Fractional Order Systems - Microsoft PowerPoint - FOSTA.pdf

‘Fractional Order Systems’
a Reality or Absurdity:
viewing mathematically & scientifically-with
applications

1. Shantanu Das

Scientist, Reactor Control Division E&l Group BARC Mumbai,
Honorary Senior Research Professor Dept. of Phys, Jadavpur Univ (JU),
[email protected], [email protected]

3. Calculus is the language that describes a Dynamic System and we are used to

Newtonian Calculus-and we know one whole order derivative is dt _ im Af
dx > Axdo Ax

Calculus is as old as three hundred plus years so is fractional calculus-this

‘Fractional calculus’ is used to describe analyze Fractional Order Systems

Why do we need to have fractional calculus ?

The differentiation a operation is taking rate per unit (infinitesimal) i.e. Ax J 0

For an element (Ax )” defining rate as 4x40 forsay 0<a<1 gives notion of
fractional differentiation

Similarly integration with respect to element (Ax)” gives notion of fractional
integration

Obviously (Ax) > Ax what does it say then ?

Fractional calculus is extension of the classical calculus-rather Generalization of
the_Classical-calculus

Is there any conjugation or parallel between the two?

Can we apply to non-differentiable functions?

Does system dynamics having memory has direct link with fractional calculus ?
Can we have circuits and systems or dynamic systems with fractional calculus?

4. Niels Henrik Abel GF B Riemann Joseph Liouville Michele Caputo
1802-1829 1826-1866 1809-1882 1927-

Hendrik Wade Bode
1905-1982

5. Subject: Book on fractional calculus
From: "Prof. Michele Caputo" <mic.caput@tiscali it
Date: Tue, April 13, 2010 12:47 am
: [email protected]

En.

Michele Caputo

6. Few Pioneering Indians in the field of Fractional Order
Systems and Fractional Calculus

Prof. Om.Prakash Agarwal
Prof. Anil.D. Gangal

Prof. Kiran .M. Kolwankar

Prof. R.K.Saxena

Prof. A.M Mathai :
V Balakrishnan

Prof. R.K.Raina
Prof. Rasajit Kumar Bera
Prof. Versha Gejji

Prof. Siddharta Sen

S C Dutta Roy Prof. Karabi Biswas Lokenáth Debhath

8. A fair coin has two sides Heads (H) and Tails (T); can be represented as ‘states’ sa
X:0=x° & X:1=x' with equal probabilities that is Y.
The ‘Probability Generating Function’ (PGF)is:

PGF(x)= tx +ixt = (14 x)
= (P(x :0)x° + P(X :1)x') P(X:0)=P(X:1)=

P(X:0)=H, P(X:1)=T fwoevents

With two such fair coins we have the PGF as following expression

PGE (x) = (PGE(x))(PGE (x)) = (40 +x))
=tx°¢ix'+tx?
= P(X :0)x° + P(X 1x! + P(X 1 2)x?
P(X:0)=HH=4, P(X:1)=HTorTH =},

P(X:2)=TT =1 three events

Similarly with three such fair coins we have the PGF as following expression
PGF,(x) =(40+x))

0 1 2 3
= FX O +HExX +Ex +1x° fourevents

9. We extend this to n coin system; and the (n+1) states with their corresponding

probabilities is expressed in following steps in PGF

PGE, (x) = (4(1+x))"

1 nr n(n-1)(n-2
"5: (I+nx + Wa x? 4 AU 2x +x")

xg MOLD >», n= Dn = 2) 3
2"2! 2"3!

ass

Here, we elucidate a possibility of having negative probability, by constructing a
hypothetical ‘half-coin’. This notion of half-coin construct was first introduced in
2005 by, Gabor. J. Szekely, and further worked on by J. A.T Machado, for possible
applications in physical systems and in relation to fractional calculus. This
concept of negative probability; though sounds absurd, is recently getting
attention of possible usage in physical systems.

What if we place n = Y mathematically can we get Half-Coin construct!

10. PGF, (x) = (4(1+ 1))4
A
V2
Loo, to 1 ,:, 1

u x xt x? 4 Ru
V2 242 82 162

giving series of infinite ‘events’ and ‘anti-events’ (we call ‘anti-events’ for state-
variables with ‘ negative probabilities’) looks absurd though!

Now with two half coins we get PGF of a one-whole (full) coin as demonstrated:

PGE (x) =(PGF,(x))(PGE, ©) PGE (x) = (PGF, (x))(PGF,(x)) = (+1
=(¿d+ 0) (4040) A +
=H+x!) Hart ti.

=p?
mx + Here the negative

"+4x +0x” +0x' +... probabilities get
cancelled out

=ix!

11. The idea of negative probability was considered by Richard P Feynman and

M.S. Burtlett, as generalization of existing theories. Burtlett wrote in 1945,

“It has been shown that orthodox probability theory may consistently be extended
to include probability numbers outside the conventional range, and in particular
negative probabilities. Random variables are correspondingly ‘generalized’ to
include extraordinary random variables; these have been defined in general,
however, only through their characteristic functions. This ‘generalized theory’
implies redundancy, and its use is a matter of convenience. Negative probabilities
must always be combined with positive ones to give an ordinary probability before

a physical interpretation is admissible.” Therefore, the notion of negative
probability is process of concept generalization

The absurdity or paradoxical situation arises due to the fact that ‘mathematics go
far beyond our physical understanding’

12. 1 n=0
(n-1)!(n) n>0

For integer the Factorialis: n!= |

which is Generalized by Gamma Function for non integers by Euler’s formula
T(x)= fe y dy, T(x +) =x0 (x) =x!=x(x-1)!

0

eS 1
T(x) = 2f ye dy T(x) = [(Int) “ay
0 0
TT)
T(d)=1 P(x) (- x) = ———_, B(p,q) = ==
(D=1, Fran sin(zx) EP F(p+q)
Beta Function B(p,q)

Ramanujan’s Formula

log (7 (1 +4n(1+ 2n))) Ñ log 7

log(n!) = nlogn-n+
g(n!) = nlog 6 2

1

5 log ( +.) log(2n) log
S Ramanujan _ _ San") log(2n) | log”
1887-1920 nlogn-n+ 6 + 2 + 2

13. Euler’s formula for Gamma function is defined as in integral form
T(x) = Joe "y" "ay

Tricomi’s incomplete Gamma function is defined as in integral form

7 (a,x) = x" fe "y dy
” T(a) +0
Whereas complementary incomplete Gamma function is defined as following
Integral forx>0, —o<a<o

T(a,x) =f e*y* dy

These binomial coefficients are defined and generalized using Euler Gamma
Function "e -(*)- n! _ T(n +1)

PX G) ¡=p! Ti+ DP (= f+)?
Some values of Gamma Function (Factorial) at non-integer points

SAA (Jen)
(=T()=47, (IT), ()!

neR

14. Poles at 0, -1, -2, -3,...

-2 0

—

Gamma-Function

“4

“6
Reciprocal Gamma-Function

15. In a letter to L'Hospital in 1695 September 30, Lei raised the possibility of
generalizing the operation of differentiation to non-integer orders, and
L’Hospital asked what would be the result of half-differentiating f(x) = x thatis:

1/2
LV =? or D%[x]=2

Leibniz replied “It leads to a paradox, from which one day useful consequences
will be drawn”.

We can say this query dated 30' September, 1695 gave birth to “Fractional
Calculus”; therefore this subject of fractional calculus with half derivative and
integrals etc, are as old as conventional Newtonian (or Leibniz’s) calculus.

Date of Birth of Fractional Calculus is 30' September, 1695

L’Hospital’s query is similar to absurd question of construction of half-coin

16. This is exactly what we would expect based on a straightforward interpolation of
the derivatives of a power of X . Recalling that the first few (whole) derivatives of
are x” following:
d(x") ma, G(x") mo, d’a")_ m-3
dx = mx ; ax? =m(m—1)x ; PTE = m(m—1)(m - 2)x
Thus we expect to find that the general form of the n -th derivative of x" is.
ig m! x"
dx” (m-n)!
Replacing the integer n with the general value © , and using the Gamma Function
to express the factorial, this suggests that the a fractional derivative of x" is
simply a” x") = m! neo Vn +1)

dx“ T(m-a@ +1) T(m-0a +1)
1!

moa

L’Hospital’s query answered: ai? (x)= x
dx? T(1-4+1)

d'? 1!

#0

For constant f(x) =1: m=0 A)

ax? © T(0-5+1)" CTO ax

Fractional derivative of constant not a zero — is it absurd?

17. Liouville gave this hypothesis-(also called the exponential approach)
de
dx*

ax a ax

=a e"; aeR*

This postulate is Liouville’s proposition conjugation to normal differentiation.

Generalization to arbitrary order:
Negative values of represent integrations (anti-derivative) and we can even extend
this to allow complex values of or even to a continuous distribution of this order in
some interval. de de area

Ex) fo) cay f(x)
dx dx?"

Continuously distributed order with k(@) as “distribution function” of order we

h
idad ee
dx

dx?

r def
For example distribution is delta then k(a) = Ad(«-2), fer r ED be = af

In a way this generalizes notion of integration and differentiation to arbitrary
order (fractional, complex, continuous)!

