Search results for: consensus control

Authors: Ran Etgar

Abstract:

The process of rounding off measurements in continuous variables is commonly encountered. Although it usually has minor effects, sometimes it can lead to poor outcomes in statistical process control using X ̅-chart. The traditional control limits can cause incorrect conclusions if applied carelessly. This study looks into the limitations of classical control limits, particularly the impact of asymmetry. An approach to determining the distribution function of the measured parameter (Y ̅) is presented, resulting in a more precise method to establish the upper and lower control limits. The proposed method, while slightly more complex than Shewhart's original idea, is still user-friendly and accurate and only requires the use of two straightforward tables.

Keywords: quality control, process control, round-off, measurement, rounding error

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Authors: Alicia C. Sanchez

Abstract:

This paper investigates the implementation of RAFU functions (radical functions) in robotics and automation. Specifically, the main goal is to show how these functions may be useful in lane-keeping control and the lateral control of autonomous machines, vehicles, robots or the like. From the knowledge of several points of a certain route, the RAFU functions are used to achieve the lateral control purpose and maintain the lane-keeping errors within the fixed limits. The stability that these functions provide, their ease of approaching any continuous trajectory and the control of the possible error made on the approximation may be useful in practice.

Keywords: automatic navigation control, lateral control, lane-keeping control, RAFU approximation

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Authors: U. Yavas, M. Gokasan

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Control of diesel engine’s air path has drawn a lot of attention due to its multi input-multi output, closed coupled, non-linear relation. Today, precise control of amount of air to be combusted is a must in order to meet with tight emission limits and performance targets. In this study, passenger car size diesel engine is modeled by AVL Boost RT, and then simulated with standard, industry level PID controllers. Finally, linear model predictive control is designed and simulated. This study shows the importance of modeling and control of diesel engines with flexible algorithm development in computer based systems.

Keywords: predictive control, engine control, engine modeling, PID control, feedforward compensation

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Authors: O. Ikpotokin

Abstract:

In any production process, every product is aimed to attain a certain standard, but the presence of assignable cause of variability affects our process, thereby leading to low quality of product. The ability to identify and remove this type of variability reduces its overall effect, thereby improving the quality of the product. In case of a univariate control chart signal, it is easy to detect the problem and give a solution since it is related to a single quality characteristic. However, the problems involved in the use of multivariate control chart are the violation of multivariate normal assumption and the difficulty in identifying the quality characteristic(s) that resulted in the out of control signals. The purpose of this paper is to examine the use of non-parametric control chart (the bootstrap approach) for obtaining control limit to overcome the problem of multivariate distributional assumption and the p-value method for detecting out of control signals. Results from a performance study show that the proposed bootstrap method enables the setting of control limit that can enhance the detection of out of control signals when compared, while the p-value method also enhanced in identifying out of control variables.

Keywords: bootstrap control limit, p-value method, out-of-control signals, p-value, quality characteristics

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Authors: Sheren H. Salah, Ahmed Y. Ben Sasi

Abstract:

The inverted pendulum system is a classic control problem that is used in universities around the world. It is a suitable process to test prototype controllers due to its high non-linearities and lack of stability. The inverted pendulum represents a challenging control problem, which continually moves toward an uncontrolled state. This paper presents the possibility of balancing an inverted pendulum system using sliding mode control (SMC). The goal is to determine which control strategy delivers better performance with respect to pendulum’s angle and cart's position. Therefore, proportional-integral-derivative (PID) is used for comparison. Results have proven SMC control produced better response compared to PID control in both normal and noisy systems.

Keywords: inverted pendulum (IP), proportional-integral derivative (PID), sliding mode control (SMC), systems and control engineering

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Authors: Sophia Fry, Mahir Irtiza, Alexa Hoffman, Yousef Sardahi

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This study presents an intelligent control algorithm for a flexible robotic arm. Fuzzy control is used to control the motion of the arm to maintain the arm tip at the desired position while reducing vibration and increasing the system speed of response. The Fuzzy controller (FC) is based on adding the tip angular position to the arm deflection angle and using their sum as a feedback signal to the control algorithm. This reduces the complexity of the FC in terms of the input variables, number of membership functions, fuzzy rules, and control structure. Also, the design of the fuzzy controller is model-free and uses only our knowledge about the system. To show the efficacy of the FC, the control algorithm is implemented on the flexible joint manipulator (FJM) developed by Quanser. The results show that the proposed control method is effective in terms of response time, overshoot, and vibration amplitude.

