Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
A Transportation Problem is
one of the
most
typical
problems being encountered in many situations
and
it
has
many
practical applic
ations. Many researches had been conducted
and
many methods
had been proposed to solve it. One of the most
difficult challenge in solving the problem deals with inputting a
very large volume of data. With the development of intelligent
technologies, compu
ters had already been used to solved this
problem. This paper presents a method using Genetic Algorithm
(GA) t
o provide a new tool that can quickly calculate the solution
to the Balanced Transportation Problem.
The test results are compared with selected o
ld methods to
confirm the effectiveness of the use of GA. A
mathematical model
was used to represent the GA and be applied to solve it. Finally,
the test results of the model were presented so show the
effectiveness.
CONSTRUCTING A FUZZY NETWORK INTRUSION CLASSIFIER BASED ON DIFFERENTIAL EVOLU...IJCNCJournal
This paper presents a method for constructing intrusion detection systems based on efficient fuzzy rulebased
classifiers. The design process of a fuzzy rule-based classifier from a given input-output data set can
be presented as a feature selection and parameter optimization problem. For parameter optimization of
fuzzy classifiers, the differential evolution is used, while the binary harmonic search algorithm is used for
selection of relevant features. The performance of the designed classifiers is evaluated using the KDD Cup
1999 intrusion detection dataset. The optimal classifier is selected based on the Akaike information
criterion. The optimal intrusion detection system has a 1.21% type I error and a 0.39% type II error. A
comparative study with other methods was accomplished. The results obtained showed the adequacy of the
proposed method
Particle Swarm Optimization in the fine-tuning of Fuzzy Software Cost Estimat...Waqas Tariq
Software cost estimation deals with the financial and strategic planning of software projects. Controlling the expensive investment of software development effectively is of paramount importance. The limitation of algorithmic effort prediction models is their inability to cope with uncertainties and imprecision surrounding software projects at the early development stage. More recently, attention has turned to a variety of machine learning methods, and soft computing in particular to predict software development effort. Fuzzy logic is one such technique which can cope with uncertainties. In the present paper, Particle Swarm Optimization Algorithm (PSOA) is presented to fine tune the fuzzy estimate for the development of software projects . The efficacy of the developed models is tested on 10 NASA software projects, 18 NASA projects and COCOMO 81 database on the basis of various criterion for assessment of software cost estimation models. Comparison of all the models is done and it is found that the developed models provide better estimation
Duality in nonlinear fractional programming problem using fuzzy programming a...ijscmcj
In this paper we have considered nonlinear fractional programming problem with multiple constraints. A
pair of primal and dual for a special type of nonlinear fractional programming has been considered under
fuzzy environment. Exponential membership function has been used to deal with the fuzziness. Duality
results have been developed for the special type of nonlinear programming using exponential membership function. The method has been illustrated with numerical example. Genetic Algorithm as well as Fuzzy programming approach has been used to solve the problem.
A Transportation Problem is
one of the
most
typical
problems being encountered in many situations
and
it
has
many
practical applic
ations. Many researches had been conducted
and
many methods
had been proposed to solve it. One of the most
difficult challenge in solving the problem deals with inputting a
very large volume of data. With the development of intelligent
technologies, compu
ters had already been used to solved this
problem. This paper presents a method using Genetic Algorithm
(GA) t
o provide a new tool that can quickly calculate the solution
to the Balanced Transportation Problem.
The test results are compared with selected o
ld methods to
confirm the effectiveness of the use of GA. A
mathematical model
was used to represent the GA and be applied to solve it. Finally,
the test results of the model were presented so show the
effectiveness.