18. Any function expressible as a sum of exponential functions can then be fractionall
differentiated in the same way-as demonstrated here

cos(x) = d É = =|! © — e _ De = er"

a a

iam 12 ix
Je

car? nix ells+24) taz)

(e + (e

2
=cos(x+a 2)

we have the nice result |,
cos(x) = cos(x +a 4)

a

but we find difficulty with this exponential approach of Liouville’s to reply
to L'Hospital's query for getting half derivative of f(x)=x ; as itis difficult to
express this as composition of exponential functions

19. With Liuoville’s exponential approach we have

a

cos(x) = cos (x +0 4)

a

Thus the generalized differential operator simply shifts the phase of the cosine
function and likewise the sine function by « x (90)° , that is in proportion to the
order of the differentiation. For differentiation the process advances the phase,
needless to say the integration makes the phase lagged.

We test by giving a sinusoid at the input of the circuit and measure the output and
record the phase lag or lead, depending on the fractional order.

Time(Sec) xd

Output of fractional order differentiator circuit for Y order

20. Fractional integration
This comes as Generalization of repeated n-fold manne formula of Cauchy i.e.
n-1
aie y)" f(y)dy; neN
Riemann-formula
v I FO)

BER) = DGSE Va Ey

= (FO) a (Ko): k(x) =x°"'/F(v) Convolution integral

dy, v>0; veR'

Fractional derivative
Riemann-Liouville (RL): „D< [f (x) ]= I dr IN gy, 0<a<1
E en dx “0 (x - y)”

D*f(x)= D (r FR); 0<a<i

a)
Caputo: ome [roy]. py LTD ay, 0<a<l

Ta-a)’ (x- y)
DE) = 1 (DOF (x)); 0<a<1
Euler's Formula differ-integration <a y P(p +1) pra

for power function: ox À — T(p+lra). “3 p>-l

21. D* f(x) =D" (1" “f(x))s m-l<a<m

a = 2.3, m =3 :f 69) = DI (D %7f(x))
Function need not be differentiable

“

INTEGRATION

fo fo

>

DIFFERENTIATION

(m-a) =0.7

22. “D* f(x) = 1" *(D" f(x); m-l<a<m
m=3, a=23 3 f)(x)=D" (D'f(x))

INTEGRATION | DIFFERENTIATION

@

(m—o) =0.7

Comments & observations:
Function needs be differentiable in Caputo case
Process is like travelling on a Number line!!

Because the integration is built in the formula of fractional derivative-the fractional
differentiation is ‘differ-integration operation’ and has memory
Like integration operation the fractional derivatives have start and end limits

23. Using the RL formula, we first apply v=1-«=1/2 half of an integration to this functio!
using equation Riemann integration formula with v=1/2, giving for f(x) =x

x_ fy)
T(v) % (x= y)"

EFG) =/ DF) = dy, v=%; f(x)=x

d 1/2
E(x) = ol.

Then we apply one whole differentiation to give the net result of a half-derivative

ara)" d d 4x3"? _ ja
dx"? ax (oa ))= > n

24. In this case the Caputo formula gives the same result. For «a =1/2 first
differentiate the function f(x) = x once to have f(x) =1, do the semi-

-integration of this f LL. =1 thatis:
E re
SD [FCO ]=

TT ra) (x — y

CD'?f(x) = 12 (D EG)

. 1 x 1
A Eee cay =,
O

—— say: f(y=y, 1%G)=1

> py 1/2 12 | 4 “1/27, 0 Ta) 0 x
SD '*[x]= pp? [La] - Di” ’Ia’]= TG)“ 2 LE

We note that Riemann-Liouvelli fractional derivative does not require function to
be differentiable (needs only to be continuous), whereas in Caputo case -
differentiability is essential.

This discussion points out to the fact about the there is in derivative an integral
operation which are in between whole number of derivative and integration, and
are unified operation, and non-local in character having memory and heredity.

25. TRE a

oDix? =
T(p+1-a)

1-
xa

TQ-a)

ay _
DER =

1 15 2

Start point of fractional derivative process at zero where function
is zero

Note that fractional derivatives require start point and end points like integration

26. ay? = T(p+1) zoo
2 T(p+1-4)
1
“fio ä
D: fr] Ta=a)*

a=0, f(x)=1

Start point of fractional derivative process at zero
where function is non-zero

RL Fractional derivative of constant is not-zero!!

c
Dr [C]=
DC] Td-a)

»Ps{C]

x? lim CD [c])=0

at -x,)* lim, ,,(,,D¢[C]) =0

£(%) =0

27. If we expand the exponential function € “ intoa power series as:
1 1 2 i 5
e* =1+— x + —x* + —x"
1 21 3.1
and apply Euler formula to determine the half-derivative, term by term, we get:
d'?(e*) 1 >, 8 3, 16

= = 1120158 xo + x++...]#e*
7
dx"? | Vax 3 15 105

T(n+1) xr.

F ses

jn = 0,1,

~T(n+1-4)

YDerivative of e

Half derivative of exponential function is not exponential contradicting Liouville!!

28. e* =1+(ax)+

(ax)? (ax)? + (ax)' _< (ax)!
2! 3! up k! "Ira
olf [e] = olf [+ of [(axy]+ 05 [2] +...
_ x? N al (2)x°*! A a’T(3a)x“'? . a’T(4)x“*
T(a+1) 1!T(a +2) 2!T(a +3) 3!T(d + 4)

a (ax)!

tao L (kK +a +1)
olf [e* ] =x*E, 0, (ax)

def 2 u

We get result in terms of Mittag-Leffler function E£ (x) = y _* _
mn 0 I(mk +n)

def „© x k

Em (9) = E,, (x) = LI)

EN e

m=1; E, (x)=E,(x)=e*

29. Illustrating general behavior of E,, ,(x) for 0<asl ,anda < B

30. Et)
0<a<l,

Ey) EM)

| —

| Monotonically Decaying WWW

ED C1 We

BAC

Oscillatory Decaying

Es)
iea<2,. Bel

Eu)

Oscillatory Growing
Eg ED
2<a<3; Bel

31. Ex) 7 + + A
GT(ak+1) Tao) T(1+2a) T(1+3a)

¿E[E,(ax7)]= QUE RON :
1 a T(+0) 20 T(1+2a)
= xo + ax + a
Td+a) T(l+a)P (1+ 2a) TA +20) (1+ 3a)

1 1 1 2,2 1
=- ax“ + ax IR ee
a\ Td+a) T(1+ 2a) T(1+30)

1 ax” ax aix 1 er
{fr Taro) T(l+20) T0430) +.)- }- lea )-1)

Observation that ,1! [en ] = IN e”dy = +e‘ - L Mittag-Leffler function

olx

is as conjugation to the exponential function

Therefore, this Mittag-Leffler function may play important role like exponential
function plays, for solving fractional differential equation.

32. Derived the following after expanding RL formula for fractional derivative

dl 1 u : w OR
ax” |, “Ta sale yy" (==, le yy" (po

0<a<l This comes from doing integration by parts of RL formula

From above we write the relation between RL and Caputo derivatives as following

arr 1 f(0) 1 x LOG)
„D ol = tal Ey dy, O<a<l

f £m)
o (xy

£(0) x, 0<a<l; .D;[C]= < A
T(-a@) nave

= ¿Di [f@)]- „Di [FO]
= Dé [f(x)-f@]; SD: [fo]= „Di [E]- ta)
Thus when the function’s value at start point of differentiation is zero; both RL and

Caputo derivatives are same-else they are not. In that case of non-zero value at the
start point RL and Caputo derivative are segregated by singularity at start.

spe [100] = E dy, 0<a<l

0%

SD [£G9] = DZ [£O)]-

In the derivation we assumed that function is differentiable

33. Fora function f(x), a<x<b

Riemann-formula for Left ur Integration is:
f
LEGO = D) = 1 Na, v>0; veR*

f
Tw)" (x- y)"
The Right fractional integral and ann fractional derivative is:
vb f
j : IL v d R
a (yx)

pf (x) = , Dy f(x) = — v>0; veR

a dp f(y)
_D®[f(x)]= N
Ecole =i meal gare Osa

ewe ED po 2
¿Dj [fx == yon 0<a<l

Df [f(«)]= Df je ch

T(l-a)

f(b)

34. Left Fractional Integration of Power Function
Je [£00] le y) (FG))dy, y=x(-2z)

1 0 aie _ a (x-a)/x
== [ Gaz)" (fia-az))(-adz)=

(x-a)/x

(EX - d
T@) | (f(x xz)) VA
a |
f(x)=x? P a (xP (1—z)P
(x)=x eee (xz) )dz
we apie T(p+D
= 2 (1-z)” dz = B(a,p+1)= pre
Te!” (a) dr T(a) (2941) T(p+a+D)"
Tol (+)

Used the formula of Beta Function i.e. f 2 (1-2) dz = B(@,p+1) = mer

Left Fractional Integration of Expopential Function
Laya 1 e”dy; z=x-y

IE e* |
0 [ == a)?
: eno; a
Ele ax = z° le "dz=xte"y (a@,ax)
mle Ira se
Used the formula of Tricomi's Gamma Function i.e. y "(a;X) =

