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IEEE ECC-CDC
1. OPTIMAL DISTURBANCE REJECTION
CONTROL DESIGN FOR ELECTRIC
POWER STEERING SYSTEMS
Naser Mehrabi
Nasser L. Azad
John McPhee
University of
Waterloo,
Canada.
2. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
CONTENTS
EPS Review
Types of EPS Systems
EPS Subsystems
EPS Architecture
EPS Characteristics
EPS System Dynamics
EPS Control Design
1. PID Control
2. Deterministic LQG Control
3. Modified-LQG Control
Conclusions and Future Works 2
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
3. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
TYPES OF EPS SYSTEMS
C-EPS (Column assist-type)
P-EPS (Pinion assist-type)
One axis pinion-type
Two axes pinion-type
R-EPS (Rack assist-type)
Alternating configuration
Parallel configuration
Coaxial configuration 3
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
4. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
C-EPS SUBSYSTEMS
An EPS system composed of 4 main subsystems:
1. Steering Subsystem
2. Assist Motor
3. Rack and Pinion
4. Tires
4
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
5. EPS System EPS Control 5
Introduction Dynamics Design
Comparison Conclusion
EPS OBJECTIVES
1. Assistance:
Sufficient assist torque to
the drivers.
2. Road Feel:
Reaction torque must be
sensitive to the necessary
information from the road.
5
Reference:http://www.zf-lenksysteme.com
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
6. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
SIMULATION SETUP
Simulation Subsystems
1. Driver
2. Steering System
3. Vehicle Dynamics
4. Control Logic
EPS Sensors
1. Vehicle longitudinal
speed
2. Steering torque
3. Steering wheel angle 6 6
4. Motor angular speed
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
7. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
FOURTEEN DOF VEHICLE MODEL
DOFs:
6 DOF for the rigid
body
4 DOF for vertical
displacement of
unsprung mass
4 DOF for wheel
rotation
Gaussian noise and
Coulomb friction is
added to the Tire-
Road force
Gaussian noise is
added to the
measurement
signals
Fiala tire model is
used to simulate the
tire-road interaction 7
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
8. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
EPS SYSTEM DYNAMICS
1. Steering Wheel
J 22 h K 2 2 N1 b2 2 f 2
sw
2. Electric Motor Shaft Steering Wheel
J 1
N 2 J 3 m b11
1
b1
m
Torque Sensor (Kc)
Electric
K 2 2 N1 K 3 N1 rack
x N f 1
rp Motor
3. Rack Dynamics
Gear Box N r2 /r1
K x
F fric xrack Fdist
mrack
x 3 N1 rack r2 r1
rp X rack Steering Linkage/ Tire
rp Steering Linkage Stiffness
mass
4. Electric Motor Dynamics Rack and
Pinion
Li Ri K e N1 U
8
9. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
2-DETERMINISTIC OPTIMAL LQR
System Dynamics
xt At xt Bt u t
y t C y t xt
Quadratic Integral Criterion
t1
J x t1 P xt1 xT t R1 t xt u T t R2u t dt
T
1
t0
The optimal input can be generated through a linear control law of the
form
u t K x
Where
K R2 B T Pt
1
Here P(t) is the solution of the matrix Riccati equation
Pt R1 PBR2 B T Pt AT Pt Pt A
1
9
P0 P1
10. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
2-LQR COST FUNCTION
The cost function used in the LQG for second interval is:
J qi im u 2 dt
2
0
2
Kc 2
q x7 a K c x1 x3 u dt
rp
0
Cost function in general quadratic form
J x Q xT u R u T dt
0
where a 2 K 2 2 N 2 q 0 a 2 K 2 2 Nq 0 0 0 aK 2 Nq
0 0 0 0 0 0 0
a 2 K 2 Nq 0 aK 2 q
2
0 0 0 qaK 2
2
Q 0 0 0 0 0 0 0 , R
0 0 0 0 0 0 0
10
0 0 0 0 0 0 0
aK 2 Nq 0 qaK 2 0 0 0 q
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
11. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
2-KALMAN FILTER OBSERVER
Dynamics of observer:
𝑇
𝑥 = 𝐴𝑥 + 𝐵𝑢 + 𝐵 𝑤 𝑤, 𝑤= 𝑤1 , 𝑤2
𝑦 = 𝐶𝑥 + 𝑣
w: “Process noise” – models uncertainty in the system model
