Fuzzy PID Control

- Reduce design choices

- Tuning, stability

- Standard nonlinearities

Design Procedure*

• Build and tune a conventional PID controller first.• Replace it with an equivalent linear fuzzy controller.• Make the fuzzy controller nonlinear.• Fine-tune the fuzzy controller.

*) Relevant whenever PID control is possible, or already implemented

Single Loop Control

u x

l n

yeRef

-

Controller Plant

Rule Base With 4 Rules

1. If error is Neg and change in error is Neg then control is NB3. If error is Neg and change in error is Pos then control is Zero7. If error is Pos and change in error is Neg then control is Zero9. If error is Pos and change in error is Pos then control is PB

PID Control

t

di

p dt

deTed

TeKu

0

1

n

j s

nndsj

inpn T

eeTTe

TeKu

1

11

sn

innpn Te

TeeKu

11

Fuzzy P controller

f

Rule base

uGU

UGE

Ee

GUneGEfnU *)(*)(

GUGEK

neGEneGEf

p *

)(*)(*

FP Rule Base

1. If E(n) is Pos then u(n) is 100

2. If E(n) is Zero then u(n) is 0

3. If E(n) is Neg then u(n) is -100

Fuzzy PD Controller

GUneGCEneGEfnU *)(*),(*)(

eGE

GCE

f

Rule base

E

CE

uGU

U

de/dt

GE

GCETGUGEK

neGCEneGEneGCEneGEf

dp

,*

)(*)(*)(*),(*

FPD Rule Base

1. If E(n) is Neg and CE(n) is Neg then u(n) is -2003. If E(n) is Neg and CE(n) is Pos then u(n) is 07. If E(n) is Pos and CE(n) is Neg then u(n) is 09. If E(n) is Pos and CE(n) is Pos then u(n) is 200

Fuzzy PD+I Controller

CE

eGE

f

PD rules

GCE

++ GU

E

GIEIE

u Ude/dt

edt

GUTjeGIEneGCEneGEfnUn

js *)()(*),(*)(

1

Fuzzy Incremental Controller

eGE

GCE

f

Rule base

E

CEGCU 1/s U

CUcu

de/dt

n

jsTGCUneGCEneGEfnU

1

**)(*),(*)(

Fuzzy - PID Gain Relation

Controller Kp 1/Ti Td

FP GE*GU

FInc GCE*GCU GE/GCE

FPD GE*GU GCE/GE

FPD+I GE*GU GIE/GE GCE/GE

Tuning

lKK

KnRef

KK

KKx

pp

p

11

u x

l n

yeRef

-

Controller Plant

Ziegler-Nichols Tuning

• Increase Kp until oscillation, Kp = Ku

• Read period Tu at this setting

• Use Z-N table for approximate controller gains

Ziegler-Nichols (freq. method)

Controller Kp Ti Td

P 0.5Ku

PI 0.45Ku Tu/1.2

PID 0.6Ku Tu/2 Tu/8

Z-N oscillation of 1/(1+s)3

PID control of 1/(1+s)3

Hand-Tuning

1. Set Td = 1/Ti = 0

2. Tune Kp to satisfactory response, ignore any final value offset

3. Increase Kp, adjust Td to dampen overshoot

4. Adjust 1/Ti to remove final value offset

5. Repeat from step 3 until Kp large as possible

Quick reference to controllers

Controller Advantage Disadvantage

FP Simple Maybe too simple

FPD Less overshootNoise sensitive, derivative kick

FInc Removes steady state error, smooths control signal

Slow

FPD+I All in oneWindup, derivative kick

Scaling

eGE

GCE

f

Rule base

E

CE

uGU

Uα

α

1/αde/dt

1********* GUneGCEneGEGUneGCEneGE

Nyquist 1/(s+1)3 with PID

-2 0 2-2

-1

0

1

2Kp = 4.8, Ti = 15/8, Td = 15/32

Tuning Map 1/(s+1)3

-2 0 2-2

0

2000

a)-2 0 2

-2

0

2001

b)-2 0 2

-2

0

2010

c)-2 0 2

-2

0

2011

d)

-2 0 2-2

0

2100

e)-2 0 2

-2

0

2101

f)-2 0 2

-2

0

2110

g)-2 0 2

-2

0

2111

h)

1/(s+1)3 with FPD+I

0 10 20 30 400

0.5

1

1.5

2

Co

ntr

olle

d o

utp

ut

y

0 10 20 30 40-2

0

2

4

6

Co

ntr

ol s

ign

al u

Time [s]

-1000

100

-1000

100-200

0

200

ECE

u

-100 -50 0 50 1000

0.2

0.4

0.6

0.8

1

Input family: Neg and Pos

Me

mb

ers

hip

Summary

1. Design crisp PID

2. Replace it with linear fuzzy

3. Make it nonlinear

4. Fine-tune it