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Articles 931 - 960 of 1304
Full-Text Articles in Electrical and Computer Engineering
Active Appearance Model Algorithm With K-Nearest Neighbor Classifier For Face Pose Estimation, Bing-Fei Wu, Chih-Chung Kao, Cheng-Lung Jen, Chia-Rong Chiang, Po-Hung Lai
Active Appearance Model Algorithm With K-Nearest Neighbor Classifier For Face Pose Estimation, Bing-Fei Wu, Chih-Chung Kao, Cheng-Lung Jen, Chia-Rong Chiang, Po-Hung Lai
Journal of Marine Science and Technology–Taiwan
In this paper, a face pose estimation (FPE) algorithm using active appearance model (AAM) with a k-nearest neighbor (kNN) classifier is presented. AAM is a model-based image representation method used to describe non-rigid visual objects, with both shape and texture variations, using a mean vector and linear combinations of a set of variation modes. Since AAM is a deformable model, it has several variations. Owing to the variations, the model is adjusted to the input test face image using iterative searching and fitting. The error, which measures the difference between the model and a test image, is minimized with the …
Design Of A Chart-Based Radar Simulator Augmented With A Multi-Agent Ship Traffic Simulation Platform, Shwu-Jing Chang, Yu-Ting Lai, Chun-Mai Hsu
Design Of A Chart-Based Radar Simulator Augmented With A Multi-Agent Ship Traffic Simulation Platform, Shwu-Jing Chang, Yu-Ting Lai, Chun-Mai Hsu
Journal of Marine Science and Technology–Taiwan
This paper presents a radar simulation system developed under a request to deliver a radar and an Electronic Chart Display and Information System (ECDIS) for the navigation bridge of a ship maneuvering simulator. It is designed to dynamically generate radar image based on the electronic navigational chart database and sensor information to provide flexibility in navigation area for exercise and maintain consistency with other sensors or systems of the bridge, especially the ECDIS. Since the successful delivery of our study, the radar simulation system has been augmented with a Multi-Agent Ship Traffic (MAST) simulation platform and evolved into a versatile …
Linear State Feedback Design For Surge Control Of Axial Flow Compressor Dynamics, Der-Cherng Liaw, Yun-Hua Huang, Wen-Ching Chung
Linear State Feedback Design For Surge Control Of Axial Flow Compressor Dynamics, Der-Cherng Liaw, Yun-Hua Huang, Wen-Ching Chung
Journal of Marine Science and Technology–Taiwan
Based on Moore and Greitzer’s model [15], the surge behavior of axial flow compression system was found to be attributed to the appearance of Andronov-Hopf bifurcationin [12]. In this paper, we focus on the design of control law to prevent and/or delay the occurrence of the bifurcations, which will then suppress the surge behavior. Both types of linear dynamic state feedback and linear washout filter controllers are proposed. The proposed washout design does not require the explicit knowledge of the operating point and keep the original equilibrium characteristic while the dynamic feedback design will change the equilibrium operating points for …
Dynamic Output Feedback Sliding Mode Controller Design For Chattering Avoidance, Jeang-Lin Chang, Jui-Che Tsai
Dynamic Output Feedback Sliding Mode Controller Design For Chattering Avoidance, Jeang-Lin Chang, Jui-Che Tsai
Journal of Marine Science and Technology–Taiwan
For a class of linear MIMO uncertain systems, a dynamic output feedback sliding mode control algorithm that avoids the chattering problem and high gain control is proposed in this paper. Without using any differentiator, we develop a modified asymptotically stable second order sliding mode control law in which the developed controller can guarantee the finite time convergence to the sliding mode and the system states asymptotically approach to zero. Finally, a numerical example is explained for demonstrating the applicability of the proposed scheme.
Development Of A Strapdown Inertial Navigation System Simulation Platform, Dah-Jing Jwo, Jyun-Han Shih, Chia-Sheng Hsu, Kai-Lun Yu
Development Of A Strapdown Inertial Navigation System Simulation Platform, Dah-Jing Jwo, Jyun-Han Shih, Chia-Sheng Hsu, Kai-Lun Yu
Journal of Marine Science and Technology–Taiwan
This paper presents a strapdown inertial navigation system (SINS) simulation platform. The numerical simulator solves the navigation equations in navigation frame where the quaternion vector is employed for attitude calculation. The measured quantities (angular body rates and specific force) are generated by the numerical simulator. The results are also confirmed with the navigation solutions given by a commercial inertial navigation system toolbox using the generated body angular rates and specific force of the SINS. The paper can provide the readers with further information for developing their own SINS simulation tools.
