National workshop “Intelligent Sensing, Instrumentation and Control” NIT Silchar Assam 30th January-3rd February, 2017
Applications of Fractional Calculus in Technological Development (Part-2: Control Science & Engineering)
Shantanu Das Scientist, RCSDS E&I Group BARC Mumbai, Senior Research Professor Dept. of Phys, Jadavpur Univ (JU), Adjunct Professor DIAT-Pune UGC-Visiting Fellow Dept. of Appl. Math Calcutta Univ.
[email protected] http://scholar.google.co.uk/citations?user=9ix9YS8AAAAJ&hl=en www.shantanudaslecture.com
Contents Part-A: Part-B: Part-C: Part-D: Part-E: Part-F: Part-G:
Part-H Part-I: Part-J:
Fractional Order PID and Bode’s Dream Laplace Inverse of Irrational Transfer Function Use of Bode’s dream of Open Loop Transfer Function to get enhanced robustness (iso-damping) for uncertain plant transfer function Using Tuned PID for governing uncertain Transfer function of Plant Using Tuned FOPID-for governing uncertain Transfer function of Plant Enhances robustness Minimization of Performance Index better by using five degree of freedom of FO-PID as compared to three degree freedom of PID is like “curve-fitting” Results on minimization of Performance Index better by using five degree of freedom of FO-PID as compared to three degree freedom of PID in minimizing Control Effort & getting Fuel/Energy Efficiency --Practical Results on DC Motor Speed Regulation More Experiments results to show Fractional Order Control & its advantage in Fuel/Energy Efficiency Circuit implementation of Fractional Order PID (analog and digital) An appeal to industry & research institutes in India
Dedicating this part to Prof H. W. Bode
Hendrik Wade Bode 1905-1982
Without Bode’s contribution we will not be perhaps talking feed back control systems H W Bode remarked in 1945-wish I have fractional integrator circuits
Part-A Fractional Order PID and Bode’s Dream
Why should we use Fractional Calculus in controls? No industry uses Fractional Calculus in control system That does not mean that we should not use: rather ask why India cannot be first to have industrial usage of Fractional Calculus based Controls ? Classical controls is in form of PID (Proportional Integral Derivative) controllers exist since 1910-in Laplace form the controller is i.e. classical lead-lag compensator is: kd s2 k ps ki ki C (s) k p kd s s s Its tuning methods i.e. “Ziglers-Nichols” is well proven and exists since 1942 Therefore why should we burden ourselves with Fractional Calculus based controls or call it Fractional Order PID (FO-PID)? That gets represented as following transfer function k s k s ki ki d p C (s) k p kd s , 0 , 2 s s with irrational Laplace variable ! PID was invented in 1910 by Elmer Sperrys for ship auto pilot So are we questioning PID ?-certainly not We will carry on the rest of the discussion in simple way-without rigor of mathematical deliberation on Fractional Calculus-but we will advance from what we already know from our knowledge of classical controls –that we have learnt in our colleges. So relax & enjoy
By using Fractional Calculus is it better way of communicating with Nature or with dynamics of system? 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) 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” Int. Conf. on Nuclear Engineering ICONE-13 Beijing China-Fuel Efficient Controls
Bode’s dream H W Bode in 1945 expressed : “wish if I could have circuits doing half differentiation or half integration “–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 R (s)
E (s)
Y (s)
U (s)
G p (s)
C (s)
Bode stated that ideal open loop response be of G O L C ( s ) G p ( s ) k s R (s)
G (s)
k s
1 .5
Y (s) Bode’s ideal loop
We note that s is Fractional Power Laplace variable-here indicating fractional integration is also irrational monomial
Functional Fractional Calculus
From fractional Laplace operator we find fractional derivative & fractional integration operation Classically we have
f (1) (t ) sF ( s ) f (0)
Say for case of RHS if we have
f (t ) F ( s )
where
s F ( s ) f (0)
for 0 1
( ) Then we relate to f ( t ) s F ( s ) f (0 ) as
th derivative of f ( t )
For - negative we have order integration i.e. with
f
( )
(t ) s F ( s )
So there is possibility that we have in between operations for integration & differentiation, 1 1 11 like half, one third etc D 1 2 f ( t ) D 3 f (t ) D 2 f (t ) D 2 f (t ) We are having thus fractional Laplace variables like s
1
2
s
1
2
etc
Assume presently that we have fractional differentiation and integration then how can we use the corresponding fractional Laplace variable s in controllers ? Let us proceed Kindergarten of Fractional Calculus
Why Bode wanted not to have ideal Open Loop Transfer Function as Classical Integrator of order about 1 or 2 ? Say G O L C ( s ) G p ( s ) k s 1 , i.e. one whole integrator the close loop response is
G CL
G OL k Y (s) 1 G OL s k R (s)
For unit step input we have R ( s ) 1 s Laplace inverse gives y ( t ) 1 e k t
1 thus Y ( s ) k 1 1 s s k s k s
Here we will never attain final-value i.e. one, and overshoot is 0% Say G O L C ( s ) G p ( s ) k s 2 i.e. Two-whole integrator the close loop response is
G CL For unit step we have Laplace inverse gives
G OL k Y (s) 2 1 G OL s k R (s)
1 k 1 s 1 thus Y ( s ) s s2 k s s s2 k 1 y (t ) 1 c o s k 2t
R (s)
Here we will have attained final value but with sustained oscillations around it, overshoot 100% Functional Fractional Calculus
Why Bode wanted ideal Open Loop Transfer Function as Fractional Integrator of order about 1.5 ? G O L C ( s ) G p ( s ) ks ; Bode’s ideal dream is have loop transfer function as That is consider tuned open loop transfer function as fractional integrator
k 0
G k s k Y (s) O L Then G CL 1 G OL 1 ks s k R (s) 1 For unit step input we have R ( s ) and the output is: s Overshoot is approximately k 1 1 y (t ) 1 E kt M p ( 1) 0.8 0.6 s k s 1.5 , M p 0.3 Mittag-Leffler Function of one-parameter is following
E ( kt )
n0
( kt ) n ( n 1)
1
E1( kt) 1 kt
2
E 2 ( kt 2 ) 1 1
kt 2 2!
