Delays deteriorate the control performance and could destabilize the overall system in the theory of discretetime
signals and dynamic systems. Whenever a computer is used in measurement, signal processing or control
applications, the data as seen from the computer and systems involved are naturally discrete-time because a
computer executes program code at discrete points of time. Theory of discrete-time dynamic signals and systems
is useful in design and analysis of control systems, signal filters, state estimators and model estimation from
time-series of process data system identification. In this paper, a new approximated discretization method and
digital design for control systems with delays is proposed. System is transformed to a discrete-time model with
time delays. To implement the digital modeling, we used the z-transfer functions matrix which is a useful model
type of discrete-time systems, being analogous to the Laplace-transform for continuous-time systems. The most
important use of the z-transform is for defining z-transfer functions matrix is employed to obtain an extended
discrete-time. The proposed method can closely approximate the step response of the original continuous timedelayed
control system by choosing various of energy loss level. Illustrative example is simulated to demonstrate
the effectiveness of the developed method.\
A Combination of Wavelet Artificial Neural Networks Integrated with Bootstrap...IJERA Editor
In this paper, an iterative forecasting methodology for time series prediction that integrates wavelet de-noising
and decomposition with an Artificial Neural Network (ANN) and Bootstrap methods is put forward here.
Basically, a given time series to be forecasted is initially decomposed into trend and noise (wavelet) components
by using a wavelet de-noising algorithm. Both trend and noise components are then further decomposed by
means of a wavelet decomposition method producing orthonormal Wavelet Components (WCs) for each one.
Each WC is separately modelled through an ANN in order to provide both in-sample and out-of-sample
forecasts. At each time t, the respective forecasts of the WCs of the trend and noise components are simply
added to produce the in-sample and out-of-sample forecasts of the underlying time series. Finally, out-of-sample
predictive densities are empirically simulated by the Bootstrap sampler and the confidence intervals are then
yielded, considering some level of credibility. The proposed methodology, when applied to the well-known
Canadian lynx data that exhibit non-linearity and non-Gaussian properties, has outperformed other methods
traditionally used to forecast it.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
A Combination of Wavelet Artificial Neural Networks Integrated with Bootstrap...IJERA Editor
In this paper, an iterative forecasting methodology for time series prediction that integrates wavelet de-noising
and decomposition with an Artificial Neural Network (ANN) and Bootstrap methods is put forward here.
Basically, a given time series to be forecasted is initially decomposed into trend and noise (wavelet) components
by using a wavelet de-noising algorithm. Both trend and noise components are then further decomposed by
means of a wavelet decomposition method producing orthonormal Wavelet Components (WCs) for each one.
Each WC is separately modelled through an ANN in order to provide both in-sample and out-of-sample
forecasts. At each time t, the respective forecasts of the WCs of the trend and noise components are simply
added to produce the in-sample and out-of-sample forecasts of the underlying time series. Finally, out-of-sample
predictive densities are empirically simulated by the Bootstrap sampler and the confidence intervals are then
yielded, considering some level of credibility. The proposed methodology, when applied to the well-known
Canadian lynx data that exhibit non-linearity and non-Gaussian properties, has outperformed other methods
traditionally used to forecast it.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcscpconf
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcsitconf
A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
On selection of periodic kernels parameters in time series predictioncsandit
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
Diabetic retinopathy is a disease, caused by alternation in the retinal blood vessels. It is a strong sign of early blindness and if it is not treated may tend to complete blindness and the vision lost once cannot be restored once again. In this paper different image processing techniques are used to differentiate between the normal and the diseased image. The attempt is made to see where the problem actually lies so that proper diagnosis of patient can be done. Pre processing of an image, optic disk detection, Blood vessels extraction, Exudates detection are some of the methods that are applied here. Other algorithms are designed to obtain the desired result. A large number of populations are affected by this disease around the world.
IRDO: Iris Recognition by fusion of DTCWT and OLBPIJERA Editor
Iris Biometric is a physiological trait of human beings. In this paper, we propose Iris an Recognition using
Fusion of Dual Tree Complex Wavelet Transform (DTCWT) and Over Lapping Local Binary Pattern (OLBP)
Features. An eye is preprocessed to extract the iris part and obtain the Region of Interest (ROI) area from an iris.
The complex wavelet features are extracted for region from the Iris DTCWT. OLBP is further applied on ROI to
generate features of magnitude coefficients. The resultant features are generated by fusing DTCWT and OLBP
using arithmetic addition. The Euclidean Distance (ED) is used to compare test iris with database iris features to
identify a person. It is observed that the values of Total Success Rate (TSR) and Equal Error Rate (EER) are
better in the case of proposed IRDO compared to the state-of-the art techniques.
The summary of the research, first is to introduce the harmonics in power system, by explains the meanings of the harmonic, causes and sources. Then the effects of the harmonics in power system. Second to analysis the harmonics during the fourier analysis, how we can calculate and analysis the harmonics. Third, the experiment part which is the results form lab work to inform the harmonics effects in power system. Fourth, which is explain how to eliminate the harmonics in power system, then the finally the conculation which is try to concluote the all parts of the research.
Seismic Response of RC Framed Buildings with Open Ground StoreyIJERA Editor
RC framed buildings are generally designed without considering the structural action of masonry infill walls. These masonry infill walls are widely used as partitions and considered as non-structural elements. But they affect both the structural and non-structural performance of RC buildings during earthquake. RC framed building with open ground storey is known as soft storey, which performs poorly during earthquake. In order to study this total 144 RC framed buildings having bare frame, full infill frame and open ground storey frame were analyzed by seismic coefficient method and response spectrum method for various seismic hazards. The present study deals with the comparison of base shear for medium rise RC framed buildings having P+5, P+7, P+9 and P+11 storeys for various seismic zones (III, IV & V) and for various soil conditions (Hard & Medium) as per IS 1893(part 1): 2002. This work helps in understanding the effect of earthquake with increase in height of RC framed buildings on base shear for various seismic zones and soil conditions. The result shows that the effect of infill stiffness on structural response is significant under lateral loads. It is found that the presence infill walls increases the base shear by 60-65% more than bare frame by both seismic coefficient method and response spectrum method.
