Search results for: Linear/nonlinear plants

Authors: Ki-Sung Park, Dae-Hee Lee, Dae-Ho Won, Yeon-Mo Yang

Abstract:

Solar power plants(SPPs) have shown a lot of good outcomes in providing a various functions depending on industrial expectations by deploying ad-hoc networking with helps of light loaded and battery powered sensor nodes. In particular, it is strongly requested to develop an algorithm to deriver the sensing data from the end node of solar power plants to the sink node on time. In this paper, based on the above observation we have proposed an IEEE802.15.4 based self routing scheme for solar power plants. The proposed beacon based priority routing Algorithm (BPRA) scheme utilizes beacon periods in sending message with embedding the high priority data and thus provides high quality of service(QoS) in the given criteria. The performance measures are the packet Throughput, delivery, latency, total energy consumption. Simulation results under TinyOS Simulator(TOSSIM) have shown the proposed scheme outcome the conventional Ad hoc On-Demand Distance Vector(AODV) Routing in solar power plants.

Keywords: Solar Power Plants(SPPs), Self routing, Quality of Service(QoS), WPANs, WSNs, TinyOS, TOSSIM, IEEE802.15.4

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Authors: Shuimiao Wan, Chao Yuan, Baoguang Tian

Abstract:

In statistics parameter theory, usually the parameter estimations have two kinds, one is the least-square estimation (LSE), and the other is the best linear unbiased estimation (BLUE). Due to the determining theorem of minimum variance unbiased estimator (MVUE), the parameter estimation of BLUE in linear model is most ideal. But since the calculations are complicated or the covariance is not given, people are hardly to get the solution. Therefore, people prefer to use LSE rather than BLUE. And this substitution will take some losses. To quantize the losses, many scholars have presented many kinds of different relative efficiencies in different views. For the linear weighted regression model, this paper discusses the relative efficiencies of LSE of β to BLUE of β. It also defines two new relative efficiencies and gives their lower bounds.

Keywords: Linear weighted regression, Relative efficiency, Lower bound, Parameter estimation.

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Authors: Hyun-Woo Cho

Abstract:

The early diagnostic decision making in industrial processes is absolutely necessary to produce high quality final products. It helps to provide early warning for a special event in a process, and finding its assignable cause can be obtained. This work presents a hybrid diagnostic schmes for batch processes. Nonlinear representation of raw process data is combined with classification tree techniques. The nonlinear kernel-based dimension reduction is executed for nonlinear classification decision boundaries for fault classes. In order to enhance diagnosis performance for batch processes, filtering of the data is performed to get rid of the irrelevant information of the process data. For the diagnosis performance of several representation, filtering, and future observation estimation methods, four diagnostic schemes are evaluated. In this work, the performance of the presented diagnosis schemes is demonstrated using batch process data.

Keywords: Diagnostics, batch process, nonlinear representation, data filtering, multivariate statistical approach

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Authors: G. Fouché, J. N. Eloff, K. Wellington

Abstract:

Acaricides are commonly used to control ticks but are toxic, harmful to the environment and too expensive to resource-limited farmers. Traditionally, many communities in South Africa rely on a wide range of indigenous practices to keep their livestock healthy. One of these health care practices includes the use of medicinal plants and this offers an alternative to conventional medicine. An investigation was conducted at the CSIR in South Africa, and selected indigenous plants used in communities were scientifically evaluated for the management of ticks in animals. 17 plants were selected from 239 plants used traditionally in South Africa. Two different organic extracts were prepared from the 17 samples, resulting in 34 plant samples. These were tested for efficacy against two tick species, namely Rhipicephalus microplus and Rhipicephalus turanicus. The plant extracts were also screened against Vero cells and most were found to have low cytotoxicity. This study has shown that there is potential for the development of botanicals as natural acaricides against ticks that are non-toxic and environmentally benign.

Keywords: Rhipicephalus microplus, Rhipicephalus turanicus, ticks, plant extracts, South Africa.

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Authors: Ruming Yin, Jian Yuan, Qiuhua Yang, Xiuming Shan, Xiqin Wang

Abstract:

Linear cryptanalysis methods are rarely used to improve the security of chaotic stream ciphers. In this paper, we apply linear cryptanalysis to a chaotic stream cipher which was designed by strictly using the basic design criterion of cryptosystem – confusion and diffusion. We show that this well-designed chaos-based stream cipher is still insecure against distinguishing attack. This distinguishing attack promotes the further improvement of the cipher.