( n 1) n !
k 2t 2 2!
kt 2!
k 3t 3 3!
k 2t4 4!
2
. ... e k t
... .
kt 4!
4
.. .. c o s
kt
These two above cases we have discussed in previous slide Functional Fractional Calculus
Part-B Laplace Inverse of Irrational Transfer Function
About poles of closed loop irrational transfer function G CL
G OL k ; 1 G OL s k
k 0
For 1 and 2 we have poles as s 1 k and s 1 , 2 i
1
2
s1 k
k
depicted as
s1 i
s i 2 For
3 2
k k
we have multiple poles in different Riemann-Sheets s k 0
e , m 0 , 1, 2 , .. .. s k ke At primary Riemann-sheet a r g s we have two roots given as
s1 k s2 k
i ( 2 m )
2
3
2
3
e i 2 e
/3
,
1
i 2 m /
Impulse response g C L ( t ) 1 s k k
Im s
m 1
plus there are infinite roots in several secondary sheets
s k
m 0
;
i 2 / 3
gives
s1 Re s
Branch Cut
k t 1 E ,
k t
s2 Kindergarten of Fractional Calculus
How do we get Laplace Inverse of Irrational Transfer Function? 1. Power Series Expansion and then using term by term Laplace inversion. The series may converge to close form function of Mittag-Leffler type or any higher transcendental function 2. Contour integration on complex plane –on branch cut Primary Riemann-Sheet Here we get integral representation in closed form
3. Berberan Santo Method of complex integration but without contour integration. Here we get integral representation in closed form
Kindergarten of Fractional Calculus
Contour integration recall k k s 2 2 n n2 ( s s1 ) ( s s 2 ) Two roots of rational polynomial in Primary sheet are s 1 0 i 0 ; s 2 0 i 0 i 1 1 y ( s ) li m y ( s ) e st d s ; F ( s ) y ( s ) e st 2 i 0 ; i 1 F (s )d s C A A 2 i From Cauchy-Riemann Theorem we write Let us find the Laplace inverse of
s1
B
s2
y(s)
C A B C
F ( s )d s 2 i
s1 , s 2
R e sid u e s
Residue Calculation is R esidue lim ( s s1 ) F ( s ) lim ( s s 2 ) F ( s ) s s1
s1 , s 2
s s2
ke s1t ke s 2 t ( s1 s 2 ) ( s 2 s1 )
C
ke ( 0 i 0 ) t ke ( 0 i 0 ) t ke 0 t sin 0 t 2 i 0 ( 2 i 0 ) 0 Integration on large semi-circle for F ( s ) is zero, as its radius goes to infinity so i
1 k 0 t y ( s ) y (t ) lim e st y ( s )ds Residues e sin 0t 2 i 0; i 0 1
Inverse Laplace of irrational function by integration in complex plane Contour integration method y (t )
1
k s ( s k )
1 2 i
i
i
k e st ds s(s k )
k e st F (s) s(s k )
k 0
2 m m 0, 1, 2,... s ke The power function of non-integer order that is s k ; a multi-valued and we have standard branch cut The standard branch cut in complex plane is (,0]
Pole at s 0 and
i
Apply Cauchy Riemann Law on closed contour Integration contour C is A B C D E F A
Im( s ) A
Branch Cut Apply Residue Theorem
C
1 y (t ) 2 i
C
B E
D
i
lim
0 ;
e st y ( s ) d s
i
F (s )d s B C 1 F (s )d s C D 2 i F ( s )d s Re( s ) D E R e sid u e s
F ( s )ds 2 i Residues of poles
On AB and EF for large R integration is zero as y (s) goes to zero at infinity
F
Kindergarten of Fractional Calculus
The result y (t )
1
k s ( s k )
1 2 i
lim
0 ,
i i
k e st d s ; s(s k )
k 0;
k e st F (s) s(s k )
D C E 1 y (t ) F ( s )d s F ( s )d s F ( s )d s Residues s j F 2 i C B D 1 e xt dx 2 i 2ik sin( ) 1 2 Residues s F 2 j 2 i x 2kx cos( ) k 0 x
k sin( ) e xt dx 1 0 x1 x 2 2kx cos( ) k 2 Residues s j F The pole at zero, is outside the cut-contour; so no contribution in the residue calculation For case 0 1 no poles appear in the primary Riemann-Sheet hence For case 1 2 that is s 1 , 2
k cos
Residues
two roots appear in the primary sheet, for m 0 , 1 say
i k s in
Residue
s1 , s2
k
sj
0
F 0 i
in this case residue is
1
e 0 t sin 0 t 0 Kindergarten of Fractional Calculus
0
Part-C Use of Bode’s dream of Open Loop Transfer Function to get enhanced robustness (iso-damping) for uncertain plant transfer function
Gain Phase Plot of ideal Bode’s open loop TF as fractional integrator 1 .5
G OL ks
2 Slope -40dB/decade k1 k2
k3 1 Slope- 20dB/decade 0dB gc 3
Slope -20 dB/decade Phase -90
g c1
degree Gain plot
gc 2