35. Right Fractional Integration of Power Function

NS poele (My y=x(2+1) dy=xdz

Cb=x)/x PTS
1

= a (£ (2 _ 2
$ [ (xz) (fxr +x))(xd2)= 5 {3 (f(x + xz))dz
xo? x? ©
[x ]= al "dz = Pz Pd
Hi E ] nal? (x +xz) ?dz ee x P(L+z) dz

pra _
ME »]-£ [+2 raz = LE @) ora
Fa); T'(p)
Used the formula of Beta Function i.e. 5 2°" (1+z) "dz =B(0,p-0) =o
Right Fractional Integration of exponen Function

f(x) =e"™ Jefe u Ta mateo "du; (u-x)=4, du ==

Jefe A mala rm À

Used the formula of Gamma Function i.e. T (a) = IN e *y* dy

36. _Fip-9) pra
T(p)
[x ED, »ı___ T@-) x 2
T(p) P-DFP-D

1

=. x p+l

Ep+Dl, P

er er
(-p) P

x

x

Le Y= fe dy =

We note that this Right Fractional Integration of exponential function returns
exponential function like for case of classical integration. Whereas we observed
that Left Fractional Integration returns higher transcendental functions on
fractionally integrating exponential functions !!

37. C-ex

„Di [es o] = Er (r (-a,-c(x -a)))
„Di [e* ] =c"e™

Df [e* ] =cx"“(E,, ,(cx))

-a

ap X DE
oy Inx = (nt) y-w(l-a@))

xo

„Di [su-0]- 2

La 3-a
: a = @x @x
oD; (sin(ox))=0* sin (ox + wee

SD: [E,Ox")]=RE,0x“)

a ws]. x” a
oD; [E.Ox Ten )

38. Used Euler formula

T(m +1)
De [x"1= moa
° de ] Tm+i-a) * 0

and apply tof (x) = Vx and get following steps:

DE [POD] > „De ie] |
_TG+D G+!) ,
"T@G+1-4)
0 „ash
Del 1] =4T (3) , @=% Critical order is half

x=0

Per (pa

oo , as Y

we get value of fractional derivative at the non-differentiable points !

39. The most fundamental approach may be to begin with the basic definition of the
one whole derivative of a function i.e.

d f(x) = lim f(x)-f(x-e)
dx eo €

Repeating n -times of this operation leads to a binomial series of following type

ER) _ li
PE Ec 1) l ros

¡20

MN al TOD o

ap TG =D"
Note that "C,=0 j>n thus summation above ends at n

j

Generalizing the binomial coefficients to real numbers we get a formula

40. Limit of a sum is integration

1-28 NE
I f@)] = De [foo] u [tay = lim (As GO + Ay — hy) + hy f(x —2hy) +... (Xp +hy))

= lim[ hy (f(x) +£(x —Ay) +£ (x —2hy) +R +hy))]

[Foo] =,, Dy" [£00] DO m) =lim © > Ns DE )]

¡20 20

Generalization of above is Fractional Integration

(=) 5 TU+@) f (x = i(5*))

Pa) 4 T(j+1D

y KE [FOO] = ,, DG" [F@)] = lim

41. The apparent paradoxes of fractional derivatives arise from the fact that, in general,
differentiation is non-unique and non-local, it is just as is integration-depends on
region of integration interval !

This shouldn’t be surprising, since the generalization essentially unifies integrals
and derivatives into a single operator.

Non-locality of Generalized Derivative implies Fractional Derivative is memorized;
unlike slope at a point in classical integer order derivative (i.e. a local property).

This is called Grunwald-Letnikov method unification of fractional differ-integration

x le [f (x) ] = lim

(a) $ EG; (x-

vial T(a) fm L(j+D

aie CSI RG
DOI Res 2 DES (r(x-

42. F(s) = L{f(x)} i=fe *£(x)dx f(x)=L'{F(s)} =

£ {D"fOo} = s"F(s)—s" 'f (0) - 5" 2f 0 (0) — =p >; neZ*

£ {eco} =£ {pr} = s"F(s)
L{ DEE} = L{D"( It #0) =s"£{, A FF) IA O) =... -D"" IFO)

=" LI FC) y De IEE(O)], n-I<a<n
i

El
aso -S'stD 4 Dr] a

10

El
I DE,
=

Example:
L(¿DPT6O] = sF(s)-[ ,D,° 160] a SF) [LG]. #=1
LADO F(x} = ES) DPTO), DC) 22
SF) DELL],

Requiring fractional initial states!!

43. LE (x) } = s"F(s) AS (0) =. FPO)
LLO) = 5 "F(s)

LX GDEE(2)]= LE IED If (A) OL SD (x)

1
= 5700) | srl) - y O)
k=0
n-1

= 5 F(s)- N str 0)

k=0

Example:
£ {SDF CG} s%7F(s)-s °%f(0), n=l

£ {D Pf] = ST E(s)=s 1 (0)=s FO), n=2

Requires integer order initial states

44. A simple Fractional Differential Equation (FDE) with RL derivative
.Dif(x)=C; 0O<a<l

c
Def =L {c}; s*F(s)- DC Uf ==
RC AS O E
cae ere
Given is fractional initial state as a constant ie. ol, fox), = Dy rol, =A

Cc A
F(s) = a =
s s

Using the Laplace Transform identity >! en») = x" we get solution as
e a Aja
x%+
T(a +1) T(a)
A simple Fractional Differential Equation (FDE) with Caputo derivative
SDiftG@j)=Cs O<a@ <1

£{GDitC P= LCR (sr (FOOL =<
Given is integer order initial state as a constant i.e. f (x Mi. = B
B

f(x) = ; x>0

a

F(s) € à
Ss

We get solution as c

OTe ap

45. FDE with RL derivative ,D“f(x)=C; 0<a<l
Fractional Order Initial Condition is ,1'-“f ol. De) _) =A implies

we have initialization function as Li (X | a „Di’A=A x*' which is singular

Tay

at start point ie. fin, CO], = ©

Uninitialized solution of ,Díf(x)=C is f,,(x)=,¿D,"C=C Xx

Total solution is f(x) =f,,(x)+ fan 001, gel tox! +A x

FDE with ne derivative SD x)=C; 0<a<1 has uninitialized solution as
f,, (xX) = C=Cr x"

un Tara X

Integer Order Initial Condition is Y (x )|,_, = B implies finn (X) =B

x20

Total solution is f(x) =f ,, (4) + f(x)... =

un

+B

C Fara) x“

Suitable initialization function is used to initialize FDE to have total solution

46. s Pk!
(sayo

£ OES on (ave) > ay

I Elo -—

Erna

-a

EHEN =

dla)
ap 3 Re(s) > la]

LUPE, (07) = = wees

Ss

a-l
L { E (1° )} = _S___ In Fractional Differential Equations we
eb s“ —] needto deal with irrational transfer function

£{E, MY a

ICT ENT)

47. Newtonian Derivative _
£(x) = lim f(x +h)-f(x)

hyo h
Fractional Derivative

(a) o 1 SIT(k-a) u a
[Feo] o eS (f(x-kh))], ()=h

Geometric Derivative

f(x) =lim

hdo

1
f(x+h) | OH)
f(x)

Biogeometric Derivative

1 1
fi (x) = lim fa +nyx h =P im f(kx) \ink
No f(x) Koll f(x)

Fractal Derivative

f(x)-f(x,)

a
XX =
0 x Xo

f(x) = lim

48. In systems involving course grained phenomenon, every thing happens as if
the elemental point is not infinitesimally small (zero) rather it exhibits some
thickness, what could be pictured by using (dx) (<a <1 ,insteadof dx

As dx Ÿ 0 we will have always (dx)” > (dx)

. » PA d(f “(f
In other words we are considering rate of variation as seo) or eto)

(dx ) (dx)*
Here we come across fractional derivative and fractional calculus

When media is non-uniform if we shrink the element to zero, then we loose its
actual picture

et |
awl = Hae Tes NT

def 1
div®J = lim —QOJendS=V%eJ
vorev V À
REV is ‘Representative Elementary Volume’ and a non-zero
Concept of fractional divergence
Similarly we have concept of fractional cross product & fractional curl that is
COR Pas AID us Frs Fa)

(Vx)? F

d, #3

51. With Fractional Curl operator we write the field equations as:
7 = = = = |2-
E, =(iky*[(V)"E] A)=d0) [ya] kK =o Vee Nyt

a=0 E, = E H, =H Original Solution « = 0
y PEC Perfect Electric Conductor

Transverse Magnetic Mode (TM) propagating between two PEC boundaries

E, =n H, = -E Dual to Original Solution «

PMC Perfect Magnetic Conductor

Transverse Electric Mode (TE) propagating between two PMC boundaries

52. We have the wave guide walls not with PEC or PMC

We have material which are in between PEC and PMC

This gives propagation inside a wave guide which is in-between original and dual
to original solutions, that is in between TM and TE modes

Case of fractional field

With fractional curl concept; it is interesting research to find radiation pattern of
antenna

53. 0°? u(x, t) =D run)

2v

or
© (-1)"x"
M = LT Tan 0 +
„= aaa us
v=0 M,(x)=e~*

Me ni y (a), ©?

