Stability of Fractional Order Systems Advisor : Dr. N. Pariz Consultant : Dr. A. Karimpour Presenter...

Stability of Fractional Order Systems

Advisor : Dr. N. Pariz

Consultant : Dr. A. Karimpour

Presenter : H. Malek

Msc. Student of Control Engineering

Ferdowsi University of Mashhad

Topics:Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order EquationsSolution of Fractional Order Equations Concepts of Fractional Order OperatorsConcepts of Fractional Order Operators Stability of Linear Fractional Order SystemsStability of Linear Fractional Order Systems Stability of Nonlinear Fractional Order SystemsStability of Nonlinear Fractional Order Systems

Stability of Fractional Order SystemsStability of Fractional Order Systems

Definition of Gamma Function:Definition of Gamma Function:

Fundamentals of F.C.Fundamentals of F.C.

1

0

; ( ) x tx x t e dt¥

- -" Î G =ò¡

(1) 0

( 1) ( )x x x

! ( 1) ,x x x=G + Î ¡

( 1)! ( 1) ! ( 1) 1!: ;

1! 0

n n n n nì + = + ´ = + ´ ´ ´ïï® íï =ïî

L¥ ¥

Definition of Factorial Function:Definition of Factorial Function:

Fundamentals of F.C.Fundamentals of F.C.

Definition of Mittag-Lefler Function:Definition of Mittag-Lefler Function:

G. M. Mittag-LeflerG. M. Mittag-Lefler

,0

( ) , , 0( )

k

k

xE x

k

0

( ) , 0( 1)

k

k

xE x

k

2 2

1,1

2 22,1 2,2

1 2,1

( )

sinh( )( ) cosh( ) , ( )

2( )

z

z t dt

z

E z e

zE z z E z

z

E z e e

Topics:Topics: Fundamentals of Fractional CalculusFundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order EquationsSolution of Fractional Order Equations Concepts of Fractional Order OperatorsConcepts of Fractional Order Operators Stability of Linear Fractional Order SystemsStability of Linear Fractional Order Systems Stability of Nonlinear Fractional Order SystemsStability of Nonlinear Fractional Order Systems


Def. of Fractional OperatorsDef. of Fractional Operators Riemann-Liouville Definition:Riemann-Liouville Definition:

11( ) ( ) ( ) , , 0

( )

t

a t

a

I f t t f d ta at t t aa

- += - ÎG ò ¡ f

10

1 ( )( ) , 1

( ) ( )

n t

a t n n

d fD f t d n n

dt n t

( )0

( ) ( )t

I f t f dx x=ò

( ) ( )2

0 0 0

( ) ( ) ( )t t s

I f t I f d f d dx x h h x= =ò òò

( ) ( ) 1

0

1( ) ( ) , , 0

( 1)!

tnnI f t t f d n t

nx x x-= - Î >

- ò ¥

M

If na +Î ® Î¡ ¥ Then !G®

( ) ( )0 ( ) ( )n n n n

t tD f t D D f t D Ia a a- - -= =Insomuch

Then

Def. of Fractional OperatorsDef. of Fractional Operators Grunwald-Letnikov Definition:Grunwald-Letnikov Definition:

[ ]

0 0

1 ( 1)( ) lim ( 1) ( ) ,

! 1

lm

h m

t aD f t f t mh l

h m m h

[ ]

0 0

( )( ) ( ) lim . ( )

!

l

h m

mI f t D f t h f t mh

m

( )0

( ) lim ( ) ( )h

f t f t h f t h®

é ù= + -ë û&

( ) ( )1 2

0 0( ) lim ( ) ( ) lim ( 2 ) 2 ( ) ( )

h h h

h hf t f t h f t h f t h f t h f t h

= =

® ®

é ù é ù= + - = + - + +ê ú ë ûë û&& & &

M( )( )

0 0

( ) lim 1 ( ) ,n

mn n

h m

nf t h f t mh n

m® =

æ ö÷ç ÷= - - Îç ÷ç ÷çè øå ¥

If na +Î ® Î¡ ¥ Then !G®

0 0( ) ( )t tI f t D f ta a-=Insomuch

Then

Caputo Definition:Caputo Definition:

Def. of Fractional OperatorsDef. of Fractional Operators

Like Riemann-Liouville Definition except :

( ) ( )0 ( ) ( )n n n n

t tD f t D D f t D Ia a a- -= =Insomuch

Then

( )1 ( )0

0

1( ) ( ) , 1 ,

( )

tn n

tD f t t f d n n nn

aa t t t aa

+ -= - - £ < ÎG - ò ¥

Miller-Ross Definition:Miller-Ross Definition:

[ ]1 21 2( ) ( ) , , , ,m

mD f t D D D f ta a aa a a a a= =L L

Def. of Fractional OperatorsDef. of Fractional Operators An Example of Fractional DerivativeAn Example of Fractional Derivative