2 0 lo g G O L
00
Phase plot
G OL 900
1 1 .5
1350
m3
m1
m 2
2 1800 With the changes in Gain of Plant ( k ) as the gain cross over shifts the phase margin remains unchanged at 45 degree, for 0 to infinite frequency. The open loop fractional integrator is a “constant phase element” CPE having constant phase of -90 ( ) degrees. Functional Fractional Calculus
Band Limiting for ideal Bode’s open loop TF as fractional integrator 1 .5
G OL ks ;
l h
s i Slope -30 dB/decade
0dB
l
gc
gc
h
Gain plot
k
00 Phase plot 900 CPE of -135degree 135
1 .5
0
m
450
2
1800 The band limiting is practical, and with this we should be shaping our open-loop transfer function gives a constant phase margin between low and high frequency limits-no worry to oscillations instability for wide gain changes or uncertainty-enhancing “robustness” Functional Fractional Calculus
How did we plot Magnitude phase angle for irrational function? G OL ks
i e x p i
put
G O L k ( i ) k G O L ( i ) k
2
c o s
2
2
co s
i s in
i sin
2
2
to get
2 0 lo g G O L ( i ) 2 0 lo g k 2 0 lo g
Slope of magnitude plot
d G O L ( i ) d lo g
2 0
3 , 2
d G O L ( i ) d lo g
30db / decdae
At gain cross over frequency we have Gain as unity or 0 dB 1 0 2 0 lo g k 2 0 lo g g c ; gc k
c o s 2 i s i n 2 s i n 2 1 ta n c o s 2 t a n 1 t a n 2 2
G O L k ( i ) k Phase angle plot
G OL
3 2
G OL 1350
CPE
Functional Fractional Calculus
Plot of step input response of closed loop system with ideal Bode’s Open-loop transfer function as one-half fractional integrator Close loop response to unit step is
3 2
y (t ) 1 E 32 k t
kt3 / 2 52
3
2
k 2t 3 4
k 3t 9 / 2 121
...
Overshoot remaining same to 30% with change in k
This is Robustness to variation/uncertainty in plant gain Overshoot remaining same to with change in k –iso-damping is an important aspect of fractional controls Overshoot is approximately y (t )
Overshoot approximate is
M p ( 1) 0.8 0.6 1.5 , M p 0.3
t M p ( 1)(0.8 0.6) 1, M p 0. 2 ,
M p 1
Uncertain Transfer Function (TF) of Plant wide variation in k TF variation with several operating points of different stack-current and compressor voltage to have control of excess Oxygen ratio –of FUEL CELL 8.35 10 7 G p (s) 2 s 3819 s 4.25 10 5
Gain uncertainty
gc P 10 2 At stack current of 198A Compressor Voltage 160V Pole uncertainty
A second order plant having uncertainty G p ( s )
k L , kU 1 a1 L , a1U s a2 L , a2 U s 2
Fuel cell Transfer Function NIT-W DeitY project
Part-D Using Tuned PID for governing uncertain Transfer function of Plant
What does PID control do?? It does stabilize a plant represented nominally by G p ( s ) , the plant may be inherently stable or absolutely unstable The parameters of the controller as chosen k p , ki , k d
1 for PID C ( s ) k p ki s k d s
to have unity gain at chosen grain cross over gc for open-loop TF , i.e. C ( s )G p ( s ) 1 gc
this deciding nature of fastness in transient response to have desired phase margin at the cross over frequency C ( s )G p ( s ) m gc
this deciding nature of damping in the response high frequency noise attenuation constraint for the upper-bound of the gain T ( i )
C ( i ) G p ( i ) 1 C ( i ) G p ( i )
A
t
dB
output disturbance rejection for the lower bound of gain is S(i)
1 B 1 C(i)Gp (i)
s
dB
In addition Minimizing the Performance Index say ITAE This is mixed frequency & time domain optimization-requires multi-objective optimization
Magnitude Phase plot of classical PID C (s) k
p
k i s 1 k d s
C ( i ) k
p
i k d k i
1
Minimum magnitude and zero-phase at
m in
m in 1 k
p
ki 1 kd
ki kd 1
ki kd 1 Minimum magnitude and zero-phase at 0 .5 1 gets shifted
m in m in k
p
kd 1,
ki
1 2
1 2
Abandoned Thesis
1 2
How does classical PID help in stabilizing the plant? C (s) k
p
k i s 1 k d s
C ( i ) k
p
i k d k i
1
Contributions of the controllers depends on the proportional integral and derivative gains.