120 " (Zn)!

0 3
Plot of the symmetrical M-Wright function for v =0 to 0.5

M-Wright function is Generalization to Gaussian Function

54. Opt im

1 ta
Hole J (G(s))e"ds

Opio
imfs]

>

KT

N

Bromwich- Path

Contour for rational function Contour for irrational function-
with Branch Cut in Primary
Riemann Sheet

55. Power Series expansion method

a (Hess) (et es)" (elas
N Leer)

12 = Ye DE )- Parar

220

(2+(2-a)n-(1-a)r)

n=O r=0

Residue Theorem-Contour integration method

k . “(kt)”
F(s)=————, 0 1, f®@=1-) ——
(s) s(s* +k) ses © u T (ma +1)

=1-E, (-kt*)

f=" E Ya [7 k = taste) e “dx

e"ds =1- - =
rio s(s% +k) m 9x (x +2kx" cos(mor) +k?)
Analytical method ‘without contour integration’-Berberan-Santos method

F(s) = | f{t)=e
s+b

fH=£ fos +b)" y= <j; ee sin(@t),
@

s(s 4k)

56. Integration on Bromwitch Path Goi
80= if G(s))e“ds; s=0o,+io, do,=0
Choose ©, such that for G(s) "ai

all singularities are on left of that Gi OS

line-in the complex plane s =o + ia = 2x J (Go, +i@))edo

ito

Writing e"” =cosot+isinot weget:

oot

g(t)= = | Î (G(o, + ¡o))cos(ot)do +i [ (6 (0, +i0))sntondo |

Writing G(o, + io) =Re[G(o, +i0)]+iIm[G(o, +i0)] and simplifying
We have formula e nen
g(t)= ——I, (Re[G]cos ot —Im[G]sin ot lo

In polar form plo)= (CA +io)| 6(@) = ZG(o, +ia)
e(t) = e” (Re [6 + io)]cos(ot) -Im [Co +io)]sin(ot) do
x 0

Got ©

= J(p(@))(cos(ot+0(@)) do

0

57. The inverse Laplace Transformation by Berberan Santos formula is

g(t)=L'{G(s)}, s=0,+i0

8(t) = — (Re [G Jcosor- Im [G ]sin oto

Condition g(t)=0; t<0

We have
g(t)=L'{G(s)}, s=0,+i0

g(t) 222 [7 (Re[O JcosorHo
T 0

g(0) = -2 = J; (m [6 ]sin ot Ho
7

58. G(s)=s*'(s* +1)! wewrite s=0,+io with o, =0

That is because we do not expect singularity in the right half plane of complex
frequency s ie. Re [s] >0 for function G(s) for its well-meaning behavior

G(s) == s=0+ 10

l+s
doy LOY oo" __ 0 @" (cos (#)+isin (+)
l+(ia)* 1+Go) (1+ 0% ee sin (& ))

ei sin (=) > 0° +cos (%)

1+20° cos(2)+0* a 1+ 2% cos()+0*

=o

n ( or! 40%"

Re[G(io)]=5 Im [G (io) ]= -

1+20* cos(

gi) = — [(Re[aco, +io)]cos(or) - Im [Go +ia)|sin(or) do

0

a ali (as ye
wd Î oe" sin($) — |cos(@t)+ +0” cos(#) sin(ot) do
7 +20" cos(%) +0?" 1+20° cos (%) +07”

1 fe sin (ot +%)+ 0°“ sin(or)
am 1+20* cos(% 2) 409

=

59. G(s)=s * does not have singularity at s > 0 Choosing o,=0 we obtain
G(o, +io) = (io) * = @* cos()-io™ sin (=)
Applying the formula of Berberan-Santos formula we get

où ©

e

g(t)= f(Re[GCo, +i@)]cos(ot) — Im[G(o, + io)]sin(ot) do
r

0

u = (CR cos (%)cos(t) +0”? sin ()sin(@r) do = + fro cos (ot -#)do

With condition g(t) = 0;

(o A cos (£)cos(o 1) Lo = £'! fs“ }

With the known Laplace transform ie. Ls“) == ir"; 120

-[e 2 cos (£2)cos(o 1) Ho = PT (a -l)=-n

1<0 wewrite

g(t)

n PO DY doma
= === Ss + s e 20
x = JG cos ( 2 Jeos(xy) My x2

This is integral representation of function g(x)=x"; x20

60. G(s)
The transfer function
function of complex
frequency

5=0,+10

In integral representation of function in time domain by
inverse Laplace transform by Berberan-Santos method

. cos(ar)de

ande

"\cos| ame -

H{u® sin 9) Jair VF

61. T,=90 ; T,=27.6 : A= - 0.03846

Comparison of experimental cooling curve for water
with theoretical solution using classical calculus

D'[TO]=-K(TO-T,) k>0; T()=T,+(T,-T,)e*

Soisit D° [T(t)] (T(t) -T, )if so then is it RL or Caputo derivative?

62. Classical law
D'T(t)=-k(T(t)-T, ); O<t<o, T(0)=T,
T(t)=T, +(T,-T, Jo" =T,0 "+T, (1-0); k>0

Fractional Law with (Caputo Derivative)

SD [T(09)]=-2(T(0-T,); O<a<l, 0<t<o
TO) =% Integer Order Initial State

(ST (5) =5"T(0))+ AT(s) = AT,

Ss

sT
T(s) = ET 0<a<l
ERA s(s*

T(t)=T, +(7, -T,)E, (41%) =T,E, (-41*)+T, (1 -E,(-At“))

63. 300ml water

œ

+ expt
— frac order
—— integer order

GS
g
2

=
2
El

E
=
5

temperature (degrees C)

1
20 40 E o 20 40

time (mins) time (mins)
Comparison of experimental cooling curves for water with Caputo Derivative,
for 40ml and 300ml. Showing fractional order of 0.79

°D¢ [T(t)]=4 (Ta) -T, 2 <0
T(t)=T, +(T)-T, )E, (21%)

64. Classical law
D'T(t) =—-k (T(t)-T, ); O<t<o, T(0)=T,
T()=T, +(T,-T, Jo" =Te" +T,(1-e“); k>0

Fractional Law with (Riemann-Liouville Derivative)
a. Fractional Order Initial States given

oD? [T(1)]+xT()=AT,

O<t<ow, 0<a<l; DT). = Co

c AT,
s’+K s(s’+K)

0

(s*7(s) -D“ DOME KT(s) = = ; T(s)=

Solution T(t) =C,(t®"E,,,(-Kt))+T, (1-E, (Ext”))
b. Integer Order Initial States given
oD? [TM] +KT (1) = 27, , 0<t<w, O<a<l TO, =T
Solution T(t) =T,I (aE, , (-Kt*) +T, (1- E,,(-Kt*))

0

65. 40m! water 300ml water
a = 0.79 œ= 0.79

I ‚or 7

=

Er

grees C

8

=

(degrees C)

integer order

temperature (de

1 1 1 1
20 30 60 20 40 60
time (mins) time (mins)

Comparison of experimental cooling curves for water with RL Derivative,
for 40ml and 300ml. Showing fractional order of 0.79

Di[TO]=K(TO-T,) «<0; TO)=T,(T@))E,. (Kt) +7, (1-E,,, (kr)

66. Oil Caputo form Oil R-L form
a = 0.88 © = 0.88
T T T T T T

> expt + expt
— frac order — frac order
—— integer order ge —— integer order

S
ES
=
2
El
E
S
£

L L
20 20 50 E 0 30 60
time (mins) time (mins)

Comparison of experimental cooling curves for mustard oil with Caputo & RL
Derivative showing fractional order of 0.88

67. Mercury Caputo form Mercury R-L form
&= 0.92 © = 0.92
T T T T T T

= expt E expt
— frac order frac order
—— integer order integer order

e
El

erature (degrees

GS
Zz
g
=
Eb
En
=
2
3
E
das

temp

L L 1
40 60 0 0 40 60
time (mins) time (mins)

Comparison of experimental cooling curves for Mercury with Caputo &
RL-Derivative, showing fractional order of 0.92

68. Liquid

Mustard Oil X 105.0

Mercury X 105.0

Constants and parameters for experiments with various liquids

D[TO0]=ATO-T,) 2<0 TO =T,+(T,-T,)E, (A)
(RUE,

Di [T]=K(T-T,) «<0 TO" M: (@))E,
D[TO]=ATO-7T,) 4<0 TOT, +{T-T,)e*

„ler“ )

69. height (cm)

Variation of constant with height of liquid column for Caputo and RL case with
order as 0.79 and area constant, the sudden change happening at same height!!

70. One complex process that has received quite amount of attention in the last few
decades is Newton’s Law of cooling. The nearly endless discussions about
Newton’s Law of cooling may be summarized by the statement of O’ Connel:
“Newton's law of cooling is the one of those empirical statements about natural

phenomena that should not work, but does”.