CONSTRUCTING A FUZZY NETWORK INTRUSION CLASSIFIER BASED ON DIFFERENTIAL EVOLU...IJCNCJournal
This paper presents a method for constructing intrusion detection systems based on efficient fuzzy rulebased
classifiers. The design process of a fuzzy rule-based classifier from a given input-output data set can
be presented as a feature selection and parameter optimization problem. For parameter optimization of
fuzzy classifiers, the differential evolution is used, while the binary harmonic search algorithm is used for
selection of relevant features. The performance of the designed classifiers is evaluated using the KDD Cup
1999 intrusion detection dataset. The optimal classifier is selected based on the Akaike information
criterion. The optimal intrusion detection system has a 1.21% type I error and a 0.39% type II error. A
comparative study with other methods was accomplished. The results obtained showed the adequacy of the
proposed method
Particle Swarm Optimization in the fine-tuning of Fuzzy Software Cost Estimat...Waqas Tariq
Software cost estimation deals with the financial and strategic planning of software projects. Controlling the expensive investment of software development effectively is of paramount importance. The limitation of algorithmic effort prediction models is their inability to cope with uncertainties and imprecision surrounding software projects at the early development stage. More recently, attention has turned to a variety of machine learning methods, and soft computing in particular to predict software development effort. Fuzzy logic is one such technique which can cope with uncertainties. In the present paper, Particle Swarm Optimization Algorithm (PSOA) is presented to fine tune the fuzzy estimate for the development of software projects . The efficacy of the developed models is tested on 10 NASA software projects, 18 NASA projects and COCOMO 81 database on the basis of various criterion for assessment of software cost estimation models. Comparison of all the models is done and it is found that the developed models provide better estimation
Duality in nonlinear fractional programming problem using fuzzy programming a...ijscmcj
In this paper we have considered nonlinear fractional programming problem with multiple constraints. A
pair of primal and dual for a special type of nonlinear fractional programming has been considered under
fuzzy environment. Exponential membership function has been used to deal with the fuzziness. Duality
results have been developed for the special type of nonlinear programming using exponential membership function. The method has been illustrated with numerical example. Genetic Algorithm as well as Fuzzy programming approach has been used to solve the problem.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Differential evolution (DE) algorithm has been applied as a powerful tool to find optimum switching angles for selective harmonic elimination pulse width modulation (SHEPWM) inverters. However, the DE’s performace is very dependent on its control parameters. Conventional DE generally uses either trial and error mechanism or tuning technique to determine appropriate values of the control paramaters. The disadvantage of this process is that it is very time comsuming. In this paper, an adaptive control parameter is proposed in order to speed up the DE algorithm in optimizing SHEPWM switching angles precisely. The proposed adaptive control parameter is proven to enhance the convergence process of the DE algorithm without requiring initial guesses. The results for both negative and positive modulation index (M) also indicate that the proposed adaptive DE is superior to the conventional DE in generating SHEPWM switching patterns.
BPSO&1-NN algorithm-based variable selection for power system stability ident...IJAEMSJORNAL
Due to the very high nonlinearity of the power system, traditional analytical methods take a lot of time to solve, causing delay in decision-making. Therefore, quickly detecting power system instability helps the control system to make timely decisions become the key factor to ensure stable operation of the power system. Power system stability identification encounters large data set size problem. The need is to select representative variables as input variables for the identifier. This paper proposes to apply wrapper method to select variables. In which, Binary Particle Swarm Optimization (BPSO) algorithm combines with K-NN (K=1) identifier to search for good set of variables. It is named BPSO&1-NN. Test results on IEEE 39-bus diagram show that the proposed method achieves the goal of reducing variables with high accuracy.
An optimal design of current conveyors using a hybrid-based metaheuristic alg...IJECEIAES
This paper focuses on the optimal sizing of a positive second-generation current conveyor (CCII+), employing a hybrid algorithm named DE-ACO, which is derived from the combination of differential evolution (DE) and ant colony optimization (ACO) algorithms. The basic idea of this hybridization is to apply the DE algorithm for the ACO algorithm’s initialization stage. Benchmark test functions were used to evaluate the proposed algorithm’s performance regarding the quality of the optimal solution, robustness, and computation time. Furthermore, the DE-ACO has been applied to optimize the CCII+ performances. SPICE simulation is utilized to validate the achieved results, and a comparison with the standard DE and ACO algorithms is reported. The results highlight that DE-ACO outperforms both ACO and DE.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
USING LEARNING AUTOMATA AND GENETIC ALGORITHMS TO IMPROVE THE QUALITY OF SERV...IJCSEA Journal
A hybrid learning automata–genetic algorithm (HLGA) is proposed to solve QoS routing optimization problem of next generation networks. The algorithm complements the advantages of the learning Automato Algorithm(LA) and Genetic Algorithm(GA). It firstly uses the good global search capability of LA to generate initial population needed by GA, then it uses GA to improve the Quality of Service(QoS) and acquiring the optimization tree through new algorithms for crossover and mutation operators which are an NP–Complete problem. In the proposed algorithm, the connectivity matrix of edges is used for genotype representation. Some novel heuristics are also proposed for mutation, crossover, and creation of random individuals. We evaluate the performance and efficiency of the proposed HLGA-based algorithm in comparison with other existing heuristic and GA-based algorithms by the result of simulation. Simulation results demonstrate that this paper proposed algorithm not only has the fast calculating speed and high accuracy but also can improve the efficiency in Next Generation Networks QoS routing. The proposed algorithm has overcome all of the previous algorithms in the literature..
