ATAL FDP of EEE Dept, NIT-Mizoram

A brief overview of DC‐Attraction Type

Levitation System with related Control
Systems

By Dr. RUPAM BHADURI

Professor & HOD, Department of Electrical & Electronics (EEE),
DAYANANDA SAGAR INSTITUTIONS & UNIVERSITY
2nd CAMPUS, DSATM,
BANGALORE, INDIA

1
CONTENTS
INTRODUCTION

MATHEMATICAL MODELLING OF DC ATTRACTION TYPE LEVITATION SYSTEM (DCALS)

DYNAMIC PERFORMANCE ANALYSIS OF SWITCHED MODE POWER AMPLIFIERS USED IN

DCALS

DESIGN FABRICATION AND TESTING OF A SINGLE ELECTROMAGNET BASED DCALS

THE EFFECT OF DIFFERENT PARAMETERS ON THE CONTROLLER PERFORMANCE FOR A

DCALS‐A SIMULATION STUDY

TWO ACTUATOR BASED DC ATTRACTION TYPE LEVITATION SYSTEM

OPTIMIZATION OF CONTROLLER PARAMETERS FOR THE DCALS USING GENETIC

ALGORITHM AND PARTICLE SWARM OPTIMISATION

2
 INTRODUCTION
Some basic concepts
Suspension of objects with no visible means of support due to magnetic
force is called magnetic levitation,

If a magnet is used to ‘float’ a ferromagnetic object in a stable position, it is

necessary for the gravitational force to be balanced precisely by the
attractive force of the magnet.

Two types:‐ (a) DC attraction type levitation system (DCALS): Uses the
high‐power solid‐state controls to regulate the current in a direct‐current
electromagnet, and achieves stability through active feedback.
(b) Electro‐dynamic repulsive system (EDS): Uses high speed super‐
conducting magnets. Produces the repulsive force due to eddy currents
produced in the aluminum guide ways. Causes levitation beyond a certain
threshold speed.

3
Attraction system is currently being
favored in many countries, due to its
design simplicity, operational
flexibility, in that it is suitable for low
and high‐speed systems.

Four major components of DCALS: (i)

magnetic actuators, (ii) power
Fig.1 Simplified closed loop system of DC
amplifiers, (iii) controllers, (iv) electromagnetic levitation system
sensors.

Different applications : Levitation of Most important applications are in

models in a wind tunnel, Vibration the field of Active magnetic bearing
isolation of sensitive machinery, (AMB) and Transportation systems (in
Levitation of molten metal in Railways).
induction furnaces, Levitation of
metal slabs during manufacture, Bio‐
medical instrumentation etc.

4
 Some Notable Features

DCALS is inherently unstable and strongly non‐

linear in nature.

Parameters of DCALS changes with the air‐gap.

DCALS critically deserves very fast response and

DC‐DC switch mode power amplifiers used for such
applications have important role to meet this
demand.

The time constants of electromagnets used for

suspension are generally large.

Generation of high electromagnetic noise is a

Fig. 2 Static characteristics of a dc
common phenomenon in DCALS. electromagnet with ferromagnetic object

It is essential to control different degrees of

freedom (DOF) movement for a DCALS utilizing
multiple actuators, sensors and power amplifiers.

5
 MATHEMATICAL MODELLING OF DCALS
SELECTION OF NOMINAL OPERATING GAP
In lower gap, for large variation of inductance (Fig.3), causes wide change in levitated system
parameters, making robust controller design difficult.

In higher gap zone system parameters change is little, as inductance change is small, hence, robust
controller design is easier, but current requirement is high (Fig.4), so, in terms of energy consumption
the selection of high gap is not desirable.

It is always advantageous to select the operating air‐gap in between lower and higher gap zone
(medium gap).

Fig.3 Typical inductance profile of DC attraction type Fig.4 Typical characteristic of pick‐up current versus
6
air‐
levitation system gap for an electromagnetic levitated system
 Force of attraction between a ferromagnetic
MODELLING :‐ mass and the magnet is non‐linear

d 1 2
F (i, x)   L ( x )i (t ) (1)
dx  2 
The overall inductance may be
approximated as

L0 x0
L( x)  LC  (2)
x
Now putting the inductance value from
equation (2) into the force equation (1) one can
write:
2
Fig.5 Simplified diagram of DCALS  i (t ) 
F (i, x)  C   (3)
 x (t ) 
Lx
where, C  0 0
2