Does Non-Newtonian Calculus i.e. Fractional Calculus works here?

We tried to reply that query experimentally

71. . lope,
v(t)hat*; O<a<l; i(t)=ImA va) #— | i(t)dt
c
A dv(t)
So i(t) #C, dr does fractional derivative relates the voltage-current?

aeéeR*?,

Is i(ty=c, SO
dr"

Not usual linear rise for V(t)
constant current

n

sefV (Volt)

“Loma,

| 30ma 30 ma 7
y

A \ Experimental Data \
30mA 4 =" Simulated Model Data

50 100 150 200 250 300 350
Time, t (sec)

Observed Non-linear charging voltage profile

This observationv(t) « 1” is more pronounced when charging rate is quick !!

72. The Curie-von Schweidler law relates to the relaxation current in dielectric when a
constant step DC voltage is applied and is given by
i(t)~ 17% whee t>0; O<a<l

and the power (exponent) © is called relaxation constant or decay constant
with 0 <a <1 - Wenotethat Y the exponent is non-integer
Note the usual Debye relaxation is i(t) ~ e At

The Curie-von Schweidler behavior has been observed in many instances, since
late 19' Century, such as those shown in dielectric studies and experiments

For a step voltage Vs to an uncharged capacitor the current is

V
i(t)=K,—82; O<a<l; 1>0
t

This is empirically & experimentally derived

This is also called Universal Di-electric Relaxation (UDL)

73. A 4

log(t)
[h]

7 T t 9
Current -vs -Time

At time zero a voltage of 100V is connected to a 0.47uF metalized paper dielectric
capacitor; in log-log scales average slope is -0.86. Thus exponent of relaxation

current is non-integer

74. The unit step at time zero is@ (1) its Laplace transform is £ [o (1) } = 1/5

Then practically on applying a step input voltage v(t) = Vs, (O(1)) to a capacitor
which is initially uncharged; we get a power-law decay of current given by empirical
Curie-von Schweidler law as i(t) -1 *; 0<a <1 . We obtain following steps :

1) =K Vag “3 1>0 K5)=L((0)=L(K Var”) use L{r"}= mis"?

= 1
ul) a#l; 0O<a<l
=

Laplace transform of Voltage input is V(s) = L{v(1)} = £{V 5 (@()} = Van /s
Use (a -1)!=T(a) towrite I(s)=K, PQ Von
Ss

Td-a) We)

= A

ek E
Transfer function of capacitor or Admittance is thus

K a)

v6) ES
Ss

=K,(Tl-a))s® =C,s* C, =K,(Td-a))

Y(s)=

75. mn SC, C,=K, (ra -a)) Farad /sec'"”

s=io I(@) = (cos + isin 2)0*C,V(o)

Y (s)=

i= (e*” y =e"? = (cos +isin =)
This means current leads voltage in fractional capacitor by angle 90° xa

Admittance expressionis Y(s)=I(s)/V(s)=C,s*, 0<a<l
With use of Generalized Laplace identity L”' {s*F(s)} = ,D’[f(n]; f(0)=0

We get fractional derivative as in expression i(t) =C, (.D? [OT 0<a<l

The ‘fractional capacity’ C, isinunitof Farad /sec'”*
Classical ideal capacitor & Fractional capacitor expressions are :

av"
di?

dv(t)

i(t) = C,(,Div()=C, 7

(= Ce (¿Dfv(1))=C,
vG) = 1 Dio = —-[lilx)ax 1 lp ° 7
= oo; "ch vo = DCi = Si) (ax)

Classical ideal capacitor Fractional order capacitor

76. We charge an ultra-capacitor to maximum limit voltage and keep for time T on float
charge, then it is kept under open-circuit. The self-discharge plot is depending on
T i.e. capacitor memorizes the time history

The open circuit self discharge
voltage is

V, dx
wipe BB
oe (t) r

Gaara) Fea *

Only possible if

— 4 hr holding i =C, (oD? [v0]), O<a<l

— 8 hr holding ; e )
—— 16 hr holding Le. due to fractional capacitor

This memory is also observed
1000 10000 in several dielectric studies
Time (s)

For ideal capacitor, we will have Vo: (1) as constant irrespective of T

77. Our relation of fractional order capacitor is (1) = Cy
Charge accumulated is therefore is:

210 = [908 = fie, Ex =C

AO)
v(t)=Va,; 1>0
pa io TED mn
qi =C, (¿DF Vas ) = Co Vs >— — Got by using ar I(m-n+l)
A-oa)T(-«)
T(m +1) = mT(m)
What the above expression says; it says that if we apply dc voltage to a capacitor
the charge accepted by the capacitor will continue to grow at all times. There will
not be at any time an equilibrium condition lim, q(t) =w

This is contrary to our classical thought where we have equilibrium charge as
classically we have _lim,,,q(t)=C,Vop

For a specific voltage there is a time after which the capacitor has acquired so
much of charge that that it breaks down due to electrostatic forces !!

Therefore even if you keep a capacitor floats on a constant voltage which is well
within the rated voltage, there will be a time that charges accumulation gives an
electrostatic force that breaks down the capacitor.

78. A new formula for charge stored in capacitor i.e. the charge stored in a capacitor as;
a function of time is not multiplication of capacity and voltage, instead is a
“convolution integral” of the two in time domain-is a new thought-applied to
classical capacitor and fractional capacitors . That is q(t) + (c(t)) v(t)) but is

q(t) = (c(t))*(v@))

= |" c(t x)v(x)dx = |" e(y)v(y Nay

This new formulation gives the same result of capacitor breakdown via
accumulation of charges as electrostatic breakdown described earlier plus other
observations regarding power law relaxation current-and loss tangent values.

This new formulation also returns fractional derivative for current voltage relation
of a fractional capacitor.

79. Capacity, charge, current for constant capacitor vis-a-vis
time varying capacitor to a step voltage excitation

Classical ideal capacitor e Fractional capacitor
y
Von Von

v(t) = Vg, 9 (1) v(t) = Vg, (1)

c(t)=K,r%, 1>0
K,=C,(Td-a)y

ra ————— q(t) = fées

q(t) = CV»

Cave

i(t) = CoVgnd (1)

1=0 t=0
1 — i —

Note that charge is lagging voltage in a Fractional Capacitor and also charge goes
to infinity without reaching a steady state charge

80. v(1)=V, cosoy appliedat 1=0 toa fractional capacitor with (1) = k .1-*
q(t) =c()* va) =(K,t)*(V,,cos@pt) — L {qi} = L {e()* v)} Q(s) =(C(s))(V(s))

a | Vas E © JETA eto; simon)

a 2: 02 2 jt
s Ss +05 % s +0 o

q) MELO (pa sin ot) Ya (D?sino,t)

Taking inverse Laplace transform we get o, 0,
0 0

Using the formula o Dé sin(x) = sin (x + #)+ 4 - =

We get > asta af x
goa Ve Ente NE (oye)

on a“ | oh Ta) TCa-2)

(CN CEE +)
„Cr a ox CAPE je
at steady-state lim... a.) = sin (oy +)=V,K,P1=0)0)" sin ($+ (oot 3+
0
= V,,K P(1=0)03" cos (ot -$*) = Q, cos(oyr- 22)
We get the steady state current as
i= aie = (Uco; ‘sin (09,1 + 2)) = V,C,of cos («opt +22)

; 2 2

For ideal capacitor c(t) =C,6(t) we get

q(t) = c(t) * v(t) = (C,5(1))* (V,, Cos @ of) = C,V,, COS Wot

Q, =V,,K.Pd- aos =V,C,0p

ic) = V,,C,@, cos (ot +4)

81. vi) = V,,cos@ ft, o(t)=K 1%, C,=K,T(I-u)

ic) =V,C,ojcos(oy+ 2); q(t) = V,,C,@9 | cos (oy - RE

ao

q(t) = 03 C, ( (+4) +c0s (2) (2s) - 1) (Cm) (vo)

C(v) = wf"C, [sce )+ cos (2) GT) V,=1V, C,,=IF/s", @,=1Rad/s

wi

—C, 220 pF
NEC, 1 F
—EDLC

0 0.1
Voltage /V

We obtain the Capacitance dependence on voltage for Fractional Capacitor

One experiment validating the formula q = c * v in simulation and experiment on
CPE (fractional capacitor) and ideal loss less capacitor-by A.S. ElWakil et al.

œ
o
3

£
S

D
©

N

LE
E
5

7

82. f =0.006Hz @ ~ 0.04Radians / sec
Y- a~09
sin81°
1x (0.04)
arad

= 18Farad /sec°!

