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ECTE441/841/941

Intelligent Control

Autumn 2020

Lecture 03

2

Subject Outline

Introduction to intelligent control and fuzzy sets

Fuzzy operations and rules

Fuzzy inference and PID control

Fuzzy controller design and tuning

Fuzzy extension

TSK fuzzy control

3

Objectives of the Lecture

Be able to use a fuzzy inference system.

Develop a better understanding of the fuzzy approach

by applying it to a specific example.

Be able to apply PID for fuzzy logic control

Mamdani Fuzzy Inference Systems

4

Structure of the lecture

Fuzzy inference system

Introduction

Graphic approach

Defuzzification

Example

PID control

Structure and function

Operation mechanism

Tuning

5

Tipping Problem

Problem: Given two numbers between 0 and 10 that

represents the quality of service and food at a restaurant

(when 10 is excellent for service and delicious for food)

what should the tip be for a specific visit?

This problem is based on a tipping custom in United

States which is on average 15% of the bill. This,

however, can change depending on the quality of the

service and food provided.

Rule 1: If service is poor or food is rancid then tip is cheap

Rule 2: If service is good then tip is average

Rule 3: If service is excellent or food is delicious then tip is generous

6

Repeat (1)~(3) for All Rules

Mark: Service =3, food=8

If service is poor or food is rancid then tip is cheap

7

1 Fuzzy Inference System

Fuzzy Inference System

Crisp

Input

Crisp

Output

8

Fuzzy Inference System

Aggregation

Method Defuzzify

4. 5.

Fuzzify

Inputs

Fuzzy

Operation

Implication

Operation

Fuzzify

Inputs

Fuzzy

Operation

Implication

Operation

…

1.

1.

2.

2.

3.

3.

For each fuzzy rule:

Rule 1: If service is poor or food is rancid then tip is cheap

Rule 2: If service is good then tip is average

Rule 3: If service is excellent or food is delicious then tip is generous

9

Fuzzy MFs

MF: Service MF: Food MF: Tip

Rule 1: If service is poor or food

is rancid then tip is cheap

Rule 2: If service is good then

tip is average

Rule 3: If service is excellent or

food is delicious then tip is

generous

Fuzzy Matrix

Each line for a rule

Each column for a fuzzy input

or output

Align each origin of fuzzy input

and output both vertically and

horizontally

10

(4) Aggregate All Outputs

11

(5) COA Defuzzify

i

A i

i

A i i

COA

Y

A

Y

A

COA

y

y y

y

y dy

y ydy

y

( )

( ).

( )

( )

12

MOM Defuzzification

1

A y

‘

‘

we have

‘

For

which the MF reach a maximum *.

average of the maximizing at

Mean of maximum is the

Y

Y

MOM

A

MOM

dy

ydy

y

Y y y

y

y

2

then

If ( ) reaches its maximum whenever

If ( ) has a single maximum at , then

left right

MOM

A left right

A MOM

y y

y

y y [y ,y ]

y y y* y y*.

13

More on Defuzzification

Definition

“It refers to the way a crisp value is extracted from a fuzzy

set as a representative value”

There are five methods of defuzzifying a fuzzy set A of a

universe of discourse Z

Centroid of area zCOA

Mean of maximum zMOM

Smallest of maximum zSOM

Largest of maximum zLOM

Bisector of area zBOA

14

More on Defuzzification

Bisector of area zBOA

this operator satisfies the following;

where = min z; z Z & = max z; z Z.

zBOA

zBOA

A (z)dz A (z)dz,

1

A y

15

Smallest of maximum zSOM

Amongst all z that belong to [z1, z2], the smallest is called zSOM

Largest of maximum zLOM

Amongst all z that belong to [z1, z2], the largest value is called

z

LOM

More on Defuzzification

16

Information Flow of Fuzzy Inference System

17

1. Fuzzify inputs:

Resolve all fuzzy statements in the antecedent to a degree of membership

between 0 and 1. If there is only one part to the antecedent, this is the degree

of support for the rule.

2. Apply fuzzy operator to multiple part antecedents:

If there are multiple parts to the antecedent, apply fuzzy logic operators and

resolve the antecedent to a single number between 0 and 1. This is the

degree of support for the rule.

3. Apply implication method:

Use the degree of support for the entire rule to shape the output fuzzy set.

The consequent of a fuzzy rule assigns an entire fuzzy set to the output. This

fuzzy set is represented by a membership function that is chosen to indicate

the qualities of the consequent. If the antecedent is only partially true, (i.e., is

assigned a value less than 1), then the output fuzzy set is truncated according

to the implication method.

4. Aggregation:

The combination of the consequents of each rule in preparation for

defuzzification.