Keywords: Stream cipher, chaos, linear cryptanalysis, distinguishing attack.

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Authors: Gemma A. Gruyal

Abstract:

Plants can contain a wide variety of substances with antioxidative properties which are associated with important health benefits. These positive health effects are of great importance at a time when the environment is laden with many toxic substances. Five selected herbal plants namely, Mimosa pudica, Phyllanthus niruri, Ceiba pentandra, Eleusine polydactyla and Trema amboinensi, were chosen for the experiment to investigate their total phenolics content and antioxidant activities using ABTS radical cation decolorization power, and ferric reducing antioxidant power. The total phenolic content of each herbal plants ranges from 0.84 to 42.59 mg gallic acid equivalent/g. The antioxidant activity in the ABTS radical cation decolorization power varies from 0.005 to 0.362 mg trolox equivalent/g and the FRAP ranges from 0.30 to 28.42 mg gallic acid equivalent/g. Among the five medicinal plants, Mimosa pudica has been an excellent performer in terms of the 3 parameters measured; it is followed by Phyllanthus niruri. The 5 herbal plants do not have equivalent antioxidant power. The relative high values for M. pudica and P. niruri supports the medicinal value of both plants. The total phenolics, ABTS and FRAP correlate strongly with one another.

Keywords: ABTS, FRAP, leaf extracts, phenol.

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Authors: Erik Delarue, David Bekaert, Ronnie Belmans, William D'haeseleer

Abstract:

This paper presents the development of an electricity simulation model taking into account electrical network constraints, applied on the Belgian power system. The base of the model is optimizing an extensive Unit Commitment (UC) problem through the use of Mixed Integer Linear Programming (MILP). Electrical constraints are incorporated through the implementation of a DC load flow. The model encloses the Belgian power system in a 220 – 380 kV high voltage network (i.e., 93 power plants and 106 nodes). The model features the use of pumping storage facilities as well as the inclusion of spinning reserves in a single optimization process. Solution times of the model stay below reasonable values.

Keywords: Electricity generation modeling, Unit Commitment(UC), Mixed Integer Linear Programming (MILP), DC load flow.

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Authors: El Kaak, Rachid, El Bikri, Khalid, Benamar, Rhali

Abstract:

This paper describes a study of geometrically nonlinear free vibration of thin circular functionally graded (CFGP) plates resting on Winkler elastic foundations. The material properties of the functionally graded composites examined here are assumed to be graded smoothly and continuously through the direction of the plate thickness according to a power law and are estimated using the rule of mixture. The theoretical model is based on the classical Plate theory and the Von-Kármán geometrical nonlinearity assumptions. An homogenization procedure (HP) is developed to reduce the problem considered here to that of isotropic homogeneous circular plates resting on Winkler foundation. Hamilton-s principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters which are found to be in a good agreement with the published results. On the other hand, the influence of the foundation parameters on the nonlinear fundamental frequency has also been analysed.

Keywords: Functionally graded materials, nonlinear vibrations, Winkler foundation.

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Authors: Takashi Shimizu, Tomoaki Hashimoto

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A class of implicit systems is known as a more generalized class of systems than a class of explicit systems. To establish a control method for such a generalized class of systems, we adopt model predictive control method which is a kind of optimal feedback control with a performance index that has a moving initial time and terminal time. However, model predictive control method is inapplicable to systems whose all state variables are not exactly known. In other words, model predictive control method is inapplicable to systems with limited measurable states. In fact, it is usual that the state variables of systems are measured through outputs, hence, only limited parts of them can be used directly. It is also usual that output signals are disturbed by process and sensor noises. Hence, it is important to establish a state estimation method for nonlinear implicit systems with taking the process noise and sensor noise into consideration. To this purpose, we apply the model predictive control method and unscented Kalman filter for solving the optimization and estimation problems of nonlinear implicit systems, respectively. The objective of this study is to establish a model predictive control with unscented Kalman filter for nonlinear implicit systems.

