on comfort, multi-vehicle scenarios. Such lane change maneuvers may lead to switching processes for multi-vehicle scenarios. Such lane change maneuvers may lead to switchingeffects processes for the the controller which often lead to undesired and intense braking with detrimental on comfort, fuel economy andoften safety. controller which often lead to to undesired undesired and and intense intense braking braking with with detrimental detrimental effects effects on on comfort, comfort, controller which lead fuel economy and safety. This paper proposes an ACC approach within a multi-vehicle scenario which considers lane fuel economy and fuel economy and safety. safety. This paper proposes an ACC approach aa multi-vehicle which considers change maneuvers of surrounding traffic within participants in a robustscenario and predictive way withlane the This paper proposes an ACC within scenario which lane This paper proposes an ACC approach approach within a multi-vehicle multi-vehicle scenario which considers considers lane change maneuvers of surrounding traffic participants in a robust and predictive way with the objective to increase driving comfort. Robustness is addressed by considering worst case profiles change maneuvers of surrounding traffic participants in a robust and predictive way with the change maneuvers ofdriving surrounding traffic participants in a robust and predictive way with the objective to increase comfort. Robustness is addressed by considering worst case profiles concerning change behavior of preceding vehicles. Simulation results compare the profiles robust objective to tolane increase driving comfort. Robustness is addressed addressed by considering considering worst case case profiles objective increase driving comfort. Robustness is by worst concerning lane change behavior of preceding vehicles. Simulation results compare the robust controller tolane a nominal one and of show significant improvement of results the proposed approach by concerning change preceding vehicles. Simulation compare the concerning lane change behavior behavior of preceding vehicles. Simulation results compare the robust robust controller to a nominal one and show significant improvement of the proposed approach by means of a comfort oriented performance index. controller to a nominal one and show significant improvement of the proposed approach controller tocomfort a nominal one performance and show significant improvement of the proposed approach by by means of a oriented index. means of a comfort oriented performance index. means of a comfort oriented performance index. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Adaptive Cruise Control, Model Predictive Control, Robust Control. Keywords: Adaptive Cruise Cruise Control, Model Model Predictive Control, Control, Robust Control. Control. Keywords: Keywords: Adaptive Adaptive Cruise Control, Control, Model Predictive Predictive Control, Robust Robust Control. 1. INTRODUCTION to an adjacent lane but misjudge the traffic situation, 1. INTRODUCTION INTRODUCTION to an an adjacent adjacent lane but misjudge misjudge the traffic situation, situation, especially vehicles behind them. As the a consequence, other 1. to lane but traffic 1. INTRODUCTION to an adjacent lane but misjudge the traffic situation, especially vehicles behind them. As a consequence, other vehicles are forced to hard braking maneuvers which may especially vehicles behind them. As a consequence, other especially vehicles behind them. Asmaneuvers a consequence, other vehicles are forced to hard braking which may In recent years a lot of effort has been put on the develop- vehicles often lead to traffic jams or even accidents. Consideration are forced to hard braking maneuvers which may vehicles aretoforced to hardorbraking maneuvers which may In recent recent years a lot lotACC of effort effort has been been put on on the developoften lead traffic jams even accidents. Consideration ment and testing of systems. Whereas in industry the In years a of has put the developof possible lane change operations of other vehicles in an often lead traffic jams or Consideration In recent years a of lotACC of effort has been put on developlead to to traffic jamsoperations or even even accidents. accidents. ment and testing systems. Whereas in the industry the often of possible possible lane change ofelement other Consideration vehicles in an an main focus is set on implementability and safety, research ment and testing of ACC systems. Whereas in industry the anticipatory way thus provides a key in increasing of lane change operations of other vehicles in ment and testing of ACC systems. Whereas in industry the of possible lane change operations of other vehicles in an main focus is set on implementability and safety, research anticipatory way thus provides a key element in increasing often tries to additional performance criteria like anticipatory main focus is set and research driving comfort means of avoiding acceleration and way thus provides aa key element in main focus is consider set on on implementability implementability and safety, safety, research wayby thus provides key high element in increasing increasing oftenthe tries to consider additional performance criteria like anticipatory driving comfort by means of avoiding avoiding high acceleration and e.g. increase of traffic flow, see Schakel etcriteria al. (2010), often tries to consider additional performance like jerk andcomfort furtherby reduces the risk of traffic flow instability, driving means of high acceleration and often tries to consider additional performance criteria like driving comfort by means of avoiding