Comparison of Windows and Linux Operating Systems in Advanced FeaturesIJERA Editor
Comparison between the Microsoft Windows and Linux computer operating systems is a long-running
discussion topic within the personal computer industry .This technical paper is mainly going to focus on the
differences between windows and linux in all fields. Both Windows and Linux Operating systems have their
own advantages and differ in functionalities and user friendliness. Linux and Microsoft Windows differ in
philosophy, cost, versatility and stability, with each seeking to improve in their perceived weaker areas. This
paper is mainly going to focus on the advanced features that are uniquely present in one operating system and
not in other one.
Optimization of Multi-band Rectangular-Triangular Slotted AntennaIJERA Editor
A multi-band microstrip patch antenna is developed and presented in this paper. The radiating elements in this antenna are composed of rectangular and triangular slots. These slots are engraved in the rectangular and triangular patch, joined together in one structure, and by single probe feed. The rectangular and triangular slots make the antenna to operate at multiband with relatively high gain. Therefore, this antenna can be used for wireless communication applications like WLAN, WiMax and radar system applications.
Dynamic Texture Coding using Modified Haar Wavelet with CUDAIJERA Editor
Texture is an image having repetition of patterns. There are two types, static and dynamic texture. Static texture is an image having repetitions of patterns in the spatial domain. Dynamic texture is number of frames having repetitions in spatial and temporal domain. This paper introduces a novel method for dynamic texture coding to achieve higher compression ratio of dynamic texture using 2D-modified Haar wavelet transform. The dynamic texture video contains high redundant parts in spatial and temporal domain. Redundant parts can be removed to achieve high compression ratios with better visual quality. The modified Haar wavelet is used to exploit spatial and temporal correlations amongst the pixels. The YCbCr color model is used to exploit chromatic components as HVS is less sensitive to chrominance. To decrease the time complexity of algorithm parallel programming is done using CUDA (Compute Unified Device Architecture). GPU contains the number of cores as compared to CPU, which is utilized to reduce the time complexity of algorithms.
Region based elimination of noise pixels towards optimized classifier models ...IJERA Editor
The extraction of the skin pixels in a human image and rejection of non-skin pixels is called the skin segmentation. Skin pixel detection is the process of extracting the skin pixels in a human image which is typically used as a pre-processing step to extract the face regions from human image. In past, there are several computer vision approaches and techniques have been developed for skin pixel detection. In the process of skin detection, given pixels are been transformed into an appropriate color space such as RGB, HSV etc. And then skin classifier model have been applied to label the pixel into skin or non-skin regions. Here in this research a “Region based elimination of noise pixels and performance analysis of classifier models for skin pixel detection applied on human images” would be performed which involve the process of image representation in color models, elimination of non-skin pixels in the image, and then pre-processing and cleansing of the collected data, feature selection of the human image and then building the model for classifier. In this research and implementation of skin pixels classifier models are proposed with their comparative performance analysis. The definition of the feature vector is simply the selection of skin pixels from the human image or stack of human images. The performance is evaluated by comparing and analysing skin colour segmentation algorithms. During the course of research implementation, efforts are iterative which help in selection of optimized skin classifier based on the machine learning algorithms and their performance analysis.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcscpconf
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHMcsitconf
A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
On selection of periodic kernels parameters in time series predictioncsandit
In the paper the analysis of the periodic kernels parameters is described. Periodic kernels can
be used for the prediction task, performed as the typical regression problem. On the basis of the
Periodic Kernel Estimator (PerKE) the prediction of real time series is performed. As periodic
kernels require the setting of their parameters it is necessary to analyse their influence on the
prediction quality. This paper describes an easy methodology of finding values of parameters of
periodic kernels. It is based on grid search. Two different error measures are taken into
consideration as the prediction qualities but lead to comparable results. The methodology was
tested on benchmark and real datasets and proved to give satisfactory results.
Diabetic retinopathy is a disease, caused by alternation in the retinal blood vessels. It is a strong sign of early blindness and if it is not treated may tend to complete blindness and the vision lost once cannot be restored once again. In this paper different image processing techniques are used to differentiate between the normal and the diseased image. The attempt is made to see where the problem actually lies so that proper diagnosis of patient can be done. Pre processing of an image, optic disk detection, Blood vessels extraction, Exudates detection are some of the methods that are applied here. Other algorithms are designed to obtain the desired result. A large number of populations are affected by this disease around the world.
IRDO: Iris Recognition by fusion of DTCWT and OLBPIJERA Editor
Iris Biometric is a physiological trait of human beings. In this paper, we propose Iris an Recognition using
Fusion of Dual Tree Complex Wavelet Transform (DTCWT) and Over Lapping Local Binary Pattern (OLBP)
Features. An eye is preprocessed to extract the iris part and obtain the Region of Interest (ROI) area from an iris.
The complex wavelet features are extracted for region from the Iris DTCWT. OLBP is further applied on ROI to
generate features of magnitude coefficients. The resultant features are generated by fusing DTCWT and OLBP
using arithmetic addition. The Euclidean Distance (ED) is used to compare test iris with database iris features to
identify a person. It is observed that the values of Total Success Rate (TSR) and Equal Error Rate (EER) are
better in the case of proposed IRDO compared to the state-of-the art techniques.