Keywords: fuzzy logic control, model-free control, flexible joint manipulators, nonlinear control

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Authors: Wael M. Elawady, Mohamed F. Asar, Amany M. Sarhan

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This paper adopts a continuous sliding mode control scheme for trajectory tracking control of robot manipulators with structured and unstructured uncertain dynamics and external disturbances. In this algorithm, the equivalent control in the conventional sliding mode control is replaced by a PID control action. Moreover, the discontinuous switching control signal is replaced by a continuous proportional-integral (PI) control term such that the implementation of the proposed control algorithm does not require the prior knowledge of the bounds of unknown uncertainties and external disturbances and completely eliminates the chattering phenomenon of the conventional sliding mode control approach. The closed-loop system with the adopted control algorithm has been proved to be globally stable by using Lyapunov stability theory. Numerical simulations using the dynamical model of robot manipulators with modeling uncertainties demonstrate the superiority and effectiveness of the proposed approach in high speed trajectory tracking problems.

Keywords: PID, robot, sliding mode control, uncertainties

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Authors: M. Bouferhane, A. Boukhebza, L. Hatab

Abstract:

In this paper, the position control of linear induction motor using fuzzy sliding mode controller design is proposed. First, the indirect field oriented control LIM is derived. Then, a designed sliding mode control system with an integral-operation switching surface is investigated, in which a simple adaptive algorithm is utilized for generalised soft-switching parameter. Finally, a fuzzy sliding mode controller is derived to compensate the uncertainties which occur in the control, in which the fuzzy logic system is used to dynamically control parameter settings of the SMC control law. The effectiveness of the proposed control scheme is verified by numerical simulation. The experimental results of the proposed scheme have presented good performances compared to the conventional sliding mode controller.

Keywords: linear induction motor, vector control, backstepping, fuzzy-sliding mode control

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Authors: B. H. Aitchanov, Sh. K. Aitchanova, O. A. Baimuratov

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This paper is focused on dynamic pulse-frequency modulation (DPFM) control systems. Currently, the control law based on DPFM control signals is widely used in direct digital control subsystems introduced in the automated control systems of technological processes. Statistical analysis of automatic control systems is reduced to its construction of functional relationships between the statistical characteristics of the errors processes and input processes. Structural and dynamic Volterra models of digital pulse-frequency control systems can be used to develop methods for generating the dependencies, differing accuracy, requiring the amount of information about the statistical characteristics of input processes and computing labor intensity of their use.

Keywords: digital dynamic pulse-frequency control systems, dynamic pulse-frequency modulation, control object, discrete filter, impulse device, microcontroller

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Authors: Robert Gold

Abstract:

Control flow graphs are a well-known representation of the sequential control flow structure of programs with a multitude of applications. Not only single functions but also sets of functions or complete programs can be modelled by control flow graphs. In this case the size of the graphs can grow considerably and thus makes it difficult for software engineers to analyse the control flow. Graph reductions are helpful in this situation. In this paper we define reductions to subsets of nodes. Since executions of programs are represented by paths through the control flow graphs, paths should be preserved. Furthermore, the composition of reductions makes a stepwise analysis approach possible.

Keywords: control flow graph, graph reduction, software engineering, software applications

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Authors: Z. Bouchama, N. Essounbouli, A. Hamzaoui, M. N. Harmas

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A new robust finite time synergetic controller is presented based on recently developed synergetic control methodology and a terminal attractor technique. A Fast Terminal Synergetic Control (FTSC) is proposed for controlling DC-DC buck converter. Unlike Synergetic Control (SC) and sliding mode control, the proposed control scheme has the characteristics of finite time convergence and chattering free phenomena. Simulation of stabilization and reference tracking for buck converter systems illustrates the approach effectiveness while stability is assured in the Lyapunov sense and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability.