high acceleration and e.g. the the increase ofAtraffic traffic flow, see and Schakel et al. al.on(2010), (2010), jerk and further reduces the risk of traffic flow instability, Naus et al. (2010). review of ACC its effect traffic e.g. increase of flow, see Schakel et see Zhou and Peng (2004). jerk and further reduces the risk of traffic flow instability, e.g. the increase of traffic flow, see Schakel et al. (2010), jerk and further reduces the risk of traffic flow instability, Nausimprovement et al. al. (2010). is A given reviewinof of ACC ACC and and its effect (2015). on traffic traffic see Zhou and Peng (2004). flow andits Borkar Naus et A review effect on and (2004). Naus et al. (2010). (2010). is A given reviewinofLunge ACC and its effect (2015). on traffic see see Zhou Zhou and Peng Peng (2004).this issue by introducing a reflow improvement Lunge and Borkar Our approach addresses flow improvement is given in Lunge and Borkar (2015). flow improvement is given in Lunge and Borkar (2015). Our approach addresses this issue issue by introducing introducing reOther approaches aim on increasing fuel economy and Our ceding horizon optimal control problem which is solved approach addresses this by aa Our approach addresses this issue by introducing a rereOther approaches aim on increasing fuel economy and ceding horizon optimal control problem which is solved driving comfort by using online optimal control techniques. Other approaches aim on increasing fuel economy and using MPC techniques. The costproblem functionwhich is designed to ceding horizon optimal control is solved Other approaches aim on increasing fuel economy and ceding horizon optimal control problem which is solved driving comfort by using online optimal control techniques. using MPC MPC techniques. The cost cost function function is while designed to In Schmied et by al.using (2015), Moser et control al. (2015) model using driving comfort online optimal techniques. minimize the vehicle’s acceleration and jerk maintechniques. The is designed to driving comfort by using online optimal control techniques. using MPC The cost function is while designed to In Schmied Schmied et al. al. (2015), (2015), Moser et et al. (2015) model model minimize thetechniques. vehicle’s acceleration and jerk mainpredictive control strategies are al. investigated with minimize In et Moser (2015) taining traffic flow. Theacceleration behavior ofand other the jerk while mainIn Schmied et al.(MPC) (2015), Moser et al. (2015) model minimize the vehicle’s vehicle’s acceleration and jerktraffic whileparticimainpredictive control (MPC) strategies are investigated with taining traffic flow. The behavior of other traffic particithe objective to improve efficiency. a variable predictive control (MPC) strategies are investigated with pants is traffic considered external disturbance. This enables flow. The behavior of participredictive control (MPC)fuel strategies areThereby investigated with taining taining flow. as The behavior of other other traffic traffic particithe objective objective toisimprove improve fuel efficiency. Thereby adegree variable pants is traffic considered as external disturbance. This enables enables spacing policy introduced which provides a of the to fuel efficiency. Thereby a variable the implementation of robust ACC by considering worst pants is considered as external disturbance. This the objective toisimprove fuel efficiency. Thereby adegree variable pants is considered as external disturbance. This enables spacing policy introduced which provides a of the implementation implementation of robust robust ACC ACC by by considering considering worst worst freedom for optimization. It iswhich shownprovides that these spacing policy is introduced aa strategies degree of case disturbance trajectories. the of spacing policy is introduced which provides degree of the implementation of robust ACC by considering worst freedom for optimization. optimization. It is is shown shown that that these strategies case disturbance trajectories. do not only decrease fuel consumption but also emissions freedom for It these strategies case trajectories. freedom for optimization. It is shown that strategies case disturbance disturbance trajectories. do anot not only decrease fuel consumption consumption butthese also emissions exemplary validation scenarios are defined where the of Diesel engine. However, those approaches doemissions consider Two do only decrease fuel but also do not only decrease fuel consumption but also emissions Two exemplary validation scenarios are to defined where one the of a Diesel engine. However, those approaches do consider robust control approach is compared a nominal Two exemplary validation scenarios are defined where the Two exemplary validationisscenarios are to defined where one the aofone lane vehicle only. of aa Diesel engine. However, those do Diesel engine. following However, scenario those approaches approaches do consider consider robust robust control approach compared a nominal a one lane vehicle following scenario only. which does not approach consider possible lane change maneuvers control is to one robust control approach is compared compared to aa nominal nominal one aa one lane vehicle following scenario only. one lane vehicle following scenario only. which does not consider possible lane change maneuvers Recently, various publications extended the application which anticipatory. Simulation reveal great potential in does consider possible lane maneuvers which does not not consider results possible lane aachange change maneuvers Recently, various publications extended the application application