Cont..
7
 Dynamics of the electromagnet is given by the following equations

d 2 x(t )
F (i, x)  mg  m (4)
dt 2
2
 i (t )  d 2 x(t ) (5)
or , C 
( )   mg  m dt 2
 x t 
Under levitation the force given by equation (4) is in equilibrium with the gravitational
downward pull,
So, at the equilibrium position(i0, x 0 )the normalized force equation is
2
i 
F0 ( i0 , x 0 )  C  0   mg (6)
 x0 
The small perturbation equation becomes
2
 i0  i (t )  d 2 ( x0  x(t ))
C    mg  m (7)
 0
x   x (t )  dt 2

2
 i(t ) 
2 1 
 i0   i0  d 2 ( x0  x(t ))
 C    mg  m (8)

 x0   1  x (t )  dt 2

 x0  Cont..
8
 2
i   2i(t ) 2x(t ) 4i(t )x(t )  d 2 x(t )
 C 0  1      mg  m (9)
 x0   i0 x0 i0 x0  dt 2
Neglecting higher order terms,
  i 2  
2
  
2
 
  mg  m x(t )
2
2 ( ) 2 ( )
  C    C   .
i i t i x t d
0 0
 C   .
0
  x0   x0  i0  x0  x0  dt 2 (10)
 
By eqn.(6) , the eqn.(10) becomes,

 i (t )i 0  x (t ).i 02 d 2  x (t )
 2C 2
 2C . 3
 m (11)
x0 x0 dt 2
Taking Laplace transform on both sides of equation and after rearranging, the transfer
function of the magnetic levitation system is,
i0 Ka
 2C
X (s) m . x 02 m
  (12)
I (s)  2 .i   2 K 
2
 s  2C . 0 3   s  X 
 m . x 0   m 

i0 .i02
where, K a  2C 2 and K X  2C. 3 , are the two force constants which are
x0 x0
Cont..
basically slopes of force vs. current and force vs. air-gap characteristics . 9
Equation (12) represents the linearised plant transfer function of the levitated system, when
the magnet-coil is excited by the controlled current source.
KX
The transfer function shows that the system is open loop unstable having one pole at
the RHS of s-plane. m

The dynamic model of the coil (winding) (modeled as a resistor and inductor in series) is
given as by taking the instantaneous voltage,
dI (t ) (13)
V (t )  RI (t )  L
dt d I (t)  I (t)
taking small perturbation model, V (t )  V (t)  RI (t)  I (t)  L (14)
dt
dI (t )
 V (t )  R.I (t )  L (15)
dt
taking Laplace transform of eqn.(15),
(16)
  V ( s )  R . I ( s )  L . s . I ( s )
I (s) 1
the transfer function of the actuator is,   (17)
V (s) ( R  Ls )
When the magnet-coil is excited by a controlled voltage source the transfer function
of the magnetic levitation system becomes:  Ka 
 
X ( s )  m 
G p (s)   (18)
V ( s )  K 
( R  sL) s 2  X  10
 m 
DYNAMIC PERFORMANCE ANALYSIS OF SWITCHED
MODE POWER AMPLIFIERS USED IN DCALS
Power Amplifier plays one of the most important role in DCALS.
The DCALS critically deserves very fast response, where the coil current needs to be
precisely controlled to meet necessary attractive force demand which calls for a faster DC‐
DC power amplifier.
In the literature Linear power supplies have drawbacks. Switched Mode Power Amplifiers
with high efficiency, increased switching speed, higher amount of voltage & current
handling capability with lower cost have been used.
The main requirement of power amplifiers in DCALS is that not only current rise and but
also fall (decay) through the magnet‐coil must be fast. Hence, during off‐time of switch,
application of negative voltage across coil is necessary.
Here, possible switched mode power circuits for both single as well as multi‐actuator
based levitation system are discussed and a comparative study have been done using
PSPICE software tool.
Effect of different parameters: Input DC link voltage, duty cycle and switching frequency,
resistance and inductance of magnet coil on the dynamic responses of amplifiers was
observed.

11
 A simple class D type chopper is the simplest
form of power amplifier for excitation of
magnet coil in DCALS.

A full bridge circuit having four controlled

switches can apply equal amount of positive
and negative voltage to the load (magnet coil)
while allowing coil current to be bi‐directional.
Fig.6 Simple class‐D chopper circuit A split DC supply is shown (Fig.8) for exciting
the magnet‐coil.This circuit has a de‐rated
power supply and the dynamic response
degrades due to lower input DC link voltage.

Fig.7 Full Bridge power amplifier

Fig.8 Half Bridge power amplifier

12
 Electromagnetic attraction force is,
however, independent of the coil‐current
direction and hence one may as well go for a
cheaper asymmetrical bridge circuit.

Asymmetrical bridge circuit allows only

one direction of load (coil) current.