Genetic algorithm guided key generation in wireless communication (gakg)IJCI JOURNAL
In this paper, the proposed technique use high speed stream cipher approach because this approach is useful where less memory and maximum speed is required for encryption process. In this proposed approach Self Acclimatize Genetic Algorithm based approach is exploits to generate the key stream for encrypt / decrypt the plaintext with the help of key stream. A widely practiced approach to identify a good set of parameters for a problem is through experimentation. For these reasons, proposed enhanced Self Acclimatize Genetic Algorithm (GAKG) offering the most appropriate exploration and exploitation behavior. Parametric tests are done and results are compared with some existing classical techniques, which shows comparable results for the proposed system.
Using particle swarm optimization to solve test functions problemsriyaniaes
In this paper the benchmarking functions are used to evaluate and check the particle swarm optimization (PSO) algorithm. However, the functions utilized have two dimension but they selected with different difficulty and with different models. In order to prove capability of PSO, it is compared with genetic algorithm (GA). Hence, the two algorithms are compared in terms of objective functions and the standard deviation. Different runs have been taken to get convincing results and the parameters are chosen properly where the Matlab software is used. Where the suggested algorithm can solve different engineering problems with different dimension and outperform the others in term of accuracy and speed of convergence.
This paper introduces a solution of the economic load dispatch (ELD) problem using a hybrid approach of fuzzy logic and genetic algorithm (GA). The proposed method combines and extends the attractive features of both fuzzy logic and GA. The proposed approach is compared with lambda iteration method (LIM) and GA. The investigation reveals that the proposed approach can provide accurate solution with fast convergence characteristics and is superior to the GA and LIM.
Optimization of Mechanical Design Problems Using Improved Differential Evolut...IDES Editor
Differential Evolution (DE) is a novel evolutionary
approach capable of handling non-differentiable, non-linear
and multi-modal objective functions. DE has been consistently
ranked as one of the best search algorithm for solving global
optimization problems in several case studies. This paper
presents an Improved Constraint Differential Evolution
(ICDE) algorithm for solving constrained optimization
problems. The proposed ICDE algorithm differs from
unconstrained DE algorithm only in the place of initialization,
selection of particles to the next generation and sorting the
final results. Also we implemented the new idea to five versions
of DE algorithm. The performance of ICDE algorithm is
validated on four mechanical engineering problems. The
experimental results show that the performance of ICDE
algorithm in terms of final objective function value, number
of function evaluations and convergence time.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Generating a custom Ruby SDK for your web service or Rails API using Smithyg2nightmarescribd
Have you ever wanted a Ruby client API to communicate with your web service? Smithy is a protocol-agnostic language for defining services and SDKs. Smithy Ruby is an implementation of Smithy that generates a Ruby SDK using a Smithy model. In this talk, we will explore Smithy and Smithy Ruby to learn how to generate custom feature-rich SDKs that can communicate with any web service, such as a Rails JSON API.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Elevating Tactical DDD Patterns Through Object CalisthenicsDorra BARTAGUIZ
After immersing yourself in the blue book and its red counterpart, attending DDD-focused conferences, and applying tactical patterns, you're left with a crucial question: How do I ensure my design is effective? Tactical patterns within Domain-Driven Design (DDD) serve as guiding principles for creating clear and manageable domain models. However, achieving success with these patterns requires additional guidance. Interestingly, we've observed that a set of constraints initially designed for training purposes remarkably aligns with effective pattern implementation, offering a more ‘mechanical’ approach. Let's explore together how Object Calisthenics can elevate the design of your tactical DDD patterns, offering concrete help for those venturing into DDD for the first time!
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Assuring Contact Center Experiences for Your Customers With ThousandEyes
Mimo system-order-reduction-using-real-coded-genetic-algorithm
1. World Academy of Science, Engineering and Technology
International Journal of Electrical, Electronic Science and Engineering Vol:5 No:3, 2011
MIMO System Order Reduction Using RealCoded Genetic Algorithm
International Science Index 51, 2011 waset.org/publications/4570
Swadhin Ku. Mishra, Sidhartha Panda, Simanchala Padhy and C. Ardil
Abstract—In this paper, real-coded genetic algorithm (RCGA)
optimization technique has been applied for large-scale linear
dynamic multi-input-multi-output (MIMO) system. The method is
based on error minimization technique where the integral square error
between the transient responses of original and reduced order models
has been minimized by RCGA. The reduction procedure is simple
computer oriented and the approach is comparable in quality with the
other well-known reduction techniques. Also, the proposed method
guarantees stability of the reduced model if the original high-order
MIMO system is stable. The proposed approach of MIMO system
order reduction is illustrated with the help of an example and the
results are compared with the recently published other well-known
reduction techniques to show its superiority.