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

( 1) ( 1)m m m m

t tD t D D t D t tm

a m a m m a m am mm a m a

- - + - -é ùG + G +é ù ê ú= = =ê úë û ê úG + - + G - +ë û

( )f t c cte= =If Then 0.5 ( )t

cD f t

tp=

( )f t t=If

Then 0.5 0.5(2)( )

(1.5)tD f t tG

=G

( )f t t=

( ) 1f t =&

0.5 ( )tD f t

Topics:Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional OperatorsFractional Operators Solution of Fractional Order EquationsSolution of Fractional Order Equations Concepts of Fractional Order OperatorsConcepts of Fractional Order Operators Stability of Linear Fractional Order SystemsStability of Linear Fractional Order Systems Stability of Nonlinear Fractional Order SystemsStability of Nonlinear Fractional Order Systems


Laplace Transformation of Fractional Derivatives Laplace Transformation of Fractional Derivatives

Fractional Order EquationsFractional Order Equations

{ }1

1

00

( ) . ( ) ( ) , 1m

k k

tk

L D f t s F s s D f t m m ma a a a-

- -

==

é ù= - - Îê úë ûå p p ¥

{ }1

1 ( )

0

( ) . ( ) (0) , 1m

k k

k

L D f t s F s s f m m ma a a a-

- -

=

= - - Îå p p ¥

( ) . ( )L D f t s F s

1- According to Reimann Def.1- According to Reimann Def.

2- According to the Caputo Def.2- According to the Caputo Def.

3-Grunwald Def.3-Grunwald Def.

Laplace Transformation of Mittag-Lefler FunctionLaplace Transformation of Mittag-Lefler Function

1, , 1

0

!( ) ( )

( )k st k k

k

k sL E at e t E at

s a

Fractional Order EquationsFractional Order Equations

Solutions of Linear Fractional Order Equations :

1

20

1

20

0

( ) ( ) 0 , 0

( )

t

t

t

D f t af t t

D f t c-

=

ìïï + = >ïïïí é ùï ê úï =ï ê úï ë ûïî

1 1

2 20

0

( ) ( ) ( ) 0t

t

s F s D f t aF s-

=

é ùê ú- + =ê úë û

1

2

( )c

F s

s a

=+

1 1

2 21 1

,2 2

( )f t ct E at- æ ö÷ç ÷= -ç ÷ç ÷çè ø

Example :

Solutions of Nonlinear Fractional Order Equations :

Adomian Method

Diethelm Method

Concepts of F.O. OperatorsConcepts of F.O. OperatorsGeometrical Concept of Fractional Integral :

( ) 1

0

1( ) ( )( ) 0

( )

t

tI f t f t d ta at t ta

-= - ³G ò

1( ) ( )

( 1)tg t ta at ta

é ù= - -ê úë ûG +

( )

0( ) ( ) ( )

t

tI f t f dga t t=ò

( )( )I f t

( , )g t t

( )( )I f ta

Concepts of F.O. OperatorsConcepts of F.O. OperatorsPhysical Concept of Fractional Integral :

1 2 3 4 5 6 7 812 3 4 5 6 7

The Fractional order Integral of velocity of a vehicle that its real time and its local time are not the same is the actual distance that it move.

Cosmic Time Homogenous Time

10 0( ) ( ) ( ) ( )O O t t

d dv t d t I v t D v t

dt dta a-= = =

The Fractional order Derivative of the local velocity of a vehicle that its real time and its local time are not the same is the actual velocity that it has.

0

( ) ( )t

Nd t v dt t=òDistance N=

( )T g t=

0

0 0

( ) ( ) ( ) ( ) ( )t t

O td t v dT v dg I v tat t t= = =ò ò =Distance O

Relation between Real time and Local time

Application of F.O. OperatorsApplication of F.O. Operators

Finding a curve such that the time it takes for P to go towards the origin is independent to the start point.

Abel Problem:

( )0.52

( )0.5

gT D f y-=

G

Water Passage Problem:

Finding a proper shape for water passage of reservoir such that the velocity of water flow be a function of height of passage.

