higher order linear differential equationsdr. rajesh kumar. phd, pdf (nus, singapore) smieee, fiete,...
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![Page 1: Higher Order Linear Differential EquationsDr. Rajesh Kumar. PhD, PDF (NUS, Singapore) SMIEEE, FIETE, MIE (I),LMCSI, SMIACSIT, LMISTE, MIAENG. Associate Professor , Department of Electrical](https://reader031.vdocuments.site/reader031/viewer/2022011921/6031b6193f564628db1a41d2/html5/thumbnails/1.jpg)
Genetic Algorithm
Dr. Rajesh Kumar
PhD, PDF (NUS, Singapore)SMIEEE, FIETE, MIE (I),LMCSI, SMIACSIT, LMISTE, MIAENG
Associate Professor , Department of Electrical EngineeringAdjunct Associate Professor, Centre of Energy
Malaviya National Institute of Technology, Jaipur, India, 302017Tel: (91) 9549654481
[email protected], [email protected]://drrajeshkumar.wordpress.com/
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Calculus
Maximum and minimum of a smooth function is reached at a
stationary point where its gradient vanishes
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Hill-climbing
Calculus based approach
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But…
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But…
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More realistically?
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•Noisy?
•Discontinuous?
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Scientists have tried to mimic the nature
throughout the history.
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The nontraditional optimization algorithms
are
Genetic Algorithms
Neural Networks
Ant Algorithms
Simulated Annealing
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Finonacci Newton
Direct methods Indirect methods
Calculus-based techniques
Evolutionary strategies
Centralized Distributed
Parallel
Steady-state Generational
Sequential
Genetic algorithms
Evolutionary algorithms Simulated annealing
Guided random search techniques
Dynamic programming
Enumerative techniques
Search techniques
Classes of Search Techniques
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General Introduction to GA’s
• Genetic algorithms (GA’s) are a technique to solve problems which need optimization
• GA’s are a subclass of Evolutionary Computing
• GA’s are based on Darwin’s theory of evolution
• History of GA’s
• Evolutionary computing evolved in the 1960’s.
• GA’s were created by John Holland in the mid-70’s.
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Biological Background (1) – The cell
• Every animal cell is a complex of many small “factories” working together
• The center of this all is the cell nucleus
• The nucleus contains the genetic information
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Biological Background (2) – Chromosomes
• Genetic information is stored in the chromosomes
• Each chromosome is build of DNA
• Chromosomes in humans form pairs
• There are 23 pairs
• The chromosome is divided in parts: genes
• Genes code for properties
• The posibilities of the genes forone property is called: allele
• Every gene has an unique position on the chromosome: locus
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Biological Background (3) – Genetics
• The entire combination of genes is called genotype
• A genotype develops to a phenotype
• Alleles can be either dominant or recessive
• Dominant alleles will always express from the genotype to the fenotype
• Recessive alleles can survive in the population for many generations, without being expressed.
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Biological Background (4) – Reproduction
• During reproduction “errors” occur
• Due to these “errors” genetic variation exists
• Most important “errors” are:
• Recombination (cross-over)
• Mutation
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Biological Background (5) – Natural selection
• The origin of species: “Preservation of favourable variations and rejection of unfavourable variations.”
• There are more individuals born than can survive, so there is a continuous struggle for life.
• Individuals with an advantage have a greater chance for survive: survival of the fittest.
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Biological Background (6) – Natural selection
• Important aspects in natural selection are:
• adaptation to the environment
• isolation of populations in different groups which cannot mutually mate
• If small changes in the genotypes of individuals are expressed easily, especially in small populations, we speak of genetic drift
• Mathematical expresses as fitness: success in life
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Genotype space =
{0,1}L
Phenotype space
Encoding
(representation)
Decoding
(inverse representation)
011101001
010001001
10010010
10010001
Genetic Algorithms
Notion of Genetic Algorithms
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Genetic Algorithms
Notion of Genetic Algorithms
0110011010011101 0011101
23 chromosomes
Human
A fetus is formed by a Male(sperm) and female(egg).ht
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01100110
10011101 10011101
01100110
+ +
101 01100
110 10011
111 00001
100 11110http
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10011101
10011001
Mutation Process
The evolutionary process can be expedited by
improving the variety of the gene pool.
It is done via mutation.
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Simple Genetic Algorithm
{
initialize population;
evaluate population;
while Termination CriteriaNotSatisfied{
select parents for reproduction;
perform recombination and mutation;
evaluate population;}
}
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Genetic algorithms are usually applied for
maximization problems.