087 a ¿Cral,., y + 25Farad
Im Z() «= 0.88Rad /sec R, = 900mQ
e a=05 a

07 o=0.88Rad/sec; a=05, Y=0.30

C, =K,T(-a)= sin E
_ sin 45°
0.3 x (0.88)°°
Cu = 3.78Farad

Re Z(o) R, =620mQ
Q

We write equivalent Farads as

= 2.51Farad /sec”

= n°1! ar
Cu =07C, esc E

Ca
to get equivalent Farads from C, = Farad/sec!*

83. q(t) =c(t)* v(t) = (Kat e y. cosoyt)

(0% (op
T(-a) T(=a-2)

a ij
= Vala d*sin on] _ VaCs {sin (r+ 98)
a La 0 2
® dt Oo

o
Mili
"sin (oot + 22)

q(t)

lim 7, q) =

Ea
Do

=V,K,T(-a)o¢" sin ($+ (0,1 -3+2))

= V„K,TU-o)og cos (oot -£ 2) =Q, cos (oot - 2)

Q, =V,K,F(-0)05” =V,C.09 |

mao

Anew way to describe Loss tangent as a phase difference between charge store
and voltage (-a)r
P= >

84. We have applied the new formula of charge storage i.e. via convolution operation,
of time varying capacity function and voltage stress for a fractional capacitor. This
new formulation is different to the earlier used formula of multiplication of capacity
and voltage function. We have discussed various results obtained for different
excitation voltages- sinusoidal, step and ramp; and also revisited the impedance
formula, and Nyquist’s diagrams and the concept of loss-tangent and memory in
fractional capacitors. We have given interpretations of the various theoretical
results that were obtained by this new formulation, thus verified the usage of this
new expression. Some of the results we discussed in this presentation.

We have not yet applied this to practical cases in our project as this theoretical
development very new, but plan to have further experimental and theoretical
studies on this new formula, like application in estimation state of charge (SOC)

in supercapacitors charge discharge applications, parameter extraction by
Hysteresis plot where use this formula for supercapacitors, the insight into new
way of defining loss-tangent as we obtained from this formula, and applications to
several dielectric relaxation experiments where memory is observed.

We have also addressed the issues about fractional law of cooling, and tried to get
The laws, by use of Riemann-Liouville and Caputo’s fractional derivatives.

We have briefly introduced application of Fractional Curl and Fractional Diffusion

87. PSE is following

(l+x)* =l+ax+

a@-D 2 a(a-lo-2%) a, _¥ (exe
2! 3! Sin

Where (a), = ala -1)...(@a -n+1) is falling factorial

CFE is following

def

(1+x)° =

l-a@

1+(4)(@ +1)

1-(4)(a-1)

x

1+(4)(a +2) m
GD

88. The fractional Laplace operator /s is semi-derivative operator will have CFE
approximation, is represented as ratio of rational polynomials in s

Another way of writing CFE is following

CFE = =
any E 1 ax (l+a)x (l-a)x (2+0)x (2-0)x
l-1+ 2+ 3+ 2+

CFE converges in the finite complex plane along negative real axis for X = —©
to x=-1

For obtaining rational approximation of Js put in CFE X = s — 1 and a =

CFE for four number term approximation for Js is

Js = 55? +10s+1

2

si+10s+5

89. No. of terms in CFE Rational approximation for alls
approximation
D

?+355+7

115° +1655* +4625° +3305’ +55s+1

s’+55s'+330s° + 4625? +1655+1

CFE gives approximation for fractional Laplace operator in terms of ratio of
rational polynomials. It has got interlaced poles and zeros and depending upon
the usage we can use the number of terms for approximations.

The above is semi-differentiation operator. The semi-integration operator will be
reciprocal of the above ratio.

This approximation in Transfer Function form can be realized electronically

90. PID Transfer function C(s)=k,+ks'+k,s', C(@)=k,+ i(k,o ko" )
Zero-phase point & minimum magnitude is 4C(w)=0, |C()|,,, =20logk, at

60dB

I
5204B /decade I

40dB

20dB
[C(o)

a

OdB £

+20dB / decade

Î 90°

zC(0) Ge

-90°

it

)

=0

Zero-phase point

0=0,

A co), =04B

kel
p

91. Let the plant transfer be
NG (s)=—“—, G,(@)=s

°42E0,8+07 2-04

The controller be classical PID i.e.
C(s) =k, ks +k,s', C(@)=k,+i(k,o-k,o"')

“unity-feed-back” control system ZC(o) =0, |C(@)| =20logk, at @=0,,

Therefore depending on plant transfer function’s gain and phase plot G , (s)
we select the PID parameters k,;k , to selectdesired o,, = Jk,/k,

and required gain at this © =©,,,,

The selected © ,;, is in the frequency range given by plant transfer function G,(@)
that is what we want as overall gain and phase plot's shape for Go, (5) =G,(s)C(s)
This gives us overall gain cross over frequency © „. adjustment i.e,C,,(0)|,, , =0dB
deciding fastness of close loop response; and at" o se the phase margin '
ZG,,(o)|, „ =-180° +9, stating the damping of close loop response.

With the PID we have limited freedom of loop shaping with controller giving gain
slopes of +20dB / decade and phase boosting or bucking as +90”

92. It does stabilize a plant represented nominally by G,(s) ‚the plant may be
inherently stable or absolutely unstable

The parameters of the controller as chosen k,,k;,k, for PID C(s)= k,+ ks! +k,s
to have unity gain at chosen grain cross over © ; for open-loop TF, i.e.

C (SG, OM =] This is deciding nature of fastness in transient response

pe

and to have desired phase margin at the cross over frequency i.e.
ZC(s)G, (s)| =-7+9, This is deciding nature of damping in the response
= 05

m

High frequency noise attenuation constraint for the upper-bound of the gain
C(io)G,(io)
|? (io )| =|—=— <A © 20,
1+C(io JG „(io ) se

Output disturbance rejection for the lower bound of gain is

|s Go )| = tt — <B o<o,

1+C(i0)G,(io) »

In addition to above points minimizing the “Performance Index” (P.l) say ITAE is
the goal

93. C(s)=k, +k * + kas" 0<(A,u)<2

Parameters for tuning are five k >> ki, ky, A, 4

Increased flexibility in shaping the open loop TF, but at the cost of mathematical
complexity!!

94. C(s)=k, tks" +k,s*; s=io
Clo)=k,+k i" oa") +k (Go)

=k, +ko™ (cos “*— isin 2) + ko? (co +isin =)

u (k, +k,0*cos22+k,0 "cos 2 )+i(ko° sin 42- ko" sin 22)

2 = u = 0.5 we get following formula

ee)

The magnitude function and phase function expressions are following for the
gainsandorderask,=k,=k,=1; A=u=0.5

ef, Eee) k,=k=k;=l; A=u=05

|C(@)),

1B

= 10108 (5)+10108((0+1+/20 y +(0-1) )-1010g0

ZC(@)= tan (eh)

95. For k, =k, =k, =1; A=u=05

The magnitude and phase angle functions are following

[C(o)|,, = 1010g (4) + 1010g (0 +1+ 20) +(0 =) )-101080

4C(o)=tan" (2 sk =k=k ; A=u=05
Will have zero phase angle i.e. 7C(@)=0 at © =1=0,,,

min

At zero phase angle we have magnitude as |Co)| ,, =\3+\2 =1010g(3+,/2)dB, CE

= (o:uVo), (0-1) js mini i i
Also we observe from C (wo) = (ento) “a (=) is minimum when imaginary pa
is zero; and that also corresponds to zero phase. Thus at © = 1 = ©,,;,

we also have minimum magnitude i.e. [Co =101og(3+V2 Jas #0dB; at w=l=0,

For classical PID with k, =k, =k, =1; A = 4 =1 wehaveC(o)=1+i(o-o")
The zero phase angle is at ©, =.[k,/k, =1 with minimum magnitude as
|C(@)|,,, =0dB

96. The magnitude and phase angle function for FOPID are

ICO) = 10log(+)+ 10108 ((o +1+V20 ) + (o -1)')-1010g 0

ZC(o) = tan ! (+) O min =1 for k k, =1,A=y=0.5
At low frequency w << 1 we have following limit

lim ¿o [C(0)|,, = lim. ( Olog (4)+ 10 log ((o +1+ V20) +(@-1) ) -10log o)

=10log (3)+ 10log2-10log® = -10log®
Low frequency <<1 asymptotic slope is slope for gain is-10dB / decade
At low frequency the asymptotic phase angleiim,,, <C(@)=lim,,, tan x) = tan) =-45
At high frequency « >> | we have following limit
lim y, [C(0)|,, = lim ur, (20108 (+)+ 10108 (o +1+420) + (0-1) ) -10log 0)

FA
=10log (+)+ 10108 (20° )-1010g@ = 10 log (+)+101og 2 + 20log@ —10log ©

=10log@ at high frequency asymptotic slope is +104B / decade

lim ,,,, ZC(@) = lim „,, tan” (y = tan”'(1)= 45° is the asymptotic phase angle]

97. C(s)=k,+ks “+k s
kK, =k, =k, =1; A,u=0.5
40dB

20dB

10dB

10108 (3+ V2 )aB

45°

98. Io)
+= Let the plant transfer be
6) = 2%

ES 3» G,(0)= ETS
“unity-feed-back” control system 5° +2£0,5+ 0, 2&0, )

The controller be FO- PID i.e.
Cs) =k, +k s+ ks,

s=i0

Clo)=(k, +k,o? cos E+k0" cos) +i

k,o* sinÆ-Ko "sinÆ)

+80’ cos&+k, m" cı

+(k,o* sin ko "sin &

=0, at ©=0, Co), =20 log(k, +k/0f,

in COS ZE + ko... COS
For 2. u =1.0 PID case we get on =3/x,/k, |C()),,,, =20logk,

For 2,1 =0.5 FOPID case we get o, =k,/k, |C(o),,, =20108(£, + +)

1 |c(o),,, = 2010, =10log(3+ V2)
With the FOPID we have unlimited freedom of loop shaping with controller giving

gain slopes of+20 udB / decade and -20 dB / decade and phase boosting or bucking
as -/90°; +90"

99. HW Bode in 1945 expressed : “wish if | could have circuits doing half -integration
(or half-differentiation)” -with reference to enhancing stability margins in feed-
back controls of amplifiers.

