Fuzzy-Dr-NY

8/10/2019 Fuzzy-Dr-NY

1/71

Fuzzy Set Theory

Classical Set:- A classical set A is a collectionof elements or objects of any kind. Two methods

describing sets:(i) Listing method ( )(ii) Membership rule

{1,2,3,4,5} A

| ( ) A x p x is a predicate stating x has property pHeight ofa person

Is the predicatethat height 1.8 x m


2/71

Contd.

Fuzzy Set:- A fuzzy set is a set with a smooth(unsharp) boundary. It is graded membershipover the interval [ 0 , 1 ] . A fuzzy set F isdefined as ordered pairs as

, ( ) | , ( ) [ 0, 1]

( ) , , ( ) [ 0, 1]

: 1, 2,3,............20 , .

0.9 0.6 0.31 11 2 3 4 50.1 0.3 0.5 0.21

1 2 5 8 9

F F

F F

small

Medium

F x x x U x

OR

x F x U x x

Example U is a Universe of discourse


3/71

Normal & Subnormal Fuzzy Set

Normal Fuzzy Set Subnormal Fuzzy Set

10 20 10 20

Subnormal Fuzzy Set

1

0 x

( ) A x

( ) A x

0

0.5

( ) A x

x

0.5

1

0

Clipping

0

0.4

( ) A x

Scaling

Generated during rule basedreasoning process

1

x

x


4/71

Convex Fuzzy Set

Convex Fuzzy Set :- A fuzzy set A is convexfuzzy if the membership values are strictlymonotonically increasing then strictly

monotonically decreasing with increasing values ofelements in the universe.

( (1 ) ) min ( ( ), ( )) A A A y z y z

0 y z x

( ) A x

Convex Fuzzy Set

1


5/71

Types of Membership Functions

Triangular MembershipFunction

This membership function is

very often used in theapplication of Fuzzy logiccontroller.

Example: x= -3:0.1: 6; y=trimf(x,[-1 1 4]); plot (x, y); xlabel(trimf, p=[-1 1 4 ])

Trapezoidal Membership

Due to simple formulas andcomputational efficiency this mf very popular in control problem .

Example:x= -3:0.1: 6; y=trapmf(x,[-1 1 4 5]); plot (x, y); xlabel(trapmf, p=[-1 1 4 5 ])


6/71

Contd.

Gaussian MembershipFunction

Example: x = -10:0.1:15;

y= gaussmf (x, [2 5]); plot (x, y); xlabel ('gaussmf, p =[2 5]')

Note: Gaussian membershipfunction achievesmoothness and it issymmetric about thecenter.

2

2

( )( ) exp

2 A x m

x

Note: &

.

m denote the the center

and width of the function


7/71

Contd.

Bell ShapedMembership Function

Adjust c & a to vary center and widthof the function & use

b to control theslopes at the crossingpoints.X = 0:.1:200;

Y = gbellmf (x,[20 4120]);Plot (x, y)

xlabel ('gbellmf, P =[204 120]')

21( )

1 A b x

x ca


8/71

Contd.

Sigmoid Function X = 0:.1:20; y = sigmf (x,[1 10]);plot (x, y)xlabel ('sigmf, P =[1 10]')

( )1( )

1 A a x c x

e

Shape of the curve dependson (a, c) parameters.

* A fuzzy set A whosesupport is a single point

in U with iscalled a fuzzy Singleton

( ) 1 A x

0 x1

( ) A x


9/71

Basic Operation on Fuzzy Sets

Let be two fuzzy sets in U withmembership function respectively.

A and B( ) , ( ) A B x x

, ( ) , ( ) [ 0, 1]

, ( ) , ( ) [ 0, 1]

( ) : ( ) max[ ( ) , ( ) ]( ) ( )

A A

B B

A B A B

A B

A x x x

B x x x

Union Disjunction x x x x x


10/71

Contd.sec ( ): ( ) min[ ( ) , ( ) ]

( ) ( )

: ( ) 1 ( )

* , ,int sec ,

,

A B A B

A B

A A

Inter tion Conjunction x x x x x

Complement x x

A fuzzy conjuction also called a t norm represrents a generalized er tion operator while a fuzzy

disconjuction also calle

, ( )

.

d a t conorm s norm represrents

a generalized union operator

Most Popular


11/71

Contd.

t-norm:(i) Fuzzy Standard Intersection

(ii) Algebraic Product

(iii) Bounded Product

( ( ) , ( )) min[ ( ) , ( ) ] A B A B A B x y x y

( ( ) , ( )) ( ) ( ) A B A B A B x y x y

( ( ), ( )) max 0, ( ) ( ) 1 A B A B A B x y x y


12/71

Contd.

t-conorm (s-norm)(i) Fuzzy Standard Union

(ii) Algebraic Sum

(iii) Bounded Sum

( ( ) , ( )) max[ ( ) , ( ) ] A B A B A B x y x y

( ( ) , ( )) ( ) ( ) ( ) ( ) A B A B A B A B x y x y x y

( ( ), ( )) min 1, ( ) ( ) A B A B A B x y x y


13/71

Fuzzy System

A fuzzy inference system (or fuzzy system) basicallyconsists of a formulation of the mapping from a givenfuzzy input set to an fuzzy output set using fuzzylogic.