Fig.9 Asymmetrical bridge converter circuit

Fig.10 Coil voltage (CH1, 50V/div) and coil current (CH2, 500mv/div) during
13
stable levitation using Asymmetrical H-bridge.
 The single switch based power circuit
may be suitable form of switched mode
power amplifier for single and multi‐
magnet based levitation system.

Fig.11 Single switch based chopper amplifier

Fig.12 (a) Coil voltage (CH2, 50V/div) and Fig.12 (b) Coil voltage (CH2, 50V/div) and
coil current (CH1, 500mV/div) during stable capacitor voltage (CH1, 50V/div) during
levitation using single switch circuit stable levitation using single switch circuit 14
Fig.14 Frequency response plot of Assymetrical Bridge Fig.15 Frequency response plot of Full Bridge
Converter at Inductance 0.03H Converter at inductance 0.03H, and
and Volt.100V, GCF=79.2 kHz . Volt.100V, GCF=64 KHz.

Fig.16 Frequency response plot of Half Bridge circuit

15
at inductance 0.03H, and Volt.100V GCF=24 KHz
 Table: Input DC link voltage vs. GCF
TABLE- 1
Fig.21 Plot of the Input DC link Voltage
FULL- ASYMM- HALF- SINGLE
with Gain crossover frequency BRG. BUCK BRG. BRG. SWT. Voltage
GCF GCF GCF GCF GCF Vin
(KHz) (KHz) (KHz) (KHz) (KHz) (Volts)
32.75 25.2 41.58 14.77 20 50
35.012 25.67 44.22 15 21.26 60
37 25.94 46.492 15.29 22.8 70
38.76 26.214 48.4 15.4 23.2 80
40.29 26.46 50.21 15.666 23.93 90
16
41.65 26.68 51.85 15.71 24.55 100
 CONCLUSION
Asymmetrical H‐Bridge converter is fastest while Half‐Bridge circuit is the
slowest.

Capacitor half‐bridge circuit’s dynamic response degrades due to lower

input DC link voltage.

Asymmetrical H‐Bridge’s response is close to Full Bridge converter & it

needs only half number of switches also.

Single switch circuit easily constructible & suitable for low power
application but slower in response than Asymmetrical Bridge.

But it does not need any gate drive isolation also there is overall reduction
in the EMI as the number of switches are less, that are all connected to
the same common point.

17
 DESIGN FABRICATION AND TESTING OF A SINGLE ELECTROMAGNET
BASED DC ATTRACTION TYPE LEVITATION SYSTEM

Only single axis levitation makes complete

suspension of any spherical object due to its
structural symmetry.

The suspension of a long cylindrical rod Fig.22 Schematic block diagram for the proposed single
without any tilting is an interesting and actuator based DCALS.
challenging task.

Aim is the stable suspension of this rod at

any operating gap. So it is necessary to
regulate the coil current accurately and quickly
with help of position feedback and associated
control circuitry.

18
Fig.23 Photograph of the levitated system.
EXPERIMENTAL DETERMINATION OF TRANSFER‐FUNCTION FOR PROPOSED DCALS

L0 x0
L( x)  Lc  (23)
x
i 2 dL x 
2 2
i 2   L0 x0  L0 x0  i  i (24)
F ( x)          C  
2 dx 2  x2  2  x  x
 Ka 
 
Lx  i   i02  X ( s )  m
C 0 0 k a  2C  02  k x  2C  3   (25)
2  x0  I ( s )  2 Kx 
 x0  s  
 m 

Fig.25 Current profile of the levitated system

Fig.24 Inductance profile of the levitated system
19
 DESIGN OF CONTROLLERS:

Fig.26 Basic block diagram of overall closed loop system

Fig.27 Block diagram of current Loop

The loop gain is written as Kp
(1  s )
K 1 K K K Ki 1 (26)
GH i ( s )  K ch K c ( K p  i )   ch c i 
s ( R  sL ) R s L
(1  s )
R
By frequency domain design method, PM=60 degree at GCF=100Hz :
 K p  1  L 
K ch K c  2 K p2  K i2
 90  tan    tan    180  60 (27) and from gain criterion GH i ( j )  1
1
o o 
(28)
 i 
K  R   R  L
2 2 2

Kp = 1.711, Ki = 707.33

20
 Natural Frequency of oscillation ( 
) =n72 Hz

Damping Ratio ( ) = 100
 0.6
 
Bandwidth = b   n  1  2  ( 2  4  4 ) 
2 2 4

 
 82 Hz (29)

Fig.28 Frequency response of the current loop

%Overshoot = 22.4%
Rise time = 0.00204 sec
Settling time = 0.0142 sec

Fig.29 Unity step response of current loop

21
 POSITION CONTROL LOOP

9.237 Fig.30 Block Diagram of position control loop

G p (s) 
( s  35.44)( s  35.44) G pd ( s )  K p  K d s (30)