anticipatory. Simulation results reveal great potential in area of the traditional ACC extended to more complex traffic anticipatory. Recently, various publications the increasing driving comfort if possible lane change operaSimulation results reveal a great potential in Recently, various publications extended the application Simulation results reveal alane great potential in area of of the the traditional ACCwith to amore more complex traffic anticipatory. increasing driving comfort if possible change operasituations. For example, ACC lane change assistant area traditional ACC to complex traffic tions of other vehicles are if incorporated within theoperacondriving comfort possible lane change area of the traditional ACCwith to amore complex traffic increasing increasing driving comfort if possible lane change operasituations. For example, ACC lane change assistant tions of of other other vehicles vehicles are are incorporated incorporated within within the the conconis presentedFor inexample, Dang etACC al. (2015) et al. tions situations. with lane change troller. situations. For ACC with aaand lane Habenicht change assistant assistant tions is presented presented inexample, Dang et etwith al. autonomous (2015) and Habenicht et al. al. troller.of other vehicles are incorporated within the con(2011). Vehicle control lane changing is is in Dang al. (2015) and Habenicht et troller. is presented in control Dang etwith al. autonomous (2015) and Habenicht et al. troller. (2011). Vehicle lane changing is The paper is structured as follows: First, a general descripdesigned in Nilsson et al.with (2014), Naranjo et al. changing (2008), Du (2011). Vehicle control autonomous lane is (2011). Vehicle control with autonomous lane changing is The The paper paper is structured structured as follows: follows: First, aa general general descripdesigned in Nilsson et al. (2014), Naranjo et al. (2008), Du tion of ACC systems is given. Afterwards, two motivating is as First, descripThe paper is structured as follows: First, a general descripet al. (2015), where usually a two level control structure designed in Nilsson et al. (2014), Naranjo et al. (2008), Du designed in Nilsson et al. (2014), Naranjo et al. (2008), Du tion of ACC systems is given. Afterwards, two motivating et introduced. al. (2015), (2015), where where usuallylevel aa two level control structure example scenarios areis defined and their mathematical detion of ACC systems given. Afterwards, two motivating tion of ACC systems is given. Afterwards, two motivating is The higher is charge of determining et al. usually two level control structure et al. (2015), where usuallylevel a two level control structure example example scenarios scenarios are defined defined and their mathematical mathematical deis introduced. The higher is charge of determining scription is introduced. Next, a robust model predictive are and their descenarios are defined and their mathematical dean optimal trajectory for the vehicle’s motion and the low example is introduced. The higher level is charge of determining is introduced. The higher level is charge of determining scription is introduced. Next, a robust model predictive an optimal optimal trajectory for the the vehicle’s controller. motion and and the the low low scription control strategy is established implemented. The last is Next, aa robust model scription is introduced. introduced. Next, and robust model predictive predictive level controller is realized as tracking an trajectory for vehicle’s motion an optimal trajectory for the vehicle’s motion and the low control strategy is established and implemented. The last level controller controller is is realized realized as as tracking tracking controller. controller. section presents results the investigated control strategy is and implemented. The control strategy simulation is established established and where implemented. The last last level level paper controller is realized as approach tracking controller. section strategy presents simulation results where the investigated investigated This presents an ACC for the opposite sce- section control is analyzed using the introduced scenarios presents simulation results where the section presents simulation results where the investigated This paper paper presents an an ACC ACC approach for the the opposite opposite sce- control strategy is analyzed using the introduced scenarios nario, i.e. considering change operations other vehiThis presents approach for sceand compared nominal using controller. strategy isaanalyzed the This paper presents anlane ACC approach for the of opposite sce- control control strategyto the introduced introduced scenarios scenarios nario, i.e. considering considering lane change operations of other vehiand compared compared toisaaanalyzed nominal using controller. cles while the ego-vehicle maintains its lane. This scenario nario, i.e. lane change operations of other vehiand to nominal controller. nario, i.e. considering lane change operations of other vehiand compared to a nominal controller. cles while the ego-vehicle maintains its lane. This scenario is motivated the fact maintains that sometimes change cles while ego-vehicle its This cles while the the by ego-vehicle its lane. lane.drivers This scenario scenario is motivated motivated by the fact fact maintains that sometimes sometimes drivers change is by the that drivers change is motivated by the fact that sometimes drivers change Copyright © 2016, 2016 IFAC 201Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 201 Copyright © 2016 IFAC 201 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 201Control. 10.1016/j.ifacol.2016.08.030