We saw in the frequency response of the controllers, the values of the gains or PID parameters is equivalent to the selection of the position and minimum value of the magnitude curve, and the slope of the phase plot at the frequency of minimum magnitude.
However at the high and the low frequencies the values of the slope in the magnitude curve and contribution in the phase curve are fixed.
This is the basis of classical PID tuning with three PID parameters
Abandoned Thesis
Open Loop TF after applying tuned PID Controller 8.35 10 7 G p ( s) 2 s 3819 s 4.25 105
gc 5 103
C ( s ) 0.0082 0.943s 1 0.000072 s
m 1350 Fuel cell Transfer Function NIT-W DeitY project
The response widely varies by changes in the gain of plant
Fuel cell Transfer Function NIT-W DeitY project
Part-E Using Tuned FO-for governing uncertain Transfer function of Plant Enhances robustness
Three parameters in classical PID vs five parameters in Fractional Order PID –gives more flexibility C (s) k
p
kis kd s
Parameters for tuning are five k
p
,
ki ,
kd ,
0 , 2
,
Increased flexibility in shaping the open loop TF, but at the cost of mathematical complexity!!
Abandoned Thesis
Magnitude Phase plot of Fractional Order PID PID
k
p
ki kd 1
FO-PID k
p
ki kd 1
0 .5
Abandoned Thesis
Derivation of Magnitude Phase plot of Fractional Order PID C ( s ) k p ki s k d s C ( ) k p ki (i )( ) k d (i )( )
k p ki cos 2 i sin
k p k d cos 2 ki k
p
ki 1;
k cos cos i k
2
d
2
i sin 2
2
d
sin 2 ki sin
0 .5
C ( ) 1 0.5
i 0.5
1 2
1 2 2 C ( )
1 2
C ( ) ta n
0.5
1 2
1 2
1 i 2
1
1
0.5
1 2
2
2
1
2
1 1 2 Abandoned Thesis
2
What does Fractional PID control do-i.e. it shapes the Bode-Loop? The parameters of the controller as chosen k p , ki , k d , , for FO- PID C ( s ) k p ki s k d s
To have unity gain at chosen grain cross over gc for open-loop TF , i.e. C ( s )G p ( s ) 0dB gc To have desired phase margin at the cross over frequency C ( s )G p ( s ) gc m Robustness to the gain uncertainty represented by a constant phase angle of open-loop transfer function near gain-cross over frequency d C (i )G (i ) 0 u u d gc Ensuring a desired phase margin for several frequencies near gc i.e. i 0 , 1 ,..., N 2 , N 1
C (ii )G (ii ) m
i min , 1 ,.... N 2 , max
The angle of open-loop transfer functions derivative
d C(ii )G(ii di i
min ,1 ,2 ,...N 1 ,max
0
High frequency t noise attenuation constraint for the upper-bound of the gain T ( i ) A Output disturbance rejection for the lower bound of gain is S (i ) B
s
Plus Minimizing the Performance Index say ITAE This is mixed frequency & time domain optimization-requires multi-objective optimization Abandoned Thesis
Bode-Loop Shaping by use of Fractional Order PID DC Motor speed control
G p (s)
Vin as 2 bs c
k p 14.67 ,
k i 0.45 ,
0.45 ,
k d 2.21
1.08
Constant phase margin for several frequencies d C (i i )G (i i d i i
0 min
,1 , 2 ,... N 1 , max
Iso-Damped response at several Gains of Plant Courtesy: VNIT Nagpur
Part-F Minimization of Performance Index better by using five degree of freedom of FO-PID as compared to three degree freedom of PID is like “curve-fitting”
How does extra freedom of tuning (five parameters in FO-PID) achieve energy/fuel efficiency-as compared to (three parameter tuning) in classical PID? It is like polynomial fitting by minimizing least square error (LSE) Say we have experimental data at points
x 1 , x 2 , x 3 , .. .... .... . x n
as a set containing
y 1 , y 2 , y 3 , ..... .... , y n 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) ax2 bx c
and minimum value of LSE is got as say
L S E m in L1 ; L1 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 L S E m i n L 2 ;
L 2 L1
Here we have given to find five parameters and this extra freedom is still minimizing LSE
Basic linear fit and minimizing LSE To get a function y f ( x ) ax b to minimize LSE for given n-points
LSE
n
yi i 1
(a xi b )
2
n L S E 2 i1 x i y i ( a x i b ) 0 a n L S E 2 i 1 y i ( a x i b ) 0 b n xi a n yi n i 1 i1 n n n 2 x i 1 x i b i 1 x i y i i i 1 Now compute LSE for values of obtained a and b, and for L S E 0 we proceed 1 With the obtained values a and b, we try f ( x ) a x b with and compute new LSE as
n
LSE new i 1 yi (axi b)
2
such that
LSE new LSE
So we may have fractional power curve which further minimized the LSE as compared to linear fit Similarly the fractional order controller i.e. polynomial in fractional power k p k d x ki x can minimize further a Performance Index (i.e. similar to LSE) than k p k d x ki x 1 read x as s (Laplace variable)for PID and FOPID; and LSE is similar to Performance Index
Example of basic linear fit and minimizing LSE and then using the fractional order irrational polynomial to still minimize LSE further We would fit y f ( x ) ax b for six given points n = 6 namely i6
xi i 1 0, 0.5,1,1.5, 2, 2.5
i6
yi i 1 0.43, 0.17, 3.12, 4.79, 4.85, 8.69
6 Using the formula to find a, b i.e. i 1 xi 6 x2 i 1 i
a 6 yi we get a 3.56; b 0.97 i 1 6 b 6 x y x i 1 i i 1 i i 6
6
Value of minimized LSE in this fit case is LSEmin yi (3.56x 0.97)2 9.47 i1 Now we choose the modified function with fractional power f new ( x ) 3.56 x 0.97 6
