Benefits of Knowledge of Road Friction

1

Active Safety

Systems (ESC, ABS, TCS, ACC)

Pre-Warning for

Slippery Road

to Driver (Vehicle-To-Vehicle

Information Sharing)

Intelligent

Transportation

System (Real-time Friction

Adaptive Speed Limit

Control)

Road Friction

Estimation

Lateral Excitation Based Algorithms

2

Vehicle Model

Steering

Model

,y za M

Measurement

ˆ,ˆy za M

Model output

Compare Sensor

signals (Estimation Algorithm)

Slip angle

Friction Coefficient

Update

Excitation

Tire

Model

Parameter and State Estimation for Nonlinear

Systems

3

0

1,0 11

2,0 22

( , , ) ( , , ),

.

x f x u f x u

h hhy

h hh

System

Estimate state x and parameter θ

Objective

100

2

ˆˆ ˆ( , , )ˆ

ˆ 0

x Lf x uy y

L

How can we determine gain L1 and L2?

Robust stability to uncertainties ˆx x and as t

Stability Robustness

Stability at An Operation State

4

0 0

( , , ) 0

ˆ ˆ( , , ) 2 2 0

T

T T

Lyapunov Stability

V e z u e Pe

V e z u e Pe e P F F L H H

for all uncertainties(ΔF and ΔH)

in error space

with respect to a given operation state.

Error Space Stable

Unstable

0 0ˆ ˆ ˆ ˆ, ( , ) ( , ) ( , )z F z u L z u H z u H z u

100

2

ˆˆ ˆ( , , )ˆ

ˆ 0

x Lf x uy y

L

Operation

state

Observer Requirement

5

If inside of the donut is all blue,

then the observer is robustly stable at the given operation state.

Error Space

d: convergence parameter

ε1: steady state error

ε2: stable error bound

Inside of donut: all error will converge to the inner circle

Observer requirement

(design parameters)

Error Space Error Space

Stability in an Operation State Space

6

Operation State Space

Error Space

Satisfied Satisfied Not Satisfied

Observer Gain for Robust Stability

7

Find L that maximizes the

blue area under the given

requirements.

1 211 2

ˆ ˆˆ

1 2

3 4

ˆ, ,

1 23 4

ˆ ˆˆ ˆ, ,

, ,

, ,

x x x x

x x x x

h hhfl l

x xx x

h

k

hl

k

k kl

1 4

* * *

1 1 4~

arg ma~ ,, xk k

Bluep k k Area

1 2

3

1

4

0, .

0 1

l

l l

p lP L

① System

② Observer

④ Gain

Optimization

Robust Observer Design Synthesis

8

1 1

0

2 2

, .h h

x f f yh h

0 0ˆ ˆ ˆ ˆ( , ) ( , ) ( , ) ( , )z F z u L z u H z u H z u

Derive Gain Matrix

Set d, ε1 and ε2

considering observer

requirements.

(Donut shape)

Set Plant

Uncertainties

Optimization to determine

k1, k2, k3, and k4.

, .F H

③

Friction/Slip Angle Estimation

9

2

1 2

3 4

1 1ˆ ˆ ˆ ˆˆ ˆ ,

ˆ ˆˆ ˆ ,

f yf yr y yf yr a a

x z x x z x

y yf yr a a

a abF F r l ma F F l

mV I V mV I V

l ma F F l

1 211 2

ˆ ˆˆ

1 2

3 4

ˆ, ,

1 23 4

ˆ ˆˆ ˆ, ,

, ,

, ,

x x x x

x x x x

h hhfl l

x xx x

h

k

hl

k

k kl

9 6

1 2

8 4

3 4

2.5 10 , 2.8 10 ,

1.8 10 , 1.9 10 .

k k

k k

Estimation Result (Test 8)

10

11