Keywords—Multi-input-multi-output (MIMO) system. Model
order reduction. Integral squared error (ISE). Real-coded genetic
algorithm.
I. INTRODUCTION
I
N system theory the approximation of higher order system
by lower order models is one of the important challenges.
Because by using the reduced lower order model, the
implementation of analysis, simulation and various system
designs become easier. Various order-reduction methods are
available for linear continuous time domain system as well as
systems in frequency domain. Further, the extension of singleinput single-output (SISO) methods to reduce multi-input
multi-output (MIMO) systems has also been carried out in [1][3].
Various order reduction methods are available in [4]-[8]
based on the minimization of the integral square error (ISE)
criterion. However, a familiar aspect in the methods explained
in [4]-[7] is that the denominator coefficients values of the
low order system (LOS) are selected arbitrarily by some
stability preserving methods such as dominant pole, Routh
Swadhin Kumar Mishra is working as Assistant Professor in Electronics
and Communication Engg. Department, NIST Berhampur, Orissa, India, Pin:
761008 (e-mail: swadhin.mishra@gmail.com ).
S. Panda is working as a Professor in the Department of Electrical and
Electronics Engineering, NIST, Berhampur, Orissa, India, Pin: 761008. (email: panda_sidhartha@rediffmail.com ).
Simanchala Padhy is working as Engineering Assistant, at HPT,
Berhampur, Prasar Bharati Broadcasting Corporation of India (e-mail :
simanchala.ddair@gmail.com)
C. Ardil is with National Academy of Aviation, AZ1045, Baku,
Azerbaijan, Bina, 25th km, NAA (e-mail: cemalardil@gmail.com).
approximation methods, etc. and then the numerator
coefficients of the LOS are determined by minimization of the
ISE. In [8], Howitt and Luss recommended a procedure, in
which both the numerator and denominator coefficients are
considered to be free parameters and are chosen to minimize
the ISE in impulse or step responses.
These days, Genetic algorithm (GA) is becoming popular to
solve the optimization problems in different fields of
application mainly because of their robustness in finding an
optimal solution and ability to provide a near optimal solution
close to a global minimum. Unlike strict mathematical
methods, the GA does not require the condition that the
variables in the optimization problem be continuous and
different; it only requires that the problem to be solved can be
computed. The present attempt is towards evolving a new
algorithm for order reduction, where all the numerator and
denominator parameters are considered to be free parameters
and the error minimization by RCGA is employed to optimize
these parameters. The proposed algorithm consists of
searching all the parameters by minimizing the integral square
error between the transient responses of original and LOS
using RCGA. The proposed approach is illustrated with the
help of an example and the results are compared with the
recently published techniques.
II. REAL-CODED GENERIC ALGORITHM
Genetic algorithm (GA) has been used to solve difficult
engineering problems that are complex and difficult to solve
by conventional optimization methods. GA maintains and
manipulates a population of solutions and implements a
survival of the fittest strategy in their search for better
solutions. The fittest individuals of any population tend to
reproduce and survive to the next generation thus improving
successive generations. The inferior individuals can also
survive and reproduce. Implementation of GA requires the
determination of six fundamental issues: chromosome
representation, selection function, the genetic operators,
initialization, termination and evaluation function. Brief
descriptions about these issues are provided in the following
sections.
A. Chromosome representation
Chromosome representation scheme determines how the
problem is structured in the GA and also determines the
genetic operators that are used. Each individual or
chromosome is made up of a sequence of genes. Various types
of representations of an individual or chromosome are: binary
17
2. World Academy of Science, Engineering and Technology
International Journal of Electrical, Electronic Science and Engineering Vol:5 No:3, 2011
International Science Index 51, 2011 waset.org/publications/4570
digits, floating point numbers, integers, real values, matrices,
etc. Generally natural representations are more efficient and
produce better solutions. Real-coded representation is more
efficient in terms of CPU time and offers higher precision with
more consistent results.
B. Selection Function
To produce successive generations, selection of individuals
plays a very significant role in a genetic algorithm. The
selection function determines which of the individuals will
survive and move on to the next generation. A probabilistic
selection is performed based upon the individual’s fitness such
that the superior individuals have more chance of being
selected. There are several schemes for the selection process:
roulette wheel selection and its extensions, scaling techniques,
tournament, normal geometric, elitist models and ranking
methods.
