Optimal disturbance rejection control design for electric power steering systems

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 22   h  K 2  2  N1   b2 2  f  2
      sw
2. Electric Motor Shaft Steering Wheel
J 1 
 N 2 J 3    m  b11 

1
 b1
m
 
Torque Sensor (Kc)

 
Electric
 K 2  2  N1   K 3  N1  rack
x  N  f 1

  rp  Motor
  
3. Rack Dynamics
Gear Box N  r2 /r1
K  x 
  F fric  xrack   Fdist
mrack
x  3  N1  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 N1  U

8

9. EPS System EPS Control
Introduction Dynamics Design
Comparison Conclusion
2-DETERMINISTIC OPTIMAL LQR
 System Dynamics
xt   At  xt   Bt u t 

y t   C y t  xt 
 Quadratic Integral Criterion
 
t1
J  x t1  P xt1    xT t R1 t xt   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 Pt 
1
Here P(t) is the solution of the matrix Riccati equation
 Pt   R1  PBR2 B T Pt   AT Pt   Pt A
 1
9
P0   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   qi  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

19. THANK YOU FOR YOUR
ATTENTION
19