The summary of the research, first is to introduce the harmonics in power system, by explains the meanings of the harmonic, causes and sources. Then the effects of the harmonics in power system. Second to analysis the harmonics during the fourier analysis, how we can calculate and analysis the harmonics. Third, the experiment part which is the results form lab work to inform the harmonics effects in power system. Fourth, which is explain how to eliminate the harmonics in power system, then the finally the conculation which is try to concluote the all parts of the research.
Seismic Response of RC Framed Buildings with Open Ground StoreyIJERA Editor
RC framed buildings are generally designed without considering the structural action of masonry infill walls. These masonry infill walls are widely used as partitions and considered as non-structural elements. But they affect both the structural and non-structural performance of RC buildings during earthquake. RC framed building with open ground storey is known as soft storey, which performs poorly during earthquake. In order to study this total 144 RC framed buildings having bare frame, full infill frame and open ground storey frame were analyzed by seismic coefficient method and response spectrum method for various seismic hazards. The present study deals with the comparison of base shear for medium rise RC framed buildings having P+5, P+7, P+9 and P+11 storeys for various seismic zones (III, IV & V) and for various soil conditions (Hard & Medium) as per IS 1893(part 1): 2002. This work helps in understanding the effect of earthquake with increase in height of RC framed buildings on base shear for various seismic zones and soil conditions. The result shows that the effect of infill stiffness on structural response is significant under lateral loads. It is found that the presence infill walls increases the base shear by 60-65% more than bare frame by both seismic coefficient method and response spectrum method.
Comparison of Windows and Linux Operating Systems in Advanced FeaturesIJERA Editor
Comparison between the Microsoft Windows and Linux computer operating systems is a long-running
discussion topic within the personal computer industry .This technical paper is mainly going to focus on the
differences between windows and linux in all fields. Both Windows and Linux Operating systems have their
own advantages and differ in functionalities and user friendliness. Linux and Microsoft Windows differ in
philosophy, cost, versatility and stability, with each seeking to improve in their perceived weaker areas. This
paper is mainly going to focus on the advanced features that are uniquely present in one operating system and
not in other one.
Optimization of Multi-band Rectangular-Triangular Slotted AntennaIJERA Editor
A multi-band microstrip patch antenna is developed and presented in this paper. The radiating elements in this antenna are composed of rectangular and triangular slots. These slots are engraved in the rectangular and triangular patch, joined together in one structure, and by single probe feed. The rectangular and triangular slots make the antenna to operate at multiband with relatively high gain. Therefore, this antenna can be used for wireless communication applications like WLAN, WiMax and radar system applications.
Dynamic Texture Coding using Modified Haar Wavelet with CUDAIJERA Editor
Texture is an image having repetition of patterns. There are two types, static and dynamic texture. Static texture is an image having repetitions of patterns in the spatial domain. Dynamic texture is number of frames having repetitions in spatial and temporal domain. This paper introduces a novel method for dynamic texture coding to achieve higher compression ratio of dynamic texture using 2D-modified Haar wavelet transform. The dynamic texture video contains high redundant parts in spatial and temporal domain. Redundant parts can be removed to achieve high compression ratios with better visual quality. The modified Haar wavelet is used to exploit spatial and temporal correlations amongst the pixels. The YCbCr color model is used to exploit chromatic components as HVS is less sensitive to chrominance. To decrease the time complexity of algorithm parallel programming is done using CUDA (Compute Unified Device Architecture). GPU contains the number of cores as compared to CPU, which is utilized to reduce the time complexity of algorithms.
Region based elimination of noise pixels towards optimized classifier models ...IJERA Editor
The extraction of the skin pixels in a human image and rejection of non-skin pixels is called the skin segmentation. Skin pixel detection is the process of extracting the skin pixels in a human image which is typically used as a pre-processing step to extract the face regions from human image. In past, there are several computer vision approaches and techniques have been developed for skin pixel detection. In the process of skin detection, given pixels are been transformed into an appropriate color space such as RGB, HSV etc. And then skin classifier model have been applied to label the pixel into skin or non-skin regions. Here in this research a “Region based elimination of noise pixels and performance analysis of classifier models for skin pixel detection applied on human images” would be performed which involve the process of image representation in color models, elimination of non-skin pixels in the image, and then pre-processing and cleansing of the collected data, feature selection of the human image and then building the model for classifier. In this research and implementation of skin pixels classifier models are proposed with their comparative performance analysis. The definition of the feature vector is simply the selection of skin pixels from the human image or stack of human images. The performance is evaluated by comparing and analysing skin colour segmentation algorithms. During the course of research implementation, efforts are iterative which help in selection of optimized skin classifier based on the machine learning algorithms and their performance analysis.
A Computational Analysis of Flow Development through a Constant Area C- DuctIJERA Editor
This paper represents the results of an experimental work with measurement of mean velocity along with total pressure contours in 2-D form and validation of the same with numerical results based on the wall y+ approach for various turbulent models like, Spalart Alamras, k-ε model, k-ω model and RSM models are used to solve the closure problem. The turbulence models are investigated in the commercial CFD code of Fluent using y+ as guidance in selecting the appropriate grid configuration and turbulence model. The experiment is carried out at mass averaged mean velocity of 40m/s and the geometry of the duct is chosen as rectangular cross-section of 90 curved constant area duct .In the present paper the computational results obtained from different turbulence models are compared with the experimental result along with the near-wall treatments are investigated for wall y+<30>30 in the fully turbulent region. It is concluded in the present study that the mesh resolving the fully turbulent region is sufficiently accurate in terms of qualitative features. Here RSM turbulence model predicts the best results while comparing with the experimental results.
Analysis and Design of Conventional Controller for Speed Control of DC Motor ...IJERA Editor
The objective of this paper is to control the speed of the motor using conventional controller; compensator is used to improve the steady state error. To evaluate the performance of the controller, time response analysis is carried out. The time response analysis consists of two type of analysis. One is unit step response analysis and other is performance indices analysis. The paper describes the designing of a closed loop model of the dc motor drive for controlling speed. Accuracy and the dynamic responses are better in a closed loop system. The compensator is used to compensate the parameter of the system in such a way to meet the specification, so that it improves the steady state response of the system and get desired response.
