HOW DOES INTELLIGENT CONTROL WORK ?

Importance• In the last few years the applications of artificial

intelligence techniques have been opening doors to convert human experience into a form understandable by computers. Advanced control based on artificial intelligence techniques is called intelligent control.

Fuzzy Logic• Fuzzy logic is a technique to embody human-like

thinking into a control system.• A fuzzy controller can be designed to emulate human

deductive thinking, that is, the process people use to infer conclusions from what they know.

USE OF FUZZY CONTROL Fuzzy control incorporates ambiguous human logic

into computer programs. It suits control problems that cannot be easily represented by mathematical models :

Weak modelParameter variation problemUnavailable or incomplete dataVery complex plantsGood qualitative understanding of plant or process

operation Because of its unconventional approach, design of

such controllers leads to faster development / implementation cycles

Traditional control approach requires formal modeling of the physical reality. Here we show two methods that may be used to describe a system’s behavior:

1. Experimental Method• By experimenting and determining how the process

reacts to various inputs. 2. Mathematical Modeling

• Mathematical model of the controlled process, usually in the form of differential or difference equations. Laplace transforms and and z-transforms are respectively used.

• Problems that can arise: – Model complexity

– Inaccurate values of various parameters

Alternative Approach: Heuristic Method The heuristic method consists of modeling and understanding in

accordance with previous experience, rules-of-thumb and often-used strategies.

Heuristic rule: It is a logical implication of the form:• If <condition> then <consequence>, or in a typical control

situation: If <condition> then <action>

Rules associate conclusions with conditions.• The heuristic method is actually similar to the experimental method

infused with the control strategies of human operators.

Intelligent control strategies may be implemented by other means. However, fuzzy implementations are very efficient for several reasons.

1965 Introduction of fuzzy sets theory by Lotfi Zadeh (USA) 1972 Toshiro Terano established the first working group on

fuzzy systems in Japan 1973 Paper about fuzzy algorithms by Zadeh (USA) 1974 Steam engine control by Ebrahim Mamdani (UK) 1977 Fuzzy expert system for loan evaluation by Hans

Zimmermann (Germany) 1980 Cement kiln control by F. - L. Smidth & Co. - Lauritz P.

Holmblad (Denmark) 1984 Water treatment (chemical injection) control (Japan) 1984 Subway Sendai Transportation system control (Japan) 1985 First fuzzy chip developed by M. Togai and H. Watanabe

in Bell Labs (USA) 1986 Fuzzy expert system for diagnosing illnesses in Omron

(Japan)

1987 Container crank control, tunnel excavation, soldering robot, automated aircraft vehicle landing • Second IFSA Conference in Tokyo • Togai Infralogic Inc. first company dedicated to fuzzy control in

Irvine (USA) 1988 Kiln control by Yokogawa

• First dedicated fuzzy controller sold - Omron (Japan) 1989 Creation of Laboratory for International Fuzzy Engineering

Research (LIFE) in Japan 1990 Fuzzy TV set by Sony (Japan)

• Fuzzy electronic eye by Fujitsu (Japan) • Fuzzy Logic Systems Institute (FLSI) by Takeshi Yamakawa

(Japan) • Intelligent Systems Control Laboratory in Siemens (Gemiany)

1991 Fuzzy AIl Promotion Centre (Japan)• USA start to get academic attention

Ten years of hype => We are getting to a ripe technology now !

BASIC PRINCIPLES OF BIVALENT AND MULTIVALENT LOGIC

Bivalence : something is either true or not true Logic of Aristotle was the cornerstone of such philosophical

ethics Multivalence : captures the mismatch between the real world and

our bivalent view of it.• Medical diagnosis

• Legal decisions The objective of fuzzy logic is to capture these shades of gray,

these degrees of truth. Fuzzy logic deals with uncertainty and the partial truth

represented by the various shades of gray in a systematic and rigorous manner.

Plato (427-347? Bc) saw degrees of truth everywhere and recoiled from them. No chair is perfect, it is only a chair to a certain degree.  

Charles Sanders Peirce (1839-1914) observed that All that exists is continuous and such continuums govern knowledge.

Bertrand Russell (1872-1970) Both vagueness and precision are features of language, not reality. Vagueness clearly is a matter of degree.