The standard block diagram representation of “unity-feed-back” control system
we take as-
R(s) Els) s Y (s)

Bode's ideal loop

We note that s “ is fractional power Laplace variable-here indicating fractional
integration

100. Say Go, =C(s)G,(s) = + Le. one whole integrator ?

The close loop response is G,, = Gor _ = AN = Y (s)
1+Go, s+k R(s)

“unity-feed-back” control system
For unit step input we have R(s) = + thus 7 ( k Il 1 ) 1

(> s+k

s stk

s

Laplace inverse of Y (s) gives y(t) =1- ent

Here we will never attain final-value i.e. one, and overshoot is 0%

Say Go, = C(s)G,(s) = + ie. two-whole integrator
The close loop response is G. = Go. _ _ik _ YG)
SC 14+Go, s'+k R(s)

For unit step we have R(s) = 1+ thus Y(s) = L L)- E A
ES so+k s +k

s s
Laplace inverse of Y (s) gives y(t) = 1- cos (Ve)

Here we will have attained final value but with sustained oscillations around it,
overshoot 100%

101. Bode's ideal dream is have loop transfer function as Go, =C(s)G,(s)=-4; @-1.5
That is consider tuned open loop transfer function as fractional integrator (1.5)
Then the closed loop response is 7

G = Go ke kG) OF {ee

1+Go, I+ks“ só +k R(s)

For unit step input we have R(s) = + el
we have the output as: 'unity-feed-back” control system

y(th=L uE TOA (res

cL

Overshoot is approximately
, M „= (a -1)(0.8« -0.6)
kr)
E, (=kt*) = y EDT a=15, M,=03
= T(an+1) #
For order as one and two i.e. integers we have
E,(-kt) =1-kt+ So- ¿e

E,(-kt?) =1- +
Vey. (vay
=1-/ a) +! a) =....= cos(vkr)

These two above cases we have discussed in previous slide

Mittag-Leffler is higher transcendental function

102. Close loop response to unit step is y(t) =1-E,

a

16_18 20
_—

103. The parameters of the controller as chosen k,.k,,k,,), u for FO- PID :C(s) =k, +k,s * +k,

To have unity gain at chosen grain cross over ®,. for open-loop TF, i.e.C(s)G ,(s )| =0dB

To have desired phase margin at the cross over frequency ZC(s)G »(8)| LOT +,

Robustness to the gain uncertainty represented by a constant phase angle of open-loop
¿[Aca y6, a) =0

transfer function at and near gain-cross over frequency
do

Ensuring a desired phase margin for several frequencies near ®,. i.€.@, e [ea 550% sd]

Z (Clio, )G, (io, )) =-T+,, o, € [o
The angle of open-loop transfer function’s derivative _d_
do,

|

(2(cio,)6,(e,))

High frequency © > o, noise attenuation constraint for the upper-bound of the gain|T(io)|< A
Output disturbance rejection for the lower bound of gain is |S(io)|<B; @<o,

Plus minimizing the Performance Index (P.) say ITAE

This is mixed frequency & time domain optimization-requires multi-objective optimization

104. Open-loop phase

“Robustness” to uncertainty in gain/phase
of Plant TF is isodamping
Mp

105. Rm añ
(st 2) (+ z we Me N Riz a
(s+ p,)(s + p,).. tt > E Lees
so 1 “ " e -vec
= | 1
a | Phase plot of semi-differentiator with phase angle
Î | 45 degree for frequency chosen band; realized
* à ! ] via six op-amp active filter circuits.
so! | ]
]
E | à =]
RER Ra
i a Pi ci - = - —
1 22597 60406 in 264070 500% sanx 500K
2 15955 42768 in 37.306 50k esk 100%
3 11295 30275 sar nan EN ak 20K
4 799.65 283 sur 1094 20k 18.39k 20K
5 5661.1 15173 1005 1051k EN 1764 20K
6 0078 107220 tok 185% 2x 2495 sok

106. Liouville’s postulate true for Steady-State _., D” cos(x) = cos (x + an)

Thus the generalized differential operator simply shifts the phase of the cosine function and
likewise the sine function by à x (90)°, that is in proportion to the order of the differentiation.
For differentiation the process advances the phase, needless to say the integration makes
the phase lagged.

We test by giving a sinusoid at the input of the circuit and measure the output and record the
phase lag or lead, depending on the fractional order.

Time(Sec)

CRO Output of fractional order differentiator
circuit for Ye order

107. Integral / chor Output
Gain de) I

Output Voltage

Input Step.
Voltage

108. C(s)=K, +5 +5°°
= 1948300 oy
5° +670s +1948300 s"+2%60,5+0,

o, ~1400; € ~ 0.25

ck” control system

Values of Close-loop PID Close-loop FO-PID
K, (gain

% Peak % Peak ove
overshoot simulation r root
simulation results ha e results

»

100 € 60

Fractional order PID are more robust to uncertain plant transfer function as compared to
classical PID

109. It is like polynomial fitting by minimizing least square error (LSE)

Say we have experimental data at points x ,,X,, as a set containing

Vis V2 V3,
Our objective is to fit a function y = f(x) so that LSE at all points is minimum; and
we restrict only to say second order polynomial
This is standard curve fitting say we got LSE minimized by f (x) = ax? +bx +c

and minimum value of LSE is gotas say LSE,,,=L,; L, #0

Here we have restricted to find three parameters namely a, b, c by restricting the powers of
the monomials of x to one and two.

We can always get a lower LSE by using fractional power monomials

g(x) =ax*+bx’ +c giving LSE,,, =L,; L,<L,

Here we have given to find five parameters and this extra freedom is still minimizing LSE

110. To get a function y=f(x)=ax+b to minimize LSE for given n-points
LSE =>" (y,- (ax, +b))

0

LSE = -2) xi: - (ax, +b))=0

Oa

0
ob

Disks on B Zi:

LSE = -2) (y; - (ax, +b))=0

b

Dur Dak A
Now compute LSE for values of obtained a and b, and for LSE + 0 and then we proceed
with the obtained values a and b,wetry f(x) = ax“ + b with aeR «-1
=) (y, —(ax7 +b)) such that (LSE). <LSE

and compute new LSE as (LSE)

So we may have fractional power curve which further minimized the LSE as compared to
linear fit
E, tk wok?

Similarly the fractional order controller i.e. polynomial in fractional power
1

can minimize further a Performance Index (i.e. similar to LSE) than k, +k,x +k,x~

Read x as s (Laplace variable) for PID and FOPID; and LSE is similar to Performance Index

111. We would fit y=f(x)=ax+b for six given points n = 6 namely

{x}, =40,0.5,1,1.5,2,2.5) y JO = {-0.43,-0.17,3.12,4.79,4.85,8.69}

Using the formula to find a, b i.e. R y
“ isl

PIE

Weget a=3.56; b=-0.97

Value of minimized LSE in this fit case is | SE, = y (y, -G.56x, — 0.97) Y =9.47

Now we choose the modified function with fractional power f.. (x)=3.56x”* - 0.97

new
New LSE ¡.e.LSE,,, =D”, br -(3.56x, "0% Mm) for some values of « close to 1.0 are following
= 1.05 LSE =10.93
a =1.00 LSE = 9.47
a = 0.95 LSE = 8.56
a = 0.90 LSE = 8.14
This example shows that LSE obtained by linear fit gets still minimized by fractional power
polynomial -in this case «=0.90 . Search program required to do so.

112. Classical PID controller C(s) =k, (1+T,s +(T,8)') Let the plant be G,(s) = N(s)/ D(s)

In our standard unity Feed-Back control system scheme for a step input in set-point r(t)

Le. R(s)=} the error Els) is: R(s) _ 1
1+C(5)G,(s) s(1+C(9)G,(5))

_ T,D(s) _ Bs)
T,sD(s) + N (s)k(TT,s°+T,s+1) A(s)

E(s)= R(s)-Y(s)=

0=(k,,T,T,)

“unity-feed-back” control system

Minimize PA Le. |, = || e*(0,0dt==35 | 7 E(0,s)E(0,-5)ds; 0=4k,,T,,T,)

Astrom's complex integral algorithm solves Parseval's relation above-for poles in LHP
of s-plane

, oJ
n=1; = à With this we minimize J, by — =

> 2dod 00;
n=2; Ju, = ido reido toget k,,T,,T,

2dod\dy
Now search for still minima withC(s) =k, (1 +7," +(T,5) 2); a,ßeR toget
k,»T,,T,,@,ß

p?