v: “Sensor noise” – models uncertainty in the measurements
Assumption:
𝐸 𝑤 𝑡1 𝑤 𝑡2 𝑇 = 𝑄𝛿 𝑡1 − 𝑡2 & 𝐸 𝑤 𝑡 =0
𝐸 𝑣 𝑡1 𝑣 𝑡2 𝑇 = 𝑅𝛿 𝑡1 − 𝑡2 & 𝐸 𝑣 𝑡 =0
𝐸 𝑤 𝑡1 𝑣 𝑡2 𝑇 =0
Objective:
𝑇
𝐽= 𝐸 𝑥 𝑡 − 𝑥𝑇 𝑡 𝑥 𝑡 − 𝑥𝑇 𝑡
Solution is a closed loop observer, where:
𝐿 𝑡 = 𝑃𝑒 𝑡 𝐶 𝑇 𝑅 −1
Can be achieved from following differential Riccati
equation:
𝑃𝑒 = 𝐴 𝑃𝑒 + 𝑃𝑒 𝐴 + 𝐵 𝑤 𝑄 𝐵 𝑤 − 𝑃𝑒 𝐶 𝑇 𝑅 −1 𝐶 𝑃𝑒
𝑇
𝑃𝑒 𝑡0 = 𝑃 𝑒0
11
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
12. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
3- MODIFIED LQG CONTROLLER
In problems with disturbances that tends to drive the state away from the zero
state, system dynamics can be shown as
𝑥 = 𝐴𝑥 + 𝐵𝑢 + 𝑣
y= 𝐶 𝑥
Disturbance can be represented as stochastic process, which we model as the
output of a linear system driven by white noise. (Shaping Filter)
𝑥𝑑 = 𝐴𝑑 𝑥𝑑+ 𝑤 𝑡
𝑣 𝑡 = 𝐶𝑑 𝑥𝑑 w vt y
H(s) G(s)
Augmented system
𝐴 𝐶𝑑 𝐵 0 𝑥
𝑥= 𝑥+ 𝑢+ , 𝑥= 𝑥
0 𝐴𝑑 0 𝑤(𝑡) 𝑑
Initial condition
𝑇
𝑥 0 = 𝑥 𝑡0 𝑥 𝑑 𝑡0
where w(t) is white noise with:
𝐸 𝑤 𝑡1 𝑤 𝑡2 𝑇 = 𝑄𝛿 𝑡1 − 𝑡2 & 𝐸 𝑤 𝑡 =0 12
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
13. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
3- MODIFIED LQG CONTROLLER
Quadratic regulator criterion
𝑡1
𝐽= 𝑥 𝑇 𝑅3 𝑥 + 𝑢 𝑇 𝑅2 𝑢
𝑡0
This criterion cannot be evaluated because of the stochastic nature of
the disturbances. Therefore, we average over all possible realizations
of the disturbances and consider the criterion.
𝑡1
𝐽= 𝐸 𝑧 𝑇 𝑅3 𝑧 + 𝑢 𝑇 𝑅2 𝑢
𝑡0
𝑃1 0 𝑅 0
𝑃1 = , 𝑅3 = 1
0 0 0 0
Linear control law
𝑢 = −𝐾𝑥
where
−1
𝐾 = 𝑅2 𝐵 𝑇 𝑃 𝑡
P(t) is evaluated from following differential Riccati equation
−𝑃 𝑡 = 𝑅 −1 − 𝑃 𝑡 𝐵𝑅2 𝐵 𝑇 𝑃 𝑡 + 𝐴 𝑇 𝑃 𝑡 + 𝑃 𝑡 𝐴
𝑃 𝑡1 = 𝑃1 13
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
14. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
3-MODIFIED LQG CONTROLLER
Driver torque
x D1 AD1 x D1 BD1 w
h C D1 x D1 DD1 w
Road force disturbance
xD 2 AD 2 xD 2 BD 2 w
wd C D 2 xD 2 DD 2 w
System dynamics
x Ax B u u B w w
where
x A B CD1 BwCD 2 Bu 0
x xD1 ,
A 0
AD1 0 , B u 0 ,
B w B DD1
xD 2
0
0 AD 2 0
Bu DD 2
14
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
15. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
EPS CONTROLLER COMPARISON
Performance
Characteristics Curve State Estimation under
Tracking (1-PID, 2-LQG, not-nominal conditions
3-Modified-LQG) (2-LQG)
6
Desired
5 PID
LQG
Motor Current (A)
4
3
2
1
0
-1 15
0 0.5 1 1.5 2 2.5 3
Torque Sensor (N.m)
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
16. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
EPS CONTROLLER COMPARISON
Performance Robustness
Under not-nominal conditions Under not-nominal conditions
2-LQG 3-Modified LQG
8 6
= 1.0 (nominal) = 0.1
6 5
= 0.1 = 0.5
4 = 0.5 4 = 1.5
Motor Current (A)
Motor Current (A)
= 1.5 = 1.0
2 3
0 2
-2 1
-4 0
-6 -1
-8 -2
-3 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3
16
Torque Sensor (N.m) Torque Sensor (N.m)
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
17. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
EPS CONTROLLER COMPARISON
High-assist gains
3-Modified LQG
10
a=5
8 a = 15
a = 25
6 a = 35
Current (A)
Desired
4
2
0
0 0.5 1 1.5 2 2.5 17
Torque Sensor (N.m)
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems
18. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
CONCLUSIONS AND FUTURE WORKS
Summary
An optimal disturbance rejection controller for EPS
systems was designed.
A driver torque and tie-rod force observer was developed.
PID and LQG controllers were compared.
Future Works
Include the estimated disturbances force into the cost
function
Assess the controller performance using higher fidelity
vehicle and steering models
Develop a Neuro-Musculoskeletal driver model to consider
the driver feel 18
Optimal Disturbance Rejection Control Design for Electric Power Steering Systems