The problem of controlling the load speed of a mechanical transmission system consisting of a belt- pulley and gear-pair is considered. The system is modeled as two inertia (motor and load) connected by a compliant transmission. If the transmission is assumed to be rigid, then using either the motor or load speed feedback provides the same result. However, with transmission compliance, due to belts or long shafts, the stability characteristics and performance of the closed-loop system are quite different when either motor or load speed feedback is employed. We investigate motor and load speed feedback schemes by utilizing the singular perturbation method. We propose and discuss a control scheme that utilizes both motor and load speed feedback, and design an adaptive feedforward action to reject load torque disturbances. The control algorithms are implemented on an experimental platform that is typically used in roll-to-roll manufacturing and results are shown and discussed.
Stability Control System for a Two-WheelerIOSRJEEE
A two-wheeler is statically unstable but as the speed increases vehicle achieves stability. At low speed, the vehicle loses its stability. In order to achieve stability, the driver has to balance the vehicle. While negotiating a curve, a vehicle has to lean to a certain angle, if this angle exceeds the certain value, the vehicle tends to skid. In this paper the stability control system is incorporated, so that a vehicle will maintain stability even at low speeds. The stability of a two-wheeler depends on weight distribution, tyre dynamics, speed and steering angle. In this paper, only two parameters are considered, one is steering effect and another one is speed. For developing a simplified model, the speed of the vehicle is kept as constant, using which the effect of steering angle is analysed and accordingly a controller is incorporated for providing stability.
This paper proposes a long-stroke linear switched reluctance machine (LSRM) with a primary and a secondary translator for industrial conveyance applications. The secondary one can translate according to the primary one so that linear compound motions can be achieved. Considering the fact that either one translator imposes a time-variant, nonlinear disturbance onto the other, the self-tuning position controllers are implemented for the compound machine and experimental results demonstrate that the absolute steady-state error values can fall into 0.03 mm and 0.05 mm for the secondary and primary translator, respectively. A composite absolute precision of less than 0.6 mm can be achieved under the proposed control strategy.
The active suspension system with hydraulic actuator for half car model analy...eSAT Publishing House
This document describes the design and simulation of an active suspension system with a hydraulic actuator for a half car model. A PID controller is designed and tuned using three different methods - heuristic tuning, Ziegler-Nichols tuning, and iterative learning algorithm tuning. The half car model and hydraulic actuator are modeled and simulated in MATLAB Simulink. Simulation results show that the PID controller tuned with the iterative learning algorithm provides the best ride quality performance compared to the other tuning methods or a passive suspension, reducing the body displacement under various road disturbances.
PNEUMATIC VEHICLE ACTIVE SUSPENSION SYSTEM USING PID CONTROLLERTushar Tambe
The slide contains the simulation of pneumatic active suspension behavior on different road surface. These results shows the active suspension with controllers works effectively,if feedback loop is provided.
Simulation of an Active Suspension Using PID ControlSuzana Avila
The document simulates an active vehicle suspension using PID control. It builds a quarter car model and analyzes the controllability and observability. Numerical simulations show the active suspension with PID controller improves performance over the passive suspension by reducing displacement peaks and settling times for step, harmonic, and noise road profiles. The active suspension improves comfort but has higher manufacturing costs than passive suspensions.
Mathematical Modeling and Simulation of Two Degree of Freedom Quarter Car Modelijsrd.com
The proposed study is to develop an active suspension system to increase the comfort for the passenger by reducing the body acceleration. The dynamic quarter car suspension system is considered for mathematical modelling and simulation is carried using MATLAB SIMULINK. The present suspension system is controlled by Proportional- Integral -Derivative controller. The system performance is analysed using the single speed bump road surface and the effectiveness is evaluated with active and passive controlled systems.
Hardware-in-the-loop based comparative analysis of speed controllers for a tw...journalBEEI
A comparative study of speed control performance of an induction motor drive system connecting to a load via a non-rigid shaft. The nonrigidity of the coupling is represented by stiffness and damping coefficients deteriorating speed regulating operations of the system and can be regarded as a two-mass system. In the paper, the ability of flatness based and backstepping controls in control the two-mass system is verified through comprehensive hardware-in-the-loop experiments and with the assumption of ideal stator current loop performance. Step-by-step control design procedures are given, in addition, system responses with classical PID control are also provided for parallel comparisons.
The formula cars need high tire grip on racing challenge by reducing rolling displacement at corner or
double change lands. In this case study, the paper clarifies some issues related to suspension system with
inerter to reduce displacement and rolling angle under impact from road disturbance on Formula SAE
Car. We propose some new designs, which have an advance for suspension system by improving dynamics.
We optimize design of model based on the minimization of cost functions for roll dynamics, by reducing the
displacement transfer and the energy consumed by the inerter. Base on a passive suspension model that we
carried out quarter-car and half-car model for different parameters which show the benefit of the inerter.
The important advantage of the proposed solution is its integration a new mechanism, the inerter, this
system can increase advance in development and have effects on the vehicle dynamics in stability vehicle.
Stability Control System for a Two-WheelerIOSRJEEE
A two-wheeler is statically unstable but as the speed increases vehicle achieves stability. At low speed, the vehicle loses its stability. In order to achieve stability, the driver has to balance the vehicle. While negotiating a curve, a vehicle has to lean to a certain angle, if this angle exceeds the certain value, the vehicle tends to skid. In this paper the stability control system is incorporated, so that a vehicle will maintain stability even at low speeds. The stability of a two-wheeler depends on weight distribution, tyre dynamics, speed and steering angle. In this paper, only two parameters are considered, one is steering effect and another one is speed. For developing a simplified model, the speed of the vehicle is kept as constant, using which the effect of steering angle is analysed and accordingly a controller is incorporated for providing stability.
This paper proposes a long-stroke linear switched reluctance machine (LSRM) with a primary and a secondary translator for industrial conveyance applications. The secondary one can translate according to the primary one so that linear compound motions can be achieved. Considering the fact that either one translator imposes a time-variant, nonlinear disturbance onto the other, the self-tuning position controllers are implemented for the compound machine and experimental results demonstrate that the absolute steady-state error values can fall into 0.03 mm and 0.05 mm for the secondary and primary translator, respectively. A composite absolute precision of less than 0.6 mm can be achieved under the proposed control strategy.
The active suspension system with hydraulic actuator for half car model analy...eSAT Publishing House
This document describes the design and simulation of an active suspension system with a hydraulic actuator for a half car model. A PID controller is designed and tuned using three different methods - heuristic tuning, Ziegler-Nichols tuning, and iterative learning algorithm tuning. The half car model and hydraulic actuator are modeled and simulated in MATLAB Simulink. Simulation results show that the PID controller tuned with the iterative learning algorithm provides the best ride quality performance compared to the other tuning methods or a passive suspension, reducing the body displacement under various road disturbances.
PNEUMATIC VEHICLE ACTIVE SUSPENSION SYSTEM USING PID CONTROLLERTushar Tambe
The slide contains the simulation of pneumatic active suspension behavior on different road surface. These results shows the active suspension with controllers works effectively,if feedback loop is provided.
Simulation of an Active Suspension Using PID ControlSuzana Avila
The document simulates an active vehicle suspension using PID control. It builds a quarter car model and analyzes the controllability and observability. Numerical simulations show the active suspension with PID controller improves performance over the passive suspension by reducing displacement peaks and settling times for step, harmonic, and noise road profiles. The active suspension improves comfort but has higher manufacturing costs than passive suspensions.
Mathematical Modeling and Simulation of Two Degree of Freedom Quarter Car Modelijsrd.com
The proposed study is to develop an active suspension system to increase the comfort for the passenger by reducing the body acceleration. The dynamic quarter car suspension system is considered for mathematical modelling and simulation is carried using MATLAB SIMULINK. The present suspension system is controlled by Proportional- Integral -Derivative controller. The system performance is analysed using the single speed bump road surface and the effectiveness is evaluated with active and passive controlled systems.
Hardware-in-the-loop based comparative analysis of speed controllers for a tw...journalBEEI
A comparative study of speed control performance of an induction motor drive system connecting to a load via a non-rigid shaft. The nonrigidity of the coupling is represented by stiffness and damping coefficients deteriorating speed regulating operations of the system and can be regarded as a two-mass system. In the paper, the ability of flatness based and backstepping controls in control the two-mass system is verified through comprehensive hardware-in-the-loop experiments and with the assumption of ideal stator current loop performance. Step-by-step control design procedures are given, in addition, system responses with classical PID control are also provided for parallel comparisons.
The formula cars need high tire grip on racing challenge by reducing rolling displacement at corner or
double change lands. In this case study, the paper clarifies some issues related to suspension system with
inerter to reduce displacement and rolling angle under impact from road disturbance on Formula SAE
Car. We propose some new designs, which have an advance for suspension system by improving dynamics.
We optimize design of model based on the minimization of cost functions for roll dynamics, by reducing the
displacement transfer and the energy consumed by the inerter. Base on a passive suspension model that we
carried out quarter-car and half-car model for different parameters which show the benefit of the inerter.
The important advantage of the proposed solution is its integration a new mechanism, the inerter, this
system can increase advance in development and have effects on the vehicle dynamics in stability vehicle.
This presentation introduces the concept of "impact reduction factor" and a new method that are both developed by Dr. Niyazi Özgür Bezgin to estimate vertical impact forces on railways due to track stiffness variations.
controlling the vibration of automobile suspension system using pid controllersiva kumar
This document presents a MATLAB/Simulink project that models and simulates an active automobile suspension system using a PID controller to reduce vibration and improve ride comfort. The author develops a quarter-car model to represent the vehicle dynamics, including sprung mass, suspension components, and disturbance from the road. A Simulink model is created and the PID controller is tuned to actively control the suspension system based on sensor feedback. Simulation results show the integrated system response is improved significantly with the PID controller compared to without it. The goal is to both enhance passenger comfort and increase vehicle stability by minimizing vibration effects on the suspension.
Comparative analysis of observer-based LQR and LMI controllers of an inverted...journalBEEI
An inverted pendulum is a multivariable, unstable, nonlinear system that is used as a yardstick in control engineering laboratories to study, verify and confirm innovative control techniques. To implement a simple control algorithm, achieve upright stabilization and precise tracking control under external disturbances constitutes a serious challenge. Observer-based linear quadratic regulator (LQR) controller and linear matrix inequality (LMI) are proposed for the upright stabilization of the system. Simulation studies are performed using step input magnitude, and the results are analyzed. Time response specifications, integral square error (ISE), integral absolute error (IAE) and mean absolute error (MAE) were employed to investigate the performances of the proposed controllers. Based on the comparative analysis, the upright stabilization of the pendulum was achieved within the shortest possible time with both controllers however, the LMI controller exhibits better performances in both stabilization and robustness. Moreover, the LMI control scheme is effective and simple.
Nicola Galluzzi - Research project presentationNicola Galluzzi
This document describes the development of a program to calculate losses in a pushbelt continuously variable transmission (CVT). It outlines the objectives, methodology, inputs, and models used, which include friction coefficient modeling, transverse belt speed influence, and adhesive friction between belt layers. The results show that accounting for these factors in the new model leads to higher total loss calculations than the traditional model.