Jan Lukasiewicz (1878-1956) proposed a formal model of vagueness, a logic where 1 stands for TRUE, 0 stands for FALSE, 1/2 stands for possible. Lukasiewicz was just one step away from Zadeh and his logic can be considered relative.

Max Black (1909-89) proposed a degree as a measure of vagueness. Albert Einstein (1879-1955): So far as the laws of mathematics refer to

reality, they are not certain. And so far as they are certain, they do not refer to reality.

Lotfi Zadeh (1923- ) introduced fuzzy sets logic theory. 'As the complexity of a system increases, our ability to make precise and significant statements about its behavior decreases until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics...

PARACONSISTENT LOGIC Although not very known in engineering applications,

Paraconsistent Logic has been under interest of Philosophers and Mathematicians before the foundations of fuzzy logic• Roughly speaking, a paraconsistent logic is a logic rejecting

the principle of non contradiction (PNC). • The philosophical focus was for a long time whether a

negation not obeying (PNC) is still a negation.

Newton da Costa assembled in 1963 the foundations based on the work of Stanislaw Jaskowski. Francisco Miró Quesada suggested this name (paraconsistent) in 1976.• An example could be a decision support system which is able

to support contradictions like someone is 0.7 tall and 0.5 short.

Latter you will observe that despite mathematical formalities, in engineering applications one can shift the membership functions to get a similar effect.

CONCEPT OF FUZZY DESCRIPTIONS A linguistic variable has values expressed in words,

representing qualitative concepts or impressions about the state of a system

The idea is to capture vague, imprecise, inexact, fuzzy verbal descriptions• Large, medium, big, not very large, etc…

To convert linguistic terms into numeric values one needs the fundamentals of set theory

In crisp set theory, an element is either a member (membership grade = 1) or not a member (membership grade =0)

Concept of a Fuzzy Number

Zero0-1-2-3 1 2 3

0-1-2-3 1 2 3

0-1-2-3 1 2 3

Almost Zero

Near Zero

Fuzzy Sets

In fuzzy set theory an element’s membership may have a value in the interval from 0 to1:

x1

x2

Universe of DiscourseSet A

Example : define speed over the range of 50 km/h and 100 km/h• What is the set of speeds equal or greater that 70 km/h ?

There is a sharp change from 0 to 1

Mem

bers

hip

grad

e

K m /h

1 .0

0 .5

5 0 7 0 1 0 0

S e t o f sp ee d e rs

Fuzzy Membership Fuzzy sets allow gradual transitions between membership and

non-membership; = any value in the interval [0,1]

Example :”a set of speeds greater than 70 km/h”M

embe

rshi

p

grad

e

K m /h

1 .0

0 .5

5 0 7 0 1 0 0

S e t o f sp ee d e rs

Concept of Universe of Discourse

FUZZIFICATION

Process of decomposing a system input into one or more fuzzy sets. Triangular or trapezoidal shaped are the most common.

With 50% of overlapping, any input will “fire” two fuzzy sets

C O L D C O O L G O O D W A R M H O T

Te m p e r a tu re ( C )o

11 o 1 4 o 1 7 o 2 0 o 2 3 o 2 6 o 3 2 o2 9 o8 o

1 .0

Translating Semantics into Numerical Values

“The temperature is cool.”• It the temperature was 13OC we would assign such

statement the truth value of 0.8. However, we would also assign a truth value of 0.2 for the following statement as well :

“The temperature is cold.” The set terminology is as follows:

• The temperature is a member of the set of the cool feeling with COOL(x) = 0.8

• The temperature is a member of the set of the cold feeling with COLD(x) = 0.2

Establishing a Logic Kernel

To establish a complete system of fuzzy logic, one needs to define some logic operations:

EMPTY EQUAL COMPLEMENT (NOT) CONTAINMENT ALPHA-CUT INTERSECTION (AND) UNION (OR)

Measure of fuzziness : it is the closeness of its elements to the membership grade of 0.5

dxxxfAAA

21

where A1/2 is the ½ cut of A(x)

-cut of a fuzzy set is the crisp set formed by elements A whose membership grade is greater than or equal to a given value a. This is denoted by:

in which X is the universe to which A belongs.

xXxA A

Intersection (AND) Given two sets A and B , A E, B E, where E is their common

universe of discourse, he intersection A B is a set of all elements x which are members of both A and B. This is illustrated in the Venn diagram.