To minimize f(x) (f(x)>0)using GAs, consider
maximization of
1
1+f(x)
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Example
Maximize y1.3+10e^(-xy)+sin (x-y)+3 in
R=[0,14]×[0,7].
x
y
0 10
7
12 142 4 6 8
(0,5)=000101
(2,7)=001111
(6,1)=011001
(10,4)=101100
(12,0)=110000
(8,6)=100110
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
21.02
21.02+15.46+4.11+9.17+13.21+13.31
= 0.276
Fitness:
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.276
Prob.
21.02
21.02+15.46+4.11+9.17+13.21+13.31
= 0.276
Fitness:
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob String
http
s://
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob String
0.269
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob String
0.269
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101
String
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101
String
0.923
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110
String
http
s://
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String
0.0
00-0
.179
0.1
80-0
.359
0.3
60-0
.539
0.5
40-0
.719
0.7
20-0
.899
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String String
0.707
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String String
http
s://
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String
000110
String
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String
000110100101
String
http
s://
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
000101100110000101001111110000101100
String
000110100101
String
http
s://
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
Mu
tati
on000101
100110000101001111110000101100
String
000110100101000111001101111100100000
String
000110100101000111001101111100100000
String
Changes if
p>0.95
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
Mu
tati
on000101
100110000101001111110000101100
String
000110100101000111001101111100100000
String
000110100101000111001101111100100000
String
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
Mu
tati
on000101
100110000101001111110000101100
String
000110100101000111001101111100100000
String
000110001101000111001101111100110000
String
http
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000101001111011001101100110000100110
(0,5)(2,7)(6,1)
(10,4) (12,0) (8,6)
String (x,y)
21.0215.46
4.119.17
13.21 13.31
f(x,y)
0.2760.2030.0540.120 0.173 0.174
Prob.
0.2760.4790.5330.653 0.826 1.000
Cum.Prob
000101100110000101001111110000101100
String
Cro
sso
ver
Mu
tati
on000101
100110000101001111110000101100
String
000110100101000111001101111100100000
String
000110001101000111001101111100110000
String
Go to iter. 2
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000110001101000111001101111100110000
String
http
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000110001101000111001101111100110000
String
(0,6)(2,5)(0,7)(2,5) (14,4) (12,0)
(x,y)
23.1711.0525.4311.059.24 13.21
f(x,y)
Continue
Previous Avg. = 12.71
New Avg. = 15.52
Previous Max. = 21.02
New Max. = 25.43ht
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TSP Example: 30 Cities
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Solution j(Distance = 800)
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Best Solution (Distance = 420)
42
38
35
26
21
35
32
7
38
46
44
58
60
69
76
78
71
69
67
62
84
94
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
y
x
TSP30 Solution (Performance = 420)
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Benefits of Genetic Algorithms
Concept is easy to understand
Modular, separate from application
Supports multi-objective optimization
Good for “noisy” environments
Always an answer; answer gets better with time
Inherently parallel; easily distributed
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Benefits of Genetic Algorithms (cont.)
Many ways to speed up and improve a GA-
based application as knowledge about problem
domain is gained
Easy to exploit previous or alternate solutions
Flexible building blocks for hybrid applications
Substantial history and range of use
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When to Use a GA
Alternate solutions are too slow or overly complicated
Need an exploratory tool to examine new approaches
Problem is similar to one that has already been successfully
solved by using a GA
Want to hybridize with an existing solution
Benefits of the GA technology meet key problem
requirements
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Some GA Application Types
Domain Application Types
Control gas pipeline, pole balancing, missile evasion, pursuit
Design semiconductor layout, aircraft design, keyboard configuration, communication networks
Scheduling manufacturing, facility scheduling, resource allocation
Robotics trajectory planning
Machine Learning designing neural networks, improving classification algorithms, classifier systems
Signal Processing filter design
Game Playing poker, checkers, prisoner’s dilemma
Combinatorial
Optimization
set covering, travelling salesman, routing, bin packing, graph colouring and partitioning
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WRITING A FUNCTION
Select New in the MATLAB File menu.
Select M-File. This opens a new M-file in the editor.
In the M-file, enter the following two lines of code:
function z = my_fun(x)
z = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + x(2)^2 -6*x(2);
Save the M-file in a directory on the MATLAB path. ht
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FUNCTION HANDLE
A function handle is a MATLAB value that
provides a means of calling a function indirectly.