Fig.31 Root locus of the uncompensated plant for Fig.32 Root locus plot of the 10 mm plant by PD
10mm gap. compensator
22
Fig.33 Root locus of the plant for 10mm gap by Fig.34 Dynamic position responses at different air‐gaps
Lead compensator with lead controllers

9.237
G p ( s) (maglev plant transfer function at 10mm) (31)
( s  35.44)(s  35.44)

G Lead ( s )  K
s  zc 
 15.3
 s  31
(lead compensato r at 10mm) (32)
s  pc  s  369 

23
Fig.35 Dynamic position responses at different air‐ Fig.36 Root locus of the plant for 10mm gap by
gaps with lag‐lead controllers Lag‐Lead compensator
K ( s  zc1 )(s  zc 2 ) 26.3( s  5)(s  40)
GLag  Lead ( s)   (lag - lead compenastor at 10mm)
s  pc1 s  pc 2  (s  0.5)(s  840) (33)

 Lead‐lag network provides both the satisfactory transient and steady‐state performance (Steady
state error is about 1% only).
 The system becomes conditionally stable system.
 Advantage is that here both pole and zero can be suitably placed to get the desired performance.

24
 PID controller transfer function at 10mm
air‐gap
K ( s  z c1 )( s  z c 2 ) 0 .0285 ( s  5)( s  35 )
Gc ( s )  
s s
(34)

Proportion al gain, K p  K ( z c1 zc 2 )

Integral gain, K I  K * zc1 * zc 2
Derivative gain, K D  K

Fig.37 Root locus of the plant for 10mm gap by PID controller

 Since the DCALS is inherently unstable system, the

application of PID controller becomes critical due to
having a pole at origin.

System is stabilized for a smaller value of gain , so,

margin of stability is also less.

Fig.38 Dynamic position responses at different25

air‐gaps
with PID controllers
 To enhance the system
performance (steady‐state as well as
transient) another loop must be
introduced into the two‐loop
structure of DCALS

 Transfer function of outer PI

Fig.39 Modified closed loop maglev system utilizing outer PI controller controller at 10mm air‐gap is
0.064( s  50)
(35)
s

Fig.40 Time response of different controllers

(outer most PI type) while switching from one gap Fig.41 Bode plot of 10mm air gap using outer PI
26
to other. controller
Fig.42 Position responses due to step input for the different controllers at 10 mm air‐gap
position

27
Fig.47 Complete hardware set-up for single actuator based dc
attraction type levitation system

Fig.48 The side view of levitation

28
 CONCLUSION

The cylindrical rod has been successfully suspended under the E‐core
electromagnet with single‐axis control.

Unstable maglev has been stabilized with different classical controllers and
the comparative study between controller’s performances has been
observed theoretically and experimentally.

29
 THE EFFECT OF DIFFERENT PARAMETERS ON THE
CONTROLLER PERFORMANCE FOR A DC ATTRACTION
TYPE LEVITATION SYSTEM‐A SIMULATION STUDY
Effect of parameter variation:
The basic parameters of DCALS:‐Air‐gap between the pole face of the Electro‐
magnet and ferromagnetic guide‐way, Mass of the payload (object),
Inductance and resistance of the actuator, Input DC link voltage, Lead controller’s
Gain and Pole, Zero locations.

All these parameters are supposed to get change in real life situation.

So the performances of the linear controller with the change in different

parameters have been studied.

Also the parameters of the designed controller itself get varied during
experimentation.

Hence, a sensitivity study of the controller itself is also important.

30
EFFECT OF CHANGE IN AIR‐GAP

Transfer function of the levitated system (12mm)

3.652
( s  23.27)( s  23.27)

Transfer function of the levitated system (10mm)

9.237
( s  35.44)( s  35.44)

Transfer function of the levitated system (7mm)

15.368
( s  46.62)( s  46.62)

Due to change in operating condition the

Fig.52 Effect of change in air‐gap performance of the controller deteriorates.

Due to inherent nonlinear characteristics of the

system, the uncertainty in the nominal linearized
model increases as one moves away from the
nominal operating point of 10 mm.