E

A

B

A BU

The overlapping portion of sets A and B and, as a result, it is always smaller than any of the individual sets A and B.

Membership function AB(u) , u U, of the intersection AB is defined point-by-point by :

AB(u) = A(u) t B(u) min [A(u), B(u)]

Considering the fundamental definition of AND operator as min [A(u), B(u)], the figure depicts the resulting membership function for A AND B

A B

U

A A

B B

Union (OR) Given two sets A and B , A E, B E, where E is their common

universe of discourse, the union A B is a set of all elements x which are members of set A or set B or both A and B. This is illustrated in the Venn diagram.

E

A BU

A

B

Union is the smallest subset of the universe of discourse E, which includes both sets A and B.

Membership function A B(u) , u U, of the union A B is defined point-by-point by :

A B(u) = A(u) s B(u) max [A(u), B(u)]

A BU

A A

B B

Considering the fundamental definition of OR operator as max [A(u), B(u)], the figure depicts the resulting membership function for A OR B

There are several T-norm and S-norm operators

UNION

INTERSECTION

PARAMETER

MIN / MAX xx BA ,max xx BA ,min None

ALGEBRAIC xxxx BABA xx BA None

SCHWEIZER & SKLAR

P

BA xx/1-P-P 1 1-1 0,max1 P

BA xx/1-P-P 1 0,max

,p

HAMACHER xxg

xxgxx

BA

BABA

11

2

xxxxgg

xx

BABA

BA

1 ,0g

FRANK

11

1log111

sss xx

S

BA

1

111log

sss xx

S

BA

,0s

YAGER

WW

BW

A xx1

,1min

WW

BW

A xx1

11,1min1 ,0w

DUBOIS & PRADE

,1,1max

1,,min

xx

xxxxxx

BA

BABABA

,,max xx

xx

BA

BA

1,0

DOMBI

1

11

11

1

1

xx BA

1

11

11

1

1

xx BA

,0

Properties

Given universe of discourse E, and three fuzzy sets A E, B E, C E , then the following applies: Commutativity properties:A B = B AA B = B A Associativity properties:(A B) C = A (B C)(A B) C = A (B C) Idempotence:A A = AA A = A Distributivity with respect to intersection: A (B C) = (A B) (A C) Distributivity with respect to union: A (B C) = (A B) (A C)

Fuzzy set and its complement(*):

A A’ 0

A A’ E

Fuzzy set and the null set:

A = 0

A = A

Fuzzy set and the universal set

A E = A

A E = E

Involution property:

( A’)’ = A

De Morgan’s theorem:

(A B)’= A’ B’

(A B)’= A’ B’

HEDGES Sometimes we need to refine descriptions, making them more

meaningful and accurate. Fuzzy sets can be modified to reflect this kind of linguistic refinement by applying hedges. Once a hedge has been applied to a fuzzy set, the degrees of membership of the members of the set are altered. The membership of Fast can be altered by VERY fast by an exponential operator like the one indicated in the figure :

Deg

ree

ofM

embe

rshi

p

1

0

V E RY F ast

F ast

S peed12 0 K m /h80 K m /h

Some examples of hedges : VERY, A(x) = [A(x)] SOMEWHAT, A(x) = [A(x)]

The hedge “Approximately” uses the concept of subsethood. It requires an auxiliary fuzzy set (distribution of weights) to evaluate assertions like

“How Much Approximately Cold is Hot ?

A P P R O X IM AT E LY C o ldC o ld

Tem peratu re

d is tr ibu tiono f w e igh ts

2 .5 Co 5 Co0 Co

1

0

Deg

ree

ofM

embe

rshi

p

Operations on the Same Universe of Discourse

Keeping simple : the operation of fuzzification requires a generic element and its membership degree

AND, OR and NOT can be performed like:(X and Y)  = MIN ((X), Y))(X or Y) = MAX ((X), (Y))(not X) = 1.0 - (X)

Let us assume an air-conditioner vent. Suppose it has blades that control the openings and can be controllable so as the inclination angle of the vent might be directed downward or upward. Such angle control sends the air-flow towards the floor or to the ceiling.

