Search results for: Linear matrix inequality

Authors: K. Kaabeche-Djerafi, N. Bendjaballah-Lalaoui, S. Semra

Abstract:

The paper aims at investigating influence of medium capacity on linear adsorbed solute dispersion into chemically heterogeneous fixed beds. A discrete chemical heterogeneity distribution is considered in the one-dimensional advectivedispersive equation. The partial differential equation is solved using finite volumes method based on the Adam-Bashforth algorithm. Increased dispersion is estimated by comparing breakthrough curves second order moments and keeping identical hydrodynamic properties. As a result, dispersion increase due to chemical heterogeneity depends on the column size and surprisingly on the solid capacity. The more intense capacity is, the more important solute dispersion is. Medium length which is known to favour this effect vanishing according to the linear adsorption in fixed bed seems to create nonmonotonous variation of dispersion because of the heterogeneity. This nonmonotonous behaviour is also favoured by high capacities.

Keywords: linear adsorption; chemical heterogeneity;dispersion; fixed bed; porous media

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Authors: M. B. Meenavathi, K. Rajesh

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Color image segmentation plays an important role in computer vision and image processing areas. In this paper, the features of Volterra filter are utilized for color image segmentation. The discrete Volterra filter exhibits both linear and nonlinear characteristics. The linear part smoothes the image features in uniform gray zones and is used for getting a gross representation of objects of interest. The nonlinear term compensates for the blurring due to the linear term and preserves the edges which are mainly used to distinguish the various objects. The truncated quadratic Volterra filters are mainly used for edge preserving along with Gaussian noise cancellation. In our approach, the segmentation is based on K-means clustering algorithm in HSI space. Both the hue and the intensity components are fully utilized. For hue clustering, the special cyclic property of the hue component is taken into consideration. The experimental results show that the proposed technique segments the color image while preserving significant features and removing noise effects.

Keywords: Color image segmentation, HSI space, K–means clustering, Volterra filter.

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Authors: M. P. Nanda Kumar, K. Dheeraj

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The design of a feedback controller, so as to minimize a given performance criterion, for a general non-linear dynamical system is difficult; if not impossible. But for a large class of non-linear dynamical systems, the open loop control that minimizes a performance criterion can be obtained using calculus of variations and Pontryagin’s minimum principle. In this paper, the open loop optimal trajectories, that minimizes a given performance measure, is used to train the neural network whose inputs are state variables of non-linear dynamical systems and the open loop optimal control as the desired output. This trained neural network is used as the feedback controller. In other words, attempts are made here to solve the “inverse optimal control problem” by using the state and control trajectories that are optimal in an open loop sense.

Keywords: Inverse Optimal Control, Radial basis function neural network, Controller Design.

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Authors: Michal Natora, Felix Franke, Klaus Obermayer

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Linear convolutive filters are fast in calculation and in application, and thus, often used for real-time processing of continuous data streams. In the case of transient signals, a filter has not only to detect the presence of a specific waveform, but to estimate its arrival time as well. In this study, a measure is presented which indicates the performance of detectors in achieving both of these tasks simultaneously. Furthermore, a new sub-class of linear filters within the class of filters which minimize the quadratic response is proposed. The proposed filters are more flexible than the existing ones, like the adaptive matched filter or the minimum power distortionless response beamformer, and prove to be superior with respect to that measure in certain settings. Simulations of a real-time scenario confirm the advantage of these filters as well as the usefulness of the performance measure.

Keywords: Adaptive matched filter, minimum variance distortionless response, beam forming, Capon beam former, linear filters, performance measure.

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Authors: Jaromír Marek, Jakub Wiener, Yan Wang

Abstract:

The main goal of this study was to find simple and industrially applicable production of ion exchangers based on nanofibrous polystyrene matrix and characterization of prepared material. Starting polystyrene nanofibers were sulfonated and crosslinked under appropriate conditions at the same time by sulfuric acid. Strongly acidic cation exchanger was obtained in such a way. The polymer matrix was made from polystyrene nanofibers prepared by Nanospider^TM technology.

Various types postpolymerization reactions and other methods of crosslinking were studied. Greatly different behavior between nano- and microsize materials was observed. The final nanofibrous material was characterized and compared to common granular ion exchangers and available microfibrous ion exchangers. The sorption properties of nanofibrous ion exchangers were compared with the granular ion exchangers. For nanofibrous ion exchangers of comparable ion exchange capacity was observed considerably faster adsorption kinetics.