Keywords: dc-dc buck converter, synergetic control, finite time convergence, terminal synergetic control, fast terminal synergetic control, Lyapunov

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Authors: Sheren H. Salah

Abstract:

The inverted pendulum is a highly nonlinear and open-loop unstable system. An inverted pendulum (IP) is a pendulum which has its mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart and pole. The characteristics of the inverted pendulum make identification and control more challenging. This paper presents the simulation study of several control strategies for an inverted pendulum system. The goal is to determine which control strategy delivers better performance with respect to pendulum’s angle. The inverted pendulum represents a challenging control problem, which continually moves toward an uncontrolled state. For controlling the inverted pendulum. The simulation study that sliding mode control (SMC) control produced better response compared to Genetic Algorithm Control (GAs) and proportional-integral-derivative(PID) control.

Keywords: Inverted Pendulum (IP) Proportional-Integral-Derivative (PID), Genetic Algorithm Control (GAs), Sliding Mode Control (SMC)

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Authors: Christian T. Lystbaek

Abstract:

Background: The importance of innovation management has traditionally been ascribed to private production companies, however, there is an increasing interest in public services innovation management. One of the major theoretical challenges arising from this situation is to understand public values justifying public services innovation management. However, there is not single and stable definition of public value in the literature. The research question guiding this paper is: What is the supposed added value operating in the public sphere? Methodology: The study takes an action research strategy. This is highly contextualized methodology, which is enacted within a particular set of social relations into which on expects to integrate the results. As such, this research strategy is particularly well suited for its potential to generate results that can be applied by managers. The aim of action research is to produce proposals with a creative dimension capable of compelling actors to act in a new and pertinent way in relation to the situations they encounter. The context of the study is a workshop on public services innovation within elderly care. The workshop brought together different actors, such as managers, personnel and two groups of users-citizens (elderly clients and their relatives). The process was designed as an extension of the co-construction methods inherent in action research. Scenario methods and focus groups were applied to generate dialogue. The main strength of these techniques is to gather and exploit as much data as possible by exposing the discourse of justification used by the actors to explain or justify their points of view when interacting with others on a given subject. The approach does not directly interrogate the actors on their values, but allows their values to emerge through debate and dialogue. Findings: The public values related to public services innovation management in elderly care were identified in two steps. In the first step, identification of values, values were identified in the discussions. Through continuous analysis of the data, a network of interrelated values was developed. In the second step, tracking group consensus, we then ascertained the degree to which the meaning attributed to the value was common to the participants, classifying the degree of consensus as high, intermediate or low. High consensus corresponds to strong convergence in meaning, intermediate to generally shared meanings between participants, and low to divergences regarding the meaning between participants. Only values with high or intermediate degree of consensus were retained in the analysis. Conclusion: The study shows that the fundamental criterion for justifying public services innovation management is the capacity for actors to enact public values in their work. In the workshop, we identified two categories of public values, intrinsic value and behavioural values, and a list of more specific values.

Keywords: public services innovation management, public value, co-creation, action research

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Authors: B. Ben Amara, H. Mhiri Sellami

Abstract:

Serious games adoption is increasing in multiple fields, including health, education, and business. In the same way, many research studied serious games (SGs) for various purposes such as classification, positive impacts, or learning outcomes. Although most of these research examine SG characteristics (SGCs) for conducting their studies, to author’s best knowledge, there is no consensus about features neither in number not in the description. In this paper, we conduct a literature review to collect essential game attributes regardless of the application areas and the study objectives. Firstly, we aimed to define Common SGCs (CSGCs) that characterize the game aspect, by gathering features having the same meanings. Secondly, we tried to identify specific features related to the application area or to the study purpose as a serious aspect. The findings suggest that any type of SG can be defined by a number of CSGCs depicting the gaming side, such as adaptability and rules. In addition, we outlined a number of specific SGCs describing the 'serious' aspect, including specific needs of the domain and indented outcomes. In conclusion, our review showed that it is possible to bridge the research gap due to the lack of consensus by using CSGCs. Moreover, these features facilitate the design and development of successful serious games in any domain and provide a foundation for further research in this area.

Keywords: serious game characteristics, serious games common aspects, serious games features, serious games outcomes

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Authors: R. Chakib, A. Essadki, M. Cherkaoui

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This paper proposes the study of a robust control of the doubly fed induction generator (DFIG) used in a wind energy production. The proposed control is based on the linear active disturbance rejection control (ADRC) and it is applied to the control currents rotor of the DFIG, the DC bus voltage and active and reactive power exchanged between the DFIG and the network. The system under study and the proposed control are simulated using MATLAB/SIMULINK.

