PI Tuning via Extremum Seeking Methods for Cruise Control

Adaptive Cruise Control via Extremum Seeking MethodsYiyao(Andy) Chang and Scott Moura Slide 1 of 17

PI Tuning via PI Tuning via ExtremumExtremum Seeking Seeking Methods for Cruise ControlMethods for Cruise Control

Yiyao(AndyYiyao(Andy) Chang) ChangScott MouraScott Moura

ME 569ME 569Control of Advanced Powertrain SystemsControl of Advanced Powertrain Systems

Professor Anna Professor Anna StefanopoulouStefanopoulou

December 6, 2007December 6, 2007

Motivation• Automated PID Tuning Method

– Online control synthesis– Achieve optimal performance in some sense– Reduces calibration time– Does not require knowledge of plant– Does not require special experiments

• Can ES be used as an adaptive control law?– Real-time controller adaptation– Varying plant parameters and dynamics– Varying disturbance inputs


Reference Paper

PID Tuning Using Extremum SeekingOnline, Model-Free Performance Optimization

Nick J. Killingsworth and Miroslav Krstic

IEEE Control Systems MagazineFebruary 2006


Literature Review


Relay Feedback Tuning

• Åström et al (1984)

• Leva (1993)

• Voda et al (1995)

Unfalsified Control

• Jun et al (1999)

• Saeki et al (2003, 2004)

Closed-loop (model free) PID parameter tuning

Iterative Feedback Tuning

• Hjalmarsson et al (1998)

• Lequin et al (1999, 2003)

Outline• Motivation & Literature Review• Paper Reproduction

– Extremum Seeking (ES) Algorithm– PID Tuning on Sample Plant– ES Parameter Sensitivity

• Case Study: Adaptive Cruise Control– Cruise Control Model– PI Tuning

• Vary Desired Velocity, Fix Road Grade• Vary Road Grade, Fix Desired Velocity• Vahidi Road Grade Profile (I-15 near San Diego)

• Summary & ConclusionsAdaptive Cruise Control via Extremum Seeking Methods

Yiyao(Andy) Chang and Scott Moura Slide 5 of 17

Extremum Seeking Algorithm


PID parameters

Cost

Washout filter cutoff freq

Perturbation freq.

Perturbation amplitude

Adaptation gain

PID parameters estimateθ

Jh

iω

iαγ

θ

1.) Initialize PID gains2.) Evaluate cost3.) Estimate cost gradient4.) Estimate new PID gains5.) Perturb PID gainsES algorithm parameters

Control Problem Formulation

( ) 2011 20

sG s es

−=+

( ) ( ) ( )e t , r t y t ,θ θ= −

Plant:

Controller:

( ) 11ri

C s KT s

⎛ ⎞= +⎜ ⎟

⎝ ⎠

( ) 11y di

C s K T sT s

⎛ ⎞= + +⎜ ⎟

⎝ ⎠Cost Function:

( ) ( )0

2

0

1 ,T

tJ e t dt

T tθ θ=

− ∫

[ ]Ti dK T Tθ =

Tuning Parameters:


0 5 10 15 200.8

1

1.2

1.4

1.6X: 0Y: 1.33

K

X: 20Y: 0.9987

0 5 10 15 2030.5

31

31.5

32

32.5

X: 0Y: 31

Ti X: 20Y: 31.68

0 5 10 15 207

7.5

8 X: 0Y: 7.74

Td

Iterations (k)

X: 20Y: 7.152

PID Tuning on Sample Plant


0 5 10 15 2010

-5

10-4

10-3

10-2

Iterations (k)

Cos

t

X: 0Y: 0.001207

X: 20Y: 5.201e-005

0 20 40 60 80 100-0.5

0

0.5

1

1.5

Time (sec)

y(t)

ZNIMCIFTES

Use Ziegler-Nichols as initial gains

PID Gain Trajectories wrt Cost

Cross-section of cost as a function of Td and Ti

Fixed KAdaptive Cruise Control via Extremum Seeking Methods

Yiyao(Andy) Chang and Scott Moura Slide 9 of 17

Ti

Td

30 30.5 31 31.5 326

6.5

7

7.5

8

8.5

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10-3

Varying ES Parameters


0 50 100 150 200 250 300 350 4000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10-3

Iterations (k)

Cos

t J( θ

)

α ,γα /2,γα ,γ/10α /2,γ/10

How sensitive are the results to the choice of the ES parameters?