2
New LSE i.e.LSEnew i1 yi (3.56x 0.97) for some values of close to 1.0 are following
1 .0 5 1 .0 0
L S E 1 0 .9 3 L S E 9 .4 7
0 .9 5 0 .9 0
L S E 8 .5 6 L S E 8 .1 4
This example shows that LSE obtained by linear fit gets still minimized by fractional power polynomial (in this case 0.90 . The kind of search program required to do so. This we can do for a case when obtained LSE by polynomial fit is not zero
Basics of minimizing performance index (P.I) Classical PID controller C ( s ) k p 1 Td s (Ti s ) 1
Let the plant be G p ( s )
N (s) D (s)
In our standard unity FB control system scheme For a step input in set-point R ( s ) E (s)
1 s
the error is
1 s 1 C ( s ) G p ( s ) Ti D ( s ) B (s) Ti sD ( s ) N ( s ) k p (TiTd s 2 Ti s 1) A ( s )
c n 1 s n 1 c n 2 s n 2 ... c1 s c0 E ( , s ) ; k p , Ti , Td d n s n d n 1 s n 1 ... d 1 s d 0
Minimize P.I i.e. J 0
0
2
e ( , t )d t
1 2 i
i
i
E ( s ) E ( s )d s ; k p , Ti , Td
Astrom’s complex integral algorithm solves Parseval’s relation above-for poles in LHP of s-plane
n 1; n 2; ...
J 0 ,1 J 0 ,2
c 02 2 d 0 d1 c12 d 0 c 02 d 2 2 d 0 d1d 2
With this we minimize J 0 by to get k p , T i , T d
J 0 0 i
i 1, 2 ,3
Now search for still minima with C ( s ) k p 1 Td s 1 (Ti s ) (1 ) ; ,
Part-G Result on minimization of Performance Index better by using five degree of freedom of FO-PID as compared to three degree freedom of PID in minimizing Control Effort & getting Fuel/Energy Efficiency --Practical Results on DC Motor Speed Regulation
Digital FO-PID for DC Motor Speed Control hardware
Implemented tuned Digital PID or FO-PID
M
G Load
Generator is Load for Motor
B ank
Courtesy: VNIT Nagpur
Change in Duty Cycle due to u ( t ) changes-to manipulate speed T CLK OSC CLK
S T
Q F/F
D G
R Ramp Signal
S
L C
COMP
Motor
T
u(t) Control Signal from Controller PID or FO-PID
Vin 220VDC
u(t)
To raise speed the Duty Cycle is increased so the armature V and armature I
Output of Comparator Reset Pulses to F/F Courtesy: VNIT Nagpur
Demonstration of extra freedom of tuning (five parameters in FO-PID) achieve energy/fuel efficiency-as compared to (three parameter tuning) in classical PID A PID and FO-PID is tuned to have minimization of Performance index (P.I) for a plant i.e. DC Motor Transfer Function series with Pulse Width Modulator Circuit Tuned PID C ( s ) 1 1 5 .6 0 .2 2 s 1 1 .6 s 1 Tuned FO-PID
C ( s ) 1 5 .2 0 .0 4 s 1 . 4 2 .4 s 1 .2
Minimized P.I
Calculated Controller Effort
Does Lesser Controller Effort translates as Energy/Fuel Efficiency ? Courtesy: VNIT Nagpur
Measured controller signal u ( t ) for a step input of speed demand PID and FO-PID Tuned PID k
p
1 1 5 .6 ,
1,
k i 0 .2 2 ,
1
IT A E
k d 1 .6 m in
3 4 .2 1
Tuned FO-PID k p 15.2 ,
1.4 ,
k i 0.04 ,
1.2
k d 2.4 ITAE min 14.94
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 ( t ) in case of PID will be higher in PID
Courtesy: VNIT Nagpur
Measured u ( t ) i.e. average controller output voltage from and FO-PID at various set speed of DC Motor
No-Load Condition
Loaded Condition
The average controller output voltage is greater in case of PID as compared to FO-PID at a setspeed for unloaded and loaded cases Courtesy: VNIT Nagpur
CRO Traces showing PWM pulses, average control output u ( t ), Average Armature Current, and Average Armature Voltage for PID & FOPID
CRO trace for PID PWM
CRO trace for FO-PID PWM
At 500RPM speed set the for tuned PID controls armature voltage is 61.2V with armature Current 0.957A, while tuned FO-PID case armature voltage is 64.2V and armature current is 0.825A, with higher duty cycle of FO-PID PWM due to lower u ( t ), at fixed clock frequency 25KHz; in PID case Armature Power 58.5W and FO-PID case 52W
Courtesy: VNIT Nagpur
Measured average u ( t ) , V-armature, I-armature and Input Power for PID & FO-PID at various set speed of DC Motor for loaded and un-loaded cases
No-Load Condition
Loaded Condition Average input armature power is less for FOPID for all speed settings Courtesy: VNIT Nagpur
Explanation of the observations for DC Motor Speed Control by PID & FOPID While we want to increase speed, the PWM Duty Cycle is increased by increasing the demand-speed-that also increases u ( t ). This gives increased armature voltage and thus also armature current. For example at set speed at 700RPM u ( t ) = 0.83V , V =88V, and I = 1.1A, and while the speed is demanded at 1300RPM u ( t ) =1.01V, V = 170 V, I = 1.6 A while controlling with PID Now we change the controller and employ FO-PID; and observe that at demanded or set speed of say 700RPM, the u ( t ) is decreased-and average u ( t ) = 0.760V. This small disturbance from the original operating point will tend to decrease the speed from set speed 700RPM. Here the feed-back mechanism will try to compensate this decreased speed due to disturbed u ( t ) by boosting the V (demanding more duty cycle); thus we observe slight increment in V from 88V to 90V Since here the speed has to remain fixed at 700RPM this slight increased V is compensated by decrease in I ( the current reduces from 1.1A to 0.87A-at 700RPM with FOPID.
