Search results for: Fully fuzzy linear equations

Authors: Young-Seok Choi

Abstract:

We present a new framework of the data-reusing (DR) adaptive algorithms by incorporating a constraint on noise, referred to as a noise constraint. The motivation behind this work is that the use of the statistical knowledge of the channel noise can contribute toward improving the convergence performance of an adaptive filter in identifying a noisy linear finite impulse response (FIR) channel. By incorporating the noise constraint into the cost function of the DR adaptive algorithms, the noise constrained DR (NC-DR) adaptive algorithms are derived. Experimental results clearly indicate their superior performance over the conventional DR ones.

Keywords: Adaptive filter, data-reusing, least-mean square (LMS), affine projection (AP), noise constraint.

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Authors: E. G. Bautista, J. M. Reyes, O. Bautista, J. C. Arcos

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In this work, we analyze the deformation of surface waves in shallow flows conditions, propagating in a channel of slowly varying cross-section. Based on a singular perturbation technique, the main purpose is to predict the motion of waves by using a dimensionless formulation of the governing equations, considering that the longitudinal variation of the transversal section obey a power-law distribution. We show that the spatial distribution of the waves in the varying cross-section is a function of a kinematic parameter,κ , and two geometrical parameters εh and w ε . The above spatial behavior of the surface elevation is modeled by an ordinary differential equation. The use of single formulas to model the varying cross sections or transitions considered in this work can be a useful approximation to natural or artificial geometrical configurations.

Keywords: Surface waves, Asymptotic solution, Power law function, Non-dispersive waves.

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Authors: Kai-Long Hsiao

Abstract:

In this study, an analysis has been performed for free convection with radiation effect over a thermal forming nonlinearly stretching sheet. Parameters n, k0, Pr, G represent the dominance of the nonlinearly effect, radiation effect, heat transfer and free convection effects which have been presented in governing equations, respectively. The similarity transformation and the finite-difference methods have been used to analyze the present problem. From the results, we find that the effects of parameters n, k0, Pr, Ec and G to the nonlinearly stretching sheet. The increase of Prandtl number Pr, free convection parameter G or radiation parameter k0 resulting in the increase of heat transfer effects, but increase of the viscous dissipation number Ec will decrease of heat transfer effect.

Keywords: Nonlinearly stretching sheet, Free convection, Finite-difference, Radiation effect.

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Authors: I. Sarras, G. Gerakios, A. Diamantis, A. I. Dounis, G. P. Syrcos

Abstract:

Global Positioning System (GPS) technology is widely used today in the areas of geodesy and topography as well as in aeronautics mainly for military purposes. Due to the military usage of GPS, full access and use of this technology is being denied to the civilian user who must then work with a less accurate version. In this paper we focus on the estimation of the receiver coordinates ( X, Y, Z ) and its clock bias ( δtr ) of a fixed point based on pseudorange measurements of a single GPS receiver. Utilizing the instantaneous coordinates of just 4 satellites and their clock offsets, by taking into account the atmospheric delays, we are able to derive a set of pseudorange equations. The estimation of the four unknowns ( X, Y, Z , δtr ) is achieved by introducing an extended Kalman filter that processes, off-line, all the data collected from the receiver. Higher performance of position accuracy is attained by appropriate tuning of the filter noise parameters and by including other forms of biases.

Keywords: Extended Kalman filter, GPS, Pseudorange

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Authors: Jiajia Pan, Hung Tao Shen

Abstract:

Freeze and thaw occurs seasonally in river banks in northern countries. Little is known on how the riverbank soil temperature responds to air temperature changes and how freeze and thaw develops in a river bank seasonally. This study presents a two-dimensional heat conduction model for numerical investigations of seasonal freeze and thaw processes in an idealized river bank. The model uses the finite difference method and it is convenient for applications. The model is validated with an analytical solution and a field case with soil temperature distributions. It is then applied to the idealized river bank in terms of partially and fully saturated conditions with or without ice cover influence. Simulated results illustrate the response processes of the river bank to seasonal air temperature variations. It promotes the understanding of freeze and thaw processes in river banks and prepares for further investigation of frost and thaw impacts on riverbank stability.

Keywords: Freeze and thaw, river banks, 2D model, heat conduction.

