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Fuzzy PID Control

Jan Jantzenjj@inference.dk

www.inference.dk2013

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Summary

• Reduce design choices• Tuning

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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

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Single Loop Control

Load Noise

The controller should preferably be able to follow the reference r, reject load changes l and noise disturbances n, but these requirements are in conflict with each other. We would like to transfer PID tuning methods to the fuzzy controller in order to have a tuning method.

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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

The four rules can handle many cases, and they are sufficient for a linear controller.

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Textbook PID Controllers

t

di

p teTe

TeKu

0 ddd1

n

j s

nndsj

inpn T

eeTTeT

eKu1

11

sn

innpn Te

TeeKu 1

1

Continuous version

Discrete version

Incremental, discrete version

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Fuzzy P controller

f

Rule base

uGU

UGE

Ee

GUneGEGUneGEfnU

)()()(

GUGEK p

Gain on error

Gain on control

Provided that the rule base acts like the identity function

By comparison with the P controller equation

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FP Rule Base

1. If E(n) is Pos then u(n) is 1002. If E(n) is Neg then u(n) is -100

With a proper choice of membership functions the controller will act like a linear P controller

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Fuzzy PD Controller

GEGCETGUGEK

neGEGCEneGUGE

GUneGCEneGEGUneGCEneGEfnU

dp

,

))()((

))()(()(),()(

eGE

GCE

f

Rule base

E

CEu

GUU

de/dt

Provided that the rule base acts like a summation

Now we know what the gains do

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FPD Rule Base1. 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

With a proper choice of membership functions the controller will act like a linear PD controller. Four rules are sufficient.

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Fuzzy PD+I Controller

CE

eGE

f

PD rules

GCE++ GU

E

GIEIE

u Ude/dt

edt

GUTjeGIEneGCEneGEfnUn

js

1

)()(),()(

It is better that the integral action bypasses the rule base. It saves rules.

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Fuzzy Incremental Controller

eGE

GCE

f

Rule base

E

CE GCU 1/s UCUcu

de/dt

n

jsTGCUneGCEneGEfnU

1

)(),()(

The output is a change to the previous state

This is an integrator. It could be a valve position, for instance.

The increment. It is a change to the sum of all previous signals.

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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/GEIt tells what each fuzzy gain does to the proportional gain, the derivative gain, and the integral gain. Conversely, given values for Kp, Ti and Td we can find one or more sets of values for the fuzzy gains. Very important table.

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Tuning

lKK

KnrKKKK

xpp

p

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Process gain

If we increase Kp, we suppress load changes.

If we increase Kp, the response will be more sensitive to noise.

If we increase Kp too much, the system might oscillate or even become unstable

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Ziegler-Nichols Tuning

• Increase Kp until oscillation, Kp = Ku

• Read period Tu at this setting• Use Z-N table for approximate controller

gains

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Ziegler-Nichols (freq. method)

Controller Kp Ti Td

P 0.5Ku

PI 0.45Ku Tu/1.2

PID 0.6Ku Tu/2 Tu/8

Given values for Ku and Tu, the table provides the gains in the three controller cases. Easy, but often the result is a poorly damped system.

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

The ultimate gain Ku = 8, and the ultimate period is Tu = 15/4 s

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PID control of 1/(1+s)3 Response to a reference step Response to a

load step

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Fuzzy FPD+I control of 1/(1+s)3 The response is the same as for PID control Trajectory on

the control surface, which is a plane

The membership functions are linear

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Hand-Tuning1. Set Td = 1/Ti = 02. Tune Kp to satisfactory response, ignore any final value

offset3. Increase Kp, adjust Td to dampen overshoot4. Adjust 1/Ti to remove final value offset5. Repeat from step 3 until Kp large as possible

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Quick reference to controllers

Controller Advantage Disadvantage

FP Simple Maybe too simple

FPD Less overshoot Noise sensitive, derivative kick

FInc Removes steady state error, smooths control signal Slow

FPD+I All in one Windup, derivative kick

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Scaling

eGE

GCE

f

Rule base

E

CEu

GUU

α

α1/α

de/dt

1 GUneGCEneGEGUneGCEneGE

The linear controller is invariant towards scaling. In the nonlinear controller we can use it to avoid saturation in the input universes.

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Summary

1. Design crisp PID2. Replace it with linear fuzzy3. Make it nonlinear4. Fine-tune it

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ADVANCED SECTION

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Nyquist 1/(s+1)3 with PID

-2 0 2-2

-1

0

1

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

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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)