The document provides details about the syllabus of a course on Kinematics of Machinery. It is divided into 5 units. Unit I discusses mechanisms, kinematic pairs, degrees of freedom and inversions. Unit II covers velocity and acceleration analysis using graphical and relative velocity methods. Unit III focuses on straight line motion mechanisms. Unit IV discusses cams and cam mechanisms. Unit V is about higher pairs like gears, gear trains, epicyclic gears and their analysis. The document also provides the session planner and question bank for the course.
Quarter model of passive suspension system with simscapeabuamo
The document summarizes key aspects of vehicle suspension systems. It defines a suspension system as using springs and shock absorbers to connect wheels/axles to the vehicle chassis. Suspension systems serve to carry weight, maximize tire traction, provide stability and handling, and ensure passenger comfort by smoothing bumps. Springs absorb shock from bumps by converting it to potential energy, while shock absorbers dissipate shock without causing undue vehicle oscillation. Passive suspensions use traditional springs and dampers, while active suspensions constantly sense the road and adjust components like shock stiffness electronically. Simscape software can be used to model and simulate multi-domain physical systems like vehicle suspensions.
This document presents a mathematical model of a vehicle suspension that was developed as a midterm project. It includes:
1) A quarter-car model to simulate the bounce of one wheel over a pothole or bump.
2) A full-car model combining 4 quarter-car models to simulate the bounce of each tire.
3) An extension of the full-car model to simulate pitching of the vehicle body when encountering road variations.
4) Consideration of rolling of the vehicle body when taking turns. Non-linear springs are proposed to improve the model.
This paper describes the design and the simulation of a non-linear controller for two-mass system using induction motor basing on the backstepping method. The aim is to control the speed actual value of load motor matching with the speed reference load motor, moreover, electrical drive’s respone ensuring the “fast, accurate and small overshoot” and reducing the resonance oscillations for two-mass system using induction motor fed by voltage source inveter with ideally control performance of stator current. Backstepping controller uses the non-linear equations of an induction motor and the linear dynamical equations of two-mass system, the Lyapunov analysis and the errors between the real and the desired values. The controller has been implemented in both simulation and hardware-in-the-loop (HIL) real-time experiments using Typhoon HIL 402 system, when the drive system operates at a stable speed (rotor flux is constant) and greater than rated speed (field weakening area). The simulation and HIL results presented the correctness and effectiveness of the controller is proposed; furthermore, compared to PI method to see the response of the system clearly.
This document discusses modeling and simulation of a semi-active suspension system for automobiles using a PID controller in MATLAB Simulink. It presents a quarter car model of a semi-active suspension system and develops state space equations to model vehicle body displacement, acceleration, wheel deflection, and other variables. The system is simulated in Simulink using PID control. Results show the PID controller improves performance over a passive system by reducing peak overshoots and settling times under both step and random road inputs. The semi-active suspension provides better ride quality and vehicle handling than a conventional passive suspension.
Robust composite nonlinear feedback for nonlinear Steer-by-Wire vehicle’s Yaw...journalBEEI
Yaw control is a part of an Active Front Steering (AFS) system, which is used to improve vehicle manoeuvrability. Previously, it has been reported that the yaw rate tracking performance of a linear Steer-by-Wire (SBW) vehicle equipped with a Composite Nonlinear Feedback (CNF) controller and a Disturbance Observer (DOB) is robust with respect to side wind disturbance effects. This paper presents further investigation regarding the robustness of the combination between a CNF and a DOB in a nonlinear environment through a developed 7-DOF nonlinear SBW vehicle. Moreover, in contrast to previous studies, this paper also contributes in presenting the validation works of the proposed control system in a real-time situation using a Hardware-in-Loop (HIL) platform. Simulation and validation results show that the CNF and DOB managed to reduce the influence of the side wind disturbance in nonlinearities.
This document summarizes an experimental study on load sharing analysis of a planetary gear box. Strain gauges were mounted on the planet pins to measure the load carried by each planet. Testing was done with different input torques both with and without intentional errors introduced to one planet pin. Results showed the actual load sharing varied from the theoretical even load sharing due to manufacturing errors. Load carried by each planet was calculated from the pin stresses measured by strain gauges and FEM analysis. With a 100um error introduced, load on the error planet increased while loads on other planets decreased compared to the no error condition. Experimental and FEM pin stress results closely matched the theoretical values.
Modern pedagogy-techniques-for-dc-motor-speed-controlRajesh Kumar
Based on a survey conducted for second and third-year students of the electrical engineering department at Maharishi Markandeshwar University, India, it was found that around 92% of students felt that it would be better to introduce a virtual environment for laboratory experiments. Hence, a need was felt to perform modern pedagogy techniques for students which consist of a virtual environment using MATLAB/Simulink. In this paper, a virtual environment for the speed control of a DC motor is performed using MATLAB/Simulink. The various speed control methods for the DC motor include the field resistance control method and armature voltage control method. The performance analysis of the DC motor is hence analyzed.
This document proposes a fuzzy sliding mode controller for speed control of induction motors. It begins with an abstract that summarizes the proposed control scheme using fuzzy logic techniques to dynamically control the sliding mode control equivalent action. It then provides background on induction motor modeling, field oriented control, and sliding mode control. The document describes developing a fuzzy sliding mode controller where fuzzy logic controllers replace the inequalities that determine the sliding mode control parameters. Simulation results show the proposed fuzzy sliding mode controller provides good performance, disturbance rejection, and robustness to parameter variations compared to classical sliding mode control.
The document analyzes a quarter-car suspension system using equations of motion, block diagrams, and Simulink modeling. It derives the equation of motion from a free body diagram. Natural frequency and damping coefficients are calculated using the given spring and damping constants. A block diagram models the system and is implemented in Simulink. Simulink plots show the displacement, velocity, and forces over time, validating the dynamic behavior of the suspension system.
Control-Integrated Design by Theoretical Simulation for a Torque-Actuated 6-S...IDES Editor
A design algorithm has been proposed for a Stewart platform with six legs, each having a ball-screw at the middle and powered by a torque motor at the bottom. When a motor shaft rotates, the leg extends or collapses and the axis could rotate about a spherical joint supporting the motor. Consequent actuation from all the legs through a universal joint at the top of each causes the platform to change its pose. The joints at each end lie on the intersection of a pitch circle and a semi-regular hexagon. An inverse model that neglects friction and leg inertia has been employed in a step-by-step simultaneous search to determine the platform height at the neutral and
the radius of the bottom pitch circle within the constraint of permissible joint angle and motor specifications. The proposed control for a basic pose demand involves a feedforward
estimation of motor torque variation, a proportional-derivative feedback and appropriate compensating demand for minimizing unwanted coupled motion. The forward modeling of the pose dynamics and its Simulink implementation have
established the control as satisfactory.
This document provides notes on kinematic mechanisms from a mechanical engineering course. It includes definitions of key terms like kinematic pairs, degrees of freedom, and Grubler's criterion. It discusses different types of kinematic chains including four bar chains, single slider crank chains, and double slider crank chains. It also summarizes inversions of these chains used in common mechanisms like the beam engine, locomotive coupling rod, and reciprocating engine. Examples of mechanisms derived from different inversions are provided like the rotary engine from the single slider crank chain.
This document provides an introduction to kinematics of machines. It defines key terms like kinematics, mechanisms, machines, degrees of freedom, and types of links and motion. It also describes different types of kinematic pairs that constrain relative motion between links, including lower pairs like sliding and revolute pairs, and higher pairs. The main topics covered are the basic concepts, definitions, and analytical tools used in the kinematic analysis of machines.
Active steering for better safety comfortÑero Lopez
This document discusses the history and concepts of active steering systems for vehicles. It describes how early systems from the 1970s and 1980s used proportional feedback of yaw rate to add a corrective steering angle for improved handling. Later systems in the 1980s-1990s introduced four-wheel steering with feedforward and feedback control. The author proposes a concept from 1990 onward that separates the driver's path following task from an automatic yaw stabilization system through integral feedback of yaw rate to front wheel steering. Experimental results on a test vehicle showed improved disturbance rejection. The document outlines the continued development of robust control algorithms and actuator technologies to enable implementation of an active steering system.
Design and Analysis of Mechanism for Dynamic Characterization of Power Transm...iosrjce
Power transmission systems are being widely used for transmission of power between two members.
Once a particular transmission system is realized it needs to be qualified before its course of application. As
part of this intended torque of the transmission systems needs to be measured and tested. Conventional means of
dynamic characterization of power transmission system has got the demerit of energy consumption to a greater
extent. Because of this more effort is to be put in terms of power for the sake of testing the intended system.
Great need exists for a system which consumes less or ideally no energy while performing test. This project
aims at evolution of a novel technique for evaluating the torque transmitting capability of power transmission
systems without consuming more energy. To start with all the subsystems of the proposed design will be
identified and each of them will be designed for getting their dimensions. Then these dimensional models will be
transformed to solid model of the assembled configuration using 3D CAD software. Functional load which will
be experienced by this design will be assessed and structural analysis will be carried out against these loads
using Finite Element Method (FEM) in commercial FEA software i.e. ANSYS
The document describes the design and analysis of a mechanism to dynamically characterize power transmission systems without consuming significant energy. It involves using an electro-mechanical clutch connected to a motor via pulleys of different sizes to measure the torque transmitting capability of a transmission system by gradually increasing the load torque until slip occurs. Structural analysis using FEA found the design to be safe with a factor of safety above the minimum required.
This presentation introduces the concept of "impact reduction factor" and a new method that are both developed by Dr. Niyazi Özgür Bezgin to estimate vertical impact forces on railways due to track stiffness variations.
controlling the vibration of automobile suspension system using pid controllersiva kumar
This document presents a MATLAB/Simulink project that models and simulates an active automobile suspension system using a PID controller to reduce vibration and improve ride comfort. The author develops a quarter-car model to represent the vehicle dynamics, including sprung mass, suspension components, and disturbance from the road. A Simulink model is created and the PID controller is tuned to actively control the suspension system based on sensor feedback. Simulation results show the integrated system response is improved significantly with the PID controller compared to without it. The goal is to both enhance passenger comfort and increase vehicle stability by minimizing vibration effects on the suspension.
Comparative analysis of observer-based LQR and LMI controllers of an inverted...journalBEEI
An inverted pendulum is a multivariable, unstable, nonlinear system that is used as a yardstick in control engineering laboratories to study, verify and confirm innovative control techniques. To implement a simple control algorithm, achieve upright stabilization and precise tracking control under external disturbances constitutes a serious challenge. Observer-based linear quadratic regulator (LQR) controller and linear matrix inequality (LMI) are proposed for the upright stabilization of the system. Simulation studies are performed using step input magnitude, and the results are analyzed. Time response specifications, integral square error (ISE), integral absolute error (IAE) and mean absolute error (MAE) were employed to investigate the performances of the proposed controllers. Based on the comparative analysis, the upright stabilization of the pendulum was achieved within the shortest possible time with both controllers however, the LMI controller exhibits better performances in both stabilization and robustness. Moreover, the LMI control scheme is effective and simple.
Nicola Galluzzi - Research project presentationNicola Galluzzi
This document describes the development of a program to calculate losses in a pushbelt continuously variable transmission (CVT). It outlines the objectives, methodology, inputs, and models used, which include friction coefficient modeling, transverse belt speed influence, and adhesive friction between belt layers. The results show that accounting for these factors in the new model leads to higher total loss calculations than the traditional model.