Keywords: doubly fed induction generator (DFIG), active disturbance rejection control (ADRC), vector control, MPPT, extended state observer, back-to-back converter, wind turbine

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Authors: Javier Fernandez De Canete, Alvaro Fernandez-Quintero

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Nowadays, adaptive control schemes are being used when model based control schemes are applied in presence of uncertainty and model mismatches. Artificial neural networks have been employed both in modelling and control of non-linear dynamic systems with unknown dynamics. In fact, these are powerful tools to solve this control problem when only input-output operational data are available. A neural network controller under SIMULINK together with the ARDUINO hardware platform has been used to perform real-time speed control of a computer case fan. Comparison of performance with a PID controller has also been presented in order to show the efficacy of neural control under different command signals tracking and also when disturbance signals are present in the speed control loops.

Keywords: neural networks, ARDUINO platform, SIMULINK, adaptive speed control

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Authors: Alexander Winkler, Jozef Suchý

Abstract:

This paper investigates simple implicit force control algorithms realizable with industrial robots. A lot of approaches already published are difficult to implement in commercial robot controllers, because the access to the robot joint torques is necessary or the complete dynamic model of the manipulator is used. In the past we already deal with explicit force control of a position controlled robot. Well known schemes of implicit force control are stiffness control, damping control and impedance control. Using such algorithms the contact force cannot be set directly. It is further the result of controller impedance, environment impedance and the commanded robot motion/position. The relationships of these properties are worked out in this paper in detail for the chosen implicit approaches. They have been adapted to be implementable on a position controlled robot. The behaviors of stiffness control and damping control are verified by practical experiments. For this purpose a suitable test bed was configured. Using the full mechanical impedance within the controller structure will not be practical in the case when the robot is in physical contact with the environment. This fact will be verified by simulation.

Keywords: robot force control, stiffness control, damping control, impedance control, stability

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Authors: Pavol Tanuska, Pavel Vazan, Michal Kebisek, Dominika Jurovata

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The paper gives the results of the project that was oriented on the usage of knowledge discoveries from production systems for needs of the hierarchical process control. One of the main project goals was the proposal of knowledge discovery model for process control. Specifics data mining methods and techniques was used for defined problems of the process control. The gained knowledge was used on the real production system, thus, the proposed solution has been verified. The paper documents how it is possible to apply new discovery knowledge to be used in the real hierarchical process control. There are specified the opportunities for application of the proposed knowledge discovery model for hierarchical process control.

Keywords: hierarchical process control, knowledge discovery from databases, neural network, process control

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Authors: Shiuh-Jer Huang, Shin-Ham Lee, Sheam-Chyun Lin

Abstract:

A two wheel inverted pendulum (TWIP) vehicle is built with two hub DC motors for motion control evaluation. Arduino Nano micro-processor is chosen as the control kernel for this electric test plant. Accelerometer and gyroscope sensors are built in to measure the tilt angle and angular velocity of the inverted pendulum vehicle. Since the TWIP has significantly hub motor dead zone and nonlinear system dynamics characteristics, the vehicle system is difficult to control by traditional model based controller. The intelligent model-free fuzzy sliding mode controller (FSMC) was employed as the main control algorithm. Then, intelligent controllers are designed for TWIP balance control, and two wheels synchronization control purposes.

Keywords: balance control, synchronization control, two-wheel inverted pendulum, TWIP

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Authors: Zhan Haiyang, Gu Miao

Abstract:

This article mainly introduces the control system architecture of space environment simulator, simultaneously also briefly introduce the automation control technology of industrial process and the measurement technology of vacuum and cold black environment. According to the volume of chamber, the space environment simulator is divided into three types of small, medium and large. According to the classification and application of space environment simulator, the control system is divided into the control system of small, medium, large space environment simulator and the centralized control system of multiple space environment simulators.

Keywords: space environment simulator, control system, architecture, automation control technology

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Authors: P. Javid, A. Aeenmehr, J. Taghavifar

Abstract:

In this paper, supply air pressure of HVAC system has been modeled with second-order transfer function plus dead-time. In HVAC system, the desired input has step changes, and the output of proposed control system should be able to follow the input reference, so the idea of using model based predictive control is proceeded and designed in this paper. The closed loop control system is implemented in MATLAB software and the simulation results are provided. The simulation results show that the model based predictive control is able to control the plant properly.