Experimental records of Energy/Fuel Efficiency for DC Motor Speed Controls-showing less input power in case of FO-PID for various speeds
No-Load Condition
Loaded Condition
A long standing conjecture/hypothesis about “Fuel/Energy Efficient Controls” is realized today: long way still to go!! Courtesy: VNIT Nagpur
Part-H More Experiments results to show Fractional Order Control & its advantage in Fuel/Energy Efficiency
Controller output u ( t ) for PID and Fractional PID for Magnetic Levitation System
r(t)
Controller PID / FO-PID
+
u(t)
Plant Magnetic-Levitation
c(t)
ki kd s s k C F O P ID ( s ) k p i k d s s C P ID ( s ) k p
r ( t ) : set point sinusoidal c ( t ) : position of the levitated ball u ( t ) : out-put of controller
We note that Mag-Lev is inherent unstable unit
Courtesy: VNIT Nagpur
Control effort in case of PID control
Voltage to coil –Controller output u ( t ) Ball position
Control is taking lot of effort to make ball position sinusoidal!! Courtesy: VNIT Nagpur
Control effort in case of Fractional Order PID Controls
Voltage to coil-Controller Output u ( t ) Ball position
Control is effortless to make ball position sinusoidal!! Courtesy: VNIT Nagpur
CRO traces for u ( t ) control voltage and c ( t ) ball position
PID case has greater RMS value for controller output voltage as compared to the FO-PID case, for same control objective of ball positioning.
This manifests as greater losses in coil impedance in PID case as compared to FO-PID case. This point is related to energy efficiency in controls !!
Courtesy: VNIT Nagpur
Why this control effort less? and its repercussion-once a conjecture To do the same job-that is to position the floating ball and slowly making its position follow sinusoidal command the output of the controller in case of first case (PID) is fluctuating severely, compared to FO-PID case (RMS value is much greater in PID case as compared to FO-PID). The reason is that fractional derivatives and fractional integrals are having (decaying memory )
inherent memory
This memory in the systems works to govern the ball position based on its previous experiencetherefore these fractional differentiation and integration gives an ideal filtering action. Whereas the classical differentiation is a point property-does not therefore has memory, and acts instantly with no previous experience. Thus in the first case the maneuvering signal is going very high instantaneously and in the next moment going very low again and again-lot of effort?
So we can see fractional calculus based system does the control action with a lesser effort than the conventional classical calculus based controllers, therefore are better efficient.
Can we say Fuel/Energy Efficient Controls?
Experimental observation on PID and FO-PID to balance a stick “Inverted Pendulum” The control of inverted pendulum via classical PID & Fractional order PID both tuned; gives interesting observations.
The movement on the X-Y plane, of the Plate, while balancing the stick via PID gives large displacements as compared to, when the control is done via FOPID The greater displacement while doing PID balancing calls for greater effortthereby asking expenditure of greater energy, as compared to FO-PID. PID action with greater X-Y movement is similar to action when you do the stick balancing for the first time-you do not have memory (experience) and thus spend lot of effort & physical energy Fractional Calculus is not a point property has memory, and the FO-PID action thus based on past experiences, and thus being done with lesser effort and XY movement-is another way to explain Courtesy: VNIT Nagpur
What we got by using fractional Laplace operator or in a way Fractional differentiation & fractional integration in Controls Helped to realize Bode’s dream of robust controls with CPE as Open Loop Transfer Function Helped to stabilize the plant with wide uncertainties in gain spread Helped to get iso-damping Helped to remove spurious plant trips due to large overshoot/undershoot due to uncertainty of plant TF Helped to operate controller without going towards saturation-avoiding controller saturation Helped to regulate the plant with lesser expenditure of control effort Helped to have Energy/Fuel efficient controls Well the fractional Laplace operator (irrational monomial ) s has done all the above but how to realize that in circuits & systems !!