The document provides details about the syllabus of a course on Kinematics of Machinery. It is divided into 5 units. Unit I discusses mechanisms, kinematic pairs, degrees of freedom and inversions. Unit II covers velocity and acceleration analysis using graphical and relative velocity methods. Unit III focuses on straight line motion mechanisms. Unit IV discusses cams and cam mechanisms. Unit V is about higher pairs like gears, gear trains, epicyclic gears and their analysis. The document also provides the session planner and question bank for the course.
Quarter model of passive suspension system with simscapeabuamo
The document summarizes key aspects of vehicle suspension systems. It defines a suspension system as using springs and shock absorbers to connect wheels/axles to the vehicle chassis. Suspension systems serve to carry weight, maximize tire traction, provide stability and handling, and ensure passenger comfort by smoothing bumps. Springs absorb shock from bumps by converting it to potential energy, while shock absorbers dissipate shock without causing undue vehicle oscillation. Passive suspensions use traditional springs and dampers, while active suspensions constantly sense the road and adjust components like shock stiffness electronically. Simscape software can be used to model and simulate multi-domain physical systems like vehicle suspensions.
This document presents a mathematical model of a vehicle suspension that was developed as a midterm project. It includes:
1) A quarter-car model to simulate the bounce of one wheel over a pothole or bump.
2) A full-car model combining 4 quarter-car models to simulate the bounce of each tire.
3) An extension of the full-car model to simulate pitching of the vehicle body when encountering road variations.
4) Consideration of rolling of the vehicle body when taking turns. Non-linear springs are proposed to improve the model.
This paper describes the design and the simulation of a non-linear controller for two-mass system using induction motor basing on the backstepping method. The aim is to control the speed actual value of load motor matching with the speed reference load motor, moreover, electrical drive’s respone ensuring the “fast, accurate and small overshoot” and reducing the resonance oscillations for two-mass system using induction motor fed by voltage source inveter with ideally control performance of stator current. Backstepping controller uses the non-linear equations of an induction motor and the linear dynamical equations of two-mass system, the Lyapunov analysis and the errors between the real and the desired values. The controller has been implemented in both simulation and hardware-in-the-loop (HIL) real-time experiments using Typhoon HIL 402 system, when the drive system operates at a stable speed (rotor flux is constant) and greater than rated speed (field weakening area). The simulation and HIL results presented the correctness and effectiveness of the controller is proposed; furthermore, compared to PI method to see the response of the system clearly.
This document discusses modeling and simulation of a semi-active suspension system for automobiles using a PID controller in MATLAB Simulink. It presents a quarter car model of a semi-active suspension system and develops state space equations to model vehicle body displacement, acceleration, wheel deflection, and other variables. The system is simulated in Simulink using PID control. Results show the PID controller improves performance over a passive system by reducing peak overshoots and settling times under both step and random road inputs. The semi-active suspension provides better ride quality and vehicle handling than a conventional passive suspension.
Robust composite nonlinear feedback for nonlinear Steer-by-Wire vehicle’s Yaw...journalBEEI
Yaw control is a part of an Active Front Steering (AFS) system, which is used to improve vehicle manoeuvrability. Previously, it has been reported that the yaw rate tracking performance of a linear Steer-by-Wire (SBW) vehicle equipped with a Composite Nonlinear Feedback (CNF) controller and a Disturbance Observer (DOB) is robust with respect to side wind disturbance effects. This paper presents further investigation regarding the robustness of the combination between a CNF and a DOB in a nonlinear environment through a developed 7-DOF nonlinear SBW vehicle. Moreover, in contrast to previous studies, this paper also contributes in presenting the validation works of the proposed control system in a real-time situation using a Hardware-in-Loop (HIL) platform. Simulation and validation results show that the CNF and DOB managed to reduce the influence of the side wind disturbance in nonlinearities.
This document summarizes an experimental study on load sharing analysis of a planetary gear box. Strain gauges were mounted on the planet pins to measure the load carried by each planet. Testing was done with different input torques both with and without intentional errors introduced to one planet pin. Results showed the actual load sharing varied from the theoretical even load sharing due to manufacturing errors. Load carried by each planet was calculated from the pin stresses measured by strain gauges and FEM analysis. With a 100um error introduced, load on the error planet increased while loads on other planets decreased compared to the no error condition. Experimental and FEM pin stress results closely matched the theoretical values.
Modern pedagogy-techniques-for-dc-motor-speed-controlRajesh Kumar
Based on a survey conducted for second and third-year students of the electrical engineering department at Maharishi Markandeshwar University, India, it was found that around 92% of students felt that it would be better to introduce a virtual environment for laboratory experiments. Hence, a need was felt to perform modern pedagogy techniques for students which consist of a virtual environment using MATLAB/Simulink. In this paper, a virtual environment for the speed control of a DC motor is performed using MATLAB/Simulink. The various speed control methods for the DC motor include the field resistance control method and armature voltage control method. The performance analysis of the DC motor is hence analyzed.
This document proposes a fuzzy sliding mode controller for speed control of induction motors. It begins with an abstract that summarizes the proposed control scheme using fuzzy logic techniques to dynamically control the sliding mode control equivalent action. It then provides background on induction motor modeling, field oriented control, and sliding mode control. The document describes developing a fuzzy sliding mode controller where fuzzy logic controllers replace the inequalities that determine the sliding mode control parameters. Simulation results show the proposed fuzzy sliding mode controller provides good performance, disturbance rejection, and robustness to parameter variations compared to classical sliding mode control.
The document analyzes a quarter-car suspension system using equations of motion, block diagrams, and Simulink modeling. It derives the equation of motion from a free body diagram. Natural frequency and damping coefficients are calculated using the given spring and damping constants. A block diagram models the system and is implemented in Simulink. Simulink plots show the displacement, velocity, and forces over time, validating the dynamic behavior of the suspension system.
Control-Integrated Design by Theoretical Simulation for a Torque-Actuated 6-S...IDES Editor
A design algorithm has been proposed for a Stewart platform with six legs, each having a ball-screw at the middle and powered by a torque motor at the bottom. When a motor shaft rotates, the leg extends or collapses and the axis could rotate about a spherical joint supporting the motor. Consequent actuation from all the legs through a universal joint at the top of each causes the platform to change its pose. The joints at each end lie on the intersection of a pitch circle and a semi-regular hexagon. An inverse model that neglects friction and leg inertia has been employed in a step-by-step simultaneous search to determine the platform height at the neutral and
the radius of the bottom pitch circle within the constraint of permissible joint angle and motor specifications. The proposed control for a basic pose demand involves a feedforward
estimation of motor torque variation, a proportional-derivative feedback and appropriate compensating demand for minimizing unwanted coupled motion. The forward modeling of the pose dynamics and its Simulink implementation have
established the control as satisfactory.
This document provides notes on kinematic mechanisms from a mechanical engineering course. It includes definitions of key terms like kinematic pairs, degrees of freedom, and Grubler's criterion. It discusses different types of kinematic chains including four bar chains, single slider crank chains, and double slider crank chains. It also summarizes inversions of these chains used in common mechanisms like the beam engine, locomotive coupling rod, and reciprocating engine. Examples of mechanisms derived from different inversions are provided like the rotary engine from the single slider crank chain.
This document provides an introduction to kinematics of machines. It defines key terms like kinematics, mechanisms, machines, degrees of freedom, and types of links and motion. It also describes different types of kinematic pairs that constrain relative motion between links, including lower pairs like sliding and revolute pairs, and higher pairs. The main topics covered are the basic concepts, definitions, and analytical tools used in the kinematic analysis of machines.
Active steering for better safety comfortÑero Lopez
This document discusses the history and concepts of active steering systems for vehicles. It describes how early systems from the 1970s and 1980s used proportional feedback of yaw rate to add a corrective steering angle for improved handling. Later systems in the 1980s-1990s introduced four-wheel steering with feedforward and feedback control. The author proposes a concept from 1990 onward that separates the driver's path following task from an automatic yaw stabilization system through integral feedback of yaw rate to front wheel steering. Experimental results on a test vehicle showed improved disturbance rejection. The document outlines the continued development of robust control algorithms and actuator technologies to enable implementation of an active steering system.
Design and Analysis of Mechanism for Dynamic Characterization of Power Transm...iosrjce
Power transmission systems are being widely used for transmission of power between two members.
Once a particular transmission system is realized it needs to be qualified before its course of application. As
part of this intended torque of the transmission systems needs to be measured and tested. Conventional means of
dynamic characterization of power transmission system has got the demerit of energy consumption to a greater
extent. Because of this more effort is to be put in terms of power for the sake of testing the intended system.
Great need exists for a system which consumes less or ideally no energy while performing test. This project
aims at evolution of a novel technique for evaluating the torque transmitting capability of power transmission
systems without consuming more energy. To start with all the subsystems of the proposed design will be
identified and each of them will be designed for getting their dimensions. Then these dimensional models will be
transformed to solid model of the assembled configuration using 3D CAD software. Functional load which will
be experienced by this design will be assessed and structural analysis will be carried out against these loads
using Finite Element Method (FEM) in commercial FEA software i.e. ANSYS
The document describes the design and analysis of a mechanism to dynamically characterize power transmission systems without consuming significant energy. It involves using an electro-mechanical clutch connected to a motor via pulleys of different sizes to measure the torque transmitting capability of a transmission system by gradually increasing the load torque until slip occurs. Structural analysis using FEA found the design to be safe with a factor of safety above the minimum required.
1) A novel gearshift system is introduced comprising a 2 degree-of-freedom electromagnetic actuator to simplify structure, increase efficiency, and improve shift quality of automated manual transmissions.
2) The gearshift process is divided into non-synchronization and synchronization phases, with different control algorithms designed for each. Extended state observer based inverse system method is used for non-synchronization, while active disturbance rejection controller is used for synchronization.
3) Comparative simulations and experiments demonstrate the effectiveness of the proposed control method in achieving good gearshift performance for the novel system. The control strategy provides a new solution for automated manual transmission applications.
This document presents an adaptive sliding-mode (ASM) controller for a vehicle steer-by-wire (SbW) system. It models the SbW system as a second-order system and regards self-aligning torque and friction as external disturbances. An ASM controller is designed that can estimate the coefficient of self-aligning torque and handle parametric uncertainties. Experiments show the ASM controller achieves better tracking of a slalom path and circular path than a sliding-mode controller or H-infinity controller under various road conditions.
Experimental verification of SMC with moving switching lines applied to hoisti...ISA Interchange
In this paper we propose sliding mode control strategies for the point-to-point motion control of a hoisting crane. The strategies employ time-varying switching lines (characterized by a constant angle of inclination) which move either with a constant deceleration or a constant velocity to the origin of the error state space. An appropriate design of these switching lines results in non-oscillatory convergence of the regulation error in the closed-loop system. Parameters of the lines are selected optimally in the sense of two criteria, i.e. integral absolute error (IAE) and integral of the time multiplied by the absolute error (ITAE). Furthermore, the velocity and acceleration constraints are explicitly taken into account in the optimization process. Theoretical considerations are verified by experimental tests conducted on a laboratory scale hoisting crane.