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Authors: Shaharin A. Sulaiman, Malcolm Lawes

Abstract:

Combustion of sprays is of technological importance, but its flame behavior is not fully understood. Furthermore, the multiplicity of dependent variables such as pressure, temperature, equivalence ratio, and droplet sizes complicates the study of spray combustion. Fundamental study on the influence of the presence of liquid droplets has revealed that laminar flames within aerosol mixtures more readily become unstable than for gaseous ones and this increases the practical burning rate. However, fundamental studies on turbulent flames of aerosol mixtures are limited particularly those under near mono-dispersed droplet conditions. In the present work, centrally ignited expanding flames at near atmospheric pressures are employed to quantify the burning rates in gaseous and aerosol flames. Iso-octane-air aerosols are generated by expansion of the gaseous pre-mixture to produce a homogeneously distributed suspension of fuel droplets. The effects of the presence of droplets and turbulence velocity in relation to the burning rates of the flame are also investigated.

Keywords: Burning Rate, Droplets, Flames, Turbulent

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Authors: Gholam Reza Koochaki

Abstract:

This work presents the highly accurate numerical calculation of the natural frequencies for functionally graded beams with simply supported boundary conditions. The Timoshenko first order shear deformation beam theory and the higher order shear deformation beam theory of Reddy have been applied to the functionally graded beams analysis. The material property gradient is assumed to be in the thickness direction. The Hamilton-s principle is utilized to obtain the dynamic equations of functionally graded beams. The influences of the volume fraction index and thickness-to-length ratio on the fundamental frequencies are discussed. Comparison of the numerical results for the homogeneous beam with Euler-Bernoulli beam theory results show that the derived model is satisfactory.

Keywords: Functionally graded beam, Free vibration, Hamilton's principle.

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Authors: Viet-Hung Truong, Seung-Eock Kim

Abstract:

This paper presents a robust software package for practical advanced analysis of space steel framed structures. The pre- and post-processors of the presented software package are coded in the C++ programming language while the solver is written by using the FORTRAN programming language. A user-friendly graphical interface of the presented software is developed to facilitate the modeling process and result interpretation of the problem. The solver employs the stability functions for capturing the second-order effects to minimize modeling and computational time. Both the plastic-hinge and fiber-hinge beam-column elements are available in the presented software. The generalized displacement control method is adopted to solve the nonlinear equilibrium equations.

Keywords: Advanced analysis, beam-column, fiber-hinge, plastic hinge, steel frame.

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Authors: L. Tani, N. El Ouzzani

Abstract:

Crosstalk among interconnects and printed-circuit board (PCB) traces is a major limiting factor of signal quality in highspeed digital and communication equipments especially when fast data buses are involved. Such a bus is considered as a planar multiconductor transmission line. This paper will demonstrate how the finite difference time domain (FDTD) method provides an exact solution of the transmission-line equations to analyze the near end and the far end crosstalk. In addition, this study makes it possible to analyze the rise time effect on the near and far end voltages of the victim conductor. The paper also discusses a statistical analysis, based upon a set of several simulations. Such analysis leads to a better understanding of the phenomenon and yields useful information.

Keywords: Multiconductor transmission line, Crosstalk, Finite difference time domain (FDTD), printed-circuit board (PCB), Rise time, Statistical analysis.

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Authors: Hamdy M. Youssef, Eman A. Al-Lehaibi

Abstract:

This paper presents an iteration method for the numerical solutions of a one-dimensional problem of generalized thermoelasticity with one relaxation time under given initial and boundary conditions. The thermoelastic material with variable properties as a power functional graded has been considered. Adomian’s decomposition techniques have been applied to the governing equations. The numerical results have been calculated by using the iterations method with a certain algorithm. The numerical results have been represented in figures, and the figures affirm that Adomian’s decomposition method is a successful method for modeling thermoelastic problems. Moreover, the empirical parameter of the functional graded, and the lattice design parameter have significant effects on the temperature increment, the strain, the stress, the displacement.

Keywords: Adomian, Decomposition Method, Generalized Thermoelasticity, algorithm, empirical parameter, lattice design.

