30th Benelux Meeting on Systems and Control

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on

Systems and Control

March 15 – 17, 2011 Lommel, Belgium

Book of Abstracts

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a Scientiﬁc Research Network of the Research Foundation - Flanders (FWO-Vlaanderen),

and supported by the

Belgian Programme on Interuniversity Poles of Attraction DYSCO

(Dynamic Systems, Control and Optimization), initiated by the Belgian State, Prime Minister’s Ofﬁce for Science.

Clara Ionescu, Robin De Keyser, Patrick Guillaume, Rik Pintelon, Johan Schoukens, Ilse Smets, Joos Vandewalle, Jan Van Impe and Jan Swevers (eds.)

Book of Abstracts 30th Benelux Meeting on Systems and Control

Universiteit Gent - Vakgroep Elektrische energie, Systemen en Automatisering Technologiepark 913, B-9052 Gent (Belgium)

Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotocopie, microﬁlm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever.

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J. Tenreiro Machado March 15-17, 2011

Application of Fractional Calculus in Engineering Sciences

2

-2 -1 0 1 2

Integer

Fractional

Integer vs fractional numbers

3

Factorial vs Gamma function () ( )

!111nnnn=−=+Γ

Γ(1) = 0!

Γ(2) = 1!

Γ(3) = 2!

() ³

^∞ −− =Γ 0

1 dtetntn

Integer Fractional 4

Integer vs Fractal dimension

d = 0.63 d= 1.58 d= 2.73d = 1 d= 2 d= 3

Fractional Integer

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5

α-complex

Integer vs Fractional derivative

D1(eax) = aeax D2(eax) = a2eax D3(eax) = a3eax .... Dn(eax) = aneax D(eax) = a eax

n-integer

Harry Potter and the Philosopher's Stone Platform 9-3/4, King's Cross Station, London 7

What is the meaning of D½y? This is an apparent paradox from which, one day, useful consequences will be drawn…

1695 Guillaumede l'Hôpital (1661–1704) Gottfried WilhelmLeibniz (1646–1716)

intimate connection between derivatives and infinite series 8

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910

Fractional Calculus

•The name "fractional calculus" is actually a misnomer •The designation "integration and differentiation of arbitrary order" is more appropriate 11

Motivation: sin( ax ) function

•D1[sin(ax)] = a1sin(ax+ 1/2) •D2[sin(ax)] = a2sin(ax+ 2/2) •D3[sin(ax)] = a3sin(ax+ 3/2) •D4[sin(ax)] = a4sin(ax+ 4/2) •... •D[sin(ax)] = asin(ax+ /2) Weylderivative α ∞−tD 12

Vector interpretation of D

for the function f = sin( x )

D1f= sin(x+ π/2) D0f= sin(x) D−1f= sin(x−π/2)

Dαf= sin(x+ απ/2) απ/2

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13

Power-Law Function

•A power lawis a relationship that exhibits the property of scale invariance

()

ℜ∈=baaxxfb ,, •For a constant c

() () () ()

xfxfccxacxfbb ~== which implies the scale invariance 14

Fractional Integral (1) () () ³

=−t a

adftfDττ1

() () () () ³

−− − −=

t a

nn adft ntfDτττ1 !1

1

Cauchy’s integral and power-law function

() ( ) () ³

−=−t a

adfttfDτττ2 15

Fractional Integral (2) () () () () ³

−− − Γ=

t a

nn adft ntfDτττ11 Replacing nby q > 0:

() () () () ³

−− −Γ=

t aqq ad tf qtfDτ ττ 11 16

Fractional Integral (3) () () () () () ()

tf q

t d t

f qtfD

qt qq *11 010 Γ= −Γ=− −−

³

τ ττ

Laplace transform and Convolution. Without loss of generality let a = 0: The symbol *denotes the convolution in the perspective of the Laplace transform:

() () {} () ³

^+∞ − == 0

0dtetftfLsFst

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17

Fractional Integral (4)

()(){}()()sGsFtgtfL=*

() ()

0,11 >=

° ¿

° ¾

½

° ¯

° ®

Γ

− q sth q

t L q

q

Denoting h(t) as the unit step function and knowing the property: The Laplace transform of the fractional integral results:

() ()

0,11 >=

° ¿

° ¾

½

° ¯

° ®

Γ

− q sth q

t L q

q

results: 18

Definitions of fractional derivatives-1

() ()() () nn

t

df dt

d ntfD

t an

n ta <α<−τ−ττ

¸ ¹

·

¨ ©

§ α−Γ=

³

+−αα 1

1 1 BernhardRiemann (1826–1866)

JosephLiouville (1809–1882)