The selection approach assigns a probability of selection
Pj to each individuals based on its fitness value. In the present
study, normalized geometric selection function has been used.
In normalized geometric ranking, the probability of
selecting an individual Pi is defined as:
Pi = q ' (1 − q )r −1
(1)
q' =
q
1 − (1 − q ) p
(2)
where,
q = probability of selecting the best individual
r = rank of the individual (with best equals 1)
P = population size
C. Genetic Operator
The basic search mechanism of the GA is provided by the
genetic operators. There are two basic types of operators:
crossover and mutation. These operators are used to produce
new solutions based on existing solutions in the population.
Crossover takes two individuals to be parents and produces
two new individuals while mutation alters one individual to
produce a single new solution. The following genetic
operators are usually employed: simple crossover, arithmetic
crossover and heuristic crossover as crossover operator and
uniform mutation, non-uniform mutation, multi-non-uniform
mutation, boundary mutation as mutation operator. Arithmetic
crossover and non-uniform mutation are employed in the
present study as genetic operators. Crossover generates a
random number r from a uniform distribution from 1 to m and
creates two new individuals by using equations:
⎧ xi ,
xi' = ⎨
⎩ yi ,
⎧ yi ,
yi' = ⎨
⎩ xi ,
if i < r
(3)
otherwise
if i < r
′
X = r X + (1 − r )Y
′
Y = rY + (1 − r ) X
(5)
Non-uniform mutation randomly selects one variable j and
sets it equal to an non-uniform random number
xi' = xi + (bi − xi ) f (G ) if r1 < 0.5
xi' = xi + ( xi + ai ) f (G ) if r1 ≥ 0.5
xi
(6)
otherwise
where
⎛ ⎛
G ⎞⎞
⎟⎟
f (G ) = ⎜ r2 ⎜1 −
⎟⎟
⎜ ⎜ G
max ⎠ ⎠
⎝
⎝
b
(7)
r1, r2 = uniform random numbers between 0 and 1.
G = current generation.
Gmax = maximum number of generations.
b = shape parameter
D. Initialization, evaluation function and stopping criteria
An initial population is needed to start the genetic algorithm
procedure. The initial population can be randomly generated
or can be taken from other methods. Evaluation functions or
objective functions of many forms can be used in a GA so that
the function can map the population into a partially ordered
set. The GA moves from generation to generation until a
stopping criterion is met. The stopping criterion could be
maximum number of generations, population convergence
criteria, lack of improvement in the best solution over a
specified number of generations or target value for the
objective function.
III. DESCRIPTION OF THE PROPOSED ALGORITHM.
Let the transfer function matrix of the HOS of order 'n'
having ‘p’ inputs and ‘m’ outputs are:
⎡ a11 ( s ) a12 ( s )
⎢
1 ⎢ a21 ( s ) a22 ( s )
[G ( s )] =
M
Dn ( s ) ⎢ M
⎢
⎢am1 ( s ) am 2 ( s )
⎣
L a1 p ( s ) ⎤
L a2 p ( s ) ⎥
⎥ (8)
L
M ⎥
⎥
L amp ( s )⎥
⎦
or, [G(s)] = [gij(s)]
i=1, 2… m and j=1, 2… p is a m x p
matrix.
The general form of gij(s) is taken as:
aij ( s )
a0 + a1s + a2 s 2 + .... + an −1s n −1
gij ( s ) =
=
Dn ( s ) b0 + b1s + b2 s 2 + .... + bn −1s n −1 + s n
(4)
(9)
otherwise
Arithmetic crossover produces two complimentary linear
combinations of the parents, where r = U (0, 1):
D(s) is the nth order denominator polynomial of the
transmission matrix for the HOS.
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3. World Academy of Science, Engineering and Technology
International Journal of Electrical, Electronic Science and Engineering Vol:5 No:3, 2011
⎡ 2( s + 5)
⎢
1
[G( s)] = ⎢ ( s +s )(s +)10)
( + 10
⎢
⎢ ( s + 1)( s + 20)
⎣
Let, the transfer function matrix of the HOS of order 'n' having
‘p’ inputs and ‘m’ outputs to be synthesized is:
⎡ b11( s ) b12 ( s )
⎢
1 ⎢ b21( s ) b22 ( s )
[ R( s )] =
M
Dr ( s ) ⎢ M
⎢
⎢bm1 ( s ) bm 2 ( s )
⎣
L b1 p ( s ) ⎤
L b2 p ( s ) ⎥
⎥ (10)
L
M ⎥
⎥
L bmp ( s )⎥
⎦
International Science Index 51, 2011 waset.org/publications/4570
or, [R(s)] = [rij(s)]
i=1, 2… m and j=1, 2… p is a m x p
matrix.