Keywords: Electrospinning, ion exchangers, nanofibers, polystyrene.

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Authors: Dipti Patra, Mridula J

Abstract:

In this paper, we propose a new image segmentation approach for colour textured images. The proposed method for image segmentation consists of two stages. In the first stage, textural features using gray level co-occurrence matrix(GLCM) are computed for regions of interest (ROI) considered for each class. ROI acts as ground truth for the classes. Ohta model (I1, I2, I3) is the colour model used for segmentation. Statistical mean feature at certain inter pixel distance (IPD) of I2 component was considered to be the optimized textural feature for further segmentation. In the second stage, the feature matrix obtained is assumed to be the degraded version of the image labels and modeled as Markov Random Field (MRF) model to model the unknown image labels. The labels are estimated through maximum a posteriori (MAP) estimation criterion using ICM algorithm. The performance of the proposed approach is compared with that of the existing schemes, JSEG and another scheme which uses GLCM and MRF in RGB colour space. The proposed method is found to be outperforming the existing ones in terms of segmentation accuracy with acceptable rate of convergence. The results are validated with synthetic and real textured images.

Keywords: Texture Image Segmentation, Gray Level Cooccurrence Matrix, Markov Random Field Model, Ohta colour space, ICM algorithm.

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Authors: Daniele Losanno, Giorgio Serino

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This paper focuses on parametric analysis of reinforced concrete structures equipped with supplemental damping braces. Practitioners still luck sufficient data for current design of damper added structures and often reduce the real model to a pure damper braced structure even if this assumption is neither realistic nor conservative. In the present study, the damping brace is modelled as made by a linear supporting brace connected in series with the viscous/hysteretic damper. Deformation capacity of existing structures is usually not adequate to undergo the design earthquake. In spite of this, additional dampers could be introduced strongly limiting structural damage to acceptable values, or in some cases, reducing frame response to elastic behavior. This work is aimed at providing useful considerations for retrofit of existing buildings by means of supplemental damping braces. The study explicitly takes into consideration variability of (a) relative frame to supporting brace stiffness, (b) dampers’ coefficient (viscous coefficient or yielding force) and (c) non-linear frame behavior. Non-linear time history analysis has been run to account for both dampers’ behavior and non-linear plastic hinges modelled by Pivot hysteretic type. Parametric analysis based on previous studies on SDOF or MDOF linear frames provide reference values for nearly optimal damping systems design. With respect to bare frame configuration, seismic response of the damper-added frame is strongly improved, limiting deformations to acceptable values far below ultimate capacity. Results of the analysis also demonstrated the beneficial effect of stiffer supporting braces, thus highlighting inadequacy of simplified pure damper models. At the same time, the effect of variable damping coefficient and yielding force has to be treated as an optimization problem.

Keywords: Brace stiffness, dissipative braces, non-linear analysis, plastic hinges, reinforced concrete.

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Authors: Hong Li, Shou-Ming Zhong, Hou-Biao Li

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In this paper, we mainly study the stability of linear and interval linear fractional systems with time delay. By applying the characteristic equations, a necessary and sufficient stability condition is obtained firstly, and then some sufficient conditions are deserved. In addition, according to the equivalent relationship of fractional order systems with order 0 < α ≤ 1 and with order 1 ≤ β < 2, one may get more relevant theorems. Finally, two examples are provided to demonstrate the effectiveness of our results.

Keywords: Fractional order systems, Time delay, Characteristic equation.

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Authors: G. Parmar, R. Prasad, S. Mukherjee

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The authors present an algorithm for order reduction of linear dynamic systems using the combined advantages of stability equation method and the error minimization by Genetic algorithm. The denominator of the reduced order model is obtained by the stability equation method and the numerator terms of the lower order transfer function are determined by minimizing the integral square error between the transient responses of original and reduced order models using Genetic algorithm. The reduction procedure is simple and computer oriented. It is shown that the algorithm has several advantages, e.g. the reduced order models retain the steady-state value and stability of the original system. The proposed algorithm has also been extended for the order reduction of linear multivariable systems. Two numerical examples are solved to illustrate the superiority of the algorithm over some existing ones including one example of multivariable system.