Riemann-Liouvilledefinition ()()() [] xofpart integer

11 lim 0 −

−

¸¸ ¹

· ¨¨ ©

§α −=

¦

» ¼

º

« ¬

ª− =αα x

khtf khtfD

hat k

k ta AlekseyLetnikov (1837-1888) AntonGrünwald (1838-1920)

Grünwald-Letnikovdefinition 19

Definitions of fractional derivatives -2

() ()

() () () nn

t

df ntfD

t an

n tC a <α≤−τ−ττ α−Γ=

³

+−αα 1

1 1

Caputo definition Laplace definition ()()()

° ¿

° ¾

½

° ¯

° ®

−=

¦

− ==−−αα−α1 00

11n kt

kk txDssXsLtxD

MicheleCaputo Pierre-SimonLaplace (1749-1827) 20

Left and Right fractional derivatives

Left-sided Right-sided

() ()() ()

³

+−αα τ−ττ

¸ ¹

·

¨ ©

§ α−Γ=

t a

n

n ta t

df dt

d ntfD 11 () ()() ()

³

+−αα τ−ττ

¸ ¹

·

¨ ©

§ − α−Γ=

b t

n

n bt t

df dt

d ntfD 11 past of f(t)future of f(t)atb()tfDtaα ()tfDbtα

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21

Properties (1) ()

αα α− −Γ=tc cD 1

The fractional derivative of the function f ( t ) = t

, >0, >0 : The fractional derivative of the constant c :

() ()

μαμα μαα− +−Γ+Γ =ttD 1

1 22

Properties (2)

βαβα−−−− =DDD βαβα+ =DDD

It is always true for >0, >0: It is not always true that:

23

Grünwald-Letnikov definition

()[]()() hhxfxf xfD h−− = →0

1 lim ()[]()()()() 30

33233 lim h

hxfhxfhxfxf xfD h−−−+−− = → ()[]()()()() 2/102/1316128121 lim h

hxfhxfhxfxf xfD h−−−−−−− = →

¸¸ ¹

· ¨¨ ©

§ − 1

3

¸¸ ¹

· ¨¨ ©

§ 2

3

¸¸ ¹

· ¨¨ ©

§ − 3

3

¸¸ ¹

· ¨¨ ©

§ 0

3 ()[]()()() 20

222 lim h

hxfhxfxf xfD h−+−− = → ()()

() ()

1!

1 11 +−Γ+Γ −=¸¸ ¹

· ¨¨ ©

§ − kkk

kk ααα 24

A probabilistic perspective…

()[]()()()() 2/102/1316128121 lim h

hxfhxfhxfxf xfD h−−−−−−− = → t

x(0) x(h)x(2h)x(t) present past x(h) γ(α,1)x(2h) γ(α,2) hα

E(X)

θ

1 16

1 8

1 2

1 =+++

non uniform time variation future

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25

Fractional-Order Integrals of Several Functions

()ℜ∈xx,ϕ

( )

()CxxI∈ℜ∈+αϕα ,, ()1− −β ax

() () ()()0Re,1 >− +ΓΓ−+ β βαββα ax x eλ()0Re,>− λλλαx e

() ()

¯® x

x λλ cos

sin

() () ()1Re,0, 2cos

2sin >> ¯® −−− αλ παλπαλ λα x

x

() ()

¯® x

x ex γγλ cos

sin

( )

() () () ()

1Re,0

arctan , cos

sin 222>>= ¯® −− +λγλγφ αφγαφγ γλαλ xxex 26

Approximations of fractional derivatives

Two methods: Frequency-based approach Discrete-time approach 27

Natural phenomenon

Differential equation of fractional-order 1<α<2:

Water mass in movementPorous obstacle 28

Frequency-based approach

Recursive circuit: Admittance: C/ηn C/ηC −V+I1InI0

I RR/εR/εn¦ ==

n iiII 0εⁱ iR R=+1 ηⁱ iC C=+1 ()

() ()

()¦ =+==

n ii

i CRj

Cj jVjI jY 0ηεωεω ωω ω

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29

Frequency-based FD approximation

ηεε loglog

log +=′m

εη ωω ωω == ′′++ ii ii11ε ωω = ′iiη ωω =′+ ii1 ′=ωεη 2

2 RC log ω

′mπ 2

π 2

ωεη 2= RC

log ω ′=ωη 1RC

ω11= RC 20m’ db/dec 20db/dec

Δdb

log η log ε20 log10 |Y(jω)| arg{Y(jω)} limited bandwidth

Recursive relationships of pole/zero frequencies: Average slope: Approach toDα(0 < α< 1): m’= α 30

Discrete-time FD approximation (1)

Grünwald-Letnikovdefinition: ()()() ()()() »» ¼º «« ¬ª − +−αΓ+Γ+αΓ −=

¦

∞ =α→α 0011

1 11 k

k hkhtx kkhlimtxD ()