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Authors: Bahaa Eltahawy, Mikko Ylihärsilä, Reino Virrankoski, Esko Petäjä

Abstract:

Sheet metal processing is automated, but the step from product models to the production machine control still requires human intervention. This may cause time consuming bottlenecks in the production process and increase the risk of human errors. In this paper we present a system, which automatically recognizes features from the CAD-model of the sheet metal product. By using these features, the system produces a complete model of the particular sheet metal product. Then the model is used as an input for the sheet metal processing machine. Currently the system is implemented, capable to recognize more than 11 of the most common sheet metal structural features, and the procedure is fully automated. This provides remarkable savings in the production time, and protects against the human errors. This paper presents the developed system architecture, applied algorithms and system software implementation and testing.

Keywords: Feature recognition, automation, sheet metal manufacturing, CAM, CAD.

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Authors: K. Boualem, T. Yahiaoui, A. Azzi

Abstract:

Numerical investigations are performed to analyze the flow behavior over NACA0015 and to evaluate the efficiency of synthetic jet as active control device. The second objective of this work is to investigate the influence of momentum coefficient of synthetic jet on the flow behaviour. The unsteady Reynolds-averaged Navier-Stokes equations of the turbulent flow are solved using, k-ω SST provided by ANSYS CFX-CFD code. The model presented in this paper is a comprehensive representation of the information found in the literature. Comparison of obtained numerical flow parameters with the experimental ones shows that the adopted computational procedure reflects nearly the real flow nature. Also, numerical results state that use of synthetic jets devices has positive effects on the flow separation, and thus, aerodynamic performance improvement of NACA0015 airfoil. It can also be observed that the use of synthetic jet increases the lift coefficient about 13.3% and reduces the drag coefficient about 52.7%.

Keywords: Active control, CFD, NACA airfoil, synthetic jet.

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Authors: Hong Li, Shou-ming Zhong, Hou-biao Li

Abstract:

In this paper, some brief sufficient conditions for the stability of FO-LTI systems dαx(t) dtα = Ax(t) with the fractional order are investigated when the matrix A and the fractional order α are uncertain or both α and A are uncertain, respectively. In addition, we also relate the stability of a fractional-order system with order 0 < α ≤ 1 to the stability of its equivalent fractional-order system with order 1 ≤ β < 2, the relationship between α and β is presented. Finally, a numeric experiment is given to demonstrate the effectiveness of our results.

Keywords: Interval fractional-order systems, linear matrix inequality (LMI), asymptotical stability.

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Authors: Chao Yuan, Bao Guang Tian

Abstract:

A relative efficiency is defined as Ridge Estimate in the general linear model. The relative efficiency is based on the Mean square error. In this paper, we put forward a parameter of Ridge Estimate and discussions are made on the relative efficiency between the ridge estimation and the General Ridge Estimate. Eventually, this paper proves that the estimation is better than the general ridge estimate, which is based on the MSE.

Keywords: Ridge estimate, generalized ridge estimate, MSE, relative efficiency.

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Authors: Seema Kapoor, Sushil K. Kansal, Baljinder K. Gill, Aarti Sharma, Swati Arora

Abstract:

Isobaric vapor-liquid equilibrium measurements are reported for the binary mixtures of Mesitylene + 1-Heptanol and Mesitylene + 1-Octanol at 97.3 kPa. The measurements have been performed using a vapor recirculating type (modified Othmer's) equilibrium still. Both the mixtures show positive deviation from ideality. The Mesitylene + 1-Heptanol mixture forms an azeotrope whereas Mesitylene + 1- Octanol form a non – azeotropic mixture. The activity coefficients have been calculated taking into consideration the vapor phase nonideality. The data satisfy the thermodynamic consistency tests of Herington, and Hirata. The activity coefficients have been satisfactorily correlated by means of the Margules, Redlich-Kister, Wilson, Black, and NRTL equations. The activity coefficient values have also been obtained by UNIFAC method.

Keywords: Binary mixture, Mesitylene, Vapor-liquid equilibrium, 1-Heptanol, 1-Octanol.

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Authors: Ming-Hui Lee, Shou-Jen Hsu

Abstract:

This study presents an active vibration control technique to reduce the earthquake responses of a retained structural system. The proposed technique is a synthesis of the adaptive input estimation method (AIEM) and linear quadratic Gaussian (LQG) controller. The AIEM can estimate an unknown system input online. The LQG controller offers optimal control forces to suppress wall-structural system vibration. The numerical results show robust performance in the active vibration control technique.