COMPARATIVE ANALYSIS OF CONVENTIONAL PID CONTROLLER AND FUZZY CONTROLLER WIT...IJITCA Journal
All the real systems exhibits non-linear nature,conventional controllers are not always able to provide good and accurate results. Fuzzy Logic Control is used to obtain better response. A model for simulation is designed and all the assumptions are made before the development of the model. An attempt has been made to analyze the efficiency of a fuzzy controller over a conventional PID controller for a three tank level control system using fuzzification & defuzzification methods and their responses are compared. Analysis is done through computer simulation using Matlab/Simulink toolbox. This study shows that the application of Fuzzy Logic Controller (FLC) gives the best response with triangular membership function and centroid defuzzification method.
A New Approach for Design of Model Matching Controllers for Time Delay System...IJERA Editor
Modeling of physical systems usually results in complex high order dynamic representation. The simulation and design of controller for higher order system is a difficult problem. Normally the cost and complexity of the controller increases with the system order. Hence it is desirable to approximate these models to reduced order model such that these lower order models preserves all salient features of higher order model. Lower order models simplify the understanding of the original higher order system. Modern controller design methods such as Model Matching Technique, LQG produce controllers of order at least equal to that of the plant, usually higher order. These control laws are may be too complex with regards to practical implementation and simpler designs are then sought. For this purpose, one can either reduce the order the plant model prior to controller design, or reduce the controller in the final stage, or both. In the present work, a controller is designed such that the closed loop system which includes a delay response(s) matches with those of the chosen model with same time delay as close as possible. Based on desired model, a controller(of higher order) is designed using model matching method and is approximated to a lower order one using Approximate Generalized Time Moments (AGTM) / Approximate Generalized Markov Moments (AGMM) matching technique and Optimal Pade Approximation technique. Genetic Algorithm (GA) optimization technique is used to obtain the expansion points one which yields similar response as that of model, minimizing the error between the response of the model and that of designed closed loop system.
Explicit model predictive control of fast dynamic systemeSAT Journals
Abstract Explicit Model Predictive Control approach provides offline computation of the optimization law by Multi Parametric Quadratic Programming. The solution is Piece wise affine in nature. It is explicit representation of the system states and control inputs. Such law then can be solved using binary search tree and can be evaluated for fast dynamic systems. Implementing such controllers can be done on microcontroller or ASIC/FPGA. DC Motor Speed Control - one of the benchmark systems is discussed here in this context. Its PWA law obtained, simulation of closed loop e-MPC is presented and its implementation approach using MPT toolbox and other such toolboxes is shown in brief. Index Terms: Model Predictive Control, explicit, Piece-wise Affine, and Multi Parametric Toolbox
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Stability and stabilization of discrete-time systems with time-delay via Lyap...IJERA Editor
The stability and stabilization problems for discrete systems with time-delay are discussed .The stability and
stabilization criterion are expressed in the form of linear matrix inequalities (LMI). An effective method
allowing us transforming a bilinear matrix Inequality (BMI) to a linear matrix Inequality (LMI) is developed.
Based on these conditions, a state feedback controller with gain is designed. An illustrative numerical example
is provided to show the effectiveness of the proposed method and the reliability of the results.
Output feedback trajectory stabilization of the uncertainty DC servomechanism...ISA Interchange
This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
SYSTEM IDENTIFICATION AND MODELING FOR INTERACTING AND NON-INTERACTING TANK S...ijistjournal
System identification from the experimental data plays a vital role for model based controller design. Derivation of process model from first principles is often difficult due to its complexity. The first stage in the development of any control and monitoring system is the identification and modeling of the system. Each model is developed within the context of a specific control problem. Thus, the need for a general system identification framework is warranted. The proposed framework should be able to adapt and emphasize different properties based on the control objective and the nature of the behavior of the system. Therefore, system identification has been a valuable tool in identifying the model of the system based on the input and output data for the design of the controller. The present work is concerned with the identification of transfer function models using statistical model identification, process reaction curve method, ARX model, genetic algorithm and modeling using neural network and fuzzy logic for interacting and non interacting tank process. The identification technique and modeling used is prone to parameter change & disturbance. The proposed methods are used for identifying the mathematical model and intelligent model of interacting and non interacting process from the real time experimental data.
SYSTEM IDENTIFICATION AND MODELING FOR INTERACTING AND NON-INTERACTING TANK S...ijistjournal
System identification from the experimental data plays a vital role for model based controller design. Derivation of process model from first principles is often difficult due to its complexity. The first stage in the development of any control and monitoring system is the identification and modeling of the system. Each model is developed within the context of a specific control problem. Thus, the need for a general system identification framework is warranted. The proposed framework should be able to adapt and emphasize different properties based on the control objective and the nature of the behavior of the system. Therefore, system identification has been a valuable tool in identifying the model of the system based on the input and output data for the design of the controller. The present work is concerned with the identification of transfer function models using statistical model identification, process reaction curve method, ARX model, genetic algorithm and modeling using neural network and fuzzy logic for interacting and non interacting tank process. The identification technique and modeling used is prone to parameter change & disturbance. The proposed methods are used for identifying the mathematical model and intelligent model of interacting and non interacting process from the real time experimental data.