31
EFFECT OF CHANGE IN MASS (PAYLOAD)

The damping ratio is related to the mass

of the system by the following equation
(neglecting the effect of the pole of the lead
controller on the transient performance)

kka
 (36)
2 m (kka z  k X )

Fig.53 Effect of change in mass of payload

With the increase of the mass the system
 Ka  open loop poles are coming closer to the
 
X ( s ) imaginary axis thereby reducing the overall
  
m
G p ( s)  (37) closed‐loop stability.
I ( s )  2 K X 
s  
 m 

32
EFFECT OF CHANGE IN COIL RESISTANCE

The coil resistance may be increased due to the

change in operating temperature for prolong
operation of the system.

It has been observed that the effect of coil

resistance on the dynamic performance of
controller is less.

Fig.54 Frequency response at 10mm air gap

with nominal resistance

Fig.55 Frequency response when resistance is increased 50%

33
EFFECT OF CHANGE IN COIL‐ INDUCTANCE

With the increase of coil‐inductance (when the

air‐gap between the electromagnet and
ferromagnetic rod reduces), the system transient
response becomes poorer showing more overshoots
and undershoots before reaching to the final steady‐
state point.

Fig.56 Effect of coil inductance variation

Fig.57 Frequency response with nominal Fig.58 Frequency response when inductance increased by
34
inductance 20%
EFFECT OF CHANGE IN CONTROLLER GAIN

The transfer function of the designed

lead controller for a operating air‐gap of
10 mm is given by

k(s  zc ) 15.3(s  31)

Gpc (s)   (39)
(s  pc ) s  369

With the increase of gain the system

dynamic response becomes poorer with
increasing overshoot and settling time.

Fig.61 Effect of change in controller gain

35
 CONCLUSION

The performance of the proposed control system for a DCALS with the
change of different parameters has been studied.

The idea of this study will give an insight for designing any corrective
measures with parametric uncertainties for such critical systems.

The effect of the similar parameters can be studied experimentally and the
comparison between the theoretical and practical results would be a future
extension of the work.

36
 TWO ACTUATOR BASED
DC ATTRACTION TYPE LEVITATION SYSTEM
Why two actuators ?
Single coil produces tilting effect
on suspended cylindrical rod, due
to non‐uniformity of the
distributed field flux.
Fig.64 Schematic block diagram of individual unit

To obtain better pitching control

where both the ends of a long rod
can be controlled independently,
by two identical controllers.

For controlling other degrees of

freedom movement at least two
actuators are necessary.

37
Fig.65 Photograph of the experimental setup
Ansys Simulation Results

Fig.66 Ansys Model(exact) of the Fig. 67Flux pattern for the system at 10mm air-gap
prototype

Fig.68 Field Intensity nodal solution under contour plot at 10mm air-gap 38
 Estimation of System Parameters & Determination of Plant Transfer
Function
Inductance vs Air-gap for Coil-1

0.2

0.192
Inductance, H

0.184

0.176

0.168

0.16
0 5 10 15 20 25
Air gap, mm
Pick-up current (Coil-1) vs Air-gap
1.1
Fig.69 Inductance profile of Coil-1
0.9

Pick-upcurrent, A
0.7

0.5

0.4

0.2

0.0
0 5 10 15 20 25
Air-gap, mm
Fig.70 Pickup Current vs. air-gap of Coil-1 39
 Current Control loop

Fig.73 Basic block diagram of overall closed loop system

K
• The Loop gain: (1  s
p
) (42)
K 1 K K K Ki 1
GH i ( s)  K ch K c ( K p  i )   ch c i 
s ( R  sL) R s L
(1  s )
• By Frequency Domain approach:    
R
K  L 
From phase angle criterion:  90  tan  K   tan  R   180  
o 1 p 1 o
• (43)
 i 
• The phase margin (PM) is assumed to be around 60 deg. for good margin of
stability (at GCF of 100Hz).
• Sample Calculations (at 10mm air‐gap): L = 0.1829 H , R = 4.1Ω.
K s  Ki  1 (44)
GH i ( s )  K ch K c 
p

s ( R  sL )
K ch K c  2 K p2  K i2  K p  L 
or , GH i ( j )   90 o  tan 1    tan 1    180  60
o o

 Ki   R 
 R 2   2 L2
G ( s) H ( s)  1 at 100Hz Kp = 8.1228 Cont…
Ki = 3194.5 40
 Fig.74 Frequency response of the
current loop

Fig.75 Unity step response for

10mm of current loop

41
 Position Control loop

Fig.76 Block Diagram of position control loop

In this two actuator based maglev system, Lag‐Lead compensator has been used for
10mm air gap and the designed transfer function is
21.2( s  25)s  5
GPc ( s)  (45)
s  665s  0.5
Design procedure is similar to that of single actuator based system described
earlier.
Cont…
42
Fig.77 Root locus for uncompensated
plant