Part-I Circuit implementation of Fractional Order PID (analog and digital)
CFE for approximation of fractional Laplace operator CFE is Continued Fraction Expansion CFE is defined as following (1 x )
1
d ef
x
1 1
12 (
x
1) 1
16 (
x
1) 1
16 (
x
2) 1
For obtaining rational approximation of
110 (
2)
x 1 ....
s put in CFE x s 1 and
CFE for four number term approximation for
s is
1 2
5s2 10s 1 s s2 10s 5
Here we have got approximation for half-differentiation in Laplace domain-is implementable We can have a transfer function of a filter (IIR) to realize fractional Laplace variable Kindergarten of Fractional Calculus
CFE for approximation of fractional semi differential Laplace Operator for various number of terms S.No
No. of terms in CFE Rational approximation for approximation
1
2
2
4
3
6
4
8
s
3s 1 s 3 5s2 10s 1 s2 10s 5 7 s 3 3 5 s 2 2 1s 1 s 3 2 1s 2 3 5 s 7 1 1s 5 1 6 5 s 4 4 6 2 s 3 3 3 0 s 2 5 5 s 1 s5 55s4 330s3 462s2 165s 11
CFE gives approximation for fractional Laplace operator in terms of ratio of rational polynomials. The above is semi-differentiation operator. The semi-integration operator will be reciprocal of the above ratio. Makes way to have fractional order analog and digital circuits & systems This gives us one method of representing irrational monomial via rational approximation Kindergarten of Fractional Calculus
Recursive Algorithm to get rational transfer function for fractional Laplace Variable to realize fractional derivative or fractional integration H (s)
1
s pT
1
m
( s z 1 ) ( s z 2 ) ( s z 3 ) ... . ( s p 1 ) ( s p 2 ) ( s p 3 ) . ...
N 1 i0 N i 0
1 1 s zi
s pi
0 m 1 [ y / 20 m ] p p 10 0 T the first pole the first zero z0 p010[ y /10(1m)]
the second pole, p1 z0 10[ y /10 m ] the second zero z p 10[ y /10(1m)] 1
1
[ y /10(1m)] z p 10 N 1 N 1 the Nth zero, the (N+1)th pole, pN zN 110[ y /10m]
z0 p010[ y /10(1m)]
Finite range of frequency, can be truncated to a finite number N, and the Slope -20m dB/decade is approximated by zig-zag lines of slopes -20 dB/dec (pole) and 0 dB/dec (zero), placed alternately with above rule, to get fractional integrator (called Fractional Power Pole) Functional Fractional Calculus
Recursive Algorithm to get rational transfer function for fractional Laplace Variable to realize fractional derivative or fractional integration
Fractional power Pole (integrator) of order half
l 1,
h 10 3
Finite range of frequency, can be truncated to a finite number N, and the CPE-45 degree is approximated by zig-zag lines of slopes -45 deg/dec (pole) and +45 deg/dec (zero), placed alternately with above rule, to get fractional integrator (called Fractional Power Pole) Similar to “Khanra” method
Realized active circuit for fractional differ-integration
s
1
2
( s z 1 ) ( s z 2 ) .. ( s p 1 ) ( s p 2 )..
50 45 40
P h a se (d B )
35
Phase plot of semi-differentiator with phase angle 45 degree for frequency chosen band; realized via six op-amp active filter circuits.
30 25 20 15 10 5 0 -1
0
1
2
3 Frequency
4
5
6
7
Rfi= Rii i
Zi
Pi
Ci
1
2.2537
6.0406
2
15.955
3
Rzi
TP
TP
1µ
264.07k
500k
443.71k
500k
42.764
1µ
37.30k
50k
62.67k
100k
112.95
302.75
680nf
11.21k
20k
18.83k
20k
4
799.65
2143.3
68nF
10.94k
20k
18.39k
20k
5
5661.1
15173
10nF
10.51k
20k
17.64
20k
6
40078
107420
1nF
14.85k
20k
24.95k
50k
Courtesy: VNIT Nagpur
How to test Fractional Order Integrator/Differentiator circuit D x cos( x ) cos x
2
is Liouville’s postulate true for Steady-State
Thus the generalized differential operator simply shifts the phase of the cosine function and likewise the sine function by (90) 0, 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.