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2. GENERAL PROBLEM FORMULATION Adaptive Cruise Control (ACC) represents an Advanced Driver Assistance System (ADAS) where the control of the longitudinal vehicle dynamics is taken over from the driver. Thereby, the ACC sets the actuator signals, i.e. gas and brake pedal, according to a desired velocity setpoint vdes selected by the driver. Additionally, the vehicle’s velocity is adapted if an object in front of the vehicle is detected which is travelling at a lower velocity than the vehicle such that a specific safe distance is kept to this object.

(a) Lane change of V1 leads to switching from following to cruising mode

This means that ACC operation can be divided into two basic modes: • Cruising mode: No object is detected and the ACC systems follows the desired velocity vdes • Following mode: The ACC adapts the vehicle’s velocity according to a detected preceding object

(b) Lane change of V2 leads to switching of the target object

In general, vehicles with an ACC system are equipped with a radar sensor to measure distance and relative velocity to other objects located ahead. The desired distance in the following mode is usually given by a constant time headway policy, see Shrivastava and Li (2000),

Fig. 1. Scenarios causing a switching process during ACC operation

∆sdes = ∆s0 + th1 v

Vehicle Dynamics: In the sequel, each vehicle Vi is assigned with its position si , velocity vi and acceleration ai , v˙ i = ai i ∈ {1, 2, 3, E} . (4) s˙ i = vi

(1)

or an extension considering also the relative velocity ∆sdes = ∆s0 + th1 v + th2 ∆v .

(2)

Here ∆s0 denotes the desired inter-vehicle distance at stand still, th1 and th2 constant time headways, v the velocity of the controlled vehicle and ∆v the relative velocity to the target object. The application target of ACC systems can be formulated also in form of an optimal control problem min J(x, u) u

J(x, u) = ξJC (x, u) + (1 − ξ)JF (x, u) + JECO (x, u) .

(3a) (3b)

its velocity according to possible or supposable lane change maneuvers of other traffic objects.