3 22 ( )Q gD f hp -=

Stability of L.F.O. SystemsStability of L.F.O. Systems

General Form of Linear Fractional Order Equations :( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

D x t A t x t B t u t

y t C t x t D t u t

aìï = +ïíï = +ïî01

01

1 0

1 0

( )( )

( )

m m

n n

m m

n n

b s b s b sY sG s

U s a s a s a s

1( )G s C s I A B D

, ,k k k ka b a a + += = Î Î¡ ¢1

, qq

ZConmesurate Rational

Stability Analysis of Linear Fractional Order Equations :1-Direct

Method : 1 11 1( ) ( ) ( ) (0)x t L X s L s I A BU s s I A x

11( ) ( )t L s I A E At

0

( ) ( ) (0) ( ) [ ( )]

( ) (0) ( ) ( )t

x t t x t Bu t

t x t Bu d

Stability of L.F.O. SystemsStability of L.F.O. SystemsStability Analysis of Linear Fractional

Order Equations :

For the system of conmesurate order systems:

arg( )2i

In the special case and for integer order systems:

arg( )2i

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.8a=

2-Eign Value Method :

0.13 2 0

0.5 0.2 0 0

0 0.13 2

0 0 0.5 0.2

A

g g

g g

+ -

- -=

- +

- -

é ùê úê úê úê úê úê úë û

( ) ( ) ( )D x t Ax ta =

( )( )( )

2

4 3 2

2 0.2 0.0671 0.97341

0.1342 1.9513 0.1307 0.9475

g l l l

l l l l

- - + ++

+ + + +

3-Argument Principle for Studying Stability3-Argument Principle for Studying Stability

( 1)1 1 0

1 1( ) , ,

n nn n

G s q na s a s a s a qa a a a +

--

= = Î+ + + +

¡L

1 2 3

1

2

3

: arg( ) , [0, )2

: lim . , ( , )2 2

: arg( ) , ( ,0)2

j

RR e

1

2

1

1.25s s

2 1

3 2

1

0.5s s

Stability of L.F.O. SystemsStability of L.F.O. SystemsStability Analysis of Linear Fractional

Order Equations :

Topics:Topics: Fundamentals of Fractional Calculus Fundamentals of Fractional Calculus Fractional Operators Fractional Operators Solution of Fractional Order EquationsSolution of Fractional Order Equations Concepts of Fractional Order OperatorsConcepts of Fractional Order Operators Stability of Linear Fractional Order SystemsStability of Linear Fractional Order Systems Stability of Nonlinear Fractional Order SystemStability of Nonlinear Fractional Order System


Stability of NL.F.O. SystemsStability of NL.F.O. Systems

Stability Analysis with linearization method:Stability Analysis with linearization method:

* 1 1 1 21 01 2 02

* 2 2 1 2

( ) ( , ), (0) , (0)

( ) ( , )

D x t f x xx x x x

D x t f x x

a

a

ìï =ï = =íï =ïî

Linearization

( ) ( )D x t Ax ta = Which22 12

21 22

, iij

j eq

a a fA a

a a x

é ù ¶ê ú= =ê ú ¶ë û

If 2 1arg( ) , arg ( )2 2

ap apl l> > Then the considered system is

asymptoticallystable.

“ The Fractional order systems, are stable at least as same as their equivalent system in integer order. ”


Example:Example:

( )

( )* 1 1 1 2

1 2

* 2 2 1

( ) ( ) ( ) ( ), 0 1 , , 0

( ) ( ) ( )

D x t x t r ax t bx tx x

D x t x t d cx t

a

aa

ìï = - -ïï < £ ³íï = - +ïïî

Equilibrium Points:Equilibrium Points:

( )0,0,0

d

b

æ ö÷ç ÷ç ÷çè ø

,d cr ad

c cb

æ ö- ÷ç ÷ç ÷çè ø

0

0

rA

d

é ùê ú=ê ú-ë û

0

brr

aAcr

da

é ùê ú- -ê ú

= ê úê ú

-ê úê úë û

0

ad bd

c cAcr ad

b

é ùê ú- -ê ú

= ê ú-ê ú

ê úê úë û

0, 1, 1, 2, 3, 0.9a b c r d a= = = = = =

1,2 2.45 jl =± ( )1,2arg 0.92 2

p pl = >


Stability Analysis with direct lyapunov method:Stability Analysis with direct lyapunov method:

Lyapunov stability is the primary method of testing the Lyapunov stability is the primary method of testing the

stability of nonlinear systems, or linear systems with stability of nonlinear systems, or linear systems with

uncertainty or reliability problems. uncertainty or reliability problems.

It is more general than other tests for stability. It does It is more general than other tests for stability. It does

not depend on testing the roots of Eigen values or of not depend on testing the roots of Eigen values or of

testing poles.testing poles.

It involves finding a “Lyapunov function” for a system. It involves finding a “Lyapunov function” for a system.

If such a function exists, then the system is stable. A If such a function exists, then the system is stable. A

related result shows that if a similar function exists, it related result shows that if a similar function exists, it

is possible to show that a system is unstable.is possible to show that a system is unstable.