Compilation and Verification of FatigueLoad Spectrum ofHighSpeed Maglev Vehic...IJRES Journal
The purpose of this study was to construct the fatigue test load spectrum for use in the high speed
maglev vehicleswing rod.Based on the measured load spectrum of the high speed maglev vehicle running
gear,the relatively poor condition and location of the swing rod was determined, according to the statistics and
analysis the load data of swing rod. With the rain-flow cycle quantile extrapolation, a high degree confidence
load spectrum was obtained in view of the differential damage among the different tests. The eight level
one-dimensional load spectrum after mean equivalent processing was established, by applying the Goodman to
correct mean stress effect on the fatigue strength and life; The stresses under the loadswere calculated using the
ANSYSsimulation.The fatigue life obtained from the programmed load spectrum is compared with that
calculated from measured load data using Miner theory,and the accuracy and validation of the programmed
load spectrum for thehigh speed maglev vehicleswing rodwere verified.
This document discusses using PI and PID controllers to control shaking in a bus suspension system. It first describes modeling a quarter-model of the bus suspension as a mass-spring-damper system. The open-loop response shows oscillations and long settling time. PI and PID controllers are then designed and applied in closed-loop control simulations in MATLAB to improve the response by reducing oscillations and shortening settling time. The PI controller uses proportional and integral terms, while the PID adds a derivative term. Both controllers are evaluated and compared based on time and frequency response characteristics.
Modeling and simulation of vehicle windshield wiper system using h infinity l...Mustefa Jibril
Vehicle windshield wiper system increases the driving safety by contributing a clear shot viewing to the
driver. In this paper, modelling, designing and simulation of a vehicle windshield wiper system with robust control
theory is done successfully. H loop shaping and robust pole placement controllers are used to improve the
wiping speed by tracking a reference speed signals. The reference speed signals used in this paper are step and sine
wave signals. Comparison of the H loop shaping and robust pole placement controllers based on the two
reference signals is done and convincing results have been obtained. Finally the comparative results prove the
effectiveness of the proposed H Loop Shaping controller to improve the wiping mechanism for the given two
reference signals.
Iaetsd estimation of frequency for a single link-flexibleIaetsd Iaetsd
This document proposes an adaptive control method for an uncertain flexible robotic arm. It uses a fast online closed-loop identification method combined with an output feedback controller called a Generalized Proportional Integral (GPI) controller. An algebraic identification method is used to identify unknown system parameters and update the GPI controller in real-time. Simulations show the robustness of the adaptive controller. The document describes the flexible manipulator model, GPI controller design, algebraic estimator development, adaptive control procedure, and simulation results showing the effectiveness of the adaptive control system.
Model Matching Control for an Active-Passive Variable Stiffness ActuatorCSCJournals
In order to increase the safety performance between human and the robot, the variable stiffness mechanism is commonly adopted due to its flexibility in various tasks. However, it makes the actuator more complex and increases the weight of the system. To deal with this problem, the active-passive variable stiffness elastic actuator (APVSEA) [1] and the variable stiffness control that uses a model matching control (MMC) with bond graph are proposed in this article. The APVSEA is modeled as a non-back drivable bond graph model. Combining the MMC with the non-back drivable bond graph model, the overall system can achieve active stiffness control. The simulations and the experiments show good control performance. They prove that the proposed method can achieve active stiffness control well. In the future, the proposed method can also be applied to the robot arm and exoskeletons for balancing safety and the control performance.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Iaetsd design of a robust fuzzy logic controller for a single-link flexible m...Iaetsd Iaetsd
This document describes the design of a fuzzy logic controller for a single-link flexible manipulator. A fuzzy-PID controller is used to control an uncertain flexible robotic arm and its internal motor dynamics parameters. The controller is tested against conventional integral and PID controllers in simulations. The results show the proposed fuzzy PID controller has better robustness under variations in motor dynamics compared to the other controllers.
Robust control strategies for an electric motor driven accumulator with elast...ISA Interchange
This document discusses robust control strategies for an electric motor driven accumulator used in industrial web processing. It first presents a nonlinear model of the accumulator dynamics, including web tension, velocity, and carriage motion. It then linearizes the model around a working point for frequency analysis. Finally, it compares multi-model PI and H1 controllers for the accumulator, finding that both allow good robustness against mechanical parameter variations.
Kineto-Elasto Dynamic Analysis of Robot Manipulator Puma-560IOSR Journals
Current industrial robots are made very heavy to achieve high Stiffness
which increases the accuracy of their motion. However this heaviness limits the robot speed and in masses the
required energy to move the system. The requirement for higher speed and better system performance makes it
necessary to consider a new generation of light weight manipulators as an alternative to today's massive
inefficient ones. Light weight manipulators require Less energy to move and they have larger payload abilities
and more maneuverability. However due to the dynamic effects of structural flexibility, their control is much
more difficult. Therefore, there is a need to develop accurate dynamic models for design and control of such
systems.This project presents the flexibility and Kineto - Elasto dynamic analysis of robot manipulator
considering deflection. Based on the distributed parameter method, the generalized motion equations of robot
manipulator with flexible links are derived. The final formulation of the motion equations is used to model
general complex elastic manipulators with nonlinear rigid-body and elastic motion in dynamics and it can be
used in the flexibility analysis of robot manipulators and spatial mechanisms. Manipulator end-effector path
trajectory, velocity and accelerations are plotted. Joint torques is to be determined for each joint trajectory
(Dynamics) .Using joint torques, static loading due to link’s masses, masses at joints, and payload, the robot
arms elastic deformations are to be found by using ANSYS-12.0 software package. Elastic compensation is
inserted in coordinates of robotic programming to get exact end-effectors path. A comparison of paths
trajectory of the end-effector is to be plotted. Also variation of torques is plotted after considering elastic
compensation. These torque variations are included in the robotic programming for getting the accurate endeffect
or’s path trajectory
1) The document discusses using a sliding mode controller for speed control of a two phase induction motor.
2) Sliding mode control is an efficient technique for speed control due to its robustness and insensitivity to parameter variations.
3) A sliding mode controller is designed for the speed control of a two phase induction motor. The controller design involves choosing a sliding surface, establishing convergence conditions, and determining the control law. Chattering reduction is also addressed.
The article gives the experimental results of the processes occurring in the combined system of traction and magnetic suspension, which was implemented on the basis of the linear switched reluctance motor. The goal of the research is to examine the possibility to combine the levitation and traction functions within one unit. The full- function physical model of the transport system with the magnetic suspension has been produced for experimental verification of the development concept for the combined system of traction and magnetic suspension. The research tests have been performed at the track structure with the limited length in order to study the processes, occurring in the most complicated start-up mode, when the discrete behavior of current in windings has the disturbance effect on the object levitation. The oscillograms of electromechanical transition processes, showing the mutual influence of traction subsystems and a suspension, are provided. The results of researches have illustrated dramatically that the development concept of the combined system of traction and magnetic suspension, based on the linear switched reluctance motor, is absolutely real. Further researches should be aimed at improving the system characteristics by reducing the mutual influence of levitation and traction processes.
Inertia identification based on adaptive interconnected Observer of Permanent...IJRES Journal
In order to achieve the inertia identification of permanent magnet synchronous motor, this paper
presents an adaptive inter connected observer based on model reference adaptive and Rhomberg observer to
observe the inertia of permanent magnet synchronous motor and build the simulation model in MATLAB. By the
simulation analysis, the observer can identify the moment of inertia of permanent magnet synchronous motor
accurately, at the same time has good robustness.
We focus a modern methodology in this paper for adding the fuzzy logic control as well as sliding model control. This combination can enhance the MLS position control robustness and enhanced performance of it.In the start, for an application in an area to control the loops placement and position for the synchronous motor what has permanent magnetic linearity we tend to control the fuzzy sliding mode control. To resolve the chattering issues a designed controller is investigated and, in this way, steady state motion in sliding with higher accuracy is obtained. In this case, method of online tuning with the help of fuzzy logic is used in order to adjust the thickness of boundary layer and switching gains.For the suggested scheme technique, the outcomes of simulation suggest that with the classical SMC the accurate state and good dynamic performance is compared due to force chattering resistance, response by quick dynamic force and external disturbance elements and robustness against them.
This document compares two control strategies - SVM based voltage oriented control and hysteresis current control - for interfacing a permanent magnet synchronous generator (PMSG) to the grid. Simulation results show that under constant and variable wind speeds:
1) The SVM control strategy regulates the grid voltage, current and DC link voltage more precisely with faster settling time compared to hysteresis control.
2) Both controllers effectively transfer maximum power from the PMSG to the grid while maintaining unity power factor.
3) Under variable wind speeds, the SVM control provides smoother transition of the generator speed, grid current, and active/reactive power compared to hysteresis control.
LMI based antiswing adaptive controller for uncertain overhead cranes IJECEIAES
This paper proposes an adaptive anti-sway controller for uncertain overhead cranes. The state-space model of the 2D overhead crane with the system parameter uncertainties is shown firstly. Next, the adaptive controller which can adapt with the system uncertainties and input disturbances is established. The proposed controller has ability to move the trolley to the destination in short time and with small oscillation of the load despite the effect of the uncertainties and disturbances. Moreover, the controller has simple structure so it is easy to execute. Also, the stability of the closed-loop system is analytically proven. The proposed algorithm is verified by using Matlab/ Simulink simulation tool. The simulation results show that the presented controller gives better performances (i.e., fast transient response, no ripple, and low swing angle) than the state feedback controller when there exist system parameter variations as well as input disturbances.
Similar to Load speed regulation in compliant mechanical transmission systems using feedback and feedforward control actions (20)
An optimal general type-2 fuzzy controller for Urban Traffic NetworkISA Interchange
This document presents an optimal general type-2 fuzzy controller (OGT2FC) for controlling traffic signal scheduling and phase succession to minimize wait times and average queue length. The OGT2FC uses a combination of general type-2 fuzzy logic sets and the Modified Backtracking Search Algorithm (MBSA) to optimize the membership function parameters. Simulation results show the OGT2FC performs better than conventional type-1 fuzzy controllers in regulating urban traffic flow.
Embedded intelligent adaptive PI controller for an electromechanical systemISA Interchange
In this study, an intelligent adaptive controller approach using the interval type-2 fuzzy neural network (IT2FNN) is presented. The proposed controller consists of a lower level proportional - integral (PI) controller, which is the main controller and an upper level IT2FNN which tuning on-line the parameters of a PI controller. The proposed adaptive PI controller based on IT2FNN (API-IT2FNN) is implemented practically using the Arduino DUE kit for controlling the speed of a nonlinear DC motor-generator system. The parameters of the IT2FNN are tuned on-line using back-propagation algorithm. The Lyapunov theorem is used to derive the stability and convergence of the IT2FNN. The obtained experimental results, which are compared with other controllers, demonstrate that the proposed API-IT2FNN is able to improve the system response over a wide range of system uncertainties.
State of charge estimation of lithium-ion batteries using fractional order sl...ISA Interchange
This paper presents a state of charge (SOC) estimation method based on fractional order sliding mode observer (SMO) for lithium-ion batteries. A fractional order RC equivalent circuit model (FORCECM) is firstly constructed to describe the charging and discharging dynamic characteristics of the battery. Then, based on the differential equations of the FORCECM, fractional order SMOs for SOC, polarization voltage and terminal voltage estimation are designed. After that, convergence of the proposed observers is analyzed by Lyapunov’s stability theory method. The framework of the designed observer system is simple and easy to implement. The SMOs can overcome the uncertainties of parameters, modeling and measurement errors, and present good robustness. Simulation results show that the presented estima- tion method is effective, and the designed observers have good performance.