CRO Output of fractional order differentiator circuit for ½ order Courtesy: VNIT Nagpur
Fractional Order Circuits and Systems Hardwired FO-PID
Hardwired Fractional Order PID connected to DC Motor Emulator Circuit Courtesy VNIT-Nagpur Dept of EE
Digital realization of fractional integrator/differentiator We have circuit realization of fractional integrator/differentiator in a choice of frequency band described by ratio of the rational polynomial as
s
( s z 1 )( s z 2 )....( s zN ) s N a N 1 s N 1 ... a1 s a 0 N ( s p 1 )( s p 2 )....( s pN ) s b N 1 s N 1 ... b1 s b 0
This is transfer function of IIR Filter We get discrete version of filter via use of Bilinear Transformation i.e. from s to z domain Let T be the sampling time based on Nyquist Rule Euler
z 1 T f (t ) F ( s )
s
1
s
1 z T f (nT ) F ( z ) z 1 F ( z ) f ( n T )
zF ( z ) f (n T )
From above Euler formula we get Tustin
2 ( z 1) s T ( z 1)
Forward shift is z Backward shift is z 1
z 1 sT e sT
z 1 1 sT e sT
2 (1 z 1 ) s T (1 z 1 )
2 sT 1 e sT z 2 sT 1 e sT
2 sT 1 e sT z 2 sT 1 e sT
Functional Fractional Calculus
In digital domain PID realization E ( z ) e(nT )
The z-Transform pair is
R(z)
E (z)
U ( z ) u (nT )
G p (z)
C (z)
Y (z)
U (z)
U (z) E (z) C ( s ) k p k i s 1 k d s C (z)
C (z) k
p
ki
T2 11 zz
1 1
k d
1 z 1 T
U ( z ) / E ( z )
T2 E ( z ) T2 z 1 E ( z )
U ( z ) z 1U ( z ) k p E ( z ) k p z 1 E ( z ) k i kd
T1 E ( z )
kd
T2 E ( z ) k d T1 z 2 E ( z )
ki
Using z inverse transformation we write time domain relation as
e ( n ) k e (n 2T ) T 1
u (n) u (n T ) k
p
k iT 2
kd T
p
k iT 2
kd T
u ( n ) u ( n 1 ) a 0 e ( n ) a 1 e ( n 1) a 2 e ( n 2 )
2 kd T
e (n T )
Realization of digital PID in FPGA/DSP PID is IIR filter of order two i.e. u ( n ) u ( n 1) a 0 e ( n ) a 1 e ( n 1) a 2 e ( n 2 )
a0 Multiplier
Shift Register -1
Set-point
r (n)
Controller output
e ( n 1)
12bits
u (n ) Adder
a1 Subtraction
Multiplier
e(n) 12bits
Shift Register -2
a2
e(n 2) Multiplier
y (n ) Feed-back CLOCK
Shift Register -3
T
u ( n 1)
Courtesy: VNIT Nagpur
Realization of digital FO-PID in FPGA/DSP Using direct discretization of the irrational monomials to have rational approximation we get z-domain transfer function
C (s) k p kis kd s T 1 z 1 k d CFE 1 1 z 2 b 0 b1 z 1 b 2 z 2 b 3 z 3 ... b M z M U (z) a 0 a 1 z 1 a 2 z 2 a 3 z 3 ... a N z N E (z)
2 C ( z ) k p ki C F E T
1 z 1 1 1 z
Rearranging and then taking z-inverse, we have higher order IIR filter as 1 u (n ) b 0 e ( n ) b1 e ( n 1) b 2 e ( n 2 ) ... a 1 u ( n 1) a 2 u ( n 2 ) .... a0 This above implementation of FO-PID requires more shift registers and more multipliers as compared to PID for DSP/FPGA realization. The same we have realized via indirect discretization, i.e. we first have rational approximation of Fractional Laplace operator in ratio of polynomials in s-parameter, then used Tustin formula Courtesy: VNIT Nagpur
Precautions The rational approximation of fractional Laplace variable should have interlaced poles and zero at LHP of s-complex plane
s
( s z 1 )( s z 2 )....( s zN ) ( s p 1 )( s p 2 )....( s pN )
Im s Re s
The rational approximation of fractional Laplace variable should have interlaced poles and zero inside the unit circle of z-complex plane
Im z
Re z
1
1
Courtesy: VNIT Nagpur
Part-J An appeal to industry & research institutes in India
Lastly So what if no industry use this ?
Why not Indian Industry be the first one ?
Well if the user does not want FO-PID let him use this as PID with setting the orders as unity, and use this as classical PID
Appeal to Indian Industry & Research Institutes to think of advancing by using this indigenous made technology (that was made right from basic mathematics)-before foreigners come out with industrial Product. So please do not throw this stating “no one uses this”.
Beyond Bode’s dream
Look at fractional complex order calculus and shape the Open Loop Transfer function as G O L G p ( s ) G c ( s ) ks i 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.
Lot of scope exists to still optimize to get robust & energy efficient controls
This was my original conjecture/hypothesis to make ‘Fuel/Energy Efficient Controls’ is realized today http://timesofindia.indiatimes.com/city/nagpur/VNIT-develops-energy-minimizing-controller-transfers-t ech-to-BARC/articleshow/55444793.cms
Fractional Calculus Engineering Laboratory was dedicated to Nation on 15-11-2016 at VNIT
Recent Email dated 19 Jan-2017 from across the boarder about Fractional Calculus Book From: "Hammad Khalil" Date: Thu, January 19, 2017 11:29 am To: Blessed to see that
[email protected] long time ago Cc: "shantanu" Priority: Normal Options: View Full Header | View Printable Version | Download this as a file
this subject is still growing that I started
Dear Director(s), I hope you would be fine. I am Dr. Hammad Khalil, Assistant Professor , at Univeristy of Education, Lahore. In the previous semester I got an opportunity to read a few papers and a good book of your department named "Functional Fractional Calculus " by Prof. Shantanu Das. I appreciate the struggle of you and yours departments collegues. The book I have mentioned, is really a great milestones in the theory as well as in the application of fractional calculus. This book have all the necessary materials and a lot of research directions. One important thing this book have, the writing style of the author is really impressive. We are studing this book and waiting to see more research in this field of calculus. Dr. Hammad Khalil Assistant Professor University of Education,Lahore, Pakistan.
Thanking you all at NIT-Silchar for this opportunity for presenting this part of “Fractional Calculus in Technological Development” End of Part-2