5. Defuzzification.

Steps of FIS

18

Fuzzy Approach Essentials

It would be ideal if we could just capture the essentials

of this problem, leaving aside all the factors that could

be arbitrary. Let’s make a list of what really matters in

this problem:

If service is poor, then tip is cheap

If service is good, then tip is average

If service is excellent, then tip is generous

The order of rules is arbitrary. We might add two more

rules:

If food is rancid, then tip is cheap

If food is delicious, then tip is generous

19

Crisp Inference System

If a<-20 and b<10 then c=2

If -20<a<0 and b<10 then c=5

If 0<a<20 and b<10 then c=20

If a<-20 and b>10 then c=0

If -20<a<0 and b>10 then c=14

If 0<a<20 and b>10 then c=-5

Rules:

Given:

a=8, b=4 Find: c=?

Answer: c=20

Only one rule will be satisfied and contribute to the answer.

20

Fuzzy Inference System

If a is tall and b is old then c=ok

If a is short and b is old then c=no

If a is normal and b is old then c=kind of

If a is tall and b is young then c=very good

If a is short and b is young then c=ok

If a is normal and b is young then c=good

Rules:

Given:

a=height, b=age Find: c=go to basket ball team?

Answer: c=?

Fuzzy inference system is a democratic system.

21

FIS

22

Adding Rues Using Rule Editor

23

Fuzzy Rules

Create rules by selecting an item in each input and

output variable box, selecting one Connection item, and

clicking Add Rule. You can choose none as one of the

variable qualities to exclude that variable from a given

rule and choose not under any variable name to negate

the associated quality.

Delete a rule by selecting the rule and clicking Delete

Rule.

Edit a rule by changing the selection in the variable

box and clicking Change Rule.

24

FIS Editor

25

Rule viewer of Tipper-From the View menu, select Rules.

26

Surface Viewer of Tipper-Select surface from the View menu

27

2 Example: Fuzzy Inference

Consider a two input-one-output Mamdani fuzzy system that is

constructed from the following two rules

If x

1 is A1 and x2 is A2, THEN y is A1

If x

1 is A2 and x2 is A1, THEN y is A2

Where A

1 and A2 are fuzzy sets in R with membership functions:

otherwise

u if u

u

otherwise

u if u

u

A A

1 1 0 2

( )

1 1 1

( )

1 2

Suppose the input to the system is (x1, x2)=(0.3, 0.6). Determine the output

of the fuzzy systems in the following situations:

a)Max for aggregation and Mean of Maximum for defuzzification

b)Max for aggregation and Centroid for defuzzification

28

Example

-1 1

1

0 2

1

1

(x1, x2)=(0.3, 0.6)

A1 A2

-1 1

1

A1

2

1

1

A2

-1

1

1

A1

2

1

1

A2

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

29

Example

-1 1

1

0 2

1

1

(x1, x2)=(0.3, 0.6)

A1 A2

-1 1

1

A1

2

1

1

A2

-1

1

1

A1

2

1

1

A2

0.7

0.6

0.6

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

30

Example

-1 -1

1

0 2

1

1

(x1, x2)=(0.3, 0.6)

A1 A2

-1 -1

1

A1

2

1

1

A2

-1

-1

1

A1

2

1

1

A2

0.7

0.6

0.6

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

31

Example

-1 1

1

0 2

1

1

(x1, x2)=(0.3, 0.6)

A1 A2

-1 1

1

A1

2

1

-1

2

1

1

1

A2

0.4 0.3

A1

0.3

1

1

A2

0.7

0.6

0.6

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

32

Example

-1 1

1

0 2

1

1

(x1, x2)=(0.3, 0.6)

A1 A2

-1 1

1

A1

2

1

-1

2

1

1

1

0.4 0.3

A1

0.3

1

1

A2

A2

0.7

0.6

0.6

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

33

Example If x1 is A1 and x2 is A2, THEN y is A1 If x1 is A2 and x2 is A1, THEN y is A2

MOM:

COA:

-1

2

1

1

0.6

0.3

Centroid by geometric decomposition The centroid of a plane figure X can be computed by dividing it into a

finite number of simpler figures. , computing the centroid Ci and area

A

i of each part, and then computing

0.355

0.84 0.3 0.3 2 / 2 0.3 0.7

0.3 0.3 1.35 0.3 0.7 1.35

0.84 0.3 0.3 2 / 2 0.3 0.7

0.84 0 0.3 0.3 (3 0.3) / 2 0.3 0.7 1.35

0.84 0.3 0.3 2 / 2 0.3 0.7

0.84 0 (1 0.3/ 3) 0.3 0.3/ 2 0.3 0.7 1.35 (2 0.3 0.3/ 3) 0.3 0.3/ 2

C

34

Example

A B

A x B x

y A B

If x1 is A1 and x2 is A2, THEN y is A1

If x1 is A2 and x2 is A1, THEN y is A2

MOM: y=0

COA:

0.355

0.84 0.3

0.84 0 0.3 1.35

-1

2

1

1

0.6

0.3

y=0.335

35

Centroid

Centroid of a finite set of points

The centroid of a finite set of points

Centroid by geometric decomposition

The centroid of a plane figure X can be computed by dividing it into a finite number of simpler figures

, computing the centroid Ci and area Ai of each part, and then computing

36

A Control System

Objective of Control:

Make a dynamic system behave in a desired manner, according

some performance specifications.