The general form of rij(s) is taken as:
bij ( s)
μ + μ1s + μ2 s 2 + .... + μ r −1s r −1
rij ( s) =
= 0
Dr ( s) b0 + b1s + b2 s 2 + .... + br −1s r −1 + s r
( s + 4) ⎤
1 ⎡ a11 ( s) a12 ( s ) ⎤
( s + 2)( s + 5) ⎥
⎥=
( s + 6) ⎥ D6 ( s) ⎢a21( s ) a22 ( s ) ⎥
⎣
⎦
( s + 2)( s + 3) ⎥
⎦
(13)
where D6(s) is the common denominator polynomial. As can
be verified the MIMO system is a 6th order system and is
given as
D6 ( s ) = ( s + 1)( s + 2)( s + 3)( s + 5)( s + 10)( s + 20)
= 6000 + 13100 s + 10060 s 2 + 3491s 3 + 571s 4 + 41s 5 + s 6
and
a11 ( s ) = 6000 + 7700 s + 3610 s 2 + 762 s 3 + 70 s 4 + 2 s 5
(11)
Dr(s) is the rth order denominator polynomial of the
transmission matrix for the reduced LOS.
The reduction process from an nth order HOS to a reduced rth
order LOS consists of the following two steps :
a12 ( s ) = 2400 + 4100s + 2182s 2 + 459 s 3 + 38s 4 + s 5
a21 ( s ) = 3000 + 3700 s + 1650 s 2 + 331s 3 + 30 s 4 + s 5
a22 ( s) = 6000 + 9100 s + 3660 s 2 + 601s 3 + 41s 4 + s 5
Step-1
The coefficients of the denominator of the reduced LOS are
found out by employing GA. The integral square error (ISE) is
found out between the transient responses between the HOS
and LOS and is given as
2
∞
E = ∫ [ y (t ) − yr (t )] dt
The proposed algorithm is applied to minimize the function E
(ISE). The ISE is calculated for each element (rij(s)) of the
transfer function matrix of the LOS and it is given by Eqn.
(14)
∞
[
where i=1,2 ; j=1,2; and gij(t) and rij(t) are the unit step
response of the original and reduced order model,
respectively.
The general form of the transfer function matrix of the
reduced second order LOS is taken as
[R( s)] =
Step-2
IV. NUMERIC EXAMPLE
To demonstrate the proposed method, one numeric example
of a higher order system (6th order) is taken from literature[9][10] and the proposed algorithm is employed to obtain a
second-order reduced model.
Consider a system having two inputs and two outputs with
transfer matrix of
(14)
0
0
Once the denominator coefficients of the reduced LOS are
found out, GA is used to find out the numerator coefficients
of LOS. Here also the Real Coded Generic Algorithm is
implemented in such a way to minimize the ISE by keeping
the values of the denominator coefficients, which are found
out in step 1, of the reduced LOS constant.
]
E = ∫ gij (t ) − rij (t ) dt
(12)
where y(t) and yr(t) are the unit step responses of original
G(s) and reduced R(s) order systems.
Real Coded Generic Algorithm is employed to minimize
the integral square error (ISE) in order to find out the
denominator coefficients of the LOS.
2
1 ⎡ b11 ( s ) b12 ( s ) ⎤
Dr ( s ) ⎢b21 ( s ) b22 ( s )⎥
⎣
⎦
(15)
Where Dr(s) is the common denominator polynomial of the
reduced model. For the implementation of RCGA normal
geometric selection is employed which is a ranking selection
function based on the normalized geometric distribution.
Arithmetic crossover takes two parents and performs an
interpolation along the line formed by the two parents. Non
uniform mutation changes one of the parameters of the parent
based on a non-uniform probability distribution. This
Gaussian distribution starts wide, and narrows to a point
distribution as the current generation approaches the
maximum generation. The follow chart of proposed approach
is shown in Fig. 1.