Concerning ego-vehicle dynamics, the normalized pedal position is considered as the input quantity u ∈ [−1, 1] where negative values indicate braking, positive values accelerating of the vehicle. The dynamics of the input u to the ego-vehicle’s jerk jE are considered by 1 jE = a˙ E = (u − aE ) (5) τE where τE represents the acceleration time constant. 3. DEFINITION OF SCENARIOS

Here, x and u denote the system state and input, JC and JF define the cost function for the cruising and the following mode, respectively. ξ ∈ {0, 1} denotes a general switching variable deciding on the ACC mode. JECO is introduced as additional cost to consider economic and comfort aspects. The switching process during ACC operation can occur because of two reasons: (a) Switching between modes ξ due to an object appearing or disappearing in the vehicle’s detection range (b) Switching of the target object if the ACC is in following mode, e.g. due to a cut-in maneuver of another vehicle Fig. 1 illustrates scenarios which induce such a switching process, the upper plot showing a switching between ACC modes and the lower plot depicting a change in the target leading vehicle. VE denotes the controlled vehicle, which in the following will be referred to as the ego-vehicle. Switching processes can lead to aggressive and uncomfortable control actions of an ACC system. Consideration of those processes in an anticipatory way may prevent such undesired behavior if the controlled vehicle adapts 202

In this work we focus on two specific scenarios with the aim to develop a control strategy which anticipates possible lane change and switching actions to increase driving comfort and economy by means of minimizing acceleration and jerk. The idea is to investigate a robust control approach that considers the worst case behavior of other traffic participants in terms of lane change and braking behavior and enables an anticipatory driving style. Scenario A: Considering the scenario depicted in Fig. 1(a), consisting of a road with 2 lanes in the same direction, the ego-vehicle VE is driving on the left lane with a velocity vE = vdes and no preceding vehicle on this lane. Another vehicle V1 with v1 < vE then changes from the right to the left lane to overtake V2 (v2 < v1 ). Thus, V1 becomes the target object for VE which in the following needs to adjust its velocity accordingly to avoid collision. Scenario B: Based on Scenario A, the situation is now extended by introducing an additional vehicle. VE is following V1 and V2 wants to overtake V3 , where the initial conditions v3 < v2 < v1 ≈ vE are assumed. Two possible evolutions are considered, namely V2 changes lane in front of or directly behind V1 . These scenarios will in the

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3.1 Mathematical Description Scenario A: Considering a possible lane change of V1 when currently no vehicle is ahead of VE a mathematical problem description for the following mode ξ = 0 can be T established using the state x = [∆s vE aE v1 ] in the form       0 0 1 0 −1 0 0 0 1  0  0 0      1 (7) x˙ =  x +  1 u +  w 0 0  0 0 −   τE τE 1 0 0 0 0 0        B

(a) Scenario B: V2 wants to overtake V3 and thus will change the lane

A

w

Bu

where the acceleration of V1 is considered as external disturbance w = a1 .

(b) Scenario B1: V2 changes lane in front of V1

(c) Scenario B2: V2 changes lane behind V1

Fig. 2. Scenario B and two possible follow-up actions following be referred to as Scenario B1 and Scenario B2, respectively. Scenario B1 forces V1 and consequently VE to reduce their velocity, whereas Scenario B2 leads to a switch of the leading vehicle of VE and thus, a switch in the distance and velocity of the ego-vehicle’s target object occurs. Fig. 2 illustrates the initial situation and the two considered evolutions of Scenario B. The time at which V1 in Scenario A or V2 in Scenario B changes the lane is denoted by τσ . Further, it is assumed that a vehicle detects a lane changing vehicle as the target object at t = τσ , which means that a transition time for the lane change maneuver is not considered explicitly. For the introduced scenarios the following assumptions are made according to their denotation (A, B, B1, B2): Assumption A. Vehicles V1 and V2 are driving with constant velocities which initially fulfill v2 < v1 < vE . Assumption B. The initial velocities fulfill v3 < v2 < v1 ≈ vE and V3 is moving with constant velocity.