Wavelet Support Vector Machine For Face Recognition, Chen-Chia Chuang, Ping-Lun Liao, Jin-Tsong Jeng
Wavelet Support Vector Machine For Face Recognition, Chen-Chia Chuang, Ping-Lun Liao, Jin-Tsong Jeng
Journal of Marine Science and Technology–Taiwan
In this paper a tool system with wavelet support vector machine (WSVM) under dimension reduction for face recognition is proposed. Eigenfaces and fisherfaces are the major methods used to reduce the dimension of face images in the proposed face recognition tool system. At the same time, noise interference image cases, namely Gaussian noise, with no ears and no eyes are also considered in this paper. Finally, the implementation of proposed system is based on objectoriented design software; the dimension reduction and wavelet kernel support vector machine are applied to the proposed tool system that can deal with face recognition problems …
Mimo Transmission In Spatially Correlated Fading, Joy Iong-Zon Chen, Chi-Tsung Huang, Yu-Hsing Chien
Mimo Transmission In Spatially Correlated Fading, Joy Iong-Zon Chen, Chi-Tsung Huang, Yu-Hsing Chien
Journal of Marine Science and Technology–Taiwan
A multiple-input, multiple-output (MIMO) system transmission over a channel characterized by spatially correlated fading is addressed in this article. Two scenarios that involved using well-known statistical fading distributions, correlatedRician and correlated-Nakagami-m fading, were deployed in analysis. The moment-generating function obtained using a stochastic process was applied to determine the ergodic, or average, channel capacity, thus by passing the difficulty traditionally associated with calculating the probability density function of the signal-to-noise ratio. The results of several numerical and Monte-Carlo comparisons are offered for validating the accuracy of the theoretically derived formulae. Furthermore, many plots were designed by combining different numbers of …
Global Localization Using Dead-Reckoning And Kinect Sensors For Robots With Omnidirectional Mecanum Wheels, Ching-Chih Tsai, Feng-Chun Tai, Yu-Cheng Wang
Global Localization Using Dead-Reckoning And Kinect Sensors For Robots With Omnidirectional Mecanum Wheels, Ching-Chih Tsai, Feng-Chun Tai, Yu-Cheng Wang
Journal of Marine Science and Technology–Taiwan
This paper presents a low-cost global pose localization method using least-squares method and extended Kalman filter (EKF) for an anthropomorphous dual-arm mobile robot (ADAMR) driven by omnidirectional Mecanum wheels in indoor environments. This method is developed by fusing measurements from the KINECT sensor and four encoders mounted on the omnidirectional Mecanum wheels. The KINECT sensor is used to recognize landmarks in the working environments of the robot, and then obtain the azimuth angles and distances between these landmarks and the robot. Based on these measured information, a static global pose initialization algorithm using least-squares method is applied to estimate both …
A Particle Swarm Optimization-Like Algorithm For Constrained Minimal Spanning Tree Problems, Chun-Chao Yeh, Ying-Che Chien
A Particle Swarm Optimization-Like Algorithm For Constrained Minimal Spanning Tree Problems, Chun-Chao Yeh, Ying-Che Chien
Journal of Marine Science and Technology–Taiwan
Previous studies have discussed various constrained minimal spanning tree (MST) problems. In this paper, we propose an efficient algorithm for solving a class of constrained MST problems. The proposed PSO (Particle Swarm Optimization)- like strategy for solving constrained MST problems identifies optimal MSTs under degree and delay constraints. The solution quality and computation time of the proposed PLCMST (PSO-Like algorithm for Constrained MST problems) algorithm is compared with two other algorithms: one based on ant colony optimization, and the other based on a genetic algorithm strategy. Our experimental results show that the PLCMST outperforms the other two approaches, particularly when …
A Wheeled Mobile Robot Path-Tracking System Based On Image Processing And Adaptive Cmac, Jih-Gau Juang, Ko-Jui Hsu, Chih-Min Lin
A Wheeled Mobile Robot Path-Tracking System Based On Image Processing And Adaptive Cmac, Jih-Gau Juang, Ko-Jui Hsu, Chih-Min Lin
Journal of Marine Science and Technology–Taiwan
