fuzzy-dr-ny
TRANSCRIPT
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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
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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
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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
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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
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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 ])
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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Architecture of the generic fuzzy controlsystem:
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Contd.
( ) ( ) nu unu t u t T d
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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
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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
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Example-1:- Stabilization of InvertedPendulum System
Obtain of Control Law (i) Modern Control approach (ii)
Fuzzy logic Approach
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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
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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
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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
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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
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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
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Simulation Results ( based on pole- placement)
Either obtain the systemresponse through Matlab program or throughSimulink.
PendulumVelocityPendulum Angle(rad.) (rad./sec)
Pole Placement Pole Placement
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Cart Position Cart Velocity (m/sec.)
Applied force to cart (N)
(m)
Pole PlacementPole Placement
Pole Placement
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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
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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.
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Contd.
For the given rulebase the fuzzy and inferenceprocess is used.
>> fuzzy
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Solution of Exam.-1 Using Fuzzy LogicController
Pendulum Angle Pendulum Velocity
(rad.) (rad./sec.)FuzzyController
FuzzyController
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Cart Position (m) Cart Velocity
Force Applied to Cart (N)
time
(m/sec.)
FuzzyController
FuzzyController
FuzzyController
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Control Surface for 11 Rule FLC
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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]) .
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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.
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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.
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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.
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Contd.
1 A 2 A
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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Th 1 (C ti Ti F
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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
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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
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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.
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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
.
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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
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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
.
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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
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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
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