Keywords: Model predictive control, unscented Kalman filter, nonlinear systems, implicit systems.

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Authors: Atefeh Hekmat, Abolfazl Barati, Ebrahim Vasheghani Frahani, Ali Afraz

Abstract:

In modern agriculture, polymeric hydrogels are known as a component able to hold an amount of water due to their 3-dimensional network structure and their tendency to absorb water in humid environments. In addition, these hydrogels are able to controllably release the fertilisers and pesticides loaded in them. Therefore, they deliver these materials to the plants' roots and help them with growing. These hydrogels also reduce the pollution of underground water sources by preventing the active components from leaching. In this study, sIPN acrylamide based hydrogels are synthesised by using acrylamide free radical, potassium acrylate, and linear polyvinyl alcohol. Ammonium nitrate is loaded in the hydrogel as the fertiliser. The effect of various amounts of monomers and linear polymer, measured in molar ratio, on the swelling rate, equilibrium swelling, and release of ammonium nitrate is studied.

Keywords: Hydrogel, controlled release, ammonium nitrate fertiliser, sIPN.

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Authors: Geng Yuan, Yiwan Guo, Fahui Zhai, Shuhua Zhang

Abstract:

In this paper, we discuss some properties of left spectrum and give the representation of linear preserver map the left spectrum of diagonal quaternionic matrices.

Keywords: Quaternionic matrix, left spectrum, linear preserver map.

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Authors: Lazim Abdullah, Nurhakimah Ab. Rahman

Abstract:

Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.

Keywords: Fuzzy linear systems, Jacobi, Jacobi Over- Relaxation, Refinement of Jacobi, Refinement of Jacobi Over- Relaxation.

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Authors: Shu-Xin Miao

Abstract:

In this paper, we present an efficient numerical algorithm, namely block homotopy perturbation method, for solving fuzzy linear systems based on homotopy perturbation method. Some numerical examples are given to show the efficiency of the algorithm.

Keywords: Homotopy perturbation method, fuzzy linear systems, block linear system, fuzzy solution, embedding parameter.

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Authors: E. Sargsyan, A. Vardanyan, L. Ghalachyan, S. Bulgadaryan

Abstract:

Our results showed that for the growth of qualitative seedling and vegetative raw material of ðó. marschallianus Willd. and T. serphyllum L. it is more profitable to use the in vitro and hydroponics combined method. In in vitro culture it is possible to do micro-propagation whole year with 98-99% rhizogenesis. 30000 micro-plants were obtained from one explant during 9 months. Hydroponic conditions provide the necessary microclimate for microplants where the survival rate without acclimatization was 93.3%. The essential oil content in hydroponic dry herb of both species in vegetative and blossom phase was 1.3% whereas in wild plants it was 1.2%, the content of extractive substances and vitamin C also exceeded wild plants. Our biochemical and radiochemical investigations indicated that the medicinal raw materials obtained from hydroponic and wild plants of Thymus species correspond to the demands of SPh XI, and the content of artificial radionuclides does not exceed the MACL.

Keywords: Hydroponics, In vitro, Micro-propagation, Thymus

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Authors: Saman M. Siddiqui, Fang Jiancheng

Abstract:

Micro electromechanical sensors (MEMS) play a vital role along with global positioning devices in navigation of autonomous vehicles .These sensors are low cost ,easily available but depict colored noises and unpredictable discontinuities .Conventional filters like Kalman filters and Sigma point filters are not able to cope with nonwhite noises. This research has utilized H∞ filter in nonlinear frame work both with Kalman filter and Unscented filter for navigation and self alignment of an airborne vehicle. The system is simulated for colored noises and discontinuities and results are compared with not robust nonlinear filters. The results are found 40%-70% more robust against colored noises and discontinuities.

Keywords: filtering, integrated navigation, MEMS, nonlinearfiltering, self alignment

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Authors: Pallabi Kalita, Hui Tag, H. N. Sarma, A. K. Das.