Step-1: Identify ranges of the inputs & outputs.Create fuzzification (fuzzy sets) of each input andoutput variable.Step-2: Application of fuzzy operator (AND, OR,

NOT) in the IF (antecedent) part of the rule.

Fuzzyoutput

Fuzzyinput

Fuzzy system


14/71

Contd.

Step-3: Implication from antecedent toconsequent (conclusion) is Then part of the

rule [ (i) crisp consequent (If < antecedent >Then y = a (ii) fuzzy consequent ( If Then y = A (A is Fuzzy Set)Mamdani model ) (iii) functional consequent

( If ThenTakagi-Sugeno model ) ].

0 1 21

( , ,... )n

i i ni

y a a x f x x x

If antecedent Then y aIf antecedent Then y a


15/71

Contd.

Step-4: Aggregation of the consequent across therules.

Step-5: Defuzzification (it is a mapping from thespace of a fuzzy set to a space of crisp values.

Step-6: Implement the fuzzy system, test it, andmodify fuzzy rules if necessary.

If antecedent Then y a If antecedent Then y a


16/71

Contd.

Architecture of Fuzzy Controller

Fuzzy Logic Controller

Data Base

RuleBase

Inference Engine

Defuzzifica-tion

Fuzzifica-tion

Dynamic Filter

Dynamic Filter Plant

( )u t

( )u t f

Fuzzy Logic Controller Scheme


17/71

Design Tips for AntecedentMembership Function

(i) Each membership function Overlaps Onlywith the closest neighboring membershipfunction.( )

For any possible input data, its membershipvalues in all relevant fuzzy sets should sumequal to 1(or nearly so).( )

( ) , 1, 1i j A A null set j i i i

( ) 1i A

i

x


18/71

Control Design Process

(i) Selection of Control Design Time Domain, PIDto LQ optimal

Frequency Domain,Classical Loop shapingto . H

(ii) Technical Design Objectives Steady stateerror, rise time, settling time and Maximum overshoot; in

case of optimal control design performance weights play an important role to modify the system response.

(iii) Development of Mathematical Model from the

Physical Laws of the Systems.


19/71


20/71

Architecture of the generic fuzzy controlsystem:


21/71

Contd.

( ) ( ) nu unu t u t T d


22/71

Fuzzy Multi-Term Controllers &Structures

Fuzzy-PI Rule :" " " " " " If e is A and e is B Then u is D

ee u uFuzzy RuleBase

+ _

1 z

uuFuzzy-PI

Fuzzy-PD Rule:

" " " " " " If e is A and e is B Then u is D

uFuzzy Rule Basee e


23/71

Contd.

Fuzzy-PID Rule:

Fuzzy Rule

Base

e

e u2e

Fuzzy-PID

u

2" " " " " " " "If e is A and e is B and e is c Then u is D


24/71

Example-1:- Stabilization of InvertedPendulum System

Obtain of Control Law (i) Modern Control approach (ii)

Fuzzy logic Approach


25/71

Mathematical Modeling of cart & pendulum System

Pendulum Dynamics:

2

2

2

sinsin cos

( ) 4 cos3

:

sin cos( )

F ml g

M m

t ml

M m

Cart Dynamics

F ml x t M m

To stabilize in upright position & this in turnimplies that can be neglected in the above expressions.

00 ( )nearly zero 2


26/71

Contd.

Linearized model

( )4

3

F g

M mt

ml

M m

Pendulum Dynamics

( )

Cart Dynamics

F ml x t

M m

Note: One can also linearize the above system by Taylorseries expansion (retaining only first order terms in theexpression.

