2010 ieee american control conference
TRANSCRIPT
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Cooperative DYC System Design for Optimal Vehicle Handling Enhancement
S.H. Tamaddoni *, S. Taheri, M. AhmadianCenter for Vehicle Systems and Safety (CVeSS)
Department of Mechanical EngineeringVirginia Tech, USA
* email: [email protected]
Virginia TechCENTER FOR VEHICLESYSTEMS & SAFETY
Virginia TechCENTER FOR VEHICLESYSTEMS & SAFETY
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Outline
Motivations
Game Theory
System Model
Control Derivation
Simulation and Results
Conclusions
GAME THEORY
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Motivations
Vehicle Stability Control (VSC) improves vehicle stability and handling performance. Ferguson (2007) has shown that VSC can reduce
• single-vehicle crashes by 30-50% in cars and SUVs,• fatal rollover crashes by 70-90% regardless of vehicle type.
© www.racq.com.au
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Interaction Model Driver / VSC interaction model:
Driver’s Processing Unit
Driver’s Action Unit
VSC Processing & Action Unit
Vehicle System
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Game Theory
The systems are governed by several controllers, i.e., decision makers or players, where each controller aims to minimize its own cost function.
No player can improve his/her payoff by deviating unilaterally from his/her Nash strategy once the equilibrium is attained.
For a game with a sufficiently small planning horizon, there is a unique linear feedback Nash equilibrium that can be computed by solving a set of so-called Nash Riccati differential equations.
© Andrew Gelman
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Primary Objectives
Driver:• Steer the vehicle through the maneuver
Controller:• Guarantee vehicle handling stabiltity where the desired
value of yaw rate is obtained from Wong (2001):
2( )(1 )x
desired FF B us x
vl l K v
ψ δ=+ +
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Evaluation Model
The evaluation vehicle model includes• longitudinal & lateral dynamics• yaw, roll, pitch motions• combined-slip Pacejka tire model• steering system model• 4-wheel ABS system Ls
Rsφ
ψ
yv
xv
yBLF
xBLFzBLF
Fl
BlyFRF
zFRFFδFRα
yFLF
xFLF
zFLF
xFRF
FδFLα
XZ
Y
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Control Model
2-DOF bicycle model• y: absolute lateral position• ψ: absolute yaw angle
[ ]
1 1 2 2 1 2
1 2
2 2
0 0 0 0 0
, ,0 1 0 0 00 0 0
, , 000 0 0 11
0 0
( )
F zc
x
F B F F B B Fx
x x
F FF F B B F F B Bz
zz x z x
u u u u Mv
C C l C l C Cvmv mv m
l Cl C l C l C l C III v I v
t y y
α α α α α
αα α α α
δ
ψ ψ
= + + = =
+ − − − − = = = − + −
=
x Ax B B
A B B
x
T
ψCG
Y
X
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Theorem 1: Certain system
Let the strategies be such that there exist solutions to the differential equations
in which,
such that,
and satisfies
( )* *,f zcMδ
( )1 2,P P
( ) ( )*
* * * * * *, , , , , , . ,ji ii f zc i f zc i
i
d x M x Mdt x u x
γδ δ
∂∂ ∂= − −
∂ ∂ ∂H HP P P
( ) ( )2 21 2 1 2, , , ,T T
i f zc i i i f i zc i f zcx M x x r r M x Mδ δ δ= + + + + +H P Q P A B B
( )* *, , , 0,if zc i
i
x Mu
δ∂=
∂*H P
* *
0 0
( ) ( ) ,
( ) .f zcx t x t M
x t x
δ = + +
=
* *1 2
*
A B B
*x
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Theorem 1: Certain system
Then,is a Nash equilibrium with respect to the
memoryless perfect state information structure, and the following equalities hold:
( )* *,f zcMδ
{ } { }* 1 ( ),
, , ,
Ti ii i i
f zc
u R t
i M u Mδ δ
−= −
∈ ∈
B P( ) ( )i t x tK
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Theorem 2: linear feedback
Suppose satisfy the coupled Riccati equations
where
Then the pair of strategies
is a linear feedback Nash equilibrium.
1 1 1 1 1 1 1 1 2 2 2 2 1 2 1 22 ,T= − − − + + + −K A K K A Q K S K K S K K S K K S K
2 2 2 2 2 2 2 2 1 1 1 1 2 1 2 11,T= − − − + + + −K A K K A Q K S K K S K K S K K S K
1 1 1, .T Ti i ii i ij j jj ij jj j
− − −= =S B R B S B R R R B
( ) ( )* * 1 111 1 1 22 2 2, ( ) , ( )T T
f zcM t x t xδ − −= − −R B K R B K
( )1 2,K K
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Simulation
Vehicle: 2-axle Van
Maneuver: standard “Moose” test at 60 kph
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Simulation
Selected Q & R matrices:
5 3
1 0 0 0 0 0 0 00 0.1 0 0 0 0.1 0 0
, ,0 0 0 . 10 0 0 0 00 0 0 0 . 0 10 0 0 110, 0,
10 , 10
M
M
MM M
δ
δδ δ
δ−
= =
= =
= =
Q Q
R RR R
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Results
unit \ strategy Nash LQR
Driver 97,363 162,060
Controller 9,734,700 16,204,000
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Conclusions
A novel cooperative optimal control strategy for driver/VSC interactions is introduced:
• The driver’s steering input and the controller’s compensated yaw moment are defined as two dynamic players of the game “vehicle stability”
• GT-based VSC is optimally more involved in stabilizing the vehicle compared to the common LQR controllers.
• GT-based VSC improves vehicle handling stability more than the common LQR controllers can do with the same driver and controller cost matrices.
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Thank You !
GAME THEORY