Assumption B1. V2 is driving with constant velocity v2 and changes lane before a safety critical distance to V3 and V1 is reached. V1 starts braking not before t = τσ and reaches steady state conditions v 1 = v2 ∆s12 = s1 − s2 = ∆s12,des (< 0)

(6a) (6b)

Scenario B: Here, VE is already following V1 and a possible lane change of V2 behind V1 needs to be considered which results in a switching leader. A variable σ ∈ {1, 2} is introduced to denote whether the target vehicle is V1 (σ = 1) or V2 (σ = 2). Defining the state vector x = T [∆s vE aE ∆s1 v1 ∆s2 v2 ] with ∆s being the distance to the actual target object and ∆s1 , ∆s2 the distance to V1 , V2 , respectively, the linear system dynamics can be written in the form  0 1 0 0 σ−2 0 1−σ 0 0 1 0 0 0 0    0 0 −τE−1 0 0 0 0    x˙ = 0 1 0 0 0 0 −1  x 0 0 0 0 0 0 0    0 1 0 0 −1 0 0  0 0 0 0 0 0 0    A   (8)  0 0 0  0   0 0  −1   0 0   τE    a     w= 1 . +  0  u +  0 0 w , a  0  2  1 0      0  0 0 0 1 0       Bu

Bw

Additionally to the switched dynamics a state jump occurs in Scenario B2 at t = τσ and V2 becomes the new leader   1 0 0 −1 0 1 0 0 0 1 . . . x+ = Φx , Φ= (9) ..  ..  ... . . 0

...

1

+

where x denotes the state vector directly after τσ . 4. CONTROL STRATEGY

Model predictive control (MPC) strategies have shown at least within a certain time interval T2ss = h(v1 , v2 , ∆s12 ). to be suited well for ACC implementations due to their ability to handle input and state constraints as well as Assumption B2. V1 is driving with constant velocity v1 information about future events directly in the control and V2 changes to the left lane maximal Tlc,max seconds law, see Kamal et al. (2015). Switching processes, like after V1 has passed it. If V2 has to reduce its velocity lane change maneuvers of preceding traffic participants to avoid collision with V3 it accelerates up to its initial can thus be considered in a predictive way while input and safety constraints are maintained. This is one main velocity at time of lane change τσ . 203

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reason why MPC is chosen as control strategy within this work, too. The optimal control problem is solved over a prediction horizon TP H at every sampling instant. For the cost definition (3) the components need to be specified, they are chosen to t0 +T  PH (v − vdes )2 dt (10a) JC = t0

2

JF = (∆s(TP H ) − ∆sdes (TP H )) t0 +T  PH JECO = uT Ru u + j T Rj j dt .

(10b) (10c)

t0

In the cruising mode deviations from the desired velocity vdes are penalized across the prediction horizon TP H (10a). Continuous distance tracking usually stays in conflict with improving comfort and economy. This is why during following mode deviations from the desired distance ddes are considered as terminal cost (10b) only which introduces a degree of freedom for improving comfort and economy (10c). To ensure safety aspects a minimum distance to preceding traffic objects must not be violated, i.e. (11) ∆s = sE − si ≤ ∆ssaf e where si , i ∈ {1, 2, . . .} denote the absolute positions of possible target objects. Note that in this work, negative values of ∆s = sE − si < 0 indicate that the ego-vehicle is driving behind another traffic object. 4.1 Robust Optimal Control Formulation The control objective is to minimize the cost function (3) while being robust against possible switching behavior in terms of lane change maneuvers of other traffic participants. In this work, robustness is addressed by means of a M inM ax optimal control formulation (see Campo and Morari (1987), Kothare et al. (1996) for fundamentals), i.e. minimization of the cost function with respect to u under worst case assumptions for disturbance w and switching signals ξ, τσ . (12a) min max {JC + JF + JECO } u

ξ,τσ ,w

s.t. x˙ = Ax + Bu u + Bw w (12b) x∈X, u∈U, w∈W The state set X considers the distance constraint (11) and boundaries to the vehicle’s minimum and maximum velocity and acceleration. Accordingly, U and W represent the admissible set for the pedal input of the ego-vehicle and the acceleration of other traffic participants. As the following mode of the ACC system is active iff a preceding vehicle is driving slower then the desired velocity vdes it can be deduced easily that maximization of the cost function (12a) with respect to ξ always leads to ξ = 0 if there is a vehicle in range which might change to the egovehicle’s lane within TP H . For Scenario A this means that when V1 is detected by VE it is considered as target vehicle because there is the 204