T l d d d li i d d l


27/71

Two coupled second-order linearized models arethen transformed into state-space form (assuming,and position of the pendulum angle & cartposition are measurable & Syst. is controllable)

1 2 3 4; ; ; x x x x x x

1 x

1 1 1

2 21 2 2 2

3 3 3

4 41 4 4 4

1 2 3 4

0 1 0 0 0

0 0 0 ( ); 1 0 0 00 0 0 1 0

0 0 0

12( ) ( ) ( ); ( )3

4

x x x

x a x b xu t y x x x

x a x b x

x

x X t A X t bu t where u t k k k k

x

x

3 x


28/71


29/71

Regulator Design Using Pole-Placement Technique

Matlab Program to obtain controller gain K

% Stabilization of inverted pendulum using pole- placement techniqueA=[0 1 0 0; 9.81 0 0 0; 0 0 0 1;-3.27 0 0 0];B=[0;-0.667;0;0.889];C=[1 0 0 0];

d=[0];%Check for controllability%Rank of controllabilty matrix(M)%=

rank_of_M=rank(ctrb(A,B))

2 1[ ......... ]n Rank B AB A B A B


30/71

Contd.

%enter the desired characteristic equationchareqn=[1 12 72 192 256]%Calculate desired closed loop polesdesired_poles=roots(chareqn)%Calculate feedback gain matrix ' K ' using

Ackermann's formula (K=[0 0 .1] )

K=acker(A,B,desired_poles)Results: rank_of_M=4; chareqn= 1 12 72 192 256; -4.0

+j 4.0; -4 j 4.0; -2 +j 2 & -2 j 2.

K=[ - 174.82 -57.12 -39.14 -29.35 ]

1 ( ) M A


31/71

Simulation Results ( based on pole- placement)

Either obtain the systemresponse through Matlab program or throughSimulink.

PendulumVelocityPendulum Angle(rad.) (rad./sec)

Pole Placement Pole Placement


32/71

Cart Position Cart Velocity (m/sec.)

Applied force to cart (N)

(m)

Pole PlacementPole Placement

Pole Placement


33/71

Solution of Exam.-1 Using Fuzzy LogicController

Two Inputs Single Output (Controller)

FuzzyController

( )u t

(i) For each input & output three linguistic terms areconsidered as NB, Z & PB.

0

NB( )

Z PB

-0.1 0.1

( ) Z PB

-0.5 0.5

NB

0


34/71


35/71

Contd.

R5: If is Z and Z Then is ZR6: If is Z and NB Then is NBR7: If is NB and PB Then is ZR8: If is NB and Z Then is NBR9: If is NB and NB Then is NBR10: If PB Then is PBR11: If NB Then is NB

f

f

f

f

f

f

f

How to Implement Rule Fuzzy Controller using Fuzzy InferenceSystem (FIS) Editor, Membership editor, Fuzzy rule editor, Rule

viewer & Surface viewer.


36/71

Contd.

For the given rulebase the fuzzy and inferenceprocess is used.

>> fuzzy


37/71


38/71


39/71


40/71

Solution of Exam.-1 Using Fuzzy LogicController

Pendulum Angle Pendulum Velocity

(rad.) (rad./sec.)FuzzyController

FuzzyController


41/71

Cart Position (m) Cart Velocity

Force Applied to Cart (N)

time

(m/sec.)

FuzzyController

FuzzyController

FuzzyController


42/71

Control Surface for 11 Rule FLC


43/71

Summarize the Steps Involve inSimulink Implementation ofFuzzy logic Control Problem

>> fuzzy (this brings up the main menu screen )

Double-clicking on the input/output iconbrings up the membership editor .Go to edit menu & then click Add MFs . Youcan select no. of MFs according to yourrequirement. Next adjust the range of variableand select parameters of each membership

function( say, trimf(x,[1,3,5]) .


44/71

After adjusting all inputs & outputs membershipfunctions, Go to the main menu & double clickingon the center box (Mamdani) will bring up the FISrule editor . Write all rules there only.

When editing is complete, go to file menu in therule editor & save to disk as a file namependulum .fis

The inverted pendulum fuzzy controller problem cannow be implemented in SIMULINK.

The fuzzy logic icon is obtained by opening the fuzzylogic tool box within the simulink Library Browser, anddragging it across.


45/71

Note, the properties of the fuzzy logic controller

have not been defined. So, at the MATLABprompt, type>> fismat=readfis (this will allow you to select

from a directory a filename pendulum .fis which is ready stored in a disk.) This means that the fuzzy logic controllerparameters have been placed in the work spaceunder fismat, and one can now proceed forsimulation.


46/71

Remarks

Note: Design of conventional Controller is basedon mathematical model of a plant. FuzzyController is basically an adaptive and non-linearcontrol which gives robust performance for alinear or non-linear plant with parametricuncertainty and moreover, the controller does

not require any knowledge of mathematicalmodel of dynamic system.


47/71


48/71

Contd.