{}

()()() ()

α−∞ =

− α

α ¸¸ ¹· ¨¨ ©§− = +−αΓ+αΓ− ≈

¦

Tz z k!kTzX

txDZ k

kk1 0

1 1

111 Fraction approximation

h≈T, T-sampling period: Series approximation 31

1.0E-06

1.0E-04

1.0E-02

1.0E+00 1.0E+001.0E+011.0E+021.0E+03 k

|γ(α

,k)|α = 0.1 α = 0.9

Discrete-time FD approximation (2)

()[]()() »» ¼º «« ¬ª −=¦^∞ =→ 0

1 0 kkhtxk, hlimtxD hαγαα ()()

() ()

1

1 1 +−Γ+Γ −= k!kk,k αα αγ 32

Discrete-time FD approximation (3)

i.Grünwald-Letnikovdefinition: ii.h≈Twhere Tis the sampling period:

()()() ()()() »» ¼

º «« ¬

ª − +−αΓ+Γ+αΓ −=

¦

∞ =α→α 0011

1 11 k

k hkhtx kkhlimtxD ()

{}

()()() ()

α−∞ =

− α

α ¸¸ ¹· ¨¨ ©§− = +−αΓ+αΓ− ≈

¦

Tz z k!kTzX

txDZ k

kk1 0

1 1

111

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33

Truncated series

Discrete-time FD approximation (4)

¸ ¹

·

¨ ©

§ −−−−−=

¸¸ ¹

·

¨¨ ©

§−−−−−− 4321 21

211 1285 161 81 21 111 zzzz TTz ()

{}

()()zXzzzz TzUtxDZ

» ¼

º

« ¬ ª −−−−−==−−−− 4321 2121 128

5 16

1 8

1 2

1 11 ()()()()[]()[] ¿¾½ ¯® −−−−−= =TkxTkxkTx TkTutxD kTt2 81 1 211 2121 34

Discrete-time FD approximation (5)

^{() ()} 1 45 83 641

1 4

7 8

7 64

7 1 123

123 21 +−+−

+−+− = −−−

−−− zzz

zzz TzX

zU

1 4

5 8

3 64

1

1 4

7 8

7 64

7 11 123

123 21

211 +−+−

+−+− ≈

¸¸ ¹

·

¨¨ ©

§− −−−

−−− − zzz

zzz TTzFraction approximation ()()()()[]()[]()[] ()[]()[]()[] ¿¾½ ¯® −+−+−+

+ ¿¾½ ¯® −−−+−−= = TkuTkuTku TkxTkxTkxkTx TkTu txD kTt 3 641 2 83 1 45

3 64

7 2 8

7 1 4

71 21 21 35

0.1

110 0.010.1110

Mod Ω

(jΩ

)1/2 H1 H2 H4

H3

00.5

11.5 00.511.522.5

(jΩ)1/2 H1

H2 H3

H4 Re

ImDiscrete-time FD approximation (6) Frequency response 36

Louis Amstrong: What a wonderful world! FC researcher: What a fractional world!

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37

Mittag-Leffler Function (1)

The Mittag-Leffler function is a generalization of the exponential function that plays an important role in fractional calculus. The function was developed by the Scandinavian mathematician G. M. Mittag-Leffler(1846- 1927). 38

Mittag-Leffler Function (2) () () ()

z kz kz zE k

k k

k exp !1 00

1== +Γ=

¦ ¦

∞ =

() ()

zzEcosh2=

In particular, when = 1 and = 2, we have

The function E(z) is defined by:

() () ¦

^∞ =+Γ= 01 k

k kz zE αα 39

Mittag-Leffler Function (3) () () ¦

^∞ =+Γ= 0k

k , kz zE βαβα

The function E,(z) is defined by: When = 1, E,(z) coincides with the Mittag-Lefflerfunction:

() ()

zEzEαα=1, 40

()

zzE, −= 1

1 10

() ( ) ()

zerfczexpzE,−=2 121

( ) ()

zcoszE,=−2 12

Mittag-Leffler Function (4) () ()

zz zEsinh 2,2=

In particular, we have:

()

z

e zE

z 1 2,1− =

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41

Mittag-Leffler Function (5) () ()

1,1 12,≠ −Γ≈α αα αα zzE

() ()

υα αυα υα≠ −Γ≈−,1 , zzE

An important characteristic of the Mittag-Lefflerfunction is its asymptotic behavior. In the case where the argument z 0, the Mittag-Lefflerfunction decreases monotonically. In particular for large values of zwe can write:

() ()

1,1 11,≠ −Γ≈−α αα α zzE 42

Mittag-Leffler Function (6) ()

asdtetaEtst #1 0,21 2121 =±−∞ −

³ ()

ass dteatEtst #α

βα α βαβ− −∞ − =±

³

0,1

Three very important related improper integrals define the Laplace transformation of the one-and two-parameter Mittag- Lefflerfunction:

() ()

ass

s dteatEst #α

α α α=±−∞

³

0 43

Integer vs Fractional Mechanics…

Spring Hooke law Viscous friction Newton fluid Mass Newton 2ndlaw

xkF= xkF=

kxF= ()α =kxF 44

Applications: Control Systems 21 DK21 Ms

SH + −

nonlinear system

mec ()[]{}()() ()()()zX Tz TrunczXz k!kTtxDZn

n k

kk

° ¿

° ¾

½

° ¯

° ®

¸¸ ¹

·

¨¨ ©

§− = »» ¼

º «« ¬

ª +−αΓ+αΓ− ≈

α− =

− αα ¦

1 0

1 1

111

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45

Control Systems γrelay mc

λ

hysteresis μmc linear system (n = 7) nonlinear system (n = 1)

nonlinear system (n = 7)

linear system (n = 7) nonlinear system (n = 7) nonlinear system (n = 1)

γrelay mc γ=1μ=1 λ=0.1 46

Fractional-Order Controllers

The fractional-order PIDcontroller: () () ()

μλ− ++==sKsKK sE

sU sCdip μ sKdpK λ− sKi()sU()sEPID PI

PD P =1

=1 0 47

Control robustness (1)

C(s)R(s) −

+ () α= sK sGRe(s) K= 0

π−π/α π−π/α

+∞←KIm(s) +∞←K

s-plane

Root-locusFeedback control system, 1<α<2 isodamping 48

Control robustness (2)

constant phase margin

Bode diagrams

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49

Mittag-Leffler Function: Example

() ()

11 +=αssRsY ())inputstep(1 ssR=

R(s)+ −

Y(s)1 sα0

1 () ()11 +=αsssY()()α α−−=tEty1¹⁻^L 50

Mittag-Leffler Function E

α

( − t

α

), 0 < α ≤ 1

0510150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 t

α E α (-t )

α=0.25 α=0.5 α=0.75 α=1 -> e-t 51

Mittag-Leffler Function E

α

( − t

α

), 1 < α ≤ 2

051015-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 t

α E α (-t )

α=1.25 α=1.5 α=1.75 α=2 -> cos(t) 52

Stability of fractional order systems (1)

A general fractional-order linear system can be described by a fractional differential equation of the form:

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

tuDbtuDbtuDb tyDatyDatyDa mm

nn mm

nn 01

01 01

01 βββ

ααα +++=+++ −− −− where Dq= 0Dq tdenotes the Riemann-Liouville or Caputo's fractional derivative.

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53

Stability of fractional order systems (2) () ( ) ()

kk nn

mm sPsQ sasasa

sbsbsb sG nn

mm α

β ααα

βββ = ++++++ = −− −− 01

01 01

01

•The corresponding transfer function of incommensurate real orders is of the following form: •The incommensurate order system can also be expressed in commensurateform by:

()

00 01 1

01 1 α

β sasasa

sbsbsb sG vvn n

vvm m ++++++ = 54

Stability of fractional order systems (3)

•In the particular case of commensurateorder systems, it holds that k= qk; k= qk; (0<q<1,k - Z), and the transfer function has the form:

() () ()

() ()

^q

q kqN kk

kqM kk sPsQ K sa

sb KsG0 00 0==

¦ ¦

== 55

Stability of fractional order systems (4)

•A commensurate order system described by a rational transfer function G(s) is stable if only if |arg(i)|>q /2, for all i,i= i-throot of P(sq). Re

Im 0<q<1 q /2 Stable region

Unstable region

Re

Im1<q<2 q /2 Stable regionUnstable region 56

A gallery of FC root locus (1) () ( )

1122235.05.05.12 −+++−−=sksssssQ

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57

A gallery of FC root locus (2) ()

ksssQ+++=5560.12385.57943.08372.05708.2 58

A gallery of FC root locus (3) ()

042.00.3 1

s kesQ− += 59

Electromagnetism: Skin effect (1)

tE rE rrE ∂∂ = ∂∂ + ∂∂ γμ1 2

2

Maxwell equations t∂∂ +=×∇D H

t∂∂ −=×∇B E ρ=⋅∇D

0=⋅∇B

() ()

⁰₀ ⁰₁

00 2

~ qrJ

qrJ r

ql Z σπ=

For a sinusoidal field we have: For a conductor of length l0results: 60

Electromagnetism: Skin effect (2)

102103104105106107100

101 ω

Mod [Z]

Z

Za1 Za2

Z Za1 Za2 Fractional slope

γπω 2 0

0~ 0 r

l Z≈→

()

j r l Z+ γωμ π≈∞→ω1 22

~ 0

0