This study entailed applying image-processing techniques and an intelligent control method to a wheeled mobile robot (WMR) for real-time trajectory recognition and tracking. The WMR was fitted with a charge-coupled device (CCD) camera to capture images of the environment, and the hue-saturationintensity color space was used to classify the color features of the captured images. The WMR calculates the relative position of a target object by using image-processing and edgedetection algorithms. This paper proposes a cerebellar model articulation controller (CMAC) and adaptive CMAC to replace conventional proportional-integral-derivative (PID) controller for guiding the WMR to track a desired path. Moreover, using …
The Development Of A Cost-Effective And Optimized Dp-Spiv System For Fluid Turbulence Researches, Yu-Chi Chang, Yi-Hung Chang, Yaw-Huei Lee, Yi-Chih Chow
The Development Of A Cost-Effective And Optimized Dp-Spiv System For Fluid Turbulence Researches, Yu-Chi Chang, Yi-Hung Chang, Yaw-Huei Lee, Yi-Chih Chow
Journal of Marine Science and Technology–Taiwan
Turbulent flows and issues of turbulence modeling have long been intriguing and important in fluid researches and practices. The mean or instantaneous velocity gradient tensors with six independent components, or other tensorial variables involving gradients, play a key role in turbulence dynamics and models. The advancement of the knowledge of these parameters relies on measurement systems built on sophisticated flow diagnostic techniques that enable both detailed and complete measurements. However, such systems seem inevitably complicated to operate and costly to build and maintain. Therefore, we develop and optimize a dual-plane stereoscopic particle image velocimetry (DP-SPIV) system that is much more …
Numerical Modeling Of Wave-Induced Rotations Of A Bottom-Hinged Flapper With A Sph Model, Da-Wei Chen, Shiaw-Yih Tzang, Chih-Min Hsieh, Yi-Chih Chow, Jiahn-Horng Chen, Chen-Chou Lin, Robert Rong-Jiann Hwang
Numerical Modeling Of Wave-Induced Rotations Of A Bottom-Hinged Flapper With A Sph Model, Da-Wei Chen, Shiaw-Yih Tzang, Chih-Min Hsieh, Yi-Chih Chow, Jiahn-Horng Chen, Chen-Chou Lin, Robert Rong-Jiann Hwang
Journal of Marine Science and Technology–Taiwan
In this study, a Smoothed Particle Hydrodynamics model for simulating wave-induced rotations of a bottom-hinged flapper was established in a 2-D numerical wave flume. The simulated rotating angles illustrated that the flapper could swing back and forth following harmonic wave loadings. The simulations were also seen to be in good agreement with experimental data, confirming the applicability of the present numerical model. The simulated hydrodynamic behaviors at different phases showed that the flapper moved downstream under the wave crest and upstream under the wave trough following the elliptical form of water particle trajectory. The energy conversions of a flapper during …
Application Of Adaptive Variable Speed Back-Stepping Sliding Mode Controller For Pmlsm Position Control, Mei-Yung Chen, Jian-Shiun Lu
Application Of Adaptive Variable Speed Back-Stepping Sliding Mode Controller For Pmlsm Position Control, Mei-Yung Chen, Jian-Shiun Lu
Journal of Marine Science and Technology–Taiwan
In this research, an adaptive variable speed back-stepping sliding mode controller (AVSBSMC) is proposed to control the permanent magnet linear synchronous motor (PMLSM) on high-precision positioning servo system platform. This controller combines the dynamic features of variable speed control (VSC) and the positioning characteristics of adaptive back-stepping sliding mode control (ABSMC). Firstly, the detailed dynamic machine model of PMLSM is established. Secondly, the proportional-integral-derivative (PID) and ABSMC controllers are proposed to compensate the uncertainties in position control. Then, the one provides the better servo performances is the position controller in the motor system. Thirdly, the variable speed controller (VSC) and …
Real Time Swim Instructor, Konrad Antoniuk
Real Time Swim Instructor, Konrad Antoniuk
Electrical Engineering
Both competitive and recreational swimmers want an efficient swim stroke but often lack consistent feedback with technical insight. Competitive swimmers rely on coaches for instruction but have to stop swimming to listen. The Real Time Swim Instructor project provides live feedback to a swimmer so they can feel their stroke correction while they’re swimming. The developing swimmer feels the difference between correct and incorrect form instantaneously.