Abstract:

Wild edible food plants contain a number of organic phytochemical that have been linked to the promotion of good health. These plants used by the local people of Arunachal Pradesh (Northeast India) are found to have high nutritional potential to maintain general balance diet. A study was conducted to evaluate the nutritional potential of five commonly found, unexplored wild food plants namely, Piper pedicellatum C. DC (leaves), Gonostegia hirta (Blume ex Hassk.) Miq. (leaves), Mussaenda roxburghii Hook.f (leaves), Solanum spirale Roxb. (leaves and fruits) and Cyathea spinulosa Wall. ex Hook. (pith portion and tender rachis) from East Siang District of Arunachal Pradesh Northeast (India) for ascertaining their suitability for utilization as supplementary food. Results of study revealed that P. pedicellatum, C. spinulosa, and S. spirale (leaves) are the most promising species which have high nutritional content out of the five wild food plants investigated which is required for the normal growth and development of human.

Keywords: Wild edible plants, Gross energy, Gonostegia hirta, Cyathea spinulosa,

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Authors: Abderrahmane El Kachani, El Mahjoub Chakir, Anass Ait Laachir, Abdelhamid Niaaniaa, Jamal Zerouaoui, Tarik Jarou

Abstract:

This paper presents a wind turbine based on the doubly fed induction generator (DFIG) connected to the utility grid through a shunt active power filter (SAPF). The whole system is controlled by a double nonlinear predictive controller (DNPC). A Taylor series expansion is used to predict the outputs of the system. The control law is calculated by optimization of the cost function. The first nonlinear predictive controller (NPC) is designed to ensure the high performance tracking of the rotor speed and regulate the rotor current of the DFIG, while the second one is designed to control the SAPF in order to compensate the harmonic produces by the three-phase diode bridge supplied by a passive circuit (rd, Ld). As a result, we obtain sinusoidal waveforms of the stator voltage and stator current. The proposed nonlinear predictive controllers (NPCs) are validated via simulation on a 1.5 MW DFIG-based wind turbine connected to an SAPF. The results obtained appear to be satisfactory and promising.

Keywords: Wind power, doubly fed induction generator, shunt active power filter, double nonlinear predictive controller.

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Authors: F. Garousi, Sz. Veres, É. Bódi, Sz. Várallyay, B. Kovács

Abstract:

Specific leaf area (SLA; cm2leaf g-1leaf) the ratio of leaf area to leaf dry mass is a key ecophysiological parameter influencing leaf physiology, photosynthesis, and whole plant carbon gain and also can be used as a rapid and diagnostic tool. In this study, two species of soluble inorganic selenium forms, selenite (Se^IV) and selenate (Se^VI) at different concentrations were investigated on maize plants that were growing in nutrient solutions during 2 weeks and at the end of the experiment, amounts of SLA for first and second leaves of maize were measured. In accordance with the results we observed that our regarded Se concentrations in both forms of Se^IV and Se^VI were not effective on maize plants’ SLA significantly although high level of 3 mg.kg-1 Se^IV had negative affect on growth of the samples that had been treated by it but about Se^VI samples we did not observe this state and our different considered Se^VI concentrations were not toxic for maize plants.

Keywords: Maize, Sodium selenate, sodium selenite, specific leaf area.

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Authors: Altayeb Qasem, Tarek Zayed, Zhi Chen

Abstract:

Statistics Canada stated that the wastewater treatment facilities in most provinces are aging and passes 63% of their useful life in 2007 the highest ratio among public infrastructure assets. Currently, there is no standard condition rating system for wastewater treatment plants that give a specific rating index that describe the physical integrity of different infrastructure elements in the treatment plant and its environmental performance. The main objective of this study is to develop a condition-rating index for wastewater treatment plants mainly activated sludge systems. The proposed WWTP CRI, is based on dividing the treatment plant into its three treatment phases; primary phase, secondary phase and the tertiary phase. The condition-rating index will reflect the infrastructures state for each phase, mainly tanks, pipes, blowers and pumps.

Keywords: Condition rating index, Wastewater treatment plants, AHP- MUAT.