ON M(M,m)/M/C/N: Interdependent Queueing ModelIOSR Journals
Abstract: This paper deals with a multiple server Queueing system in which arrivals and services are
intredependent and follow a bivariate Poisson process and having startup delay. Using the Supplementary
Varible Techniue these models are analyzed. The expected length of the Dorment period, the busy period,
expected number of units in the queue are derived and analyzed in the light of the dependence paramater
Big Bang- Big Crunch Optimization in Second Order Sliding Mode ControlIJMTST Journal
In this article, Second order sliding mode with Big Bang- Big Crunch optimization technique is employed
for nonlinear uncertain system.The sliding surface describes the transient behavior of a system in sliding
mode. Frequently, PD- type sliding surface is chosen as a hyperplane in the system state space.An integral
term incorporated in the sliding surface expression that resulted in a type of PID sliding surface as hyperbolic
function for alleviating chattering effect. The sliding mode control law is derived using direct Lyapunov
stability approach and asymptotic stability is proved theoretically. Here, novel tuning scheme is introduced for
estimation of PID sliding surface coefficients, due to which it reduces the reaching time as well as disturbance
effect.The simulation results are presented to make a quantitative comparison with the traditional sliding
mode control. It is demonstrated that the proposed control law improves the tracking performance of system
dynamic model in case of external disturbances and parametric uncertainties.
Controller Design Based On Fuzzy Observers For T-S Fuzzy Bilinear Models ijctcm
This article is devoted to the design of a fuzzy-observer-based control for a class of nonlinear systems with bilinear terms. The class of systems considered is the Takagi-Sugeno (T-S) fuzzy bilinear model. A new procedure to design the observer-based fuzzy controller for this class of systems is proposed. The aim is to design the fuzzy controller and the fuzzy observer of the augmented system separately in order to guarantee that the error between the state and its estimation converges faster to zero. By using the Lyapunov function, sufficient conditions are derived such that the closed-loop system is globally asymptotically stable. Moreover, sufficient conditions for controller design based on state estimation for robust stabilization of TS FBS with parametric uncertainties is proposed. The observer and controller gains are obtained by solving some linear matrix inequalities (LMIs). An example is given to illustrate the effectiveness of the proposed approach.
Multivariable Parametric Modeling of a Greenhouse by Minimizing the Quadratic...TELKOMNIKA JOURNAL
This paper concerns the identification of a greenhouse described in a multivariable linear system
with two inputs and two outputs (TITO). The method proposed is based on the least squares identification
method, without being less efficient, presents an iterative calculation algorithm with a reduced
computational cost. Moreover, its recursive character allows it to overcome, with a good initialization, slight
variations of parameters, inevitable in a real multivariable process. A comparison with other method s
recently proposed in the literature demonstrates the advantage of this method. Simulations obtained will be
exposed to showthe effectiveness and application of the method on multivariable systems.
TEST GENERATION FOR ANALOG AND MIXED-SIGNAL CIRCUITS USING HYBRID SYSTEM MODELSVLSICS Design
In this paper we propose an approach for testing time-domain properties of analog and mixed-signal circuits. The approach is based on an adaptation of a recently developed test generation technique for hybrid systems and a new concept of coverage for such systems. The approach is illustrated by its application to some benchmark circuits.
Test Generation for Analog and Mixed-Signal Circuits Using Hybrid System Mode...VLSICS Design
In this paper we propose an approach for testing time-domain properties of analog and mixed-signal circuits. The approach is based on an adaptation of a recently developed test generation technique for hybrid systems and a new concept of coverage for such systems. The approach is illustrated by its application to some benchmark circuits.
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In this study the controller for three tank multi loop system is designed using coefficient Diagram method. Coefficient Diagram Method is one of the polynomial methods in control design. The controller designed by using CDM technique is based on the coefficients of the characteristics polynomial of the closed loop system according to the convenient performance such as equivalent time constant, stability indices and stability limit. Controller is designed for the three tank process by using CDM Techniques; the simulation results show that the proposed control strategies have good set point tracking and better response capability.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
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• Remote control system for accessing CCR and allied system over serial or TCP.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
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A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
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This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Discretizing of linear systems with time-delay Using method of Euler’s and Tustin’s approximations
1. Bemri H’mida et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -2) March 2015, pp.83-89
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Discretizing of linear systems with time-delay Using method of
Euler’s and Tustin’s approximations
Bemri H’mida1
, Mezlini Sahbi2
and Soudani Dhaou3
Tunis El Manar University
Laboratory of Research on Automatic (LARA)
National Engineers School of Tunis
BP 37, Belvedere 1002 Tunis. Tunisia.
ABSTRACT
Delays deteriorate the control performance and could destabilize the overall system in the theory of discrete-
time signals and dynamic systems. Whenever a computer is used in measurement, signal processing or control
applications, the data as seen from the computer and systems involved are naturally discrete-time because a
computer executes program code at discrete points of time. Theory of discrete-time dynamic signals and systems
is useful in design and analysis of control systems, signal filters, state estimators and model estimation from
time-series of process data system identification. In this paper, a new approximated discretization method and
digital design for control systems with delays is proposed. System is transformed to a discrete-time model with
time delays. To implement the digital modeling, we used the z-transfer functions matrix which is a useful model
type of discrete-time systems, being analogous to the Laplace-transform for continuous-time systems. The most
important use of the z-transform is for defining z-transfer functions matrix is employed to obtain an extended
discrete-time. The proposed method can closely approximate the step response of the original continuous time-
delayed control system by choosing various of energy loss level. Illustrative example is simulated to demonstrate
the effectiveness of the developed method.
Keywords - Discretization, Time-delay systems, Z-transform.