Keywords: Active vibration control, AIEM, LQG, Optimal control

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Authors: H. Sediki, Dj. Ould Abdeslam, T. Otmane-cherif, A. Bechouche, K. Mesbah

Abstract:

This paper explores steady-state characteristics of grid-connected doubly fed induction motor (DFIM) in case of unity power factor operation. Based on the synchronized mathematical model, analytic determination of the control laws is presented and illustrated by various figures to understand the effect of the applied rotor voltage on the speed and the active power. On other hand, unlike previous works where the stator resistance was neglected, in this work, stator resistance is included such that the equations can be applied to small wind turbine generators which are becoming more popular. Finally the work is crowned by integration of the studied induction generator in a wind system where an open loop control is proposed confers a remarkable simplicity of implementation compared to the known methods.

Keywords: DFIM, equivalent circuit, induction machine, steady state

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Authors: Robert Niese, Ayoub Al-Hamadi, Bernd Michaelis

Abstract:

In modern human computer interaction systems (HCI), emotion recognition is becoming an imperative characteristic. The quest for effective and reliable emotion recognition in HCI has resulted in a need for better face detection, feature extraction and classification. In this paper we present results of feature space analysis after briefly explaining our fully automatic vision based emotion recognition method. We demonstrate the compactness of the feature space and show how the 2d/3d based method achieves superior features for the purpose of emotion classification. Also it is exposed that through feature normalization a widely person independent feature space is created. As a consequence, the classifier architecture has only a minor influence on the classification result. This is particularly elucidated with the help of confusion matrices. For this purpose advanced classification algorithms, such as Support Vector Machines and Artificial Neural Networks are employed, as well as the simple k- Nearest Neighbor classifier.

Keywords: Facial expression analysis, Feature extraction, Image processing, Pattern Recognition, Application.

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Authors: Mahmoud E. Mohamed, Ahmed F. Shalash, Hanan A. Kamal

Abstract:

In communication systems, frequency jump is a serious problem caused by the oscillators used. Kalman filters are used to detect that jump, despite the tradeoff between the noise level and the speed of the detection. In this paper, an improvement is introduced in the Kalman filter, through a nonlinear change in the bandwidth of the filter. Simulation results show a considerable improvement in the filter speed with a very low noise level. Additionally, the effect on the response to false alarms is also presented and false alarm rate show improvement.

Keywords: Kalman Filter, Innovation, False Detection.

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Authors: Hamid Goharnejad, Amir Hossein Eghbali

Abstract:

Recent investigations have demonstrated the global sea level rise due to climate change impacts. In this study, climate changes study the effects of increasing water level in the strait of Hormuz. The probable changes of sea level rise should be investigated to employ the adaption strategies. The climatic output data of a GCM (General Circulation Model) named CGCM3 under climate change scenario of A1b and A2 were used. Among different variables simulated by this model, those of maximum correlation with sea level changes in the study region and least redundancy among themselves were selected for sea level rise prediction by using stepwise regression. One of models (Discrete Wavelet artificial Neural Network) was developed to explore the relationship between climatic variables and sea level changes. In these models, wavelet was used to disaggregate the time series of input and output data into different components and then ANN was used to relate the disaggregated components of predictors and input parameters to each other. The results showed in the Shahid Rajae Station for scenario A1B sea level rise is among 64 to 75 cm and for the A2 Scenario sea level rise is among 90 t0 105 cm. Furthermore, the result showed a significant increase of sea level at the study region under climate change impacts, which should be incorporated in coastal areas management.

Keywords: Climate change scenarios, sea-level rise, strait of Hormuz, artificial neural network, fuzzy logic.

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Authors: Ubonwan Chaiyo, Savitri Garivait, Kobsak Wanthongchai

Abstract:

The study site was located in Ratchaburi Province, Thailand. Four experimental plots in dry dipterocarp forest (DDF) and four plots in mixed deciduous forest (MDF) were set up to estimate the above-ground biomass of tree, sapling and bamboo. The allometry equations were used to investigate above-ground biomass of these vegetation. Seedling and other understory were determined using direct harvesting method. Carbon storage in above-ground biomass was calculated based on IPCC 2006. The results showed that the above-ground biomass of DDF at 20-40% slope, <20% slope and MDF at <20% slope were 91.96, 30.95 and 59.44 ton/ha, respectively. Bamboo covers about half of total aboveground biomass in MDF, which is a specific characteristic of this area. The carbon sequestration potential in above-ground biomass of plot slope range 20-40% DDF, <20% DDF and <20% MDF are 43.22, 14.55 and 27.94 ton C/ha, respectively.