Fuzzy sets DOWNWARD and UPWARD describe the position of the vent blades. If the blades are totally rotated to –45 degrees in respect to the horizontal then they are completely downward. If the blades are totally rotated to +45 degrees, they are completely upward.

U p w a rd1 .0

-45 0 4 5

D o w n w a rd

U p w a rdO R

D o w n w a rd

U p w a rdA N D

D o w n w a rd

-45 0 4 5

-45 0 4 5-45 0 4 5

1 .0

1 .0 1 .0

Statements

Statements assert facts or states of affairs; they give descriptions that can be organized in several rules of reasoning. Application of mathematical descriptions and the use of logical rules for formulating hypotheses was developed by several philosophers.

They have been influenced by the earlier syllogistic logic, in which premises were manipulated to produce true conclusions. A typical syllogistic rule of inference that goes by the Latin name modus ponens (affirmative mode) can be given as• If A is true• And A implies B• Then B is true

where the connectives and, or, and not are essential to derive the truth of above rules.

Operations in Different Universes of Discourse

What are the operations between sets belonging to different universes of discourse ?

In control systems, mappings between input and output are our main concern. These mappings are between input variable sets A( u) U and output variable sets B(v) V through the conditional statement of inference:

A B or: IF A(u) THEN B(v) Such map links the antecedent (condition) set A (defined

by its membership vector A(u), u U with the consequent (result or action) set B (defined by its membership vector B(v), v V).

Suppose A and B are crisp sets. Defining P(u, v) = A(u) X B(v), where the symbol ”X” stands for the Cartesian product operator. Since both sets A and B are characterized by their respective membership vectors , their Cartesian product will be a matrix of crisp numbers.

Cartesian product of fuzzy sets => In this case, the membership grades, , are in M = [0,1]. If the spaces are, for example, U1 = {x} and U2 = {y}, then the Cartesian product is P{x, y} = U1 X U2 with membership function c(x, y) where each ordered pair is in [0,1]. In fuzzy set theory, the component sets of a Cartesian product are always universes of discourse, hence they are always crisp.

In practice, the t-norms min and algebraic product are mostly used .

A fuzzy set A E1 will induce another fuzzy set B E2 whose membership function will beB(y/x). Then one can write:

B(y) = A(x) ° R(x,y)

where “°” is the compositional operator which indicates a compositional rule of inference. For the purpose of practical computation, it can also be written in terms of the membership functions of the respective sets:

Max-min composition:

B(y) = MAX[MIN(A(x), R(x,y))] x E1

Max-product composition:

B(y) = MAX[A(x) . R(x,y))] x E1

A multi-input multi-output system can be characterized by a set of rules :

IF var1 = A <connective> var2 = B <connective> … THEN Out1 = C <connective>…

<connective’>

IF var1 = D <connective> var2 = E <connective> … THEN Out2 = E <connective>…

<connective’>

……

……

where A, B, C, D and E are crisp or fuzzy sets, and <connective> represents the particular fuzzy operator chosen to express the fuzzy inference or fuzzy implication desired.

Fuzzy Rules Rules of inference : IF <conditions> THEN <consequences>

Rule 1 If the distance between two cars is medium and the speed of the car is medium Then brake medium for speed reduction

Rule 2 If the distance between two cars is small and the speed of the car is medium Then brake hard for speed reduction

M EDIUM

M EDIUM M EDIUM

M EDIUM

SM ALL

HARD

Defuzzification

Brake action forspeed reduction

Center o fgravity

Speed (S)Distance (D)

M e m b e rsh ip fu n c tio ncu rv es fo r in p u ts

M e m b e rsh ip fu n c tio ncu rv es fo r o u tp u ts

T h e lo w er o f th e tw oIF c o n d it io n o u tp u ts

a re se le cte d

Rule-based fuzzy controllers have several advantages

Fuzzy control rules are easy to understand by maintenance personnel

Control functions associated with a rule can be tested individually. This improves maintainability because the simplicity of the rules allows the use of less skilled personnel.

Individual rules combine to form a structured complex control; parallel processing allows fuzzy logic to control complex systems using simple expressions.

Rules can be added for alarm conditions, both linear and nonlinear control functions can be implemented by a rule-based system, using expert knowledge formulated in linguistic terms.

Fuzzy controllers are inherently reliable and robust. A partial system failure may not significantly degrade the controller’s performance.