Keywords: air conditioning system, GPC, dead time, air supply control

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Authors: Tomoaki Hashimoto

Abstract:

Recently, feedback control systems using random dither quantizers have been proposed for linear discrete-time systems. However, the constraints imposed on state and control variables have not yet been taken into account for the design of feedback control systems with random dither quantization. Model predictive control is a kind of optimal feedback control in which control performance over a finite future is optimized with a performance index that has a moving initial and terminal time. An important advantage of model predictive control is its ability to handle constraints imposed on state and control variables. Based on the model predictive control approach, the objective of this paper is to present a control method that satisfies probabilistic state constraints for linear discrete-time feedback control systems with random dither quantization. In other words, this paper provides a method for solving the optimal control problems subject to probabilistic state constraints for linear discrete-time feedback control systems with random dither quantization.

Keywords: optimal control, stochastic systems, random dither, quantization

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Authors: Yi Ming Zhang, Zhan Xi Wang, Hang Chen, Gang Wang

Abstract:

Manual polishing has problems such as high labor intensity, low production efficiency and difficulty in guaranteeing the consistency of polishing quality. It is more and more necessary to replace manual polishing with robot polishing. Polishing force directly affects the quality of polishing, so accurate tracking and control of polishing force is one of the most important conditions for improving the accuracy of robot polishing. The traditional force control strategy is difficult to adapt to the strong coupling of force control and position control during the robot polishing process. Therefore, based on the analysis of force-based impedance control and position-based impedance control, this paper proposed a new type of adaptive controller. Based on force feedback control of active compliance control, the controller can adaptively estimate the stiffness and position of the external environment and eliminate the steady-state force error produced by traditional impedance control. The simulation results of the model shows that the adaptive controller has good adaptability to changing environmental positions and environmental stiffness, and can accurately track and control polishing force.

Keywords: robot polishing, force feedback, impedance control, adaptive control

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Authors: H. Abderrezek, M. N. Harmas

Abstract:

DC-DC converters are widely used as reliable power source for many industrial and military applications, computers and electronic devices. Several control methods were developed for DC-DC converters control mostly with asymptotic convergence. Synergetic control (SC) is a proven robust control approach and will be used here in a so-called terminal scheme to achieve finite time convergence. Lyapunov synthesis is adopted to assure controlled system stability. Furthermore particle swarm optimization (PSO) algorithm, based on an integral time absolute of error (ITAE) criterion will be used to optimize controller parameters. Simulation of terminal synergetic control of a DC-DC converter is carried out for different operating conditions and results are compared to classic synergetic control performance, that which demonstrate the effectiveness and feasibility of the proposed control method.

Keywords: DC-DC converter, PSO, finite time, terminal, synergetic control

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Authors: Goodarz Sabetian, Sajjad Moshfe

Abstract:

Control system for hi-tech industries could be realized generally and deeply by a special document. Vast heavy industries such as power plants with a large number of I/O signals are controlled by a distributed control system (DCS). This system comprises of so many parts from field level to high control level, and junior instrument engineers may be confused by this enormous information. The key document which can solve this problem is “control configuration system diagram” for each type of DCS. This is a road map that covers all of activities respect to control system in each industrial plant and inevitable to be studied by whom corresponded. It plays an important role from designing control system start point until the end; deliver the system to operate. This should be inserted in bid documents, contracts, purchasing specification and used in different periods of project EPC (engineering, procurement, and construction). Separate parts of DCS are categorized here in order of importance and a brief description and some practical plan is offered. This article could be useful for all instrument and control engineers who worked is EPC projects.

Keywords: control, configuration, DCS, power plant, bus

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Authors: Inuwa Abdu Ibrahim

Abstract:

Public officials are required to posses certain values to adequately protect public interest, by being leaders that are servants of the people. The reality in Nigeria is that leaders rule as masters of the people rather than servants. The paper looked at control and control systems of administration in Nigeria, its resultant consequences and ways of achieving true control of administrators and administration. Secondary source of data was adopted for the research. It concludes that the keys to administrative efficiency and effectiveness through control are implementation of the already existing procedures and laws, as well as commitment on the part of public officials.