I. INTRODUCTION
Sampling a continuous-time system is a
fundamental problem in a variety of scientific
areas, such as computer control, system
identification and signal processing. It is
becoming even more conspicuous in light of the
huge success of computer-aided processing and
networking [9, 12]. There are many intriguing
problems related to sampling. Time delay is one
of the key factors influencing the overall system
stability and performance. In particular, as the
different effects of actuator, sensor and
controller exist in control systems, delays are
often formulated as state time delays, input
time delays as well as output time delays in a
continuous-time or discrete-time framework [8,
14]. To digitally simulate and design a
continuous-time delayed control system, it is
often required to obtain an equivalent discrete-
time model. The digital modeling of
continuous-time systems with input delays can
be found in a standard textbook .To improve the
performance of a continuous-time system with
multiple time delays, several advanced control
theories and practical design techniques have
been proposed [7,10]. Most control systems are
formulated in a continuous-time framework, for
which many analysis tools and control
methodologies are well-established. With the
rapid advances in digital technology and
computers, digital control provides various
advantages over its analog counterpart for better
reliability, lower cost, smaller size, more
flexibility and better performance. The resulting
digitally controlled continuous time system
becomes a sampled-data system.
Systems including time delay, due to the
system dynamics, are widely present in the
industry which imposes a lot of constraints that
make the control and computer programming of
such system difficult [1].Then the control of a
system with a time delay is generally difficult;
due to the constraints imposed by the time
delay. These constraints can cause performance
deterioration that leads the process to instability
especially when operating in closed loop [2, 3,
4, 11].Most physical systems, a macroscopic
point of view, are continuous. In modern
control systems, information is digitally
processed which requires sampling signals [3,
RESEARCH ARTICLE OPEN ACCESS
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www.ijera.com 84 | P a g e
5]. One speaks in this case of sampled or
discrete systems. If necessary, it is always
possible to obtain the state equations from the
transfer-function matrix [13]. Another
advantage of using the transfer-function matrix
is that one can decompose the multivariable
system into p subsystems, each with one output
and m inputs. For this reason we need the
discretization of continuous time-delay systems.
The objective of this paper is to extend and
analyze the ideas in the just cited references; we
consider three methods for obtaining the
discrete-time delay approximation. These are:
(i) Backward difference method.
(ii) Forward difference method.
(iii) Bilinear z-transformation method.
The paper is organized as follows: The next
section discusses discretization of systems with
external point delay; Section 3 provides
numerical examples; Section 4 includes a
comparison between the three methods
followed by a conclusion in the end of the
paper.
II. DISCRETIZATION OF SYSTEMS WITH
EXTERNAL POINT DELAY
In a digital computer, time cannot flow
continuously as it is perceived in the physical
world. The time is defined on a discrete set of
times, which are separated by a regular time
interval known by one sampling period. It is
therefore necessary to define new mathematical
tools adapted to discrete time, to represent the
sampled signals and systems and to adapt tools
and methods for automatic analog continuous
time in the design of digital controllers.
Then our problem may be stated as the
determination of a discrete time approximation
corresponding to the following state-space
equations:
0 1
.
( ) ( ) ( ) ( )
( ) ( )
t A x t A x t h Bu t
y t Cx t
x (1)
Where:
0
( )A x t : Original Term state.
1
( )A x t h : Delayed Term state.
With h qT : a multiple delay the sampling
period is an integer q .
T : The sampling period assumed chosen
suitably.
,x u andy Respectively are the state vector, the
vector and the control vector output.
0 1
, ,A A BandC are matrices of suitable
dimensions.
Calculating the Laplace transform of the
system (1), we get the following equation
model:
0 1
( ) X(0) A ( ) ( ) B ( )
( ) ( ) ; X(0) 0
hp
pX p X p A e X p U p
Y p CX p
(2)
This is equivalent to this equation:
0 1
( ) ( )hp
X p pI A A e BU p (3)
This equation can be written as:
1
0 1
( ) ( )hp
X p pI A A e BU p (4)
Then:
1
0 1
( ) ( ) ( )hp
Y p CX p C pI A A e BU p (5)
From (5) we can get:
1
0 1
( )
( )
( )
hpY p
H p C pI A A e B
U p
(6)
Implementation of continuous-time control
and filtering functions in a computer program,
in some cases we need to find a discrete-time z-
transfer function matrix from a given
continuous-time p -transfer function. In
accurate model based design of a discrete
controller for a process originally in the form of
a continuous-time p -transfer function,
( )H p .The latter should be discredited to get a
discrete-time process model before the design is
started. There are several methods for
discretization of a p -transfer function. The
discretization can be realized in several ways by
calculating the z-transfer function matrix for
( )H p .The methods can be categorized as
follows, and they are described in the following
section:
A. Forward difference method
This method is also called Forward
rectangular rule. Here we apply the technique of
the first order approximation by linear
transformation. These approximations on the z-
transform function matrix exploit the
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www.ijera.com 85 | P a g e
relationship: pT
z e .The idea is to approximate
this relationship by a linear relationship
between z and p .