1 A 2 A


49/71

Contd.

1 1 1 1 1 1( ) 2 2 If x is A Then y f x x Rule-1:

Rule-2: 1 2 1 2 1 1( ) 1 4 If x is A Then y f x x

1 1 1 1 2 1 2 1 1 1 1 11 1 2 2

( ) ( ) ( ) ( ) 1 1 12 2 1 4

( ) ( ) 2 2 2 A A

A A

x f x x f x y x x x x

x x

Defuzzification (weighted Average):

21 1 1

1 31 12 2 x x if x

12 2 1 x if x

11 4 1 x if x


50/71


51/71

Contd.

:thi Rule1 1 2 2

( ) ( ) ( )

( ) ( )

...

( ) ( ) ( )

( 1) ( ) ( ); 1, 2,...

i i n in

i i i

i i

IF x is A And x is A And x is A

THEN X t A X t B U t

Or X k F X k G U k i L

A non-linear interpolation between linear systems is

presented as (Algebraic Product is used as inference process)

' ' L

( ) ( )

1

1

1

( ( ))[ ( ) ( )]( ) , ( ( )) ( )

( ( ))ij

Li i

i ni

i A j L j

i

i

X t A X t B U t X t where X t x

X t


52/71

Contd.( ) ( )

1 21 1

( ) ( ( )) ( ) ( ( )) ( ), .... L L

T i ii i L

i i

X t A X t X t B X t U t where

( ) ( ) ( ) X t A X t B U t , , 1, 2,...... , Note for i L wehave

1 21

0, 1, .... 0, 1 L T

i i Li

and

A system model given by ( ) ( ) ( ) X t A X t B U t

is also referred to as in the literature as the polytopic system.

f ll


53/71

Design of Fuzzy Logic Controller(T-S Fuzzy Controller):

It is assumed that is measurable and thecontroller is another T-S fuzzy system with Lrules (same number of rules as was used todescribe the plant) of the form

( ) X t

1 1 2 2

( ) ( )

( ) ( )

...

( )

( 1) ( ); 1, 2,... ,

i i n in

i i

i i

IF x is A And x is A And x is A

THEN U K X t

Or U k K X k i L and K .is the Control Matrix

Note : The designed fuzzy controller shares the same fuzzysets with the fuzzy model in the premise parts.


54/71

Contd.

In this case,( )

1

1

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

( ( ))

L j j

j j L j

j j

X t U t K X t X t where

X t

Using the above control law in ( ) ( ) ( ) X t A X t B U t

we get, the closed loop system as

( ) ( )

1 1( ) ( ( )) ( ) ( ( ))

L L

i ii i

i i X t A X t X t B X t

( )

1( ( )) ( )

L

j j

j K X t X t

which is in the form of ( ) ( ( )). X t f X t Our problem islog

.

now to study the stability of the fuzzy ic based

control system

S bili A l i B d


55/71

Stability Analysis Based onLyapunovFunction

Let is a Lyapunov function withP>0. To check the Fuzzy closed-loop system isglobally asymptotically stable,

( ( )) ( ) ( )T V X t X t PX t

( ( )) 0, ( )V X t X t

( ( )) ( ) ( ) ( ) ( )T T V X t X t P X t X t P X t

1 1

1 1

1

1 1

1 1

1

( ( )) ( ( ))( ) ( ( )) ( ( )) ( )

( ( ))

( ( )) ( ( ))( ) ( ( )) ( ( ))

( ( ))

L Li

i j L LT i ji J

i j Li j

j J

L Li

i j L LT i ji J

i j Li j

j J

A X t X t X t P B X t K X t X t

X t

A X t X t X t B X t K X t

X t

( )

T

PX t


56/71

Contd.

,

( ) ( ( )) ( ( )) ( )T T i i j i i j

i ji j

X t X t X t A B K P P A B K X t

After simplification we get,

, ( ) ( ) , 0 ( ( )) ( ( )) 1T T i i j i i j

i ji j

X t A B K P P A B K X t where X t X t

For asymptotic stability the following condition must be satisfied,

0, 1, 2, ...... & 1, 2, ....T i i j i i j

A B K P P A B K i L j L Note: We need to find out common matrix such that the Lyapunovequations are negative definite. Linear matrix inequality (LMI)methods can be used to find P if it exists. A brief outline of LMI is

given below.

P


57/71

LMI- Problems

Definition: A LMI is linear matrix inequalityexpression of the form

where variables and symmetric matricesare given. The feasibility problem is to determinethe variables so that the aboveInequality holds. Multiple LMIs

can be expressed as a single LMI

00

( ) 0m

i ii

F X F x F

i x i F

.