Fractional order PID for tracking control of a parallel robotic manipulator t...ISA Interchange
This paper presents the tracking control for a robotic manipulator type delta employing fractional order PID controllers with computed torque control strategy. It is contrasted with an integer order PID controller with computed torque control strategy. The mechanical structure, kinematics and dynamic models of the delta robot are descripted. A SOLIDWORKS/MSC-ADAMS/MATLAB co-simulation model of the delta robot is built and employed for the stages of identification, design, and validation of control strategies. Identification of the dynamic model of the robot is performed using the least squares algorithm. A linearized model of the robotic system is obtained employing the computed torque control strategy resulting in a decoupled double integrating system. From the linearized model of the delta robot, fractional order PID and integer order PID controllers are designed, analyzing the dynamical behavior for many evaluation trajectories. Controllers robustness is evaluated against external disturbances employing performance indexes for the joint and spatial error, applied torque in the joints and trajectory tracking. Results show that fractional order PID with the computed torque control strategy has a robust performance and active disturbance rejection when it is applied to parallel robotic manipulators on tracking tasks.
Fuzzy logic for plant-wide control of biological wastewater treatment process...ISA Interchange
The application of control strategies is increasingly used in wastewater treatment plants with the aim of improving effluent quality and reducing operating costs. Due to concerns about the progressive growth of greenhouse gas emissions (GHG), these are also currently being evaluated in wastewater treatment plants. The present article proposes a fuzzy controller for plant-wide control of the biological wastewater treatment process. Its design is based on 14 inputs and 6 outputs in order to reduce GHG emissions, nutrient concentration in the effluent and operational costs. The article explains and shows the effect of each one of the inputs and outputs of the fuzzy controller, as well as the relationship between them. Benchmark Simulation Model no 2 Gas is used for testing the proposed control strategy. The results of simulation results show that the fuzzy controller is able to reduce GHG emissions while improving, at the same time, the common criteria of effluent quality and operational costs.
Design and implementation of a control structure for quality products in a cr...ISA Interchange
In recent years, interest for petrochemical processes has been increasing, especially in refinement area. However, the high variability in the dynamic characteristics present in the atmospheric distillation column poses a challenge to obtain quality products. To improve distillates quality in spite of the changes in the input crude oil composition, this paper details a new design of a control strategy in a conventional crude oil distillation plant defined using formal interaction analysis tools. The process dynamic and its control are simulated on Aspen HYSYS dynamic environment under real operating conditions. The simulation results are compared against a typical control strategy commonly used in crude oil atmospheric distillation columns.
Model based PI power system stabilizer design for damping low frequency oscil...ISA Interchange
This paper explores a two-level control strategy by blending a local controller with a centralized controller for the low frequency oscillations in a power system. The proposed control scheme provides stabilization of local modes using a local controller and minimizes the effect of inter-connection of sub-systems performance through a centralized control. For designing the local controllers in the form of proportional-integral power system stabilizer (PI-PSS), a simple and straight forward frequency domain direct synthesis method is considered that works on use of a suitable reference model which is based on the desired requirements. Several examples both on one machine infinite bus and multi-machine systems taken from the literature are illustrated to show the efficacy of the proposed PI-PSS. The effective damping of the systems is found to be increased remarkably which is reflected in the time-responses; even unstable operation has been stabilized with improved damping after applying the proposed controller. The proposed controllers give remarkable improvement in damping the oscillations in all the illustrations considered here and as for example, the value of damping factor has been increased from 0.0217 to 0.666 in Example 1. The simulation results obtained by the proposed control strategy are favorably compared with some controllers prevalent in the literature.
A comparison of a novel robust decentralized control strategy and MPC for ind...ISA Interchange
This document summarizes a research article that compares a novel decentralized control strategy based on override control to a model predictive controller (MPC) for controlling an industrial high purity methanol distillation column. Both controllers were able to maintain tight product purity and high recovery specifications under disturbances. The MPC provided tighter control of product purity but used more energy, while the proposed override control provided tighter recovery control and had lower costs. An economic analysis showed the optimal choice depends on factors like energy costs.
Fault detection of feed water treatment process using PCA-WD with parameter o...ISA Interchange
This research article proposes a new fault detection algorithm called PCA-WD that combines wavelet denoising (WD) with principal component analysis (PCA) to improve fault detection performance for feed water treatment processes (FWTP). The algorithm is applied to operational data from a FWTP sustaining two 1000 MW coal-fired power plants. Parameter selection for the PCA-WD algorithm is formulated as an optimization problem solved using particle swarm optimization to determine optimal parameters automatically rather than relying on individual experience. Results show that WD effectively reduces noise in PCA statistics, improving fault detection. The optimized PCA-WD algorithm outperforms classical PCA and a related method in detecting various faults in the FWTP data.
Model-based adaptive sliding mode control of the subcritical boiler-turbine s...ISA Interchange
As higher requirements are proposed for the load regulation and efficiency enhancement, the control performance of boiler-turbine systems has become much more important. In this paper, a novel robust control approach is proposed to improve the coordinated control performance for subcritical boiler-turbine units. To capture the key features of the boiler-turbine system, a nonlinear control-oriented model is established and validated with the history operation data of a 300 MW unit. To achieve system linearization and decoupling, an adaptive feedback linearization strategy is proposed, which could asymptotically eliminate the linearization error caused by the model uncertainties. Based on the linearized boiler-turbine system, a second-order sliding mode controller is designed with the super-twisting algorithm. Moreover, the closed-loop system is proved robustly stable with respect to uncertainties and disturbances. Simulation results are presented to illustrate the effectiveness of the proposed control scheme, which achieves excellent tracking performance, strong robustness and chattering reduction.
A Proportional Integral Estimator-Based Clock Synchronization Protocol for Wi...ISA Interchange
Clock synchronization is an issue of vital importance in applications of wireless sensor networks (WSNs). This paper proposes a proportional integral estimator-based protocol (EBP) to achieve clock synchronization for wireless sensor networks. As each local clock skew gradually drifts, synchronization accuracy will decline over time. Compared with existing consensus-based approaches, the proposed synchronization protocol improves synchronization accuracy under time-varying clock skews. Moreover, by restricting synchronization error of clock skew into a relative small quantity, it could reduce periodic re-synchronization frequencies. At last, a pseudo-synchronous implementation for skew compensation is introduced as synchronous protocol is unrealistic in practice. Numerical simulations are shown to illustrate the performance of the proposed protocol.
An artificial intelligence based improved classification of two-phase flow patte...ISA Interchange
Flow pattern recognition is necessary to select design equations for finding operating details of the process and to perform computational simulations. Visual image processing can be used to automate the interpretation of patterns in two-phase flow. In this paper, an attempt has been made to improve the classification accuracy of the flow pattern of gas/ liquid two- phase flow using fuzzy logic and Support Vector Machine (SVM) with Principal Component Analysis (PCA). The videos of six different types of flow patterns namely, annular flow, bubble flow, churn flow, plug flow, slug flow and stratified flow are re- corded for a period and converted to 2D images for processing. The textural and shape features extracted using image processing are applied as inputs to various classification schemes namely fuzzy logic, SVM and SVM with PCA in order to identify the type of flow pattern. The results obtained are compared and it is observed that SVM with features reduced using PCA gives the better classification accuracy and computationally less intensive than other two existing schemes. This study results cover industrial application needs including oil and gas and any other gas-liquid two-phase flows.
New Method for Tuning PID Controllers Using a Symmetric Send-On-Delta Samplin...ISA Interchange
In this paper we present a new method for tuning PI controllers with symmetric send-on-delta (SSOD) sampling strategy. First we analyze the conditions that produce oscillations in event based systems considering SSOD sampling strategy. The Describing Function is the tool used to address the problem. Once the conditions for oscillations are established, a new robustness to oscillation performance measure is introduced which entails with the concept of phase margin, one of the most traditional measures of relative stability in closed-loop control systems. Therefore, the application of the proposed robustness measure is easy and intuitive. The method is tested by both simulations and experiments. Additionally, a Java application has been developed to aid in the design according to the results presented in the paper.
Load estimator-based hybrid controller design for two-interleaved boost conve...ISA Interchange
This paper is devoted to the development of a hybrid controller for a two-interleaved boost converter dedicated to renewable energy and automotive applications. The control requirements, resumed in fast transient and low input current ripple, are formulated as a problem of fast stabilization of a predefined optimal limit cycle, and solved using hybrid automaton formalism. In addition, a real time estimation of the load is developed using an algebraic approach for online adjustment of the hybrid controller. Mathematical proofs are provided with simulations to illustrate the effectiveness and the robustness of the proposed controller despite different disturbances. Furthermore, a fuel cell system supplying a resistive load through a two-interleaved boost converter is also highlighted.
Effects of Wireless Packet Loss in Industrial Process Control SystemsISA Interchange
Timely and reliable sensing and actuation control are essential in networked control. This depends on not only the precision/quality of the sensors and actuators used but also on how well the communications links between the field instruments and the controller have been designed. Wireless networking offers simple deployment, reconfigurability, scalability, and reduced operational expenditure, and is easier to upgrade than wired solutions. However, the adoption of wireless networking has been slow in industrial process control due to the stochastic and less than 100% reliable nature of wireless communications and lack of a model to evaluate the effects of such communications imperfections on the overall control performance. In this paper, we study how control performance is affected by wireless link quality, which in turn is adversely affected by severe propagation loss in harsh industrial environments, co-channel interference, and unintended interference from other devices. We select the Tennessee Eastman Challenge Model (TE) for our study. A decentralized process control system, first proposed by N. Ricker, is adopted that employs 41 sensors and 12 actuators to manage the production process in the TE plant. We consider the scenario where wireless links are used to periodically transmit essential sensor measurement data, such as pressure, temperature and chemical composition to the controller as well as control commands to manipulate the actuators according to predetermined setpoints. We consider two models for packet loss in the wireless links, namely, an independent and identically distributed (IID) packet loss model and the two-state Gilbert-Elliot (GE) channel model. While the former is a random loss model, the latter can model bursty losses. With each channel model, the performance of the simulated decentralized controller using wireless links is compared with the one using wired links providing instant and 100% reliable communications. The sensitivity of the controller to the burstiness of packet loss is also characterized in different process stages. The performance results indicate that wireless links with redundant bandwidth reservation can meet the requirements of the TE process model under normal operational conditions. When disturbances are introduced in the TE plant model, wireless packet loss during transitions between process stages need further protection in severely impaired links. Techniques such as re-transmission scheduling, multi-path routing and enhanced physical layer design are discussed and the latest industrial wireless protocols are compared.
Fault Detection in the Distillation Column ProcessISA Interchange
Chemical plants are complex large-scale systems which need designing robust fault detection schemes to ensure high product quality, reliability and safety under different operating conditions. The present paper is concerned with a feasibility study of the application of the black-box modeling method and Kullback Leibler divergence (KLD) to the fault detection in a distillation column process. A Nonlinear Auto-Regressive Moving Average with eXogenous input (NARMAX) polynomial model is firstly developed to estimate the nonlinear behavior of the plant. Furthermore, the KLD is applied to detect abnormal modes. The proposed FD method is implemented and validated experimentally using realistic faults of a distillation plant of laboratory scale. The experimental results clearly demonstrate the fact that proposed method is effective and gives early alarm to operators.