199

∆si , vi

max J

τσ , w

τσ ,w

MPC

Radar u

Vehicle

s E , v E , aE

Fig. 3. Block diagram of control structure. possibility that V1 changes the lane within TP H . The particular time of lane change τσ has no influence on the maximization operation as only the final inter-vehicle distance error is penalized.Thus, the M inM ax problem (12a) yields to min max {JF + JECO } . (13) u

w

Regarding Scenario B, the value of τσ has a huge impact on the evolution of the scenario. Qualitatively speaking, in Scenario B1 for example, a later time of lane change leads to more intense braking of V1 because as stated in Assumption B1, V1 starts braking not until τσ . The special structure of the cost function (12a) penalizing the distance error at the end of the prediction horizon only, enables to derive the worst case combinations of τσ and w. This is due to the fact that the dynamics of the other vehicles V1 , V2 are modeled in a simple way. Together with Assumption B, B1 and B2 this allows the derivation of positions and inter-vehicle distances at the end of the prediction horizon t0 + TP H depending on the initial state conditions at t0 . For the solution of the robust optimal control problem, this implies that the maximization process is independent from the control input u and can be done separately. Step (1)

max {JF + JECO } w

s.t. x˙ = Ax + Bu u + Bw w w∈W Step (2)

min {JF + JECO }

(14)

u

s.t. x˙ = Ax + Bu u + Bw w x∈X, u∈U

(15)

Hence, the control implementation reduces to the design of a nominal MPC, considering worst case values ξ, τσ and profiles w(t), see Fig. 3 showing a block diagram of the resulting control structure. The states of preceding traffic objects are captured by the ego-vehicle’s radar sensor and used for deriving the worst case scenarios and the control signal u. Implementation of the MPC requires discretization of the optimal control problem (12a) and the system dynamics (12b). The MPC itself has been implemented in MATLAB using YALMIP, L¨ofberg (2004), and the solver quadprog with a sampling time of Ts = 0.1s and a prediction horizon of TP H = 5s. The weighting matrices have been chosen to Ru = 0.05 and Rj = 0.02 which represents a good compromise between driving comfort and inter-vehicle distance tracking, as it will be shown in the following.

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5. SIMULATION RESULTS AND DISCUSSION

τ

σ

−10

∆ s [m]

−30 nominal

−40 −50 0

5

10

15

20

robust

25

30

26 nominal

robust

22 20 18 0

(16)

vehicle 1

24

5

10

15

20

25

30

5

10

15

20

25

30

0.5

0

using the weightings Ru , Rj according to (10c).

u[]

0

All scenarios are investigated also using Monte-Carlo simulation where initial conditions and time of lane change τσ are chosen randomly within certain bounds. 20 simulation experiments are run to investigate the performance of the control approaches under different conditions. Tab. 1 shows the results by means of the average final perfor¯ end ) and the percentage of improvement of the mance C(t robust control strategy compared to the nominal one. Concerning real time capability it should be mentioned that the average MPC execution time in the Monte-Carlo experiment is around 0.038s, maximum execution time was 0.41s. This means that the MPC strategy as it is implemented now cannot be used for real time systems with the chosen sampling time but there are several possibilities like using fast QP solvers, increasing sampling time, implementing abort criteria in case of time critical events to enable implementation of the control strategy on a real time system. As for most robust control approaches, the main disadvantage of the robust MPC is its conservativeness. Here this means that the MPC always expects lane change actions 205

C(t)

0.3 0.2

C(t) nom.

C(t) rob.

0.15

c(t) nom.

c(t) rob.

0.1

0.1 0 0

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10

15 time [s]

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0 30

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Fig. 4. Simulation results for Scenario A comparing nominal and robust control approach τ

σ

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∆ s [m]

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30

26 Velocity [m/s]

The anticipation of possible lane change maneuvers and the resulting smooth acceleration and velocity profile also has a positive effect on traffic flow stability because sudden lane change actions of other vehicles induce much smaller propagation of acceleration shock waves, see Zhou and Peng (2004).