The syntax is
handle = @function name
For example
h = @my_funht
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USING THE TOOLBOX
There are two ways to use genetic algorithm with
the toolbox :
Calling the genetic algorithm function ga at the
command line
Using the genetic algorithm tool , a graphical
interfaceht
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AT COMMAND LINE
The syntax is
[ x fval ] = ga( @fitnessfun , nvars , options )
Where
• @fitnessfun is a function handle to the fitness function
• nvars is the number of independent variables for the fitness function
• options is a structure containing options for the genetic algorithm
• fval is the final value of the fitness function
• x is the point at which final value is attainedht
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USING GENETIC
ALGORITHM TOOL
To open we have to write following syntax at
the command line
gatool
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PAUSING AND STOPPING
Click Pause to temporarily suspend the algorithm.
To resume the algorithm using the current
population at the time you paused, click Resume.
Click Stop to stop the algorithm. The Status and
results pane displays the fitness function value
of the best point in the current generation at the
moment you clicked Stop.
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SETTING STOPPING
CRITERIA Generations -- The algorithm reaches the specified
number of generations.
Time -- The algorithm runs for the specified amount of time in seconds.
Fitness limit -- The best fitness value in the current generation is less than or equal to the specified value.
Stall generations -- The algorithm computes the specified number of generations with no improvement in the fitness function.
Stall time limit -- The algorithm runs for the specified amount of time in seconds with no improvement in the fitness function.
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DISPLAYING PLOTS
The Plots pane, shown in the following figure, enables you to
display various plots of the results of the genetic algorithm.
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RESULTS
For the function we wanted to minimise i.e my_fun
z = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + x(2)^2 - 6*x(2);
The final result that will be obtained after using genetic
algorithm will be
x = [ 0.25836 3.26145]ht
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Optimization of both the structure and the associated parameters of
four cascaded controllers for an IM drive. The GA tests and compares
controllers having different orders. The GA optimizes both the
number and the location of controller’s zeros and poles.
While conventional design strategies firstly tune the current controller
and then the speed controller (thus leading to a potentially sub-optimal
couple of controllers), the GA simultaneously optimizes the two
controllers, to obtain the best overall two cascaded- loops control
systems
We developed a system based on MATLAB/SIMULINK environment
and dSPACE control boards to run the optimization on-line, directly
on the hardware
Control of Induction Motor Drives
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Induction motor drive block diagram
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Crossover
•Binary flags: multi-point crossover
•Real valued chromosomes: either heuristic or arithmetical crossover is applied with equal probability.
Mutation
•Binary flags: binary mutation
•Real valued chromosomes: uniform mutation
Selection
•Tournament selection
Crossover, Mutation, and Selection
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The ability to work on heterogeneous solutions makes the considered
GA similar to other evolutionary computation techniques, such as
Genetic or Evolutionary Programming (GP or EP).
A significant difference between EP and GAs lays in the rate of
application of the mutation.
In the preliminary investigation for optimal occurrence rate of
operators, a much faster convergence was obtained with higher
mutation rates.
Due to our final choice of emphasizing the rate of mutation (80%)
with respect to crossover (20%), the algorithm can be viewed as a
hybrid GA-EP evolutionary algorithm.
Main Details of the Evolutionary
Algorithm
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The objective function is a weighted sum of five performance
indices that are directly measured on system’s response to the a
sequence of speed steps of different amplitude.
Choice of the fitness function
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Steady state speed response
Overview of the single indexes
This index measures the speed
error along segments of the
speed response settling around
the steady state.
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Speed overshoot
Overview of the single indexes
This index measures the
speed overshoot when a
speed step change is
required
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Transient speed response duration
Overview of the single indexes
This index measures the
duration of transient
conditions, that is the sum
of all the settling times
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Voltage reference oscillations for constant speed reference
Overview of the single indexes
This index accounts for
undesired ripples and
oscillations of the voltage
that would increase losses
and acoustic noise,
contrasting with constant
torque operations
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Current response
Overview of the single indexes
This index measures the
sum of absolute current
errors
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Experimental Setup
Induction Motor
PWM Inverter
PC with ds1104
Controller
Current & Voltage Sensor
Load Bank
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Speed response at the end of the GA optimization
On-line optimization with standard GA:
Induction Motor Drive
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Rotor flux
response at
the end of
the GA
optimization
On-line optimization with standard GA:
Induction Motor Drive
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Conclusions
Question: ‘If GAs are so smart, why ain’t they
rich?’
Answer: ‘Genetic algorithms are rich - rich in
application across a large and growing
number of disciplines.’
- David E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning
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