Keywords: Genetic algorithm, Integral square error, Orderreduction, Stability equation method.

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Authors: Hans H. Diel

Abstract:

This paper describes a computer model of Quantum Field Theory (QFT), referred to in this paper as QTModel. After specifying the initial configuration for a QFT process (e.g. scattering) the model generates the possible applicable processes in terms of Feynman diagrams, the equations for the scattering matrix, and evaluates probability amplitudes for the scattering matrix and cross sections. The computations of probability amplitudes are performed numerically. The equations generated by QTModel are provided for demonstration purposes only. They are not directly used as the base for the computations of probability amplitudes. The computer model supports two modes for the computation of the probability amplitudes: (1) computation according to standard QFT, and (2) computation according to a proposed functional interpretation of quantum theory.

Keywords: Computational Modeling, Simulation of Quantum Theory, Quantum Field Theory, Quantum Electrodynamics

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Authors: Hassan Berbia, Faissal Elbouanani, Rahal Romadi, Mostafa Belkasmi

Abstract:

This paper introduces two decoders for binary linear codes based on Metaheuristics. The first one uses a genetic algorithm and the second is based on a combination genetic algorithm with a feed forward neural network. The decoder based on the genetic algorithms (DAG) applied to BCH and convolutional codes give good performances compared to Chase-2 and Viterbi algorithm respectively and reach the performances of the OSD-3 for some Residue Quadratic (RQ) codes. This algorithm is less complex for linear block codes of large block length; furthermore their performances can be improved by tuning the decoder-s parameters, in particular the number of individuals by population and the number of generations. In the second algorithm, the search space, in contrast to DAG which was limited to the code word space, now covers the whole binary vector space. It tries to elude a great number of coding operations by using a neural network. This reduces greatly the complexity of the decoder while maintaining comparable performances.

Keywords: Block code, decoding, methaheuristic, genetic algorithm, neural network

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Authors: Houda Bdiri Gabbouj, Néjib Hassen, Kamel Besbes

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This paper presents a low-voltage low-power differential linear transconductor with near rail-to-rail input swing. Based on the current-mirror OTA topology, the proposed transconductor combines the Flipped Voltage Follower (FVF) technique to linearize the transconductor behavior that leads to class- AB linear operation and the virtual transistor technique to lower the effective threshold voltages of the transistors which offers an advantage in terms of low supply requirement. Design of the OTA has been discussed. It operates at supply voltages of about ±0.8V. Simulation results for 0.18μm TSMC CMOS technology show a good input range of 1Vpp with a high DC gain of 81.53dB and a total harmonic distortion of -40dB at 1MHz for an input of 1Vpp. The main aim of this paper is to present and compare new OTA design with high transconductance, which has a potential to be used in low voltage applications.

Keywords: Amplifier class AB, current mirror, flipped voltage follower, low voltage.

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Authors: A. Pedram, M. R. Jamali, T. Pedram, S. M. Fakhraie, C. Lucas

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Local Linear Neuro-Fuzzy Models (LLNFM) like other neuro- fuzzy systems are adaptive networks and provide robust learning capabilities and are widely utilized in various applications such as pattern recognition, system identification, image processing and prediction. Local linear model tree (LOLIMOT) is a type of Takagi-Sugeno-Kang neuro fuzzy algorithm which has proven its efficiency compared with other neuro fuzzy networks in learning the nonlinear systems and pattern recognition. In this paper, a dedicated reconfigurable and parallel processing hardware for LOLIMOT algorithm and its applications are presented. This hardware realizes on-chip learning which gives it the capability to work as a standalone device in a system. The synthesis results on FPGA platforms show its potential to improve the speed at least 250 of times faster than software implemented algorithms.

Keywords: LOLIMOT, hardware, neurofuzzy systems, reconfigurable, parallel.

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Authors: Ramaswamy Palaniappan

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This paper reports a new approach on identifying the individuality of persons by using parametric classification of multiple mental thoughts. In the approach, electroencephalogram (EEG) signals were recorded when the subjects were thinking of one or more (up to five) mental thoughts. Autoregressive features were computed from these EEG signals and classified by Linear Discriminant classifier. The results here indicate that near perfect identification of 400 test EEG patterns from four subjects was possible, thereby opening up a new avenue in biometrics.