Neural Network-Based Actuator Fault Diagnosis for a Non-Linear Multi-Tank SystemISA Interchange
The paper is devoted to the problem of the robust actuator fault diagnosis of the dynamic non-linear systems. In the proposed method, it is assumed that the diagnosed system can be modelled by the recurrent neural network, which can be transformed into the linear parameter varying form. Such a system description allows developing the designing scheme of the robust unknown input observer within H1 framework for a class of non-linear systems. The proposed approach is designed in such a way that a prescribed disturbance attenuation level is achieved with respect to the actuator fault estimation error, while guaranteeing the convergence of the observer. The application of the robust unknown input observer enables actuator fault estimation, which allows applying the developed approach to the fault tolerant control tasks.
A KPI-based process monitoring and fault detection framework for large-scale ...ISA Interchange
Large-scale processes, consisting of multiple interconnected sub-processes, are commonly encountered in industrial systems, whose performance needs to be determined. A common approach to this problem is to use a key performance indicator (KPI)-based approach. However, the different KPI-based approaches are not developed with a coherent and consistent framework. Thus, this paper proposes a framework for KPI-based process monitoring and fault detection (PM-FD) for large-scale industrial processes, which considers the static and dynamic relationships between process and KPI variables. For the static case, a least squares-based approach is developed that provides an explicit link with least-squares regression, which gives better performance than partial least squares. For the dynamic case, using the kernel re- presentation of each sub-process, an instrument variable is used to reduce the dynamic case to the static case. This framework is applied to the TE benchmark process and the hot strip mill rolling process. The results show that the proposed method can detect faults better than previous methods.
An adaptive PID like controller using mix locally recurrent neural network fo...ISA Interchange
Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled using classical proportional integral derivative (PID) controller. To enhance the effectiveness of the conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller which consists of at most three hidden nodes which act as proportional, integral and derivative node. The gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initi- alized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than assuming randomly. A sequential learning based least square algorithm is then investigated for the on- line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme is tested against the plant parameters uncertainties and external disturbances for both links of the two link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID) controller in order to establish the effectiveness of the MLRNNPID controller.
A method to remove chattering alarms using median filtersISA Interchange
Chattering alarms are the most found nuisance alarms that will probably reduce the usability and result in a confidence crisis of alarm systems for industrial plants. This paper addresses the chattering alarm reduction using median filters. Two rules are formulated to design the window size of median filters. If the alarm probability is estimated using process data, one rule is based on the probability of alarms to satisfy some requirements on the false alarm rate, or missed alarm rate. If there are only historical alarm data available, the other rule is based on percentage reduction of chattering alarms using alarm duration distribution. Experimental results for industrial cases testify that the proposed method is effective.
Infrastructure Challenges in Scaling RAG with Custom AI modelsZilliz
Building Retrieval-Augmented Generation (RAG) systems with open-source and custom AI models is a complex task. This talk explores the challenges in productionizing RAG systems, including retrieval performance, response synthesis, and evaluation. We’ll discuss how to leverage open-source models like text embeddings, language models, and custom fine-tuned models to enhance RAG performance. Additionally, we’ll cover how BentoML can help orchestrate and scale these AI components efficiently, ensuring seamless deployment and management of RAG systems in the cloud.
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Climate Impact of Software Testing at Nordic Testing DaysKari Kakkonen
My slides at Nordic Testing Days 6.6.2024
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Load speed regulation in compliant mechanical transmission systems using feedback and feedforward control actions
1. Load speed regulation in compliant mechanical transmission systems
using feedback and feedforward control actions
P.R. Raul a
, R.V. Dwivedula b
, P.R. Pagilla c,n
a
Mechanical & Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078, United States
b
Department of Mechanical Engineering, Sree Vidyanikethan Engineering College, Tirupati, Andhra Pradesh 517102, India
c
Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, United States
a r t i c l e i n f o
Article history:
Received 19 February 2015
Received in revised form
19 November 2015
Accepted 13 March 2016
Available online 25 April 2016
This paper was recommended for publica-
tion by Didier Theilliol
Keywords:
Adaptive feedforward
Load speed regulation
Belt–pulley mechanical transmission
a b s t r a c t
The problem of controlling the load speed of a mechanical transmission system consisting of a belt-
pulley and gear-pair is considered. The system is modeled as two inertia (motor and load) connected by a
compliant transmission. If the transmission is assumed to be rigid, then using either the motor or load
speed feedback provides the same result. However, with transmission compliance, due to belts or long
shafts, the stability characteristics and performance of the closed-loop system are quite different when
either motor or load speed feedback is employed. We investigate motor and load speed feedback
schemes by utilizing the singular perturbation method. We propose and discuss a control scheme that
utilizes both motor and load speed feedback, and design an adaptive feedforward action to reject load
torque disturbances. The control algorithms are implemented on an experimental platform that is
typically used in roll-to-roll manufacturing and results are shown and discussed.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Mechanical transmissions are widely used in various industries
where the mechanical power is typically transmitted from motor
shafts to load shafts by utilizing transmission systems. Examples
include manufacturing, power generation, and transportation
systems. Power transmission with speed reduction and variable
torque requirement is made possible with mechanical transmis-
sion systems. Belt–pulley and gear transmission systems are
commonly used. In many applications, a mechanical transmission
system containing a combination of belt–pulley and a gear-pair is
very convenient over a purely gear transmission system. When the
center distance between the driving (motor) shaft and the driven
(load) shaft is too large for use of a single gear-pair, using a belt to
transmit motion/power may be the only practical alternative.
Further, such an arrangement is advantageous because coupling
the drive motor directly to the process end mandates very accurate
collinearity of the axes and takes considerable amount of time;
also, there is no guarantee that collinearity is maintained over
extended period of time due to load disturbances. Belt driven
transmission systems offer considerable flexibility as small
inaccuracies in alignment can be absorbed into compliance of the
belt. However, compliance of the belt introduces additional
dynamics into the system. The belt driven power transmission
system is common in roll to roll manufacturing. The presence of
compliance from transmissions and the stiffness of web material
[1] will pose different levels of severity in properly transporting
the web.
Control of load speed is essential in many applications. When
rigid transmissions are employed, there is no dynamic relation
between the motor shaft and the load shaft, and typically the
motor shaft speed is controlled to control the speed of the load
shaft. However, due to the transmission dynamics, resulting from
the compliance of belt as well as long shafts in the transmission,
regulating load shaft speed is not the same as regulating motor
shaft speed. In the presence of such a transmission, practicing
engineers are often confronted with the question of whether to
use (i) motor speed feedback to control load speed as is done in
conventional practice, or (ii) use load speed feedback, or (iii) use a
combination of motor and load speed feedback.
There is a large body of literature on the characteristics of belt
drives and design of mechanisms using belt drives. Much of this
work focused on the mechanism of motion/power transfer, loca-
tion and extent of slip-arc, nature of frictional contact, efficiency
limit of the belt-drive system, and methodology of design/selec-
tion of belt-drive components [2–11]. In [12], modeling and con-
trol of a belt-drive positioning table is discussed, and in [13], direct
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.03.005
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding author. Tel.:þ1 979 458 4829; fax: þ1 979 845 3081.
E-mail addresses: pramod.raul@okstate.edu (P.R. Raul),
ramamurthy@vidyanikethan.edu (R.V. Dwivedula),
ppagilla@tamu.edu (P.R. Pagilla).
ISA Transactions 63 (2016) 355–364
2. drive control of X–Y table is presented. However, no specific model
is reported for including the effect of compliance of the belt;
system identification techniques were used to obtain the system
dynamics, to be later used in tuning of the feedback gains. Simi-
larly in [14], a composite fuzzy controller, consisting of a feedback
fuzzy controller and a feed-forward acceleration compensator, is
proposed to control a belt drive precision positioning table; the
effects of belt compliance were not included in this paper. In [15],
a robust motion control algorithm for belt-driven servomechanism
is reported. In this paper, the belt-stretch dynamics is assumed to
contribute a pair of purely imaginary poles to the transfer function
of the system; the fact that the belt serves as an interconnection
from load-side to the motor-side is ignored in this paper. Modeling
of belt–pulley and gear-pair transmission system with gear back-
lash is given in [16]. Analysis and control of speed drive systems
with torsional loads is reported in [17–20].
The following are the contributions of this work: based on the
model of the two inertias (motor and load) connected by a belt–
pulley and gear-pair transmission system, we have investigated
the effect of using either motor or load feedback to control the
load speed by utilizing the singular perturbation method. In each
case, we consider a PI controller that is typical in the industry for
the feedback controller. The small parameter in the singular per-
turbation method is proportional to the reciprocal of the square
root of the belt compliance. The singular perturbation analysis
revealed that the controller using pure load feedback results in an
unstable system. Therefore, use of pure load feedback must be
avoided. To directly control the load speed, we also propose a
control scheme that uses both the motor speed and load speed
feedback and show that such scheme results in a stable closed-
loop system. Since feedback action is not sufficient in rejecting
periodic disturbances that commonly act on the load, we also
consider an adaptive feedforward compensation action that is
based on adaptive estimation of the coefficients of the periodic
disturbance as suggested in [21]. This adaptive feedforward action
is quite suitable for this application because it preserves closed-
loop stability achieved with the feedback controller. Experiments
were conducted to evaluate the performance of the various control
schemes on an industrial grade transmission system that is com-
mon in roll-to-roll manufacturing.
The remainder of the paper is organized as follows. The model
of the system is described in Section 2. Sections 3 and 4 describe
the motor speed feedback only and load speed feedback only
cases, respectively. A control scheme that utilizes both motor and
load speed feedback is discussed in Section 5. An add-on adaptive
feedforward compensation to reject load speed disturbances is
discussed in Section 6. Section 7 provides a description of the
experimental platform and a comparison of the results with the
various control schemes. Conclusions are given in Section 8.
2. Model of the system
A schematic of the belt–pulley and gear transmission system
connecting the motor with the load is shown in Fig. 1. In the
schematic, Ji denote the inertias, bi denote the viscous friction
coefficients, Ri denote the radii of the pulleys and gears, θi denote
the angular displacements of the inertias, τm denotes the motor
torque, τL denotes the torque disturbance on the load, and Kb
denotes the stiffness of the belt.