−1 0

vehicle 1

24

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robust

22 20 18 0

5

10

15

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25

30

5

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0.5 0 u[]

The simulation results for Scenario B1 and Scenario B2 presented in Fig. 5 and Fig. 6 show similar effects. Anticipation of possible lane changes and braking of the preceding object results in a deceleration some seconds before τσ and hence increases driving comfort and economy significantly.

−0.5

−0.5 −1 0 0.4

0.2

0.3 C(t)

First, Scenario A is investigated where a vehicle (V1 ) on the adjacent lane that is driving with a lower velocity than the controlled ego-vehicle (VE ) changes lane and thus forces VE to brake. Fig. 4 shows simulation results for this scenario and compares a nominal to a robust control approach. It can be seen that the robust MPC reduces the velocity anticipatory before τσ because it assumes a lane change of V1 . This leads to a smooth braking profile avoiding high values of the jerk. On the other side, the nominal controller reacts on the lane change of V1 at τσ and consequently has to brake harder. Considering the performance index it can be seen that a high increase in driving comfort can be achieved with the robust MPC.

c(t)

ci (t) = Ru u(t)2 + Rj j(t)2  t Ci (t) = ci (t)dt , i ∈ {nom, rob} .

−20

Velocity [m/s]

The MPC strategy is now tested using the scenarios explained in 2. For each scenario, the robust MPC is compared to a nominal one which does not consider lane change maneuvers of surrounding vehicles in a predictive way and reacts on changing leaders at t = τσ . To enable a fair comparison of the two approaches, the following performance indices are introduced

0.2

C(t) nom.

C(t) rob.

c(t) nom.

c(t) rob.

0.1 0 0

0.15 0.1

c(t)

IFAC AAC 2016 200 June 19-23, 2016. Norrköping, Sweden

0.05 5

10

15 time [s]

20

25

0 30

Fig. 5. Simulation results for Scenario B1 comparing nominal and robust control approach of preceding objects even if they do not occur. Including additional rules, e.g. considering possible lane changes only if a vehicle on the adjacent lane is driving faster than its predecessor, could help to counteract this disadvantage. 6. CONCLUSION In this paper a strategy for robust adaptive cruise control with the target to increase driving comfort by anticipa-

IFAC AAC 2016 June 19-23, 2016. Norrköping, Sweden

Roman Schmied et al. / IFAC-PapersOnLine 49-11 (2016) 196–201

τσ −10

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26 vehicle 2

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u[]

0 −0.5

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0.8 0.4 0 0

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c(t) nom.

c(t) rob.

20

25

0.4 c(t)

−1 0

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Fig. 6. Simulation results for Scenario B2 comparing nominal and robust control approach Table 1. Comparison of mean final perfor¯ end ) mance index C(t Scenario

A

B1

B2

¯ end ) nominal C(t ¯ end ) robust C(t

0.32 0.08

0.24 0.07

0.95 0.13

Mean Improvement

75%

71%

86%

tion of possible lane change maneuvers of other traffic participants is investigated. Model predictive control techniques are used to solve an optimal control problem in receding horizon manner. Input and safety constraints are addressed directly in the optimal control formulation. Two different traffic scenarios are then introduced and analyzed in detail by comparing the proposed robust ACC approach with a nominal one. Simulation results show that the robust consideration of possible lane change maneuvers obtain great benefits concerning driving comfort and may have positive effects on traffic flow stability, too. ACKNOWLEDGEMENTS This work has been partially supported by the Linz Center of Mechatronics (LCM) in the framework of the Austrian COMET-K2 program. REFERENCES Campo, P.J. and Morari, M. (1987). Robust Model Predictive Control. In American Control Conference, 1987, 1021–1026. Dang, R., Wang, J., Li, S., and Li, K. (2015). Coordinated Adaptive Cruise Control System With Lane-Change Assistance. Intelligent Transportation Systems, IEEE Transactions on, 16(5), 2373–2383. 206

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