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Authors: Sajjad Farashi

Abstract:

Neurons in the nervous system communicate with each other by producing electrical signals called spikes. To investigate the physiological function of nervous system it is essential to study the activity of neurons by detecting and sorting spikes in the recorded signal. In this paper a method is proposed for considering the spike sorting problem which is based on the nonlinear modeling of spikes using exponential autoregressive model. The genetic algorithm is utilized for model parameter estimation. In this regard some selected model coefficients are used as features for sorting purposes. For optimal selection of model coefficients, self-organizing feature map is used. The results show that modeling of spikes with nonlinear autoregressive model outperforms its linear counterpart. Also the extracted features based on the coefficients of exponential autoregressive model are better than wavelet based extracted features and get more compact and well-separated clusters. In the case of spikes different in small-scale structures where principal component analysis fails to get separated clouds in the feature space, the proposed method can obtain well-separated cluster which removes the necessity of applying complex classifiers.

Keywords: Exponential autoregressive model, Neural data, spike sorting, time series modeling.

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Authors: Bin Yu, Minghui Wang, Juntao Zhang

Abstract:

By the real representation of the quaternionic matrix, an iterative method for quaternionic linear equations Ax = b is proposed. Then the convergence conditions are obtained. At last, a numerical example is given to illustrate the efficiency of this method.

Keywords: Quaternionic linear equations, Real representation, Iterative algorithm.

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Authors: M. Zamurad Shah, M. Kemal Özgören, Raza Samar

Abstract:

This paper presents a 3D guidance scheme for Unmanned Aerial Vehicles (UAVs). The proposed guidance scheme is based on the sliding mode approach using nonlinear sliding manifolds. Generalized 3D kinematic equations are considered here during the design process to cater for the coupling between longitudinal and lateral motions. Sliding mode based guidance scheme is then derived for the multiple-input multiple-output (MIMO) system using the proposed nonlinear manifolds. Instead of traditional sliding surfaces, nonlinear sliding surfaces are proposed here for performance and stability in all flight conditions. In the reaching phase control inputs, the bang-bang terms with signum functions are accompanied with proportional terms in order to reduce the chattering amplitudes. The Proposed 3D guidance scheme is implemented on a 6-degrees-of-freedom (6-dof) simulation of a UAV and simulation results are presented here for different 3D trajectories with and without disturbances.

Keywords: Unmanned Aerial Vehicles, Sliding mode control, 3D Guidance, Path following, trajectory tracking, nonlinear sliding manifolds.

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Authors: Wudhichai Assawinchaichote, Sing Kiong Nguang

Abstract:

This paper investigates the problem of designing a robust state-feedback controller for a class of uncertain Markovian jump nonlinear systems that guarantees the L2-gain from an exogenous input to a regulated output is less than or equal to a prescribed value. First, we approximate this class of uncertain Markovian jump nonlinear systems by a class of uncertain Takagi-Sugeno fuzzy models with Markovian jumps. Then, based on an LMI approach, LMI-based sufficient conditions for the uncertain Markovian jump nonlinear systems to have an H performance are derived. An illustrative example is used to illustrate the effectiveness of the proposed design techniques.

Keywords: Robust H, Fuzzy Control, Markovian Jump Systems, LMI.

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Authors: Sarun Phibanchon, Michael A. Allen

Abstract:

A variational method is used to obtain the growth rate of a transverse long-wavelength perturbation applied to the soliton solution of a nonlinear Schr¨odinger equation with a three-half order potential. We demonstrate numerically that this unstable perturbed soliton will eventually transform into a cylindrical soliton.

Keywords: Soliton, instability, variational method, spectral method.

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Authors: F. Garousi, S. Veres, É. Bódi, S. Várallyay, B. Kovács

Abstract:

Extracting and determining chlorophyll pigments (chlorophyll a and b) in green leaves are the procedures based on the solvent extraction of pigments in samples using N,Ndimethylformamide as the extractant. In this study, two species of soluble inorganic selenium forms, selenite (SeIV) and selenate (SeVI) at different concentrations were investigated on maize plants that were growing in nutrient solutions during 2 weeks and at the end of the experiment, amounts of chlorophyll a and b for first and second leaves of maize were measured. In accordance with the results we observed that our regarded Se concentrations in both forms of SeIV and SeVI were not effective on maize plants’ chlorophyll a and b significantly although high level of 3 mg.kg-1 SeIV had negative affect on growth of the samples that had been treated by it but about SeVI samples we did not observe this state and our different considered SeVI concentrations were not toxic for maize plants.

Keywords: Maize, sodium selenate, sodium selenite, chlorophyll a and b.