Keywords: Autoregressive, Biometrics, Electroencephalogram, Linear discrimination, Mental thoughts.

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Authors: Nur Nadiah Abd Hamid, Ahmad Abd. Majid, Ahmad Izani Md. Ismail

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Linear two-point boundary value problem of order two is solved using extended cubic B-spline interpolation method. There is one free parameters, λ, that control the tension of the solution curve. For some λ, this method produced better results than cubic B-spline interpolation method.

Keywords: two-point boundary value problem, B-spline, extendedcubic B-spline.

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Authors: Ching-Sheng Wang, Xin-Mao Huang, Ming-Yu Hung

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In this paper, we present a method named Signal Level Matrix (SLM) which can improve the accuracy and stability of active RFID indoor positioning system. Considering the accuracy and cost, we use uniform distribution mode to set up and separate the overlapped signal covering areas, in order to achieve preliminary location setting. Then, based on the proposed SLM concept and the characteristic of the signal strength value that attenuates as the distance increases, this system cross-examines the distribution of adjacent signals to locate the users more accurately. The experimental results indicate that the adaptive positioning method proposed in this paper could improve the accuracy and stability of the positioning system effectively and satisfyingly.

Keywords: RFID positioning, localization, indoor, location-aware.

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Authors: Wisuwat Plodpradista

Abstract:

Tubular linear induction motor (TLIM) can be used as a capsule pump in a large pneumatic capsule pipeline (PCP) system. Parametric performance evaluation of the designed 1-meter diameter PCP-TLIM system yields encouraging results for practical implementation. The capsule thrust and speed inside the TLIM pump can be calculated from the combination of the PCP fluid mechanics and the TLIM equations. The TLIM equivalent circuits derived from those of the conventional three-phase induction motor are used as a model to predict the static test results of a small-scale PCP-TLIM system. In this paper, additional dynamic tests are performed on the same small-scale PCP-TLIM system with two capsules of different diameters. The behaviors of the capsule inside the pump are observed and analyzed. The dynamic performances from the dynamic tests are compared with the theoretical predictions based on the TLIM equivalent circuit model.

Keywords: Pneumatic capsule pipeline, Tubular linear induction motor

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Authors: Javad Abdalkhani

Abstract:

Convergence of power series solutions for a class of non-linear Abel type equations, including an equation that arises in nonlinear cooling of semi-infinite rods, is very slow inside their small radius of convergence. Beyond that the corresponding power series are wildly divergent. Implementation of nonlinear sequence transformation allow effortless evaluation of these power series on very large intervals..

Keywords: Nonlinear transformation, Abel Volterra Equations, Mathematica

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Authors: Myat Su Mon Win

Abstract:

The pseudorandom number generators based on linear feedback shift registers (LFSRs), are very quick, easy and secure in the implementation of hardware and software. Thus they are very popular and widely used. But LFSRs lead to fairly easy cryptanalysis due to their completely linearity properties. In this paper, we propose a stochastic generator, which is called Random Feedback Shift Register (RFSR), using stochastic transformation (Random block) with one-way and non-linearity properties.

Keywords: Linear Feedback Shift Register, Non Linearity, R_Block, Random Feedback Shift Register

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Authors: Duy Cuong Nguyen, Ali Makke, Guillaume Montay

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This work presents an improved single fiber pull-out test for fiber/matrix interface characterization. This test has been used to study the Inter-Facial Shear Strength ‘IFSS’ of hemp fibers reinforced polypropylene (PP). For this aim, the fiber diameter has been carefully measured using a tomography inspired method. The fiber section contour can then be approximated by a circle or a polygon. The results show that the IFSS is overestimated if the circular approximation is used. The Influence of the molding temperature on the IFSS has also been studied. We find that a molding temperature of 183◦C leads to better interfacial properties. Above or below this temperature the interface strength is reduced.

Keywords: Interface, pull-out, processing, temperature, hemp, polypropylene, composite.

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Authors: Benedek Kovacs, Janos Toth

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Solutions are proposed for the central problem of estimating the reaction rate coefficients in homogeneous kinetics. The first is based upon the fact that the right hand side of a kinetic differential equation is linear in the rate constants, whereas the second one uses the technique of neural networks. This second one is discussed deeply and its advantages, disadvantages and conditions of applicability are analyzed in the mirror of the first one. Numerical analysis carried out on practical models using simulated data, and our programs written in Mathematica.

