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![Page 1: 1 Challenge the future M.Wang, W.Daamen, S. P. Hoogendoorn and B. van Arem Driver Assistance Systems Modeling by Optimal Control Department of Transport.](https://reader031.fdocuments.in/reader031/viewer/2022032606/56649e8a5503460f94b8f7bd/html5/thumbnails/1.jpg)
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
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2Challenge the future
Outline
• Context
• Control framework for car-following support
• Adaptive Cruise Control (ACC) model
• EcoACC control model
• Simulation results
• Summary and outlook
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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.
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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
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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.
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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
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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
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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
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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.
![Page 10: 1 Challenge the future M.Wang, W.Daamen, S. P. Hoogendoorn and B. van Arem Driver Assistance Systems Modeling by Optimal Control Department of Transport.](https://reader031.fdocuments.in/reader031/viewer/2022032606/56649e8a5503460f94b8f7bd/html5/thumbnails/10.jpg)
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
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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
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12Challenge the future
Simulation results
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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
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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
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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.
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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
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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
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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
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19Challenge the future
Questions?