Stability Analysis with direct lyapunov method:Stability Analysis with direct lyapunov method: The most important part of this approach is finding a The most important part of this approach is finding a

Lyapunov function that it should be satisfy some conditions:Lyapunov function that it should be satisfy some conditions:

( ) 0,V x >1-

2- ( ) 0V x £&

Comment:Comment:

If in the If in the Then the E.P. is asymptotic stable.Then the E.P. is asymptotic stable.( ) 0V x <& { }0D -

If in the If in the Then the E.P. is stable.Then the E.P. is stable.( ) 0V x £& { }0D -



( ) ( ), 0 1D x f xa a= < <14444444444244444444443

( ) 1( )

0

1( ) (0) ( ) (0) ( , ( ))

( )

t

x t x I f x x t f x daa t t t t

a-

= + = + -G ò

( )( ) 2

0

1( ) ( , ( ))

( ) 1

t

x t t f x da

t t t ta a

-= -

G - ò&

Nonlinear Fractional Order Systems =Nonlinear Fractional Order Systems =

Lyapunov Theorem Lyapunov Theorem can’t be applied, can’t be applied,

because of its orderbecause of its order

= Second kind of= Second kind of

Convolution Volterra Integral EquationConvolution Volterra Integral Equation



Lyapunov function candidate:Lyapunov function candidate:

( )( )

( ) ( ) ( )2

0 0

1( ) ( ) 2 ( ) ( ) 0

2 1 ( )

t t

V t F x t t s f x f x s d dsa

t t ta a

-= - - - >

- G òò

( )( )( ) ( ) ( )

( )( ) ( ) ( )

3

0 0

2

0

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

( ) 1 2

1( ) ( )

( ) 1

t t

x

t

V t F x x t t s f x s f x d ds

t f x t f x s ds

a

a

t t ta a a

ta a

-

-

= - - -G - -

- -G -

òò

ò

64444744448& &

( )( ) ( )f x t x t6444447444448

&

( )( )

( ) ( )2

0

( )( )

( ) 1

tf x tt s g x s ds

a

a a-

-G - ò

( )( )( ) ( ) ( )3

0 0

1( ) 2 ( ) ( ) 0

( ) 1 2

t t

V t t s f x f x s d dsa

t t ta a a

-=- - - <

G - - òò&



Example:Example:(0.9)

1 2

0 3( ) ( ) , (0) 1, (0) 2

2 0D x t x t x x

é ù-ê ú= = =ê úë û

( )2

1.1

2 0 0

30 31 ( ) 1

( ) 2 ( ) ( ) 022 020 (0.9)

( ) 2

t tx tV t t s x x s d ds

x tt t t

-é ù

é ù-ê ú- ê ú= ê ú+ - - >ê úê ú G ë ûê úë û

òò

( )

( )

2.1

0 0

1.1

0

0 3 ( ) 0 31.1( ) ( ) 2 ( ) ( )

2 ( ) 0 2 010 (0.9)

0 31( ) ( )

2 010 (0.9)

t t

t

x tV t x t t s x x s d ds

x t

t x t x d

t t t

t t t

-

-

é ù é ù- -ê ú ê ú= - - -ê ú ê úGë û ë û

é ù-ê ú+ -ê úG ë û

òò

ò

& &

( ) 2.1

0 0

0 31.1( ) 2 ( ) ( ) 0

2 0(0.9)

t t

V t t s x x s d dst t t- é ù-

ê ú=- - - <ê úG ë û

òò&



Theorem:Theorem:

If If ( )( ) ,f x CÎ - ¥ ¥ and and 0; ( ) 0x xf x" ¹ > and and

0

( ) ( )x

F x f dx x® ¥òB

whenwhenx ® ¥ Then the E.P. ofThen the E.P. of ( ) ( ) ( )D x t f xa = is asymptotic stable.is asymptotic stable.

Example:Example:

( )2(0.8) ( ) 3 ( ) 1 ( )D x t x t x t= -

( )2 2( ) 3 1 0xf x x x= - >1-

2- 4 3 25 10 5( ) 0

4 3 2F x x x x= - + >

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

0.1x =1.21x =

SuggestionSuggestion

In the following of applying the lyapunov theorem on the nonlinear fractional order systems, some subject can be suggested :

1.Finding the region(s) of attraction in the nonlinear fractional systems.

2.Proving the instability theorem, global stability theorem, … and other theorems that related to the nonlinear integer order systems.

3.Finding the controller based on the lyapunov function.

4.Applying this approach to the linear fractional order systems.

Thank you!Thank you!

Special Thanks to Dr. Pariz

Special Thanks to Dr. Karimpour

Special Thanks to Prof. Vahidian

Thanks to Dr. Chen

Thanks to Prof. Podlubny

Thanks to Prof. Diethelm

To be Continued…!To be Continued…!

Stability of Fractional Order Systems Advisor : Dr. N. Pariz Consultant : Dr. A. Karimpour Presenter...

Documents

Transcript of Stability of Fractional Order Systems Advisor : Dr. N. Pariz Consultant : Dr. A. Karimpour Presenter...