Keywords: Neural networks, parameter estimation, linear regression, kinetic models, reaction rate coefficients.

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Authors: Osama Yusuf Ababneh

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For the last years, the variants of the Newton-s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we present a new Newton-s method based on contra harmonic mean with cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.

Keywords: Third-order convergence, non-linear equations, root finding, iterative method.

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Authors: A. Elsayed Ahmed, A. Hafez, A. N. Ouda, H. Eldin Hussein Ahmed, H. Mohamed Abd-Elkader

Abstract:

Unmanned aircraft systems (UAS) are playing increasingly prominent roles in defense programs and defense strategies around the world. Technology advancements have enabled the development of it to do many excellent jobs as reconnaissance, surveillance, battle fighters, and communications relays. Simulating a small unmanned aerial vehicle (SUAV) dynamics and analyzing its behavior at the preflight stage is too important and more efficient. The first step in the UAV design is the mathematical modeling of the nonlinear equations of motion. . In this paper, a survey with a standard method to obtain the full non-linear equations of motion is utilized, and then the linearization of the equations according to a steady state flight condition (trimming) is derived. This modeling technique is applied to an Ultrastick-25e fixed wing UAV to obtain the valued linear longitudinal and lateral models. At the end the model is checked by matching between the behavior of the states of the nonlinear UAV and the resulted linear model with doublet at the control surfaces.

Keywords: Equations of motion, linearization, modeling, nonlinear model, UAV.

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Authors: Pitsanu Tongkhow, Pichet Jiraprasertwong

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This research investigates risk factors for defective products in autoparts factories. Under a Bayesian framework, a generalized linear mixed model (GLMM) in which the dependent variable, the number of defective products, has a Poisson distribution is adopted. Its performance is compared with the Poisson GLM under a Bayesian framework. The factors considered are production process, machines, and workers. The products coded RT50 are observed. The study found that the Poisson GLMM is more appropriate than the Poisson GLM. For the production Process factor, the highest risk of producing defective products is Process 1, for the Machine factor, the highest risk is Machine 5, and for the Worker factor, the highest risk is Worker 6.

Keywords: Defective autoparts products, Bayesian framework, Generalized linear mixed model (GLMM), Risk factors.

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Authors: Norman Wu, Darren Sun

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Key performance indicators (KPIs) are used for post result evaluation in the construction industry, and they normally do not have provisions for changes. This paper proposes a set of dynamic key performance indicators (d-KPIs) which predicts the future performance of the activity being measured and presents the opportunity to change practice accordingly. Critical to the predictability of a construction project is the ability to achieve automated data collection. This paper proposes an effective way to collect the process and engineering management data from an integrated construction management system. The d-KPI matrix, consisting of various indicators under seven categories, developed from this study can be applied to close monitoring of the development projects of aged-care facilities. The d-KPI matrix also enables performance measurement and comparison at both project and organization levels.

Keywords: Aged-care project, construction, dynamic KPI, healthcare system.

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Authors: Justin Leo Cheang Loong, Khazaimatol S Subari, Muhammad Kamil Abdullah, Nurul Nadia Ahmad, RosliBesar

Abstract:

Heart sound is an acoustic signal and many techniques used nowadays for human recognition tasks borrow speech recognition techniques. One popular choice for feature extraction of accoustic signals is the Mel Frequency Cepstral Coefficients (MFCC) which maps the signal onto a non-linear Mel-Scale that mimics the human hearing. However the Mel-Scale is almost linear in the frequency region of heart sounds and thus should produce similar results with the standard cepstral coefficients (CC). In this paper, MFCC is investigated to see if it produces superior results for PCG based human identification system compared to CC. Results show that the MFCC system is still superior to CC despite linear filter-banks in the lower frequency range, giving up to 95% correct recognition rate for MFCC and 90% for CC. Further experiments show that the high recognition rate is due to the implementation of filter-banks and not from Mel-Scaling.