The reduced 2nd order common denominator polynomial is
found by implementing our method as:
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4. World Academy of Science, Engineering and Technology
International Journal of Electrical, Electronic Science and Engineering Vol:5 No:3, 2011
Dr ( s ) = 1.0459s 2 + 4.1138s + 2.9920
0.5
and
0.4
Amplitude
b11( s ) = 1.3935s + 2.9918
b12 ( s ) = 1.0773s + 1.1985
b21( s) = 0.6247 s + 1.4963
b22 ( s ) = 1.8935s + 3.0069
0.2
0.1
The step responses of original and reduced order models are
compared in Figs. 2(a-d). A comparison of proposed method
with the other well-known order-reduction techniques
available in literature is given in the Table 1. The comparison
is made by calculating the ISE given by Eqn. (14) pertaining
to a step input. The ISE is calculated for each element of the
transfer function matrix of the LOS with respect to the
corresponding original system.
0
Original 6th order system
Reduced 2nd order system
0
2
4
Time (sec)
6
8
(b)
0.6
Amplitude
0.5
Start
0.4
0.3
0.2
Original 6th order system
Reduced 2nd order system
0.1
Specify the parameters for GA
0
0
2
Generate initial population
4
Time (sec)
6
8
(c)
Gen.=1
1
Time-domain simulation
Amplitude
0.8
Find the fittness of each individual
in the current population
0.6
0.4
Gen.=Gen.+1
Gen. > Max. Gen.?
0.2
Stop
Yes
0
No
0
2
4
Time (sec)
6
8
Fig. 2 Comparison of step response (a) Output 1 for input 1 (b)
Output 1 for input 2
(c) Output 2 for input 1 (d) Output 2 for input
2
Fig. 1 Flowchart of the proposed approach
V. CONCLUSION
1
0.8
0.6
0.4
0.2
0
Original 6th order system
Reduced 2nd order system
(d)
Apply GA operators:
selection,crossover and mutation
Amplitude
International Science Index 51, 2011 waset.org/publications/4570
0.3
Original 6th order system
Reduced 2nd order system
0
2
4
Time (sec)
(a)
6
8
An approach based on the error minimization technique
employing real-coded genetic algorithm has been presented, to
derive stable reduced order models for linear time invariant
multi-input multi-output dynamic systems. The algorithm has
been implemented in MATLAB 7.0.1 on an Inetl Core 2 Duo
processor and the computation time is negligible being less
than 1 minute. The matching of the step response is assured
reasonably well in the method. The ISE in between the
transient parts of original and reduced order systems is
calculated and compared in the tabular form as given in Tables
1, from which it is clear that the proposed algorithm compares
well with the other existing techniques of model order
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5. World Academy of Science, Engineering and Technology
International Journal of Electrical, Electronic Science and Engineering Vol:5 No:3, 2011
reduction. The method also preserves the model stability and
avoids any steady-state error between the time responses of
the original and reduced systems.
TABLE I
COMPARISON OF REDUCTION METHODS
r11
r12
r21
r22
Proposed
method
Vishwakarma
and Prasad [9]
Parmar et al.
[10]
Prasad and Pal
[11]
Safonov and
Chiang [12]
Prasad et al.
[13]
International Science Index 51, 2011 waset.org/publications/4570
Reduction
method
4.0656x10-4
7.772x10-5
3.2448x10-5
0.0068
0.001515
7.845x10-5
2.9984x10-4
0.0047
0.014498
0.008744
0.002538
0.015741
0.136484
0.002446
0.040291
0.067902
0.590617
0.037129
0.007328
1.066123
0.030689
0.000256
0.261963
0.021683
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
S. Mukherjee and R.N. Mishra,“Reduced order modelling of linear
multivariable systems using an error minimization technique”, Journalof
Franklin Inst., Vol. 325, No. 2 , pp. 235-245,1998.
S. S. Lamba, R. Gorez and B. Bandyopadhyay “New reduction
technique by step error minimization for multivariable systems”, Int. J.
Systems Sci., Vol. 19, No. 6, pp. 999-1009,1988.
R. Prasad, A. K. Mittal and S. P. Sharma “ A mixed method for the
reduction of multivariable systems”, Journal of Institute of Engineers,
India, IE(I) Journal-EL, Vol. 85, pp 177-181,2005.
4. C. Hwang “ Mixed method of Routh and ISE criterion approaches for
reduced order modelling of continuous time systems”, Trans. ASME, J.
Dyn. Syst. Meas. Control, Vol. 106,pp. 353-356. 1984.
S. Mukherjee, and R.N. Mishra, “Order reduction of linear systems
using an error minimization technique”, Journalof Franklin Inst., Vol.
323, No. 1, pp. 23-32, 1987.
N.N. Puri, and D.P. Lan, “Stable model reduction by impulse response
error minimization using Mihailov criterion and Pade’s approximation”,
Trans. ASME, J. Dyn. Syst. Meas. Control, Vol. 110, pp. 389-394, 1988.