The project takes on the responsibility of measuring the moving body, accurately determining stroke improvement, functioning under water, and providing recognizable feedback. Additionally, the user can review their swim from data stored in memory. …
Automated Foosball Table, Jim R. Stefani, Alex J. Herpy, Brett Gordon Jaeger, Kevin S. Haydon, Derek Alan Hamel
Automated Foosball Table, Jim R. Stefani, Alex J. Herpy, Brett Gordon Jaeger, Kevin S. Haydon, Derek Alan Hamel
Mechanical Engineering
This project is the second iteration of an automated foosball table for Yaskawa America as a trade show display. The table is meant to provide an interactive experience which highlights the speed and precision of the Yaskawa hardware. The first iteration of the project was mainly focused on creating the physical hardware for the system and to begin the basic programming for the system. This phase of the project was focused on finalizing the physical hardware of the system, implementing the vision system and to continue the basic programing of the system AI. A third team will be assigned to …
Variable Precision Tandem Analog-To-Digital Converter (Adc), Colton A. Parsons
Variable Precision Tandem Analog-To-Digital Converter (Adc), Colton A. Parsons
Master's Theses
This paper describes an analog-to-digital signal converter which varies its precision as a function of input slew rate (maximum signal rate of change), in order to best follow the input in real time. It uses Flash and Successive Approximation (SAR) conversion techniques in sequence.
As part of the design, the concept of "total real-time optimization" is explored, where any delay at all is treated as an error (Error = Delay * Signal Slew Rate). This error metric is proposed for use in digital control systems. The ADC uses a 4-bit Flash converter in tandem with SAR logic that has variable …
Automated Pet Food Dispenser, Kevin M. Shibley
Automated Pet Food Dispenser, Kevin M. Shibley
Electrical Engineering
Electronics revolutionize the world and simplify life. The Automated Pet Food Dispenser (APFD) revolutionizes the pet industry and provides unprecedented food capacity and delivery capability, giving pet owners worry-free vacationing. Sensors and timers optimize functionality and reduce waste while improving reliability and usability. User-defined settings and customization gives the device adaptability to meet every customer’s needs.
Non-Linear Electromechanical System Dynamics, Shathiyakkumar Ganapathy Annadurai
Non-Linear Electromechanical System Dynamics, Shathiyakkumar Ganapathy Annadurai
LSU New Orleans Theses and Dissertations
Electromechanical systems dynamics analysis is approached through nonlinear differential equations and further creating a state space model for the system. There are three modules analyzed and validated, first module consists two magnet coupled with a mass spring damper system as a band-pass system, Low-pass equivalent system and Low-pass equivalent system through perturbation analysis. Initially Band Pass frameworks for the systems are formulated considering the relation between the mechanical forcing and current. Using Mathematical tools such as Hilbert transforms, Low-Pass equivalent of the systems are derived. The state equations of the systems are then used to design a working model in …
Implementation Of Prediction Algorithms To Reduce Latency In Control Of Switching Converters, Brent Bell
Implementation Of Prediction Algorithms To Reduce Latency In Control Of Switching Converters, Brent Bell
Electrical Engineering Undergraduate Honors Theses
Using pulse-width modulation (PWM) to control switching converters such as a dual active bridge introduces latency into a system [1]. Latency in a switching converter decreases the effective bandwidth that the converter can be switched at and for the case of a closed loop control system decreases the effective closed loop system stability. This delay can be attributed to sampling of the control signal and calculation time for the PWM. If a prediction scheme is implemented in this calculation, the next switching time could be predicted to remove any latency. There are effectively two steps in order to achieve this …
An Adaptive Kalman Filter For Voltage Sag Detection In Power Systems, Hisham Odeh Alrawashdeh
An Adaptive Kalman Filter For Voltage Sag Detection In Power Systems, Hisham Odeh Alrawashdeh
Dissertations
The exact values of the noise covariance matrix of the Kalman filter state vector Q and the measured signal noise covariance matrix R must be obtained in order to achieve the optimal performance of the Kalman filter. There have been many techniques and assumptions made to evaluate and compute Q and R. The effects of Q and R are investigated in detail in this dissertation. Based on these investigations, the Kalman filter has been modeled to detect the fundamental signal amplitude variations of power system signals. This technique helps in evaluating voltage sags in power systems.