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Authors: Ahmed Bensenouci

Abstract:

This paper provides the design steps of a robust Linear Matrix Inequality (LMI) based iterative multivariable PID controller whose duty is to drive a sample power system that comprises a synchronous generator connected to a large network via a step-up transformer and a transmission line. The generator is equipped with two control-loops, namely, the speed/power (governor) and voltage (exciter). Both loops are lumped in one where the error in the terminal voltage and output active power represent the controller inputs and the generator-exciter voltage and governor-valve position represent its outputs. Multivariable PID is considered here because of its wide use in the industry, simple structure and easy implementation. It is also preferred in plants of higher order that cannot be reduced to lower ones. To improve its robustness to variation in the controlled variables, H∞-norm of the system transfer function is used. To show the effectiveness of the controller, divers tests, namely, step/tracking in the controlled variables, and variation in plant parameters, are applied. A comparative study between the proposed controller and a robust H∞ LMI-based output feedback is given by its robustness to disturbance rejection. From the simulation results, the iterative multivariable PID shows superiority.

Keywords: Linear matrix inequality, power system, robust iterative PID, robust output feedback control

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Authors: H. Y. Jung, Ju H. Park, S. M. Lee

Abstract:

Sampled-data controller is presented for solid oxide fuel cell systems which is expressed by a sector bounded nonlinear model. The proposed control law is obtained by solving a convex problem satisfying several linear matrix inequalities. Simulation results are given to show the effectiveness of the proposed design method.

Keywords: Sampled-data control, Sector bound, Solid oxide fuel cell, Time-delay.

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Authors: Hidetoshi Konno, Akio Suzuki

Abstract:

The paper presented a transient population dynamics of phase singularities in 2D Beeler-Reuter model. Two stochastic modelings are examined: (i) the Master equation approach with the transition rate (i.e., λ(n, t) = λ(t)n and μ(n, t) = μ(t)n) and (ii) the nonlinear Langevin equation approach with a multiplicative noise. The exact general solution of the Master equation with arbitrary time-dependent transition rate is given. Then, the exact solution of the mean field equation for the nonlinear Langevin equation is also given. It is demonstrated that transient population dynamics is successfully identified by the generalized Logistic equation with fractional higher order nonlinear term. It is also demonstrated the necessity of introducing time-dependent transition rate in the master equation approach to incorporate the effect of nonlinearity.

Keywords: Transient population dynamics, Phase singularity, Birth-death process, Non-stationary Master equation, nonlinear Langevin equation, generalized Logistic equation.

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Authors: L. Almulla

Abstract:

Native plants are better adapted to the local environment providing a more natural effect on landscape projects; their use will both conserve natural resources and produce sustainable greenery. Continuation of evaluation of additional native plants is essential to increase diversity of plant resources for greenery projects. Therefore, in this project an effort was made to study the mass multiplication of further native plants for greenery applications. Standardization of vegetative propagation methods is essential for conservation and sustainable utilization of native plants in restoration projects. Moreover, these simple propagation methods can be readily adapted by the local nursery sector in Kuwait. In the present study, various treatments were used to mass multiply selected plants using vegetative parts to secure maximum rooting and initial growth. Soft or semi-hardwood cuttings of selected native plants were collected from mother plants and subjected to different treatments. Pennisetum divisum can be vegetatively propagated by cuttings/off-shoots. However, Tamarix aucheriana showed maximum number of rooted cuttings and stronger vigor seedlings with the lowest growth hormone concentration. Standardizing the propagation techniques for the native plant species will add to the rehabilitation and landscape revegetation projects in Kuwait.

Keywords: Kuwait desert, landscape, rooting percentage vegetative propagation.

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Authors: Nisha Goyal, R.K. Gupta

Abstract:

Einstein vacuum equations, that is a system of nonlinear partial differential equations (PDEs) are derived from Weyl metric by using relation between Einstein tensor and metric tensor. The symmetries of Einstein vacuum equations for static axisymmetric gravitational fields are obtained using the Lie classical method. We have examined the optimal system of vector fields which is further used to reduce nonlinear PDE to nonlinear ordinary differential equation (ODE). Some exact solutions of Einstein vacuum equations in general relativity are also obtained.

Keywords: Gravitational fields, Lie Classical method, Exact solutions.

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