, 1, 2, ...i x i m( ) 0, 1, 2,...... ,i F i p

(1) (2) ( )

..... 0 . p

diag F F F


58/71

Application of LMI Problems

Fuzzy Control Problem ( Discrete System)Let us consider a discrete-time system (for one rule

only) is(quadratically) stable if such that

( 1) ( ) ( ) ( ) ( ) X k AX k BU k for U k FX k

0 P

0T A BF P A BF P The control problem is to find such that the closedloop system is stable. This stabilization problem can berecast as an LMI problem in the following way.

' ' F

1 1 1 0T P A BF P A BF P P


59/71

Th 1 (C ti Ti F


60/71

Theorem:-1 (Continuous Time Fuzzy TS System)

If for some the conditions0 P

0, 1, 2,......T i i i i i i A B K P P A B K i L

0,T ij ijG P PG i j L and

,T i i j j j i

ijG A B K P P A B K i j L Where,

are satisfied, then the closed-loop system modeled by ( ) ( )

1 1

( ) ( ( )) ( ) ( ( )) L L

i ii i

i i

X t A X t X t B X t

( )1

( ( )) ( ) L

j j

j

K X t X t

is asymptotically stable.

D i f t bl f t ll f


61/71

Design of stable fuzzy controller fordiscrete time system:

TS Model:-1

( 1) ( ( )) ( ) ( ) L

i ii

i

X k X k A X k B U k ( )

1

( ) ( ( )) ( ) L

ii

j

U k X k F X k TS Controller:-

1 1

( 1) ( ( )) ( ( )) ( ) L L

i i ji j

i j

X k X k X k A B F X k

Combining the above two equations we get,

1 1

( ( )) ( ( )) ( ) 2 ( ( )) ( ( )) ( ) L L L

i i ii i i i ij

i i j i

X k X k A B F X k X k X k G X k

,2

i i j j j i

ij

A B F A B F G

Where, i j such that 0i j

Th 2( ) Th ilib i f f t l


62/71

Theorem:2(a) The equilibrium of a fuzzy controlsystem is globally asymptotically stable if

a common s.t the following two conditions

are satisfied. 0 P

0 , 1, 2,....T i i i i i i A B F P A B F P i L

0, , . 0 0T ij ij i j i jG PG P i j L s t or and

2

i i j j j i

ij

A B F A B F G

Where,

The above two nonlinear inequalities can now be convertedto LMIs using the Schur complement to check the stabilityof the overall fuzzy system.


63/71

More specifically, find , 1, 2,......i Z and M i L satisfying the resulting LMIs

1 0 Z P

0 , 1, 2,....

T i i i

i i i

Z A Z B M for i L

A Z B M Z

0;T

ij

ij

Z G Z for i j L

G Z Z

The feedback gain

can be obtained as i

F and P

1 Z P

1 P Z and 1.i i F M Z

.


64/71

Theorem-2 (b)

If for some symmetric positive definite thefollowing conditions Z

0 , 1, 2,....

T i i i

i i i

Z A Z B M for i L

A Z B M Z

and

0;

T

ij

ij

Z G Z for i j L

G Z Z

are satisfied, then the closed-loop system (see below) is stable

1 1

( 1) ( ( )) ( ( )) ( ) 2 ( ( )) ( ( )) ( ) L L L

i i ii i i i ij

i i j i

X k X k X k A B F X k X k X k G X k

F S stem Stabilit Via Inter al


65/71

Fuzzy System Stability Via IntervalMatrix Method:

Discrete time system: 0( 1) ( ) ( ) , ( ) X k A G k X k X k X

Stable matrixUnknown Time-Varying Perturbed Matrix

( ) mG k G for all k where, & inequality holdselement-wise.

Theorem: The time-varying discrete-time system describedabove asymptotically stable if

1 11 : max .....m n A G note A lim ( ) 0k X k

.


66/71


67/71


68/71


69/71

Nonlinear stability theory should be explored tostudy the stability of Mamdani type fuzzy

control system. Describing Function method isattractive because it is simple and gives betterinsight of effects which the fuzzy element can

have on the stability of the closed loop system.

The stability of the interval matrix can be

checked using the Lemma.

Stability Analysis of Mamdani Type Fuzzy

Control System


70/71

Different fuzzification techniques

Center of Gravity Method or Centroid method. Weighted Average MethodMean of Maximum method (MOM)Center of Sums (COS)Center of Largest Area Method (for non-convex)

Fast ( or Last) Maximum MethodMean-Max Method


71/71

Documents

Fuzzy-Dr-NY