Two algorithms are …
A Fast Algorithm For The Inversion Of Quasiseparable Vandermonde-Like Matrices, Sirani M. Perera, Grigory Bonik, Vadim Olshevsky
A Fast Algorithm For The Inversion Of Quasiseparable Vandermonde-Like Matrices, Sirani M. Perera, Grigory Bonik, Vadim Olshevsky
Publications
The results on Vandermonde-like matrices were introduced as a generalization of polynomial Vandermonde matrices, and the displacement structure of these matrices was used to derive an inversion formula. In this paper we first present a fast Gaussian elimination algorithm for the polynomial Vandermonde-like matrices. Later we use the said algorithm to derive fast inversion algorithms for quasiseparable, semiseparable and well-free Vandermonde-like matrices having O(n2) complexity. To do so we identify structures of displacement operators in terms of generators and the recurrence relations(2-term and 3-term) between the columns of the basis transformation matrices for quasiseparable, semiseparable and well-free polynomials. Finally we …
Applying Spiking Neural Network Simulation To Neuromodulatory Autonomous Robot Control, Cameron Muhammad
Applying Spiking Neural Network Simulation To Neuromodulatory Autonomous Robot Control, Cameron Muhammad
Phi Kappa Phi Research Symposium (Archived)
In this paper, simulation of the brain based on an artificial spiking neuron model is used to create a self-learning algorithm. The spiking neuron simulation is used to demonstrate a neuromodulation program in which the reward seeking properties of dopamine, the risk-adverse effects of serotonin, and the attention-focusing effects of the cholinergic and noradrenergic systems are applied to a mobile robotic platform as it moves autonomously throughout an environment. External stimuli is recorded by the program as spiking “events” that result in corresponding amounts of dopamine and serotonin influenced spiking patterns. These spiking patterns affect how the robot adapts to …
Filtered-Dynamic-Inversion Control For Fixed-Wing Unmanned Aerial Systems, Jon Mullen
Filtered-Dynamic-Inversion Control For Fixed-Wing Unmanned Aerial Systems, Jon Mullen
Theses and Dissertations--Mechanical and Aerospace Engineering
Instrumented umanned aerial vehicles represent a new way of measuring turbulence in the atmospheric boundary layer. However, autonomous measurements require control methods with disturbance-rejection and altitude command-following capabilities. Filtered dynamic inversion is a control method with desirable disturbance-rejection and command-following properties, and this controller requires limited model information. We implement filtered dynamic inversion as the pitch controller in an altitude-hold autopilot. We design and numerically simulate the continuous-time and discrete-time filtered-dynamic-inversion controllers with anti-windup on a nonlinear aircraft model. Finally, we present results from a flight experiment comparing the filtered-dynamic-inversion controller to a classical proportional-integral controller. The experimental results show …
Model Analysis And Predictive Control Of Double Electrode Submerged Arc Welding Process For Fillet Joints With Root Opening, Yi Lu
Theses and Dissertations--Electrical and Computer Engineering
Submerged Arc Welding (SAW) for fillet joints is one of the major applications in the shipbuilding industry. Due to the requirement for the weld size, a sufficient amount of metal must be deposited. In conventional SAW process, the heat input is proportional to the amount of metal melted and is thus determined by the required weld size. To meet this requirement, an excessive amount of heat is applied causing large distortions on the welded structures whose follow-up straightening is highly costly. In order to reduce the needed heat input, Double-Electrode (DE) technology has been practiced creating the Double-Electrode SAW (DE-SAW) …
Multi-Domain, Multi-Objective-Optimization-Based Approach To The Design Of Controllers For Power Electronics, Jing Shang
Multi-Domain, Multi-Objective-Optimization-Based Approach To The Design Of Controllers For Power Electronics, Jing Shang
Theses and Dissertations--Electrical and Computer Engineering
Power converter has played a very important role in modern electric power systems. The control of power converters is necessary to achieve high performance. In this study, a dc-dc buck converter is studied. The parameters of a notional proportional-integral controller are to be selected. Genetic algorithms (GAs), which have been widely used to solve multi-objective optimization problems, is used in order to locate appropriate controller design. The control metrics are specified as phase margin in frequency domain and voltage error in time-domain. GAs presented the optimal tradeoffs between these two objectives. Three candidate control designs are studied in simulation and …
Virtualized Welding Based Learning Of Human Welder Behaviors For Intelligent Robotic Welding, Yukang Liu
Virtualized Welding Based Learning Of Human Welder Behaviors For Intelligent Robotic Welding, Yukang Liu
Theses and Dissertations--Electrical and Computer Engineering
Combining human welder (with intelligence and sensing versatility) and automated welding robots (with precision and consistency) can lead to next generation intelligent welding systems. In this dissertation intelligent welding robots are developed by process modeling / control method and learning the human welder behavior.
