what is fuzzy logic

Fuzzy Logic Control • Fuzzy Logic Control FLC or sometimes known as Fuzzy Linguistic Control is a knowledge based control strategy that can be used - when either a sufficient accurate

CS 561, Sessions 22-23 1

What is fuzzy logic?

• A super set of Boolean logic

• Builds upon fuzzy set theory

• Graded truth Truth values between True and

black/white, on/off etc.

• Grades of membership Class of tall men,

class of far cities, class of expensive things,

etc.

• Lotfi Zadeh, UC/Berkely 1965 Introduced FL

to model uncertainty in natural language

Tall, far, nice, large, hot, …

• Reasoning using linguistic terms Natural to express expert knowledge

clothing

CS 561, Sessions 22-23 2

Why use fuzzy logic?

Pros:

• Conceptually easy to understand w/ “natural” maths

• Tolerant of imprecise data

• Universal approximation: can model arbitrary nonlinear functions

• Intuitive

• Based on linguistic terms

• Convenient way to express expert and common sense knowledge

Cons:

• Not a cure-all

• Crisp/precise models can be more efficient and even convenient

• Other approaches might be formally verified to work

Issues of Fuzzy logic Control

• Requires tuning of membership functions

• Fuzzy Logic control may not scale well to large or complex problems

• Deals with imprecision, and vagueness, but not uncertainty

CS 561, Sessions 22-23 4

Limitations of fuzzy logic

• How to determine the membership functions? Usually requires fine-tuning of parameters

• Defuzzification can produce undesired results

Application (2)

Fuzzy Logic Control

Conventional Control System

The dynamic filter compute all the system dynamics: x (state variables) consists of selected elements of e = r – y, de/dt, or ∫edτ

.

Plant

Conventional PID Controller Architecture

y u

Dynamic Filter

r x

PID Controller Algorithm

Mathematical Model

Fuzzy Logic Control

• Fuzzy Logic Control (FLC) or sometimes known as Fuzzy Linguistic

Control is a knowledge based control strategy that can be used

- when either a sufficient accurate and yet not unreasonably

complex model of the plant is unavailable, or

- when a (single) precise measure of performance is not meaningful

or practical.

• FLC design is based on empirically acquired knowledge regarding the

operation of the process

• This knowledge, cast into linguistic, or rule-based form, is the core of

the FLC system.

• FLC is particularly useful when the plant model is unknown or difficult

to develop

FLC Architecture

The rule base (knowledge base) provides nonlinear transformations without any built-in dynamics

Decoder (Defuzzifier)

Knowledge Base

Encoder (Fuzzifier)

Plant Inference

Engine

Fuzzy Logic Controller Architecture

y u

Dynamic Filter

r x

Fuzzification

Fuzzy quantization of the state variables For example, the state

variable "Angle" may be quantified into a set of linguistic variables,

with two parameters, polarity and size:

Fuzzification converts a crisp sensor reading (value of state variable),

x = xo, into the membership to these linguistic variables:

Control Rule Base

• General form of rule:

IF x1 is A1 AND • • • AND xM is AM, Antecedent THEN y1 is B1 AND • • • AND yN is BN Consequent

• The rule base form a fuzzy partition of the

multi-dimensional space

A3 A2

A1 B1

Fuzzy Inference

• When input variables are fuzzified, each rule in the rule base

will be used to determine its degree of activation using fuzzy logic operations, e.g union, intersection, and complement

• For rules which has a non-zero activation value, the output fuzzy variables will be combined (fuzzy union) yielding a resultant fuzzy set

0 0.1

C

z is 2 (0.2)

0 0.5 1

C

z is 3 (0.5)

Z Z

• The resultant fuzzy set needs to be quantified into a crisp

signal as the control output

• For example, a certain voltage should be determined to drive

a motor

Fuzzy Set Crisp Value

Example: automotive driving

Design a control to keep distance between cars.

distance Current speed

Adjust

Speed?

Example: automotive driving

Keep distance between cars:

Step 1: construction of control rules:

Rule 1: IF distance between cars is short

AND speed is slow

THEN hold the gas pedal steady (maintain the speed).

Rule 2: IF distance between cars is short

AND speed is fast

THEN step on the brake (reduce the speed)

Rule 3: IF distance between cars is long

AND speed is slow

THEN step on the gas pedal (increase the speed)

Rule 4: IF distance between cars is long

AND speed is fast

THEN hold the gas pedal steady (maintain the speed)

Example: automotive driving

Step 2: Define fuzzy sets and rewrite the control rules accordingly

Fuzzy sets:

• Distance X(0-40m): A1 - “short”; A2 - “long”

• Speed Y(0-100km/h): B1 - “slow”; B2 - “fast”

• Driving Z(-20-20km/h2): C1 - “maintain”; C2 - “reduce”; C3:

Example: automotive driving

Step 3: fuzzy inference for control

1) Fuzzification: based on the sensor inputs, measure the

adaptability of the premise of rules

2) Fuzzy inference: from the adaptability, infer the conclusion

of each rule

3) Defuzzification: aggregate the individual conclusions to

obtain the overall conclusion

Fuzzy controller

Inference engine

Rule

Defuzzi-fication

Control Signals

fication Sensor

Fuzzi-Signals

Example: automotive driving

Reasoning process (Mamdani’s Method):

Example: automotive driving

Defuzzification:

The fuzzy control set has to be converted into a definite

value

Take the centre of gravity of the fuzzy set for conclusion

Action: “Reduce speed somewhat”- between “keep speed”

and “reduce speed”

0

( ) ( )

c

c

z zdz z

z dz

µ µ

= ∫

Centre of gravity: -4.45 km/h 2

Fuzzy PI Control

• Introduction

• PI-like Fuzzy Logic Controller

• PI Fuzzy Logic Controller

• Conclusions

• Input of PID controller:

• Output of PID controller:

• PID FLC:

– The rules base is about

– The gains are tuned on-line with fuzzy reasoning.– This requires more experience with the system

) ,

, ( ),

, ( ),

K 1

z

z K

K ( )

k (

− +

=

Fuzzy PID control: Introduction

e e

e , ∆ , ∆2

e e

e , ∆ , ∆2

Fuzzy Rules for Computation of

r +

- ∆e

Ge G

e

e

∆

Fuzzy PI control

PI Fuzzy Logic Control

• PI controller:

• The proportional gain and integral time constant are adjusted on-line by fuzzy reasoning

Fuzzy reasoning

e

e

∆

Fuzzy reasoning PI

z T

1 1 ( K 1 z

z K K ) z (

H

i p

i p

− +

=

− +

=

p

PI Fuzzy Logic Control

• Fuzzy rules for

\ ) k (

\ ) k (

e ∆

CS 561, Sessions 22-23 25

Fuzzy Rules

• Example: “If our distance to the car in front is small, and the distance is decreasing slowly, then decelerate quite hard”

– Fuzzy variables in blue

– Fuzzy sets in red

• QUESTION: Given the distance and the change in the distance,

what acceleration should we select?

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Fuzzification: Set Definitions

distance

v small small perfect big v big

Delta (distance change)

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Fuzzification: Instance

• Distance could be considered small or perfect

• Delta could be stable or growing

Distance is small, then you slow down.

Question: What is the weight to slow down?

acceleration

slow present fast fastest brake

CS 561, Sessions 22-23 30

Rule Evaluation

distance

small o.55

Clipping approach (others are possible):

Clip the fuzzy set for “slow” (the consequent) at the height given by our belief in the

premises (0.55)

We will then consider the clipped AREA (orange) when making our final decision

Rationale: if belief in premises is low, clipped area will be very small

But if belief is high it will be close to the whole unclipped area

acceleration

slow

How do we make a final decision? From each rule we have

Obtained a clipped area But in the end we want a single

Number output: our desired acceleration

In the rule aggregation step, we merge all clipped areas into

One (taking the union).

Intuition: rules for which we had a strong belief that their premises were satisfied

Will tend to “pull” that merged area towards their own central value, since their

Clipped areas will be large

In the last step, defuzzification, we return as our acceleration

Value the x coordinate of the center of mass of the merged area

-2.3m/s^2

CS 561, Sessions 22-23 37

Rule Aggregation: Another case

• Convert our belief into action

– For each rule, clip action fuzzy set by belief in rule

CS 561, Sessions 22-23 38

Rule Aggregation: Another case

• Convert our belief into action

– For each rule, clip action fuzzy set by belief in rule

acceleration

present slow fast

CS 561, Sessions 22-23 39

Matching for Example

• Relevant rules are:

– If distance is small and delta is growing, maintain speed

– If distance is small and delta is stable, slow down

– If distance is perfect and delta is growing, speed up

– If distance is perfect and delta is stable, maintain speed

CS 561, Sessions 22-23 40

Matching for Example

• For first rule, distance is small has 0.75 truth, and delta is

growing has 0.3 truth

– So the truth of the and is 0.3

• Other rule strengths are 0.6, 0.1 and 0.1

slow present fast fastest brake

Instead of clipping, another approach is to scale the fuzzy set

By the belief in the premises

Clipping:

Scaling:

2 Apply fuzzy operator:

If there are multiple parts, apply logical operators to

determine the degree of support for the rule

CS 561, Sessions 22-23 46

Summary: If-Then rules

3 Apply implication method:

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

How do we then combine several rules?

CS 561, Sessions 22-23 47

Multiple rules

• We aggregate the outputs into a single fuzzy

set which combines their decisions

• The input to aggregation is the list of truncated fuzzy sets and the output is a single fuzzy set for each variable.

• Aggregation rules: max, sum, etc.

• As long as it is commutative then the order of rule exec is irrelevant.

CS 561, Sessions 22-23 48

Defuzzify the output

• Take a fuzzy set and produce a single crisp

number that represents the set.

• Practical when making a decision, taking an