Keywords: Carbon storage, aboveground biomass, tropical deciduous forest, dry dipterocarp forest, mixed deciduous forest.

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Authors: Rajneesh Rani, Renu Dhir, G. S. Lehal

Abstract:

Script identification is one of the challenging steps in the development of optical character recognition system for bilingual or multilingual documents. In this paper an attempt is made for identification of English numerals at word level from Punjabi documents by using Gabor features. The support vector machine (SVM) classifier with five fold cross validation is used to classify the word images. The results obtained are quite encouraging. Average accuracy with RBF kernel, Polynomial and Linear Kernel functions comes out to be greater than 99%.

Keywords: Script identification, gabor features, support vector machines.

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Authors: Nga-Viet Nguyen, Georgy Shevlyakov, Vladimir Shin

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To solve the problem of multisensor data fusion under non-Gaussian channel noise. The advanced M-estimates are known to be robust solution while trading off some accuracy. In order to improve the estimation accuracy while still maintaining the equivalent robustness, a two-stage robust fusion algorithm is proposed using preliminary rejection of outliers then an optimal linear fusion. The numerical experiments show that the proposed algorithm is equivalent to the M-estimates in the case of uncorrelated local estimates and significantly outperforms the M-estimates when local estimates are correlated.

Keywords: Data fusion, estimation, robustness, M-estimates.

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Authors: A. Ghafouri, A. Falavand Jozaei, M. Salari

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In this paper, effects of using Alumina-water nanofluid on the rate of heat transfer have been investigated numerically. Physical model is a square enclosure with insulated top and bottom horizontal walls, while the vertical walls are kept at different constant temperatures. Two appropriate models are used to evaluate the viscosity and thermal conductivity of nanofluid. The governing stream-vorticity equations are solved using a second order central finite difference scheme, coupled to the conservation of mass and energy. The study has been carried out for the Richardson number 0.1 to 10 and the solid volume fraction 0 to 0.04. Results are presented by isotherms lines, average Nusselt number and normalized Nusselt number in different range of φ and Ri for forced, combined and natural convection dominated regime. It is found that higher heat transfer rate is predicted when the effects of nanoparticle is taken into account.

Keywords: Nanofluid, Heat Transfer Enhancement, Square Enclosure, Nusselt number.

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Authors: Nurul Diyana, A., Zainul Abidin, N.

Abstract:

Social cognitive theory explains the power to inaugurate change is determined by the mutual influence of personal proclivity and social factors which will shape ones- motivations and expectations. In construction industry, green concept offers an opportunity to leave a lighter footprint on the environment. This opportunity, however, has not been fully grasped by many countries. As such, venturing into green construction for many practitioners would be their maiden experience. Decision to venture into new practice such as green construction will be influenced by certain drivers. This paper explores these drivers which is further expanded into motivational factors and later becomes the platform upon which expectation for green construction stands. This theoretical concept of motivation and expectations, which is adapted from social cognitive theory, focus on developers- view because of their crucial role in green application. This conceptual framework, which serves as the basis for further research, will benefit the industry as it elucidate cognitive angles to attract more new entrants to green business.

Keywords: Developers, Green Construction, Motivation, Expectation, Social Cognitive Theory.

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Authors: Subir Kundu, Sukhendra Singh, Sumedha Ojha, Kanika Kundu

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The continuous decline of petroleum and natural gas reserves and non linear rise of oil price has brought about a realisation of the need for a change in our perpetual dependence on the fossil fuel. A day to day increased consumption of crude and petroleum products has made a considerable impact on our foreign exchange reserves. Hence, an alternate resource for the conversion of energy (both liquid and gas) is essential for the substitution of conventional fuels. Biomass is the alternate solution for the present scenario. Biomass can be converted into both liquid as well as gaseous fuels and other feedstocks for the industries.

Keywords: Bioenergy, Biomass conversion, Biorefining, Efficient utilisation of night soil.