Weld penetration and 3D weld pool surface are first accurately controlled for an automated Gas Tungsten Arc Welding (GTAW) machine. Closed-form model predictive control (MPC) algorithm is derived for real-time welding applications. Skilled welder response to 3D weld pool surface by adjusting the welding current is then modeled using Adaptive Neuro-Fuzzy Inference System (ANFIS), and compared …
Measurement And Modeling Of Humidity Sensors, Jingbo Tong
Measurement And Modeling Of Humidity Sensors, Jingbo Tong
Theses and Dissertations--Electrical and Computer Engineering
Humidity measurement has been increasingly important in many industries and process control applications. This thesis research focus mainly on humidity sensor calibration and characterization. The humidity sensor instrumentation is briefly described. The testing infrastructure was designed for sensor data acquisition, in order to compensate the humidity sensor’s temperature coefficient, temperature chambers using Peltier elements are used to achieve easy-controllable stable temperatures. The sensor characterization falls into a multivariate interpolation problem. Neuron networks is tried for non-linear data fitting, but in the circumstance of limited training data, an innovative algorithm was developed to utilize shape preserving polynomials in multiple planes in …
Multifrequency Averaging In Power Electronic Systems, Fei Pan
Multifrequency Averaging In Power Electronic Systems, Fei Pan
Theses and Dissertations--Electrical and Computer Engineering
Power electronic systems have been widely used in the electrical power processing for applications with power levels ranging from less than one watt in battery-operated portable devices to more than megawatts in the converters, inverters and rectifiers of the utility power systems. These systems typically involve the passive elements such as inductors, capacitors, and resistors, the switching electronic components such as IGBTs, MOSFETS, and diodes, and other electronic circuits. Multifrequency averaging is one of the widely used modeling and simulation techniques today for the analysis and design of power electronic systems. This technique is capable of providing the average behavior …
Neuromodulation Based Control Of Autonomous Robots On A Cloud Computing Platform, Cameron Muhammad
Neuromodulation Based Control Of Autonomous Robots On A Cloud Computing Platform, Cameron Muhammad
College of Graduate Studies: Theses & Dissertations
In recent years, the advancement of neurobiologically plausible models and computer networking has resulted in new ways of implementing control systems on robotic platforms. The work presents a control approach based on vertebrate neuromodulation and its implementation on autonomous robots in the open-source, open-access environment of robot operating system (ROS). A spiking neural network (SNN) is used to model the neuromodulatory function for generating context based behavioral responses of the robots to sensory input signals. The neural network incorporates three types of neurons- cholinergic and noradrenergic (ACh/NE) neurons for attention focusing and action selection, dopaminergic (DA) neurons for rewards- and …
Identifiability Of Additive Actuator And Sensor Faults By State Augmentation, Suresh M. Joshi, Oscar R. Gonzalez, Jason M. Upchurch
Identifiability Of Additive Actuator And Sensor Faults By State Augmentation, Suresh M. Joshi, Oscar R. Gonzalez, Jason M. Upchurch
Electrical & Computer Engineering Faculty Publications
A class of FDI (fault detection and identification) methods for bias-type actuator and sensor faults was explored from the point of view of fault identifiability. The methods use banks of Kalman-Bucy filters (KBFs) to detect faults, determine the fault pattern, and estimate the fault values. A complete characterization of conditions for identifiability of bias-type actuator faults, sensor faults, and simultaneous actuator and sensor faults was presented. It was shown that FDI of simultaneous actuator and sensor faults is not possible using these methods when all sensors have unknown biases. The fault identifiability conditions were demonstrated via numerical examples. The analytical …