When the sampling period T is small, the
linear approximation of the first order function
exponential gives: 1pT
z e pT , so we can
write:
1
1
1 1z z
p
T Tz
(7)
This technique of discretization results from
the approximation of the derivative of two
sampling instants:
1 1( ) ( ) ( ) 1
( ) ( )
dx t x t T x t z
L pX p z X z
dt T T
(8)
It can be shown that the discrete-time
transfer function matrix of the transfer function
matrix given in (6) is obtained by using the
Forward difference method after substituting
for
1z
p
T
:
1
1
0 1
1
( )
z
h
Tz
H z C I A A e B
T
(9)
From (7) we can write:
1
1
0 1
1
( )
z
h
Tz
H z C I A A e B
T
(10)
By applying an approximation Taylor
expansion near 0 of the term
1z
h
T
e to the first
order approximation, we can write:
1
1
1
z
h
T z
e h
T
(11)
The expression of ( )H z becomes:
1
1 1
1 0 1 1( ) ( )H z CTz I A h I A T A T A h z B (12)
B. Backward difference method
This method is also called backward
rectangular rule. To simulate the entire system
as a discrete-time system one needs to find the
discrete equivalent. Here we apply the
technique of the first order approximation. This
Backward discretization results from
approximation (8) of the derivative that can be
made between two sampling times:
1 1( ) ( ) ( ) 1
( ) ( )
dx t x t x t T z
L pX p z X z
dt T zT
(13)
Then, we can write:
1
1 1z z
p
zT T
(14)
It can be shown that the discrete-time
transfer function matrix of the transfer function
matrix given in (6) is obtained by using the
backward difference method after substituting
for
1z
p
zT
:
1
1
0 1
1
( )
z
h
zTz
H z C I A A e B
zT
(15)
By applying an approximation Taylor
expansion near 0 of the term
1z
h
zT
e to the first
order approximation, we can write:
1
1
1
z
h
zT z
e h
zT
(16)
The expression of ( )H z becomes:
1
11 1
0 1
( )
A h A hI I
H z C A A z B
T T T T
(17)
C. Bilinear z-transformation method
This method is also called Tustin’s method
also the trapezoid rule in digital control
community, there is a good discrete-time
approximation for a continuous-time linear
system is obtained through the bilinear z-
transformation if the sampling interval T is
selected suitably so that 0.5wT , where w is
the magnitude of the pole of the transfer
function of the continuous-time system far from
the origin of the P-plane [6]. However, while
the bilinear z-transformation leads to a
realization in the form of a transfer-function
matrix, the method based on the trapezoidal rule
leads directly to a state space realization, more
suitable for digital simulation.
We apply a second-order approximation by
bilinear transformation homographic: Tustin
discretization. The linear approximation of the
second order function exponential gives:
1
2
1
2
T
P
z
T
P
(18)
Therefore:
1
1
2 1 2 1
1 1
z z
p
T z T z
(19)
This approximation results from the
approximate numerical integration by the
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trapezoidal rule, between two sampling times,
indeed, either:
1
( ) ( ) ( ) ( )y t x t dt Y p X p
p
(20)
From where:
(p)
( )
X
p
Y p
(21)
By approximating between two seconds
sampling, we get:
1
1( )
2
k k
k k
X X
y y kT y T
(22)
The z-transform function gives then:
1 1
1 ( ) 1 ( )
2
T
z Y z z X z
(23)
From where:
( ) 2 1
( ) 1
X z z
Y z T z
(24)
Then, we can write:
1
1
2 1 2 1
1 1
z z
p
T z T z
(25)
It can be shown that the discrete-time
transfer function matrix of the transfer function
matrix given in (6) is obtained by using the
bilinear transformation method after
substituting for
1
1
2 1
1
z
p
T z
:
1
1
1
2 11
1
0 11
2 1
( )
1
h z
T z
z
H z C I A A e B
T z
(26)
By applying an approximation Taylor
expansion near 0 of the term
1
1
2 1
1
h z
T z
e to the first
order approximation, we can write:
1
1
2 1 1
1
1
2 1
1
1
h z
T z
z
e h
T z
(27)
The expression of ( )H z becomes:
1
1 1( ) (1 ) (I )
1 0 1 1 0 12 2 2 2 2
T T T T T
H z C z I A h A A A h A A z B (28)
III. NUMERICAL EXAMPLES
In this section an example of application of
the proposed approach are presented. This
example shows the validity of those
methodologies. As the system is considered by
this equation:
0 1 0 0 0
x( ) ( ) ( 0.2) ( )
2 3 1 2 1
( ) 1 1 ( )
t x t x t u t
y t x t
(29)
In this section we are treating an example of
a continuous linear system. A delay is only in
the state which was chosen a sampling period
T=0.2s and applying the Forward difference
method, the following discrete time-delay
system is obtained:
1
1 1
1 0 1 1( ) ( )H z CTz I A h I A T A T A h z B (30)
With:
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
The operator 1
z is here a time-step delay
operator, and it can be regarded as an operator
of the time-step delay, 1
z is also the transfer
function of a time-step delay.
The simulation result in the step
responses of the continuous system and
approximate discrete system is depicted in
figure1.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Step response of the continuous and discrete time delay system
Continuous
forward Discretization
Fig. 1. Step response of the continuous and discrete
time delay system.
This method is not recommended because
the discrete-time equivalent may become
unstable; which is commonly used in
developing simple simulators, higher distortion
with the forward rule.
Applying the method based on Backward
difference method to the system considered in
the state-space equations (29), and keeping the
same sampling period T=0.2s. The following
discrete time-delay system is obtained:
1
11 1
0 1
( )
A h A hI I
H z C A A z B
T T T T
(31)
5. Bemri H’mida et al. Int. Journal of Engineering Research and Applications www.ijera.com
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With:
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
The simulation result in the step responses
of the continuous system and approximate
discrete system is depicted in figure2.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Step response of the continuous and discrete time delay system
Continuous
backward Discretization
Fig. 2. Step response of the continuous and discrete
time delay system.
The forward rectangular rule could cause a
stable continuous filter to be mapped into an
unstable digital filter, contrary to the rule of
backward rectangular rule that keeps the system
stable conditions except that the answer discrete
time not following the continuous time.
Applying the method based on Bilinear z-
transformation method to the system considered
in the state-space equations (29), and keeping
the same sampling period T=0.2s.The following
discrete time-delay system is obtained:
1
1 1
( ) (1 ) (I )
1 0 1 1 0 12 2 2 2 2
T T T T T
H z C z I A h A A A h A A z B (32)
With:
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
The simulation result in the step responses of
the continuous system and approximate discrete
system is depicted in figure3.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Amplitude
Step response of the continuous and discrete time delay system
Continuous
Tustin Discretization
Fig. 3. Step response of the continuous and
discrete time delay system.
It is clear that the value of the delay is shown
by the simulation by the three discretization
methods knowing that it did not affect the
evolution of the system. The evolution of the
response the field below presents a delay that
does not degrade the performance of the
system.