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Authors: M. Raciti Castelli, G. Grandi, E. Benini

Abstract:

This paper presents a study of laminar to turbulent transition on a profile specifically designed for wind turbine blades, the DU91-W2-250, which belongs to a class of wind turbine dedicated airfoils, developed by Delft University of Technology. A comparison between the experimental behavior of the airfoil studied at Delft wind tunnel and the numerical predictions of the commercial CFD solver ANSYS FLUENT® has been performed. The prediction capabilities of the Spalart-Allmaras turbulence model and of the γ-θ Transitional model have been tested. A sensitivity analysis of the numerical results to the spatial domain discretization has also been performed using four different computational grids, which have been created using the mesher GAMBIT®. The comparison between experimental measurements and CFD results have allowed to determine the importance of the numerical prediction of the laminar to turbulent transition, in order not to overestimate airfoil friction drag due to a fully turbulent-regime flow computation.

Keywords: CFD, wind turbine, DU91-W2-250, laminar to turbulent transition.

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Authors: Yanan Yang, Zhigang Wang, Xiang Chen

Abstract:

This paper presents the use of Legendre pseudospectral method for the optimization of finite-thrust orbital transfer for spacecrafts. In order to get an accurate solution, the System-s dynamics equations were normalized through a dimensionless method. The Legendre pseudospectral method is based on interpolating functions on Legendre-Gauss-Lobatto (LGL) quadrature nodes. This is used to transform the optimal control problem into a constrained parameter optimization problem. The developed novel optimization algorithm can be used to solve similar optimization problems of spacecraft finite-thrust orbital transfer. The results of a numerical simulation verified the validity of the proposed optimization method. The simulation results reveal that pseudospectral optimization method is a promising method for real-time trajectory optimization and provides good accuracy and fast convergence.

Keywords: Finite-thrust, Orbital transfer, Legendre pseudospectral method

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Authors: Jožef Ritonja

Abstract:

The biochemical industry is one of the most important modern industries. Biochemical reactors are crucial devices of the biochemical industry. The essential bioprocess carried out in bioreactors is the fermentation process. A thorough insight into the fermentation process and the knowledge how to control it are essential for effective use of bioreactors to produce high quality and quantitatively enough products. The development of the control system starts with the determination of a mathematical model that describes the steady state and dynamic properties of the controlled plant satisfactorily, and is suitable for the development of the control system. The paper analyses the fermentation process in bioreactors thoroughly, using existing mathematical models. Most existing mathematical models do not allow the design of a control system for controlling the fermentation process in batch bioreactors. Due to this, a mathematical model was developed and presented that allows the development of a control system for batch bioreactors. Based on the developed mathematical model, a control system was designed to ensure optimal response of the biochemical quantities in the fermentation process. Due to the time-varying and non-linear nature of the controlled plant, the conventional control system with a proportional-integral-differential controller with constant parameters does not provide the desired transient response. The improved adaptive control system was proposed to improve the dynamics of the fermentation. The use of the adaptive control is suggested because the parameters’ variations of the fermentation process are very slow. The developed control system was tested to produce dairy products in the laboratory bioreactor. A carbon dioxide concentration was chosen as the controlled variable. The carbon dioxide concentration correlates well with the other, for the quality of the fermentation process in significant quantities. The level of the carbon dioxide concentration gives important information about the fermentation process. The obtained results showed that the designed control system provides minimum error between reference and actual values of carbon dioxide concentration during a transient response and in a steady state. The recommended control system makes reference signal tracking much more efficient than the currently used conventional control systems which are based on linear control theory. The proposed control system represents a very effective solution for the improvement of the milk fermentation process.

Keywords: Bioprocess engineering, biochemical reactor, fermentation process, modeling, adaptive control.

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Authors: Syeda Maria Ali Shah, Asif Mansoor, Talat Sharafat Rehmani, Safia Mirza

Abstract:

The stock market of any country acts as a predictor of the economy. The spread of the COVID-19 pandemic has severely impacted the global financial markets. Besides, it has also critically affected the economy of Pakistan. In this study, we consider the role of the Karachi Stock Exchange (KSE) with regard to the Pakistan Stock Exchange and quantify the impact on macroeconomic variables in presence of COVID-19. The suitable macroeconomic variables are used to quantify the impact of COVID-19 by developing the stochastic model. The sufficiency of the computed model is attained by means of available techniques in the literature. The estimated equations are used to forecast the impact of pandemic on macroeconomic variables. The constructed model can help the policymakers take counteractive measures for restricting the influence of viruses on the Karachi Stock Market.

Keywords: COVID-19, Karachi Stock Market, macroeconomic variables, stochastic model, forecasting.

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