43
Fig.78 Root locus with Lag‐Lead comp.
Fig.79 Unit step response of Lag-
Lead Compensators at
different gaps

Fig.80 Bode plot for the overall closed loop frequency response
with Lag-Lead compensator. 44
 Power Amplifiers

Fig.81 Schematic diagram of the asymmetrical bridge (H-bridge) converter circuit

Fig.82 Proposed simplified chopper circuit for two-coil levitation system

Cont…
45
Fig.86 Complete hardware set‐up for the levitation system 46
Fig.87 Photograph of the levitated rod 47
Fig.88 Demonstration of pitching control of the rod 48
 Fig.91 Gate pulses (CH-3 & CH-4) and coil
voltages (CH-1 & CH-2) during stable levitation of
cylindrical rod

Fig.92 Coil voltage (CH-1), capacitor voltage (CH-

2), coil current (CH-3), Gate pulse (CH-4) during
stable levitation of cylindrical rod 49
Fig.93 Two position signals (CH1 & CH2) and two
current signals (CH3 & CH4) during stable
levitation of cylindrical rod

Fig. 94 Dynamic position responses of two corners

of rod
50
 CONCLUSION

The prototype has been successfully tested and its stable levitation has
been demonstrated at the desired gap position.

Two independent SISO controllers designed for controlling and

maintaining a fixed air‐gap at the two side of the rod are found to work
successfully in unison.

By using single switch based switched mode power amplifier the

overall hardware circuit becomes simpler, compact & cost effective.

The use of two actuator based controller demonstrates better pitching

movement than single controller based system.

51
OPTIMIZATION OF CONTROLLER PARAMETERS FOR THE
DCALS USING GENETIC ALGORITHM AND PARTICLE
SWARM OPTIMISATION
Why optimisation ?
Designed compensators/controllers were tuned by ‘trial and error’
process on the s‐plane.
Satisfactory performance was achieved. But the controllers /
compensators were not optimized to operate in other air‐gaps for
which it is not designed, i.e. they have restricted zone of operations.
Any change in position reference will result in severe deterioration of
transient performance.
Thus an optimal control study of the proposed two actuator based
DCALS scheme is necessary.
Two different optimisation techniques (GA & PSO) have been used and
a comparative study is done.

52
 Some facts about Genetic Algorithm
The Genetic algorithms (GA) belonging particularly to the family of evolutionary
computational algorithms which have been widely used in many control
engineering applications.

Main advantage of evolutionary computation techniques is that they do not have

much mathematical requirements, but they need to evaluate the objective
function.

The Genetic Algorithm utilizes Darwinian evolutionary processes.

In GA, the population that consists of multiple individuals (chromosomes) is set in

the program.

By applying GA operators (for instance, crossover, mutation) the new solution is

created and by repeating such a process several times (generations), appearance
of better individuals can be expected. Thus, optimum solution is finally obtained.

GA finds the optimal solution through cooperation and competition among the
potential solutions.
53
Fig.95 The basic cycles of genetic algorithms 54
 Genetic Operators
Selection: Chromosomes with better fitness are selected as parents for
reproduction of better offsprings for next generation.
Reproduction: After evaluation of objective function it is required to
create a new population from the current generation. Selection process
of fittest chromosomes are done for reproduction operation.
Crossover: To exploit the potential of the current gene pool, it is
necessary to use crossover process to generate new chromosomes that
is hoped of retaining good features from the previous generation.
Elitism: Some time a policy of always keeping a certain number of best
members when each new population is generated; this is elitism.
Mutation: If the population doesn’t contain the solution within the
required solution area to optimize a particular problem, no amount of
gene mixing can produce a satisfactory solution. For this reason, a
mutation operation capable of spontaneously generating new
chromosomes is included.

55
 56
Fig.96 Flow Chart for the GA programming
• DCALS Controller parameters initialization technique for
Genetic Algorithm

• The optimal values of the parameters of the Lead,

Lag‐Lead and PID controller of the nominal operating
air‐gap of 10mm are to be obtained through GA.
There are three parameters (named: one DC gain, one
pole and one zero for Lead and one DC gain and two
zeros for PID) for both Lead and PID controllers and
five parameters (named: one DC gain, two zeros &
two poles) for the Lag‐Lead controller.

57
• classically designed Lead
controller / compensator’s
K ( j)
( s  z ( j)
c )
C (s) 
( j)

 
(47)
s  pc( j )
Fig.97 Block diagram of EMLS with signals
• for stabilizing Gp(j)(s),
for j=1,2,3 where,
• K(j) = 6.7, 7.3, 8.1;
• zc(j) = 57, 25, 21;
• pc(j) = 572, 323, 293.