The bilinear transformation is motivated by
considering the trapezoidal approximation of an
integrator. It allows the passage of a Laplace
transform in a z-transform. It uses the trapezoid
method to calculate integral. This is a first-order
approximation of the natural logarithm function
that is an exact mapping of the z-plane to the P-
plane. When the Laplace transform is
performed on a discrete-time signal with each
element of the discrete-time sequence attached
to a correspondingly delayed unit impulse. The
trapezoid rule maps the stable region in the P-
plane exactly into the stable region of the z-
plane. This method also offers the most
accurate phase relative to the continuous
system. It is clear that both digital controllers
designed by emulation perform slightly worse
than the continuous controller. As expected
however, the trapezoidal integration method
performs better than the forward and the
backward rectangular method.
IV. COMPARISON BETWEEN EULER AND
TUSTIN’S APPROXIMATIONS
We have discussed three different methods
for obtaining discrete time approximations for
continuous-time delay systems. Note how the
discrete equivalent model predicts the
continuous output at the sample instances. The
facts that the second and third graph lines
generate the same result show that the formula
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we have derived to calculate the discrete
equivalent transfer function model is correct,
Euler’s backward differentiation method, which
is commonly used in discretizing simple signal
filters and industrial controllers. The backward
differentiation method may give problems since
it results in an implicit equation for the output
variable, while the Forward differentiation
method always gives an explicit equation. The
Forward differentiation method is somewhat
less accurate than the backward differentiation
method, but it is simpler to use. In signal
processing one is interested in making the
difference between the original and the
reconstructed signal as small as possible.
For continuous systems, the Laplace
transform has played a major role. In fact, this
linear transformation allows algebraically treat
the linear operators, in the case of sampled
systems; it has to introduce a transformation,
called z, which has similar properties. The z-
transfer function matrix will substitute for the
p-transfer function matrix end of the Laplace
transform. We can see very clearly the
importance of the sampling period; a lower
sampling period provides monitoring of the
response of faithful continuous system with a
longer period. The consequence of this is that
the system with the lowest sampling period will
be the one that will reach stability soon. So for
the most accurate results possible relative to the
continuous time it will be beneficial to not take
too much sampling period. For better
monitoring of the output in continuous time
then choose a sampling period that will not be
too great: but keep in mind that too little time
require great precision the calculator. In signal
processing literature the Tustin’s method is
frequently denoted the bilinear transformation
method. The term bilinear is related to the fact
that the imaginary axis in the complex P-plane
for continuous-time systems is mapped or
transformed onto the unity circle for the
corresponding discrete-time system. In addition,
the poles are transformed so that the stability
property is preserved; this is not guaranteed in
the Euler’s Forward and Backward methods.
Tustin’s method is the most accurate of these
three methods, so I suggest it is the default
choice. However, typically it is not much
difference between the Tustin’s method and the
Euler’s backward method.
The Euler’s forward method is the least
accurate method. Tustin’s method and Euler’s
backward methods are implicit methods since
the output variable, appears on both the left and
the right side of the discretized expression, and
it is in general necessary to solve the discretized
expression for at each time-step, and this may
be impractical.
V. CONCLUSION
In this paper we have proposed three methods
for the discrete time approximation of
continuous-time delay systems, using an
integral numerical approximation, typically
Euler’s forward method, Euler’s backward
method or Tustin’s method. These methods
should be used when a continuous-time
controller transfer function or filter transfer
function are discretized to get an algorithm
ready for programming. In such cases the input
signal is a discrete-time signal with no holding.
The discretization is based on some numerical
method so that the behaviour of the discrete-
time transfer function is similar to that of the
continuous-time transfer function.
Usually we try to choose the sampling
period an integer sub multiple of delay in
addition to its forced choice for dynamic
systems without delay. Finally it appears that
the model using the Tustin’s method is best
suited for numerical simulation in the case of
the continuous system is defined by a state
model and in the case of a suitable choice of the
sampling period. In a next job I am interested to
study the discretization of these types of
systems by the same methods taking into
consideration the delay this time is manifold
which appears in the state and control.
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Authors’ information
Bemri H'mida was born in Tunisia, in
January1979.He received the scientific
degree in mathematics, physics and
computer science within the Faculty of
Sciences of Tunis (FST) in 2003.He received
the degree of MA within the Faculty of
Sciences of Tunis (FST) in 2005.He received
the certificate of Aptitude Teaching of
Second Degree (CAPES) in 2008.He has been working as a
teacher in 2 Mars1934 secondary school in Siliana since 2008.
He received the degree of Master of Research in automatic and
signals processing (ATS) within the National Engineering
School of Tunis (ENIT) in 2013. His research interests include
modeling and control systems sampled to delays.
Mezlini Sahbi was born in Tunisia, in
March 1972.He received the scientific
degree in mathematics and physics
preparatory cycle of the National School
of Engineers of Gabes (ENIG) in 2000.He
received the engineering degree in
electrical engineering in the National
School of Engineers of Monastir (ENIM)
in 2003. He received the degree of Master of research in
automatic and signal processing (ATS) within the National
Engineering School of Tunis (ENIT) in 2005. His research
interests include the command by internal model of discrete
system.
Dhaou Soudani was born in Tunisia, in July 1954. He received the Master
degree in Electrical and Electronic Engineering and the “Diplôme des
Etudes Approfondies” in Automatic Control from the “Ecole Normale
Supérieure de l’Enseignement Technique” Tunisia in 1982 and 1984,
respectively. He obtained both the Doctorat in 1997, and the “Habilitation
Universitaire” in 2007 in Electrical Engineering from the “Ecole Nationale
d’Ingénieurs de Tunis” (ENIT) Tunisia. He is currently a professor in
Automatic Control and a member of the unit research “Laboratoire de
Recherche en Automatique” (LA.R.A).