P. Vilbe, and L.C. Calvez, “On order reduction of linear systems using
an error minimization technique”, Journal of Franklin Inst., Vol. 327,
pp. 513-514, 1990.
G.D. Howitt, and R. Luus, “Model reduction by minimization of
integral square error performance indices”, Journal of Franklin Inst.,
Vol. 327, pp. 343-357, 1990.
Vishwakarma and Prasad, “MIMO System Reduction Using Modified
Pole Clustering and Genetic Algorithm”, Hindawi Pub. Corp. Mod. and
Sim. in Engineering,
G. Parmar, R. Prasad, and S. Mukherjee, “Order reduction of linear
dynamic systems using stability equation method and GA,”
International Journal of Computer, Information, and Systems Science,
and Engineering, vol. 1, no. 1, pp. 26–32, 2007.
R. Prasad and J. Pal, “Use of continued fraction expansion for stable
reduction of linearmultivariable systems,” Journal of the Institution of
Engineers, vol. 72, pp. 43–47, 1991.
M. G. Safonov and R. Y. Chiang, “Model reduction for robust control: a
schur relative errormethod,” International Journal of Adaptive Control
and Signal Processing, vol. 2, no. 4, pp. 259–272, 1988.
R. Prasad, J. Pal, and A. K. Pant, “Multivariable system approximation
using polynomial derivatives,” Journal of the Institution of Engineers,
vol. 76, pp. 186–188, 1995.
Swadhin Kumar Mishra is working as an Asst. Prof. at National institute of
Science and Technology (NIST), Berhampur, Orissa. He has done his M.Tech
at Indian Institute of Technology, Madras, India in 2009 and his B.E. degree
in 2000. Earlier he worked as an Assistance Professor at RIT, Berhampur, as
Senior Lecturer at JITM, Paralakhemundi, Orissa, India and as lecturer at
SMIT, Berhampur, Orissa, India. His areas of research include MIMO
communication and optical communication.
Dr. Sidhartha Panda received Ph.D. degree from Indian Institute of
Technology (IIT), Roorkee, India in 2008, M.E. degree from VSS University
of Technology, (erstwhile UCE, Burla) in 2001 and. B.E. degree from
Bangalore University in 1991 all in Electrical Engineering. Presently he is
working as a Professor in the EEE department at National Institute of Science
and Technology (NIST), Berhampur, Orissa. Earlier he worked as an
Associate Professor in KIIT Deemed University and also in various other
engineering colleges for about 15 years. He has published about 60 papers in
various International Journals. Presently, he is acting as a reviewer of some
International Journals namely; IEEE Transactions on Industrial Electronics,
Applied Soft Computing (Elsevier), International Journal Electric Power and
Energy Systems (Elsevier), International Journal Simulation Modelling
Practice and Theory (Elsevier), International Journal of Control and Intelligent
Systems (ACTA Press). The biography of Dr Panda has been included “Who's
Who in the World”: 2010 edition, and “Who's Who in Science and
Engineering”: 2011-2012, by in Marquis', USA, “2000 Outstanding
Intellectuals of The 21st Century”, and “2000 Outstanding Scientists: 2010”,
and nominated for “Top 100 Engineers: 2010”, by International Biographical
Centre, Cambridge, England.. His areas of research include
MATLAB/SIMULINK, Flexible AC Transmission Systems (FACTS),
Modeling of Machines, Power Systems and FACTS, Controller Design,
Power System Stability, Genetic Algorithm, Particle Swarm Optimization,
Differential Evolution, Multi-objective Optimization, Economic Operation of
Power System, Fuzzy Logic, Model Order Reduction, Distributed Generation
and Wind Energy.
Simanchala Padhy received B.E. degree in Applied Electronics &
Instrumentation Engineering from R.E.C.(now NIT) Rourkela ,Orissa, India in
the year 2001. He has also done the Post Diploma in Computer Application
from U.C.P. Engineering School, Berhampur, Orissa, India in the year 1997
and Diploma in Electronics & Telcom. from U.C.P. Engineering School,
Berhampur ,Orissa, India in the year 1995. Earlier, he has worked as a lecturer
in V.I.T.A.M Engineering College, Parvathipuram, Andhrapradesh, India
from 2002-2003. Presently he is working as an Engineering Assistant in
Prasar Bharati, Broadcasting Corporation of India (Public Broadcaster of
India) in HPT Berhampur, Orissa, India.
C. Ardil is with National Academy of Aviation, AZ1045, Baku, Azerbaijan,
Bina, 25th km, NAA
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