Killingsworth’s Answer“fairly insensitive”

Our Answer• Insensitive for more conservative values

• VERY sensitive for more aggressive values

Unstable Region

Stable Region

iαγ

Perturbation amplitude

Adaptation gain

Recall

Case Study

Adaptive Cruise Control


Vehicle Dynamics Model


ji

k

( )ˆ ˆcos sinmg j iθ θ− +

ˆdragF i−

ˆfNi−

ˆNj

ˆxF i

ˆ sinˆ cos 0

x dragm i F mg fN F

m j N mg

θ

θ

⋅ = − − −

⋅ = − =

a

a

( )20.5

cosdrag d wF AC v v

N mg

ρ

θ

= +

=

( )2sin cos 0.5x d wdvm F mg fmg AC v vdt

θ θ ρ= − − − +

Assumptions

• Rolling disk dynamics

• No wheel slip

• No actuator dynamics

• Coulomb frictionθ

Wind Speed

Linearization and Ziegler-Nichols


Variable Nominal ValueTraction Force, Fx 293 N

Road Grade, θ 0 radians

Wind Speed, vw 2 m/s

Vehicle Speed, v 20 m/s ≈ 45 mph

( ) [ ]0 0 01 1 1 cos sind w xv AC v v v F mg fmgm m m

δ ρ δ δ θ θ δθ= − + + + − +

( )( )

0.075875.75 1 1

Y s KU s s sτ

= =+ +

Step Response

Time (sec)

Ampl

itude

0 50 100 150 200 250 300 350 400 4500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

System: sysRise Time (sec): 167

System: sysFinal Value: 0.0758

PI Controller:

( ) 1PIPI

sC s Ks

τ += ( ) ( ) 43 10.3845 sC s

s+

=Ziegler-Nichols

Open Loop

Step Response

PI Tuning for Cruise Control


0 200 400 600 800 1000 1200 140019

20

21

22

23

24

25

26

Act

ual S

peed

[m/s

]

Time (sec)

0 200 400 600 800 1000 1200 14000.2

0.3

0.4

0.5

0.6

0.7

PI G

ain:

Kpi

ESES FilteredZN FixZN GS

0 200 400 600 800 1000 1200 1400-1.5

-1

-0.5

0

0.5

1

1.5

Spe

ed E

rror[m

/s]

ESZN FixZN GS

0 200 400 600 800 1000 1200 140019

20

21

22

23

24

25

26

Act

ual S

peed

[m/s

]

Time (sec)

ESZN FixZN GSDesired

0 200 400 600 800 1000 1200 140034

36

38

40

42

44

PI G

ain:

τPI

Iterations (k)

ESZN FixZN GS

0 200 400 600 800 1000 1200 1400200

250

300

350

400

450

Trac

tion

Forc

e [N

]

Time (sec)

ESES FilteredZN FixZN GS

Control Methods

• Extremum Seeking (ES)

• Fixed Ziegler-Nichols Gains (ZN Fix)

• Gain Scheduling (ZN GS)

Summary & Conclusions• Tunes PID controllers by minimizing a cost function

characterizing the desired closed-loop behavior

• Achieves better or comparable results relative to other popular tuning methods

• Improper ES parameters may produce instability

• Successfully performs adaptation for time-varying systems

• Online adaptation does not require models



References[1] N.J. Killingsworth, M. Krstic, “PID tuning using extremum seeking,”

IEEE Control Systems Magazine, pp 70-79, Feb. 2006.

[2] K.J. Åström, B. Wittenmark, Computer Controlled Systems: Theory and Design, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1997.

[3] O. Lequin, E. Bosmans, T. Triest, “Iterative feedback tuning of PID parameters: Comparison with classical tuning rules,” Contr. Eng. Pract., vol. 11, no. 9, pp. 1023-1033, 2003.

[4] A. Vahidi, A. Stefanopoulou, H. Peng, “Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments,” International Journal of Vehicle Mechanics and Mobility, Jan 2005.


QUESTIONS?

Adaptive Cruise Control via Extremum Seeking MethodsYiyao(Andy) Chang and Scott Moura

APPENDIX SLIDES

Results: Road Grade Steps

0 2000 4000 6000 8000 10000 12000-0.01

0

0.01

0.02

0.03

Roa

d G

rade

[rad

]

0 2000 4000 6000 8000 10000 1200015

20

25

Act

ual S

peed

[m/s

]

0 2000 4000 6000 8000 10000 12000200

300

400

500

600

Trac

tion

Forc

e [N

]

Distance (m)0 100 200 300 400 500 600 700

0

0.05

0.1

0.15

0.2

Cos

t

Iterations (k)

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

PI G

ain:

Kpi

ESES FilterZN

0 100 200 300 400 500 600 70042.5

43

43.5

PI G

ain:

τPI


Results: Vahidi Road Grade Profile (Linear Model)


0 100 200 300 400 500 6000

0.5

1

1.5

Cos

t

ESZN

0 100 200 300 400 500 6000

1

2

3

4

PI G

ain:

Kpi

0 100 200 300 400 500 60040

45

50

55

60

PI G

ain:

τPI

Iterations (k)

0 2000 4000 6000 8000 10000 12000-0.04

-0.02

0

0.02

0.04

0.06

Roa

d G

rade

[rad

]

0 2000 4000 6000 8000 10000 12000-15

-10

-5

0

5

10

Act

ual S

peed

[m/s

]

0 2000 4000 6000 8000 10000 12000-400

-200

0

200

400

Trac

tion

Forc

e [N

]

Distance (m)

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PI Tuning via Extremum Seeking Methods for Cruise Control

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