113. DC Motor ... Implemented tuned Digital PID or Digital FO-PID

ome ¡BERG

Assembly

|

Generator Load
Bank

Generator is Load for Motor

114. A PID and FO-PID is tuned to have minimization of Performance index (P.) for a plant i.e.
DC Motor Transfer Function series with Pulse Width Modulator Circuit

Tuned PID C(s)=115.6+0.22s '+1.6s'
Tuned FO-PID C (s) =15.2 + 0.045 '*+2.45'*

Minimized P. SN. Controllers ITAE IAE ISE

1 PID 3421 46.75 7454
2 FO-PID 14.94 20.85 6086

Calculated Controller Effort SN. Controllers ITACE IACE ISCE

1 PID 46.56 51.35 10080
2 FO-PID 20.85 50.63 6663

Does Lesser Controller Effort translates as Energy/Fuel Efficiency ?

115. LS.) == | Tuned PID
JE | k,=115.6, k,=022, k,=1.6

u=1, À=1 ITAE pj = 34.21

Tuned FO-PID

k,=152, k,=0.04, k,=24
u=l4, À=12 ITAE,, =1494

‘nin

05 Time(s) 1 15

Observation is larger control output to a same step demand in case of PID. This is recorded
for a manual step demand. Say at a set speed 1000RPM, there will always be this feed back
control action due to deviation in speed due to continuous disturbance at input voltage, field
voltage, non-linearity of load inertia friction etc; thus average u (1) in case of PID will be
higher than in case of FOPID Control

116. CONTROL EFFORTS
CONTROL SIGNAL VS. SPEED

CONTROL
CONTROL SIGNAL V;

with PID with FO-PID ES rm

700 900 110

No-Load Condition Loaded Condition

The average controller output voltage is greater in case of PID as compared to
FO-PID at a set-speed for unloaded and loaded cases

117. Referenc 500 rpm 700 rpm 900 rpm 1100 rpm 1300 rpm
Controller PID FOPID PID FOPID PID FOPID PID FOPID PID FOPID
vw) 612 642 387.8 TS 118 144 147 175
1A) 0957 0747 1.08 77 12 0.916 137 0989 15 12
Paw) 585 47 9482 7893 1416 10808 197.28 14538 2684 210
UY) 0743 06 0.829 \760 0908 0.841 0974 0882 10 0921

18.03% less Pand 16,75% less Pand 23 sPand 26.30% less Pand 21 ss P and
Remark 8.20% less u(t) 8.32%less ult) % less ult) 9.44% less u(t) 8:81% less uit)
in FO-PID in FO-PID in FO-PID in FO-PID in FO-PID

No-Load Condition

300 rpm 700 rpm 300 rpm T100 rpm 1300 rpm
2
vr 63 662 968 952 160
LÍA) 1,85 1.52 220 176 : 9 239
POW) 120.81 100.62 21296 167.55 242 3824
0). 0.197 0314 0265 9 0510 0436
ss Pand 21.32% less Pand 22.34% 19.05% less Pand 18.51% less P and
Remark 59% less u(t) 15.61% less u(t) 15.88% less u(t) 14,51% less uit) 14,35% less u(t)
in FO-PID in FO-PID in FO-PID in FO-PID in FO-PID

Loaded Condition

Average input Armature Power (P) is less for FOPID for all speed settings

118. ENERGY EFFICIENCY ENERGY EFFICIENCY
Power Vs. SPEED POWER VS. SPEED
with PID_ —with FO-PID with PID —with FO-PID

1100 1300 5 700 900 1100

No-Load Condition Loaded Condition

Possible to have Fuel/Energy efficient controls by fractional calculus

119. Demonstrated this FOPID & Fuel/Energy Efficiency

E = Magnetic Levitation Control
+ x | Inverted Pendulum Control

Magnetic Levitation Brush-less DC Motor Speed Control

Inverted Pendulum

120. Many experimental evidence show that natural dynamics follow fractional calculus
Feed-back-Control system is “servo” mechanism. That is the plant being controlled follows
instruction from controller-thus the plant being controlled is servant to the controller
Representing a plant with classical Newtonian dynamics with classical calculus is far from
reality though it serves the purpose-but is prone to uncertainties in gain and phase

Thus if the plant dynamics do not actually follow the language of Newtonian calculus, how
can plant obey (efficiently) the command of the master i.e. controller; if we use classical
calculus (as commanding language) for controller design?

In France talk in French-that will give you ease and efficiency in people to people interaction!
Like if a plant works on exponential law of growth gets better stabilized via use of logarithmic}

logic was first conjectured in 2004 Vision 2020, and at ICONE-13 2005 (Beijing)-this leads to
Non-Newtonian Calculus i.e. Ratio-metric Derivative!

In a way using; Fractional Calculus gives a better way to communicate with the plant
dynamics (which usually is of non-integer order) therefore yielding greater robustness,
lesser control effort and higher efficiency in doing the stabilization job-leads also to
“Energy/Fuel Efficiency” and greater “robustness”

121. Look at fractional complex order calculus and shape the Open Loop Transfer
Function as G,, = G,(s)G,(s) = ks“?

Here the real part and the imaginary part provide the capability of independent
design freedom to have ability to deal with uncertainties in gain and uncertainty in
phase of the plant transfer function.

Can work on new sets of Performance Indices as fractional order integral time

absolute error, or fractional order integral time absolute square control effort, plus
many others different from classical ones.

Development of separate theoretical control sets for fractional/complex order
controllers only for Energy/Fuel Efficiency and applications in Power Electronics
Circuits.

123. Int. J. Electron. Commun. (AEÜ) 78 (2017) 274-280

Jaques, C. (1889) Annales de Chimie et de Physique , 17, 384-434.

Schweidler, E.R. (1907) Annalen der Physik, 329, 711-770.

Jonscher, A.K. (1983) Dielectric Relaxation in Solids. Chelsea Dielectrics Press Limited.

IEEE Trans on Dielectrics and Insulation, 1,826. (1994), S Westerlund, L Ekstam

American Chemical Society-Memory of Electric Field in Laponite and How It Affects Crack Formation: Modeling through
Generalized Calculus: pubs.acs.org/Langmuir-DO!: 10.1021/acs.langmuir.7b02034,

Physica Scripta. Vol. 43, 174-179, 1991, S. Westerlund

Physiologists 12, 919 (1969)

Notices of AMS 48, No. 1, 9-16 (2001), A. Daigneault and A. Sangalli.

Phys. Teach. 37, 5512 (1999), J. O’ Connonel

Functional Fractional Calculus 24 Edition, Springer-Verlag

Times of India Nov, 15, 2016: https://timesofindia.indiatimes.com/ city/nagpur/ VNIT-develops-energy-minimizing
-controller-transters-tech-to-BARO articleshow/55444793.cms

A New Look at Formulation of Charge Storage in Capacitors and Application to Classical Capacitor and Fractional
Capacitor Theory; Asian Journal of Research and Reviews in Physics; 1(3): 1-18, 2018.

Theoretical Verification of the Formula of Charge function in Time of Capacitor (q = c* v) For Few Cases of Excitation
Voltage; Asian Journal of Research and Review in Physics (2019)

Theoretical verification of formula for charge function in time q = ¢ * v in RC circuit for charging/discharging of fractional &
ideal capacitor; Asian Journal of Research and Review in Physics (2019)

A new method of getting rational approximations for fractional order Differ-integrals, Asian Journal of Controls, 2016.
Berberan-Santos Mario. N, Analytical inversion of Laplace transforms without contour integration: application to
luminescence decay laws and other relaxation functions, Journal of mathematical Chemistry, Vol. 38, No.2 August 2005.
Energy/Fuel Efficient and Enhanced Robust Systems Demonstrated with Developed Fractional Order PID Controller, Energy
8 Research Volume-7, Issue-1 1000182 (2018).

Szekely G J. Half of a coin: Negative Probabilities, Wilmott Magazine pp 66-68, 2005

An insight into Newton's Cooling Law using Fractional Calculus" Journal of Applied Physics, 123, 064901 (2018)

J.A. Tenreiro Machado. Entropy Analysis of Systems Exhibiting Negative Probabilities, Communications in Nonlinear
Science and Numerical Simulation, vol. pp. 58-64, 2016.

Nonlinear charge-voltage relation in constant phase element; A S Elwakil et al.

Kindergarten of Fractional Calculus

126. MoU with YASKAWA-Japan & VNIT-Nagpur using Developed
FOPID for “Robust & Fuel/Energy Efficient Controls” for
Industrial Servo Drive Application

Ñ MoU signing at Nagpur on 13-12-2018

Really blessed to see Industry Coming to use this conjecture of mine-for Industrial
Drives about Robust &Fuel efficient Controls in my living life-time !!

127. aes

Thank you NIT-Silchar
for
giving the opportunity to not highly qualified
person like me
to present this academic topic

Now I hope Fractional Order System
is not an absurdity instead a real reality
This is tip of ice-berg
I have to learn what is fractional d / dx is...and
learn again and again