1 challenge the future m.wang, w.daamen, s. p. hoogendoorn and b. van arem driver assistance systems...

1Challenge the future

M.Wang, W.Daamen, S. P. Hoogendoorn and B. van Arem

Driver Assistance Systems Modeling by Optimal Control

Department of Transport & PlanningDelft University of Technology

2Challenge the future

Outline

• Context

• Control framework for car-following support

• Adaptive Cruise Control (ACC) model

• EcoACC control model

• Simulation results

• Summary and outlook

3Challenge the future

Context

• Global interests in Advanced Driver Assistance Systems

(ADAS).• ACC are earliest ADAS in market.• Public concern on environment stimulates Eco-driving

assistance systems, i.e. EcoACC.• Needs for model EcoACC and evaluate the effects on driving

behavior.

4Challenge the future

Existing ACC

• Feedback controller, not optimal behavior

• Often be switched off at low speeds

• Cannot satisfy multiple control objectives

• Not able to model Eco-driving

5Challenge the future

This paper

An optimal control framework to model ACC/EcoACC

systems based on assumptions that:• Other vehicles driving at constant speed within a prediction

horizon;• Accelerations are controlled to minimize a cost function;• Costs are chosen to reflect multiple control objectives.

6Challenge the future

On-board system

On-board sensors

V2V&V2I Comm.

State estimation & prediction

Optimization at vehicle level

Reference control signal

Vehicle maneuver

Local traffic state

Other sensors

Vehicle actuactor

Schematic diagram for vehicle followingcontrol

7Challenge the future

Control framework

• (Local traffic) system state:

x = (x1, x2)T = (si , Δvi)T

si - following gap

Δvi - relative speed to predecessor

• State dynamics:

ui-1 - follower acceleration

ui - follower acceleration

si

Δvi = vi-1 – vi

i-1i

1

i i

i i i

s vd d

v u udt dt

x

8Challenge the future

Control framework2

• Objective function

s.t. state dynamics

• Applying Pontryagin’s Minimum Principle entails solving coupled ODE:

1) state dynamics with initial conditions x(t0)

2) co-state dynamics with terminal conditions λ(t0+T)

λ : co-states or marginal costs of the state x

0

0

*0min , ( )

t T

tuJ L u dt G t T

x x

9Challenge the future

ACC model

Functional requirements:

•Maintaining desired speed at cruising mode;

•Maintaining desired time gap at following mode.

Control objectives:

•Maximize travel efficiency;

•Minimizing risk;

•Maximizing comfort.

10Challenge the future

ACC running cost

with s*: desired gap, s*= v t* + s0; v0 : desired speed.

• The controller aims to:1) Minimize accelerations

2) Maintain a gap close to some desired gap s*

3) Match the speed of the predecessor.

• Applying the solution approach yields:

The optimal control law equals the marginal cost of relative speed.

3* 2 2 2 21 2

0

1( ) ( )

2 2 2 2ComfortSafety Efficiency

L s s v v v u

* vu

11Challenge the future

Tuning of prediction horizon

• Leader with constant speed of 72 km/h

• Initial gap: s (0) = 50 m

• Initial speed difference: Δv (0) = 0 km/h

• Desired time gap: t* = 1.5 s

• Desired speed: v0 = 120 km/h

• Prediction horizon: T = [2:20] s

s

Δv

LeaderControlled vehice

12Challenge the future

Simulation results

13Challenge the future

Choice of prediction horizon

• Large enough to ensure expected behavior;

• Not too large to avoid computational complexicity.

• A prediciton horizon of 5 s is recommanded from the results.

Intel Core 2, 2.4 GHz

14Challenge the future

EcoACC model

Control objectives:

•ACC controller objectives + minimizing fuel consumption

Running cost:

•ACC controller running cost + Eco cost

Calculation of Eco cost:

•Spatial fuel consumption rate•Microscopic fuel consumption model from ARRB

15Challenge the future

Comparison of ACC/EcoACC

0 5 10 15 20 25 3050

55

60

65

70

75

time (s)

v (k

m/h

)

Simulation setup:

•Disturbance in leader speed;

•Initial speed difference: Δv (0) = 0 km/h;

•Initial gap: s (0) = 30 m; 100 m;

•Desired time gap: t* = 1.5 s;

•Desired speed: v0 = 120 km/h;

•Prediction horizon: T = 5 s.

•Comparison

• 1) ACC;

• 2) EcoACC1, Eco cost weight = 5;

• 3) EcoACC2, Eco cost weight = 10.

16Challenge the future

Simulation: Ｉ nitial gap 100 m

0 5 10 15 20 25 3030

40

50

60

70

80

90

100

time (s)

s (m

)

0 5 10 15 20 25 30-30

-20

-10

0

10

time (s)

v

(km

/h)

0 5 10 15 20 25 3050

60

70

80

90

100

110

time (s)

v (k

m/h

)

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

time (s)

u (m

/s2 )

leader

ACC

EcoACC1 with 4 = 5

EcoACC2 with 4 = 10

17Challenge the future

Results (with reference to ACC)

EcoACC1 EcoACC2

Scenario 1(30m)

Mean speed -0.5% -1.2%

Fuel consumed -3.5% -5.1%

VKT* -0.5% -1.2%

Scenario 2(100m)

Mean speed -0.3% -0.8%

Fuel consumed -9.9% -15.2%

VKT -0.3% -0.8%

*VKT: Vehicle Kilometers Travelled

18Challenge the future

Summary

• An optimal control framework to model/design ADAS and Eco-DAS.

• Flexible state and running cost specifications reflecting control objectives.

• In our simple examples, the Eco costs result in higher fuel efficiency and similar distance

traveled.

• Stochastic case

• Local and string stability

• Cooperation between vehicles M. Wang, S.P. Hoogendoorn, W. Daamen, R.G. Hoogendoorn and B. van Arem. Driver Support and Cooperative Systems

Control Design. 2012 American Control Conference. Montreal, Canada.

Outlook

19Challenge the future

Questions?