58
 Objective function for GA

Here controller’s performance is evaluated in terms

of integral time absolute error (ITAE) error criteria.
(ITAE = )
The limits for the equation from time, t = 0 to τ = Ts,
where Ts is the settling time of the system to reach
steady state condition for a unit step input. Here the
value of Ts = 0.0281sec(for Lead at10mm),
0.254sec(for Lag‐Lead at10mm), 0.433sec(for PID
at10mm).

59
 Results obtained by GA:‐ Table:‐Comparison of Time Domain
Specifications for Lead Comp.
Lead
Contr:‐ Specifications Trial & GA
Air Gap Error
10mm Peak Overshoot(%) 27.5 0
at Settling Time(sec) 0.0767 0.127
3mm Rise Time(sec) 0.00252 0.00563
S.S.Error 0.16 0.37
10mm Peak Overshoot(%) 16 0.443
at Settling Time(sec) 0.0281 0.0172
10mm Rise Time(sec) 0.00536 0.0111
S.S.Error 0.1 0.22
10mm Peak Overshoot(%) 13.2 2.26
at Settling Time(sec) 0.0229 0.0381
17mm Rise Time(sec) 0.00653 0.014
S.S.Error 0.1 0.22
Fig. 98 Comparative transient responses for
Classical & GA based Lead controllers Table :‐ Parameters of Lead Comp. at 10mm
gap.
Lead Contr:‐ Trial & GA
Parameters Error
Gain (K) 7.3 4.0109
Zero (Z) 25 24.9409
Pole (P) 323 349.4228
60
 Convergence graphs of PID contr. parameters
PID contr. Z1 vs iteration
PID contr. K vs iteration 7.02
0.045
7
0.0445 6.98
K Z1
6.96
0.044

1st zero(Z1)
6.94
Gain(K)

0.0435 6.92
6.9
0.043 6.88
6.86
0.0425
6.84
0.042 6.82
0 20 40 60 80 100 0 20 40 60 80 100
No.of iterations No.of iterations

Fig.101 PID controller’s gain K vs Fig.102 PID controller’s zero Z1 vs iteration

iteration
PID contr. Z2 vs iteration
24.5

24

Z2
23.5
2nd zero(Z2)

23

22.5

22

21.5

21
0 20 40 60 80 100
No.of iterations

Fig.103 PID controller’s zero Z2 vs iteration Fig.104 Objective function convergence with
iterations for GA based PID controller61
 Some drawbacks of GA
1) GA have a tendency to converge towards local optima rather than the global
optima of the problem;

2) It can be seen that the parameters are becoming saturated after 30/35
iterations since the chromosomes in the population of GA is having similar
structure and having high value of fitness so that, no amount of gene mixing is
now producing better result, this is a drawback of GA. Hence the results are
not fully optimal but sub‐optimal.

3) For specific optimization problems, and given the same amount of computation
time, simpler optimization algorithms such as PSO, may find better solutions
than GAs

4) The crossover and mutation operations cannot ensure better fitness of

offspring.

62
 Particle Swarm Optimization (PSO)

Introduced and developed by Dr.

Eberhart and Dr. Kennedy in 1995,
inspired by social behavior of bird
flocking (Fig.)or fish schooling (Fig.)

PSO has a flexible and well‐balanced

mechanism to enhance the global and Fig.105 Photograph of bird flocking
local exploration abilities.

PSO is an algorithm modeled on

swarm intelligence that finds a
solution to an optimization problem
where swarm intelligence is based on
social‐psychological principles.

Fig.106 Photograph of fish schooling

63
In the PSO algorithm, instead of using evolutionary (or, genetic) operators
such as mutation and crossover, to manipulate chromosomes, for a d‐
variabled optimization problem, a flock of particles are put into the d‐
dimensional search space.

With randomly chosen velocities and positions knowing their best values
so far (Pbest) and the position in the d‐dimensional space. The velocity of
each particle, adjusted according to its own flying experience and the
other particle’s flying experience.

64
 Advantages of PSO over GA

PSO is easier to implement and there are fewer parameters to

adjust. (only particle velocity and position adjustments are
required)

PSO does not implement any complex genetic operators.

In PSO, every particle remembers its own previous best value as

well as the neighborhood best; therefore, it has a more effective
memory capability than the GA.

PSO is more efficient in maintaining the diversity of the swarm.