Keywords: Accountability, Fraud, Administration, Nigeria

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Authors: Deepak Choudhary

Abstract:

COEP-Microgrid, a project by the students of College of Engineering Pune aims at establishing a micro grid in the college campus serving as a living laboratory for research and development of novel grid technologies. Proposed micro grid has an AC-bus and DC-bus, interconnected together with a tie line DC-AC converter. In grid-connected mode AC bus of microgrid is synchronized with utility grid. Synchronization with utility grid requires grid and AC bus to have synchronism in frequency, phase sequence and voltage. Power flow requires phase difference between grid and AC bus. Control System is required to effectively regulate power flow between the grid and AC bus. The grid synchronizing control system is composed of frequency and phase control for regulated power flow and voltage control system for reduction of reactive power flow. The control system involves automatic active power flow control. It takes the feedback of DC link Capacitor and changes the power angle accordingly. Control system incorporating voltage, phase and power control was developed for grid-tie inverter. This paper discusses the design, simulation and practical implementation of control system described in various micro grid scenarios.

Keywords: microgrid, Grid-tie inverter, voltage control, automatic power control

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Authors: Syed Irfan Raza Naqvi, Zheng Jiangbin, Ahmad Moshin, Pervez Akhter

Abstract:

Currently, the IoT system depends on a centralized client-servant architecture that causes various scalability and privacy vulnerabilities. Distributed ledger technology (DLT) introduces a set of opportunities for the IoT, which leads to practical ideas for existing components at all levels of existing architectures. Blockchain Technology (BCT) appears to be one approach to solving several IoT problems, like Bitcoin (BTC) and Ethereum, which offer multiple possibilities. Besides, IoTs are resource-constrained devices with insufficient capacity and computational overhead to process blockchain consensus mechanisms; the traditional BCT existing challenge for IoTs is poor scalability, energy efficiency, and transaction fees. IOTA is a distributed ledger based on Direct Acyclic Graph (DAG) that ensures M2M micro-transactions are free of charge. IOTA has the potential to address existing IoT-related difficulties such as infrastructure scalability, privacy and access control mechanisms. We proposed an architecture, SLDBI: A Scalable, lightweight DAG-based Blockchain Design for Intelligent IoT Systems, which adapts the DAG base Tangle and implements a lightweight message data model to address the IoT limitations. It enables the smooth integration of new IoT devices into a variety of apps. SLDBI enables comprehensive access control, energy efficiency, and scalability in IoT ecosystems by utilizing the Masked Authentication Message (MAM) protocol and the IOTA Smart Contract Protocol (ISCP). Furthermore, we suggest proof-of-work (PoW) computation on the full node in an energy-efficient way. Experiments have been carried out to show the capability of a tangle to achieve better scalability while maintaining energy efficiency. The findings show user access control management at granularity levels and ensure scale up to massive networks with thousands of IoT nodes, such as Smart Connected Buildings (SCBDs).

Keywords: blockchain, IOT, direct acyclic graphy, scalability, access control, architecture, smart contract, smart connected buildings

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Authors: Kendouci Khedidja, Mazari Benyounes, Benhadria Mohamed Rachid, Dadi Rachida

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In recent years, direct torque control (DTC) has become an alternative to the well-known vector control especially for permanent magnet synchronous motor (PMSM). However, it presents a problem of field linkage and torque ripple. In order to solve this problem, the conventional DTC is combined with space vector pulse width modulation (SVPWM). This control theory has achieved great success in the control of PMSM. That has become a hotspot for resolving. The main objective of this paper gives us an introduction of the DTC and SVPWM-DTC control theory of PMSM which has been simulating on each part of the system via Matlab/Simulink based on the mathematical modeling. Moreover, the outcome of the simulation proved that the improved SVPWM- DTC of PMSM has a good dynamic and static performance.

Keywords: PMSM, DTC, SVM, speed control

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Authors: Hossein Kheiri, Bashir Naderi, Mohamad Reza Niknam

Abstract:

In this paper we study the optimal synchronization of chaotic T-system with complete uncertain parameter. Optimal control laws and parameter estimation rules are obtained by using Hamilton-Jacobi-Bellman (HJB) technique and Lyapunov stability theorem. The derived control laws are optimal adaptive control and make the states of drive and response systems asymptotically synchronized. Numerical simulation shows the effectiveness and feasibility of the proposed method.

Keywords: Lyapunov stability, synchronization, chaos, optimal control, adaptive control

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