To derive the governing equations for this system we consider
the action of the belt in transmitting power. For a given direction
of rotation of the pulley, the belt has a tight side and a slack side as
shown in Fig. 1. It is assumed that the transmission of power is
taking place on the tight side and the transport of the belt is taking
place on the slack side. Under this assumption, the net change in
tension on the slack side will be much smaller than that in the
tight side and thus may be ignored. The tight side of the belt can
then be modeled as a spring with spring constant of Kb. For given
angular displacements θm and θL, the net elongation of the tight
side of the belt can be written as ðR1θm ÀGRR2θLÞ. Because of this
elongation, the driving pulley experiences a torque of ðR1θm ÀGR
R2θLÞKbR1 and the driven pulley experiences a torque of
ðR1θm ÀGRR2θLÞGRR2Kb. Under the assumption that the inertias of
the pulleys and gears are much smaller than the motor and the
load, the governing equations of motion for the motor-side inertia
and the load-side inertia are given by
Jm
€θm þbm
_θm þR1Kb R1θm ÀGRR2θL
À Á
¼ τm; ð1aÞ
JL
€θL þbL
_θL ÀGRR2Kb R1θm ÀGRR2θL
À Á
¼ τL: ð1bÞ
A block diagram representation of the system given by (1) is
provided in Fig. 2; note that this block diagram represents the
open-loop system and the two “loops” appearing in the block
diagram that represent the interconnections in (1). The open-loop
transfer functions from the motor torque signal τm to the motor
speed ωm and load speed ωL are given by
Gτmωm
ðsÞ9
ωm sð Þ
τm sð Þ
¼
JLs2
þbLsþG2
RR2
2Kb
D sð Þ
; ð2aÞ
Gτmωm ðsÞ9
ωL sð Þ
τm sð Þ
¼
GRR1R2Kb
D sð Þ
; ð2bÞ
where
DðsÞ ¼ JmJLs3
þ bLJm þJLbm
À Á
s2
þ KbJeq þbmbL
sþKbbeq; ð3aÞ
Jeq ¼ G2
RR2
2Jm þR2
1JL; ð3bÞ
beq ¼ G2
RR2
2bm þR2
1bL: ð3cÞ
θ2
Sprocket 1
Jm
τm
Belt
Kb
2R2
Sprocket 2
Gear 1
Gear2
bm
JL
bL
τL
θL
2R1
θm
2Rg1
2Rg2
Slackside
Tightside
Fig. 1. Schematic of a belt–pulley and gear-pair transmission system.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364356
3. The goal is to control load speed. In the following we will
discuss the closed-loop control systems that consider three sce-
narios: (i) pure motor speed feedback or (ii) pure load speed
feedback or (iii) a combination of motor and load speed feedback.
3. Motor speed feedback control scheme
It is common to control load speed by using measurement of
motor speed ωm as feedback. This control scheme is shown in
Fig. 3. The control structure is designed to regulate motor speed
ωm to the reference ωrm, and thereby indirectly regulate load
speed ωL.
We consider the often used Proportional–Integral (PI) control
action which is widely used in industrial environments. The
feedback law is given by
τm ¼ Kpm ωrm Àωmð ÞþKim
Z
ωrm Àωmð Þ dτ: ð4Þ
With this control law, the closed-loop transfer function from ωrm
to ωL is obtained as
ωLðsÞ
ωrmðsÞ
¼
ðGRR1R2Kb=JmJLÞðsKpm þKimÞ
ψmðsÞ
; ð5Þ
where
ψmðsÞ ¼ s4
þc3s3
þc2s2
þc1sþc0;
c3 ¼
ðbmJL þJmbL þKpmJLÞ
JmJL
;
c2 ¼
ðKbJeq þbmbL þKpmbL þKimJLÞ
JmJL
; ð6Þ
c1 ¼
ðKbbeq þG2
RR2
2KbKpm þKimbLÞ
JmJL
;
c0 ¼
G2
RR2
2KbKim
JmJL
:
Note the the coefficients c0 to c3 depend on the controller gains.
We consider the singular perturbation method for analyzing such
a system with the small parameter proportional to the reciprocal
of the square root of the belt stiffness Kb. For conduction singular
perturbation analysis, we need to express the equations in the
form
_x ¼ A11xþA12z; x t0ð Þ ¼ x0
ð7aÞ
ϵ_z ¼ A21xþA22z; z t0ð Þ ¼ z0
ð7bÞ
where x and z are the states of the slow and the fast subsystems,
respectively, and ε is the small parameter; for our system we will
consider ε2
¼ 1=Kb. The elements of matrices Aij may depend on ε.
However, to use the singular perturbation method, the matrix A22
needs to be nonsingular [22] at ε ¼ 0. A natural choice of the state
variables for the singular perturbation analysis is θm, _θm, θL and _θL.
However, with this choice of the state variables, the matrix A22
becomes singular at ε ¼ 0. To obtain a state-space representation
in the form that would enable the use of the singular perturbation
method, we consider the following transformation of variables:
θc 9
Jmθm þJLGRðR2=R1ÞθL
Jm þJL
; ð8aÞ
θs 9θm ÀGRðR2=R1ÞθL: ð8bÞ
The variable θc is a weighted average of angular displacements
(θm and θL) referred to the motor side and the variable θs is dif-
ference between the angular displacements (θm and θL) referred to
the motor side; transformations similar to these have been used in
prior studies of two inertia systems, see for example [23]. The idea
of the weighted average of the displacements arises naturally in
the case of a translational system wherein θc represents the
position of the centroid of the masses. Now, choosing the state
variables as x ¼ ½θc; _θcŠ
and z ¼ ½θs=ε2
; _θs=εŠ
, the state space
representation of the system is obtained in the form given by (7)
where
A11 ¼
0 1
f 1 f 3
#
; A12 ¼
0 0
ϵ2
f 21 þf 22 ϵf 4
#
;
+
-
+
-
Jms + bm JLs + bLm
1 +
-
1
BRKbR2
1
s
BR
ωm
ωL
τm
τL
Fig. 2. Block diagram of the belt–pulley and gear transmission system; BR denotes the overall speed ratio, BR ¼ ðR2=R1ÞGR.
+
-
+-
1 +
-
1
s
Controller
+
-
BR
Motor Speed Feedback
BR
Jms + bm
KbR2
1
JLs + bLm ωL
τL
ωm
τm
ωrm
Fig. 3. Motor speed feedback control scheme.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364 357
4. A21 ¼
0 0
g1 g3
#
; A22 ¼
0 1
ϵ2
g21 þg22 ϵg4
#
; ð9Þ
f 1 ¼ ÀKim=J0; f 21 ¼ ÀKimJL=J2
0;
f 22 ¼ G2
RR2
2 ÀR2
1
=J0; f 3 ¼ À Kpm þbm þbL
À Á
=J0;
f 4 ¼ bLJm ÀbmJL ÀKpmJL
À Á
=J2
0;
g1 ¼ ÀKim=Jm; g21 ¼ ÀKimJ2
L = JmJLJ0
À Á
;
g22 ¼ À R2
1JL þG2
RR2
2Jm
= JmJL
À Á
;
g3 ¼ bLJm ÀbmJL ÀKpmJL
À Á
= JmJL
À Á
;
g4 ¼ À KpmJ2
L þbmJ2
L þbLJ2
m
= JmJLJ0
À Á
;
where J0 ¼ Jm þJL, and 1=ε2
¼ Kb. Notice that det A22 ϵð Þ ϵ ¼ 0j Þ ¼ð
Àg22 a0, thus satisfying the requirement of non-singularity of the
matrix A22 at ε ¼ 0. The characteristic equation for the system
given by (9) can be factored as [22]
ψmðs; εÞ %
1
ε2
ψmsðs; εÞψmf ðp; εÞ ¼ 0 ð10Þ
with
ψmsðs; εÞ9det½sI2 ÀðA11 ÀA12LðεÞÞŠ ð11aÞ
ψmf ðp; εÞ9det½pI2 ÀðA22 þεLðεÞA12ÞŠ ð11bÞ
where ψmsðs; εÞ is the characteristic polynomial for the slow sub-
system and ψmf ðp; εÞ is the characteristic polynomial of the fast
subsystem exhibited in the high-frequency scale p ¼ εs. The matrix
LðεÞ is obtained using the iterative scheme given in [22].
Using the matrices given by Eq. (9), the slow and the fast
characteristic polynomials are obtained as
ψmsðs; εÞ % s2
þα1sþα0; ð12aÞ
ψmf ðp; εÞ % p2
þα0
1pþα0
0 ð12bÞ
where
α1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2Kpm
G2
RR2
2Jm þR2
1JL
;
α0 ¼
G2
RR2
2Kim
G2
RR2
2Jm þR2
1JL
; ð13Þ
α0
1 ¼
G2
RR2
2KpmJL
JmðG2
RR2
2Jm þR2
1JLÞ
ε;
α0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Eq. (12) indicates that both the fast and the slow subsystems are
stable for all Kpm; Kim 40. The result is true even without the
approximation introduced by LðεÞ as shown in Theorem 1 in the
Appendix.
4. Load speed feedback control scheme
One can employ the load speed feedback scheme shown in
Fig. 4, where the measured variable is ωL. This seems to have the
advantage of directly controlling load speed and attenuating the
effect of the disturbance τL. The feedback law is given by
τm ¼ KpL ωrL ÀωLð ÞþKiL
Z
ωrL ÀωLð Þ dτ; ð14Þ
and the closed-loop transfer function from ωrL to ωL is obtained as
ωLðsÞ
ωrLðsÞ
¼
ðGRR1R2Kb=JmJLÞðsKpL þKiLÞ
ψLðsÞ
ð15Þ
where
ψLðsÞ ¼ s4
þd3s3
þd2s2
þd1sþd0;
d3 ¼
ðbmJL þJmbLÞ
JmJL
;
d2 ¼
ðKbJeq þbmbLÞ
JmJL
; ð16Þ
d1 ¼
ðKbbeq þGRR1R2KbKpLÞ
JmJL
;
d0 ¼
GRR1R2KbKiL
JmJL
:
eqno
rightlefthskip 12pt
Singular perturbation analysis pertaining to this control
scheme results in the following slow and fast characteristic poly-
nomials:
ψlsðs; εÞ % s2
þβ1sþβ0 ð17aÞ
ψlf ðp; εÞ % p2
Àβ0
1pþβ0
0 ð17bÞ
where
β1 ¼
G2
RR2
2bm þR2
1bL þGRR2R1KpL
G2
RR2
2Jm þR2
1JL
;
β0 ¼
GRR2R1KiL
G2
RR2
2Jm þR2
1JL
;
β0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpL
G2
RR2
2Jm þR2
1JL
ε; ð18Þ
β0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Note that the slow subsystems are stable for all KpL, KiL 40.
However, the fast subsystem is unstable for all KpL 40 and KiL 40.
Also, note that the characteristic polynomials given by Eqs. (12b)
and (17b) are identical when ε ¼ 0. Thus, analyzing the limiting
+
-
+-
1 +
-
1
s
τ
L
Controller
+
-
BR
Load Speed Feedback
BR
ωrL τm
Jms + bm
ωm
KbR2
1
JLs + bL
ωL
Fig. 4. Load speed feedback control scheme.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364358
5. case of an infinitely stiff belt, that is, ε ¼ 0 will not reveal the
instability exhibited by (17b).
Remark 1. Eq. (1b) may be used to give an interpretation of the
foregoing analysis. Differentiating (1b), we obtain
JL €ωL þbL _ωL þR2
2KbωL ¼ R1R2Kbωm; ð19Þ
which indicates that ωL can attain steady-state only when ωm
attains steady-state first. Even after ωm attains steady-state, ωL
continues to exhibit damped oscillations. Thus, by measuring only
ωL and using the control law given by the (14), it will be difficult to
distinguish whether the oscillations in ωL are due to fluctuations
in motor speed or, the oscillations are indeed damped oscillations.
In such a situation, the controller attempts to react to the damped
oscillations also, and in this process, changes ωm, which in turn
affects ωL because of the dynamics given by the (19). Thus, the
control law given by (14) does not present a desirable situation.