65
 Design Approach
In the present approach of implementation the velocity of each
particle is adjusted according to its own flying experience and
the other particle’s flying experience. The best previous position
of the i th particle is recorded and represented as:
Pbest i = (Pbest i,1 , Pbest i,2,..., Pbest i,d )

In this variant of PSO a constant Inertia weight factor has been

considered [Shi & Eberhert, 1998], advantage is that the both
explorative and exploitative search mechanism is emphasized.

66
 The index of best particle among all of the particles in the group is
gbest d . The velocity for ‘i‐th’ particle is represented as vi = (vi,1,
vi,2,….,vi,d).
The modified velocity and position of each particle can be
calculated using the current velocity and the distance from Pbest i,d
to gbest d as shown in the following formulas,

vi(,tm1)  w.vi(,tm)  c1 * rand 1 ( ) * ( pbest i , m  xi(,tm) )  c2 * rand 2 ( ) * ( gbest m  xi(,tm) )

(50)

xi(,tm1)  xi(,tm)  vi(,tm1) (51)

for, i = 1,2,….n and m = 1,2,….,d.

Cont..
67
• n Number of particles
• d Dimension
• t Pointer of iterations (generations)
(t)
• v i, m Velocity of particle i at tth iteration,
• w Inertia weight factor
• c1, c2 Acceleration constant
• rand 1( ), Random number between 0 and 1
• rand 2( )
• x i,(t) m Current position of particle i at tth iteration
• pbesti Best previous value of the ith particle
• gbest d Best particle among all the particles in the population

The PSO parameters used for this project are as follows:

Population size: 50
Iterations: 100
Inertia weight factor, w = 0.6;
Acceleration constants, c1, c2 = 1.70;

68
 Objective Function for PSO
• For GA, ITAE = , was used.
• Disadvantage of this criteria is that its minimization can result in a
response with relatively small overshoot but a long settling time
also the derivation processes of the analytical formula are complex
and time‐consuming.
β β
• For PSO:‐ minK:stabilizing W(K)  (1 e ).(MP  ESS )  e .(tS  t r ) (52)

• K is the controller gain which is d‐dimensional matrix , [Z. L. Gaing,

2004]
• β is the weightening factor. Normally the desired performance can
be achieved by choosing a proper value of the weightening factor β.
• Here, β=0.6;

69
Fig.107 Flow Chart for the PSO program 70
 SIMULATION RESULTS:‐
• Transient response is improving since optimized parameters of
the controllers are occurring more closely to the global optima
of the entire search space in the possible solution area.

• PSO parameters of the controllers are providing better

robustness in terms of transient response specifications since it
has a more effective memory capability than the GA, as all the
particles use the information related to the most successful
particle in order to improve themselves.

Cont…
71
 Lead
Contr:‐ Specifications Trial & GA PSO
Air Gap Error
10mm Peak Overshoot(%) 27.5 0 5.2
at Settling Time(sec) 0.0767 0.127 0.114
3mm Rise Time(sec) 0.00252 0.00563 0.00386
S.S.Error 0.16 0.37 0.29
10mm Peak Overshoot(%) 16 0.443 1.85
at Settling Time(sec) 0.0281 0.0172 0.0138
10mm Rise Time(sec) 0.00536 0.0111 0.0104
S.S.Error 0.1 0.22 0.18
10mm Peak Overshoot(%) 13.2 2.26 2.51
at Settling Time(sec) 0.0229 0.0381 0.0306
17mm Rise Time(sec) 0.00653 0.014 0.0116
S.S.Error 0.1 0.22 0.18

Fig.108 Comparative transient responses for PSO

based Lead controllers.

Lead Trial PSO

Com‐ & GA
Paramet
ers Error
Gain (K) 7.3 4.0109 5.202
Zero (Z) 25 24.9409 24.7062
Pole (P) 323 349.4228 380.5204
Fig.109 Comparative transient responses for Classical, GA and PSO72
based
Lead controller at 10mm air‐gap.
 Objective function Convergence for PSO based PID
controller at 10mm air‐gap

Fig.114 Objective function convergence with Fig.115 Objective function convergence with
iteration of PSO based PID controller iteration of GA based PID controller

73
 Conclusion
Sub‐optimal performce has been observed by implementing GA.

But improved transient response has been achieved through PSO.

All the particles of PSO use the information related to the most successful
particle in order to improve themselves, whereas in GA, the worse solutions
are discarded and only the good ones are saved; therefore, in GA the
population evolves as a whole group towards the optimal area.

The GA‐tuned lead, lag‐lead and PID controllers outperformed the “trial &
error” method of designing in s‐domain for the proposed EMLS, furthermore
the PSO algorithm supersede the GA.

Improvement of the work can be made by designing an online adaptive

controller based on either GA or PSO for the present electromagnetic
levitation system.

74
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