Keywords: Biometric, Phonocardiogram, Cepstral Coefficients, Mel Frequency

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Authors: Basheer M. Al-Maqaleh, Kamal K. Bharadwaj

Abstract:

Automated discovery of hierarchical structures in large data sets has been an active research area in the recent past. This paper focuses on the issue of mining generalized rules with crisp hierarchical structure using Genetic Programming (GP) approach to knowledge discovery. The post-processing scheme presented in this work uses flat rules as initial individuals of GP and discovers hierarchical structure. Suitable genetic operators are proposed for the suggested encoding. Based on the Subsumption Matrix(SM), an appropriate fitness function is suggested. Finally, Hierarchical Production Rules (HPRs) are generated from the discovered hierarchy. Experimental results are presented to demonstrate the performance of the proposed algorithm.

Keywords: Genetic Programming, Hierarchy, Knowledge Discovery in Database, Subsumption Matrix.

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Authors: M. S. Said, J. A. Ghani, Che Hassan C. H., N. N. Wan, M. A. Selamat, R. Othman

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Metal matrix composites (MMCs) attract considerable attention as a result from its ability in providing a high strength, high modulus, high toughness, high impact properties, improving wear resistance and providing good corrosion resistance compared to unreinforced alloy. Aluminium Silicon (Al/Si) alloy MMC has been widely used in various industrial sectors such as in transportation, domestic equipment, aerospace, military, construction, etc. Aluminium silicon alloy is an MMC that had been reinforced with aluminium nitrate (AlN) particle and become a new generation material use in automotive and aerospace sector. The AlN is one of the advance material that have a bright prospect in future since it has features such as lightweight, high strength, high hardness and stiffness quality. However, the high degree of ceramic particle reinforcement and the irregular nature of the particles along the matrix material that contribute to its low density is the main problem which leads to difficulties in machining process. This paper examined the tool wear when milling AlSi/AlN Metal Matrix Composite using a TiB2 (Titanium diboride) coated carbide cutting tool. The volume of the AlN reinforced particle was 10% and milling process was carried out under dry cutting condition. The TiB2 coated carbide insert parameters used were at the cutting speed of (230, 300 and 370m/min, feed rate of 0.8, Depth of Cut (DoC) at 0.4m). The Sometech SV-35 video microscope system used to quantify of the tool wear. The result shown that tool life span increasing with the cutting speeds at (370m/min, feed rate of 0.8mm/tooth and DoC at 0.4mm) which constituted an optimum condition for longer tool life lasted until 123.2 mins. Meanwhile, at medium cutting speed which at 300m/m, feed rate of 0.8mm/tooth and depth of cut at 0.4mm we found that tool life span lasted until 119.86 mins while at low cutting speed it lasted in 119.66 mins. High cutting speed will give the best parameter in cutting AlSi/AlN MMCs material. The result will help manufacturers in machining process of AlSi/AlN MMCs materials.

Keywords: AlSi/AlN Metal Matrix Composite milling process, tool wear, TiB2 coated cemented carbide tool.

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Authors: I. Habi, M. Bouguerra, D. Ouahdi, H. Meglouli

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In this paper, we apply the PQ theory with shunt active power filter in an unbalanced and distorted power system voltage to compensate the perturbations generated by non linear load. The power factor is also improved in the current source. The PLL system is used to extract the fundamental component of the even sequence under conditions mentioned of the power system voltage.

Keywords: Converter, power filter, harmonies, non-linear load, pq theory, PLL, unbalanced voltages, distorted voltages.

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Authors: Mahdi Kiani, Hassan Salarieh, Aria Alasty, S. Mahdi Darbandi

Abstract:

The design and implementation of the hybrid control method for a three-pole active magnetic bearing (AMB) is proposed in this paper. The system is inherently nonlinear and conventional nonlinear controllers are a little complicated, while the proposed hybrid controller has a piecewise linear form, i.e. linear in each sub-region. A state-feedback hybrid controller is designed in this study, and the unmeasurable states are estimated by an observer. The gains of the hybrid controller are obtained by the Linear Quadratic Regulator (LQR) method in each sub-region. To evaluate the performance, the designed controller is implemented on an experimental setup in static mode. The experimental results show that the proposed method can efficiently stabilize the three-pole AMB system. The simplicity of design, domain of attraction, uncomplicated control law, and computational time are advantages of this method over other nonlinear control strategies in AMB systems.

Keywords: Active magnetic bearing, three pole AMB, hybrid control, Lyapunov function.

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