Note:

Complex system (many inputs and many outputs, dynamic coupling,

nonlinear, etc.)

Unknown disturbances

Unknown dynamics

37

Basic Elements of Control System

Controller

(Actuator)

Plant

Output

Input + –

Feedback (from sensors)

Error

Ctrl. Signal

Plant: part of a control system to be controlled

Output: variable of the plant to be controlled

Input (reference/set point) — the desired output

Error signal: the difference between the set point and the output

Sensors: the eyes of control enabling one to see what is going

on

Controller (designed by the engineer): acts on the error signal

and issues a control signal to the plant in order to achieve the

desired output

38

Conventional Control

Methods for representing control systems

Transfer Functions

Block Diagrams

State Space Model

PID Control

In process control >90% controllers are of PI(D)-type

Good for linear process control

Relatively easy to understand (important reason for wide

popularity)

39

About Project 1

Controller

u(t)

Plant

y(t)

r(t) + e(t)

40

1. Linear Partition

x

x1

x2

3

x4

x5

a1 a2 a3 a4

y f (x)

Non-linear equation

a x

a x a

a x a

a x a

x a

k x x f x

k x x f x

k x x f x

k x x f x

k x x f x

y

4

3 4

2 3

1 2

1

5 5 5

4 4 4

3 3 3

2 2 2

1 1 1

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

Represented by 5 linear

equations

41

PID Controller

1 ( )

( ) ( ( ) ( ) ) p d

i

de t

u t K e t e d T

T dt

Controller

u(t)

Plant

y(t)

r(t) + e(t) The idea continuos PID controller

K

p : proportional gain

T

i : integral time

T

d : derivative time

t

42

A Matlab PID Controller

1 ( )

( ) ( ( ) ( ) )

/

p d

i

i p i

d p d

de t

u t K e t e d T

T dt

K K T

K K T

t

43

Functions of P, I and D

P should be dominant for reducing large error

D should comes into play to slow the output down to

reduce overshoot

I comes into play to reduce steady

state error. Its effects on the

transient phase should be as

small as possible.

1 ( )

( ) ( ( ) ( ) ) p d

i

de t

u t K e t e d T

T dt

t

44

Poles at Origin in Open Loop TF

1/s

1/(s2+2s+1)

–

R(s) C(s)

+

B(s)

E(s)

U(s)

( ) 0

( ) 1( )

e

r t t

45

Poles at Origin in Open Loop TF

K

1/(s2+2s+1)

–

R(s) C(s)

+

B(s)

E(s)

U(s)

( ) 0

( ) 1( )

e

r t t

46

Proportional

High value of gain

makes system more

insensitive to load

disturbance

Too large gain makes

system more

sensitive to

measurement noise

Steady-state error

decreases when gain

increases

Oscillation however

often increases

K

P

47

Integral

Integral term removes

steady state error

Short integration time

often leads to oscillation

Long integration time

common in process

control

K

i

48

Derivative

Derivative term can

predict output

Fast and stable

response

Noise can make

derivative control

problematic

Also long delays are

problematic when using

derivative term

K

d

49

The Characteristics of P, I, and D controllers

A proportional controller (Kp) will have the effect of reducing the rise time and

will reduce, but never eliminate, the steady-state error.

An integral control (Ki) will have the effect of eliminating the steady-state error,

but it may make the transient response worse.

A derivative control (Kd) will have the effect of increasing the stability of the

system, reducing the overshoot, and improving the transient response.

50

Proportional Control

By only employing proportional control, a steady state error occurs.

Proportional and Integral Control

The response becomes more oscillatory and needs longer to settle, the

error disappears.

Proportional, Integral and Derivative Control

All design specifications can be reached.

The Characteristics of P, I, and D controllers

51

CL RESPONSE RISE TIME OVERSHOOT SETTLING TIME S-S ERROR

Kp

Decrease

Increase

Small Change

Decrease

Ki

Decrease

Increase

Increase

Eliminate

Kd

Small Change

Decrease

Decrease

Small Change

The Characteristics of P, I, and D controllers

52

Tips for Designing a PID Controller

1. Obtain an open-loop response and determine what needs to be improved

2. Add a proportional control to improve the rise time

3. Add a derivative control to improve the overshoot

4. Add an integral control to eliminate the steady-state error

5. Adjust each of Kp, Ki, and Kd until you obtain a desired overall response.

Lastly, please keep in mind that you do not need to implement all

three controllers (proportional, derivative, and integral) into a

single system, if not necessary. For example, if a PI controller

gives a good enough response, then you don’t need to

implement derivative controller to the system. Keep the

controller as simple as possible.

53

Filtered Derivatives

Fast changes in reference signal result high derivatives which

cause control signal saturation

Fixes:

Computing derivatives from process output

Using filtered derivative term (this is used often in real applications)

Benefits:

Easy to implement in practice

More insensitive to noise than normal derivative term

Corresponds with derivation of low pass filtered signal

By choosing small τ, system has same response as by using normal

derivation in low frequency band

54

Example: PID

sltank

sltank2

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