5. Simultaneous motor and load speed feedback control
scheme
In this scheme, the load speed control corrects directly the
torque input to the system as shown in Fig. 5. The closed-loop
transfer function from ωrL to ωL is given by
ωLðsÞ
ωrLðsÞ
¼
αmLt
ψmLtðsÞ
ð20Þ
where
αmLtðsÞ ¼ a1sþa0;
a1 ¼
ðGRR1R2KbKpL þGrR2Kpm=R1Þ
JmJL
; ð21Þ
a0 ¼
ðGRR1R2KbKiL þGrR2Kim=R1Þ
JmJL
:
ψmLtðsÞ ¼ s4
þf 3s3
þf 2s2
þf 1sþf 0;
f 3 ¼
ðbmJL þJmbL þJLKpmÞ
JmJL
;
f 2 ¼
ðKbJeq þbmbL þKpmbL þJLKimÞ
JmJL
; ð22Þ
f 1 ¼
ðKbbeq þG2
RKbKpm þKimbL þGRR1R2KbKpLÞ
JmJL
;
f 0 ¼
KimG2
RR2
2Kb þGRR1R2KbKiL
JmJL
:
Note that the coefficients f0 to f3 depend on the gains of the
control law. Singular perturbation analysis for this case results in
the slow and fast characteristic polynomials as
ψmlsðs; εÞ % s2
þγ1sþγ0 ð23aÞ
ψmlf ðp; εÞ % p2
þγ0
1pþγ0
0 ð23bÞ
where
γ1 ¼
G2
RR2
2bm þR2
1bL þGRR2R1KpL þG2
RR2
2Kpm
G2
RR2
2Jm þR2
1JL
;
γ0 ¼
G2
RR2
2Kim þGRR2R1KiL
G2
RR2
2Jm þR2
1JL
;
γ0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpL þG2
RR2
2KpmðJL=JmÞ
G2
RR2
2Jm þR2
1JL
ε; ð24Þ
γ0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Therefore, the slow and fast subsystems are stable for all
positive controller gains. Note that the outputs of both load speed
and motor speed controller combine to form a torque input to the
motor; this is typically referred to as the torque mode in practice
when multiple loops such as this are employed. Another strategy
is to use the output of the load speed controller as the motor speed
reference correction; a block diagram of such a scheme is provided
in Fig. 11 in Appendix B. This strategy results in an unstable system
which is shown in Appendix B.
6. Adaptive Feedforward (AFF) compensation to reject load
disturbances
The use of feedforward compensation to reject known dis-
turbances by direct cancelation or unknown disturbances by their
estimation has been known to be effective in attenuating dis-
turbances. We consider the rejection of periodic disturbances on
the load by using an adaptive feedforward action based on load
speed error. The control scheme that utilizes the feedforward
action is shown in Fig. 6. We use an adaptive feedforward algo-
rithm given in [21] that is particularly applicable in this situation
as the feedforward action preserves the stability of the overall
system with the feedback controller with simultaneous motor and
load speed feedback. The approach is briefly discussed as applic-
able to this problem; the details are given in [21]. The idea is to
estimate the amplitude and phase of the disturbance for a known
frequency of the disturbance. The disturbance can be expressed in
the form
d ¼ θÃ
1 cos ðωtÞþθÃ
2 sin ðωtÞ
≔ϕðtÞwÃ
0 ð25Þ
where ω is a known frequency, θÃ
1 and θÃ
2 are unknown
+-
+-
1 +-
1
s
+-
ωrm
PI
Controller
+
-
PI
Controller
+
+
BR
Motor Speed Feedback
Load Speed Feedback
BR
Jms + bm
ωm
KbR2
1
JLs + bL ωL
τL
ωrL
τm
Fig. 5. Simultaneous motor and load speed feedback scheme: torque mode.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364 359
6. parameters. The adaptation laws for the unknown parameters θÃ
1
and θÃ
2 are given by the following simple pseudo-gradient algo-
rithm:
_θ1 ¼ γeðtÞ cos ðωtÞ; ð26aÞ
_θ2 ¼ γeðtÞ sin ðωtÞ; ð26bÞ
where θ1 and θ1 are the parameter estimates, eðtÞ ¼ ωrL ÀωL is the
load speed error, and γ is the adaptation gain. Using the estimated
parameters, the feedforward control action is given by
uf ¼ Àθ1 cos ðωtÞÀθ2 sin ðωtÞ: ð27Þ
The estimation of the disturbance and its cancelation when the
load speed error contains a sinusoidal component with frequency
ω may be intuitively explained as follows. If the load speed error is
eðtÞ ¼ eðtÞþeθ1
sin ðωtÞ, in the adaptive algorithm the product eðtÞ
sin ðωtÞ will generate a positive eθ1
sin
2
ðωtÞ term. This will result
in a parameter drift which results in the attenuation of dis-
turbance until it reaches its nominal value. At this point the load
speed error is free of the sinusoidal component as the disturbance
is compensated by feedforward control uf. With the compensation,
the product term eðtÞ sin ðωtÞ in the parameter adaptive law
becomes zero and the parameters converge. Since the regressor
vector ϕðtÞ is persistently exciting, the parameter vectors converge
to zero. In fact, this adaptive feedforward action with estimation of
disturbance parameters using the pseudo-gradient algorithm has
been shown to be equivalent to the use of the internal model of
the disturbance in [21].
7. Experiments
A picture of experimental setup is shown in Fig. 7. It consists of an
AC motor shaft connected to the load shaft (roll) via a belt–pulley and
gear-pair transmission. A 15 HP (11.19 KW) AC motor with a rated
speed of 1750 RPM is employed. The belt ratio (BR ¼ ðR2=R1ÞGR) for
the transmission is 3.825. An encoder on the motor shaft is employed
to measure the motor shaft speed and a laser sensor is used to
measure the load shaft speed. The real-time hardware, including the
drives, controller, and communication network, was provided by
Rockwell Automation (Allan-Bradley). All the real-time hardware
components of the machine are connected through a ControlNet
communication network. The network is updated every 5 ms (Net-
work Update Time) and data is communicated to the network every
10 ms (Request Package Interval). A brake is attached on the other
side of the load shaft to inject periodic torque disturbances; a mag-
netic clutch brake (Magpower GBC 90) that can apply 26 lb-ft torque
is used.
7.1. Design and implementation guidelines for the proposed load
speed regulation scheme
The following guidelines should provide assistance to practi-
cing engineers in the design and implementation of the proposed
load speed regulation scheme:
1. Select the PI gains for the motor speed PI controller by tuning
them for the desired regulation performance without the load.
These PI gains will provide a baseline for the motor speed PI
controller.
2. Select the PI gains for the load speed feedback PI controller by
placing stable closed-loop poles for the fast and slow dynamics
given in Section 5. It may be necessary to re-tune the PI gains of
the motor speed PI controller to obtain the desired closed-
loop poles.
3. Adaptive feedforward compensation is generated based on the
disturbance parameter estimates given in Section 6. Although
some tuning of the adaptation gain γ may provide better
Load Speed
Reference Two Inertia
System
Feedforward
Compensation
Motor Speed Feedback
PI
Controller
Disturbance
+
−
PI
Controller
+
+
−
+
Load Speed Feedback
Motor Speed
Reference
Load Speed
Output
+
+
Fig. 6. Control scheme with feedback and feedforward compensation.
Gear Box
Laser Sensor
Load
Brake
AC Motor
Timing Belt
Gear Box
Laser Sensor
Load
Brake
AC Motor
Fig. 7. Picture of the experimental platform. Top view: load side. Bottom view:
motor side.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364360
7. transient response, in theory any positive gain and reasonable
initial conditions for the estimates would work. For simplicity,
we have chosen the values of γ ¼ 1 and the initial conditions for
both estimates to be zero; this approach has worked in a
number of experiments we have conducted with different
disturbances.
4. Ensure that the the bandwidth of the drive system is sufficient
enough to attenuate the targeted disturbance frequency.
5. The adaptive feedforward compensation requires the knowl-
edge of the disturbance frequency. This can be obtained in
multiple ways. One can obtain the frequency spectrum of the
load speed to determine all the disturbance frequencies in the
frequency region of interest. It is possible to attenuate more
than one frequency by simply cascading multiple adaptive
feedforward compensation blocks with each targeting a parti-
cular frequency. In the case of a roll-to-roll system, the dis-
turbance frequency may be simply obtained from the line speed
and the radius of the roller.
The PI controller gains for the motor speed loop were chosen to
be Kpm ¼ 15 and Kim ¼ 3:09 and for the load speed loop to be KpL
¼ 0:07 and KiL ¼ 0:001. A number of experiments were conducted
at different reference speeds to evaluate the performance of pro-
posed control scheme. In each experiment, the brake provides an
external periodic disturbance torque of the form AþB sin ðωdtÞ
ðA ¼ 2; B ¼ 1:5Þ. The following disturbance frequencies were
injected to evaluate the control schemes: ωd ¼ 0:05; 0:15; 0:25 Hz.
These disturbances are typical of the disturbances that are
observed in roll-to-roll manufacturing machines where such
transmission systems are typically employed. The adaptation gain
γ ¼ 1 is chosen and the initial values of the estimates are set
to zero.
Fig. 8 shows the evolution of the load speed (reflected to the
motor side) in the presence of disturbance with frequency 0.25 Hz
when the reference speed is 719 RPM without the use of the
adaptive feedforward action. Fig. 9 shows the Fast Fourier Trans-
form (FFT) of the load speed for the two cases. It is evident that the
control scheme with the AFF action (shown in Fig. 6) can provide
significantly improved load speed regulation. Fig. 10 controls tor-
que input corresponding to the two cases, without and with
adaptive feedforward compensation. It is evident that the torque
input is larger when the adaptive feedforward is employed. Table 1
shows the standard deviation of the load speed signal from its
reference for the various schemes. It is clear that the employing
load speed feedback in addition to motor speed feedback can
improve performance. Further, use of the adaptive feedforward
0 20 40 60
712
714
716
718
720
722
724
726
Time (Sec)
Speed(RPM)
Load Speed Response without AFF
Reference Speed
Load Speed
0 20 40 60
712
714
716
718
720
722
724
726
Time (Sec)
Speed(RPM)
Load Speed Response with AFF
Reference Speed
Load Speed
Fig. 8. Load speed response with 0.25 Hz torque disturbance. Top: without AFF.
Bottom: with AFF.
0.2 0.4 0.6 0.8 1
0
0.5
1
1.5 X: 0.2522
Y: 1.603
Frequency (Hz)
SpeedVariation(RPM)
FFT of Load Speed Response without AFF
0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
X: 0.24
Y: 0.2559
Frequency (Hz)
SpeedVariation(RPM)
FFT of Load Speed Response with AFF
Fig. 9. FFT of load speed response with and without AFF.
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364 361
8. action based on load speed feedback can significantly improve the
regulation performance.
8. Conclusions
We have investigated the problem of regulating load speed in a
mechanical transmission with a compliant belt. Several speed control
strategies that rely on either motor speed feedback or load speed
feedback or both are investigated. A singular perturbation approach
with the inverse of the belt compliance as the small parameter is
employed to analyze different control strategies. It is shown that the
system is unstable when pure load speed feedback is employed. A
control system that considers both motor speed feedback and load
speed feedback in the torque mode is stable and can provide improved
closed-loop performance. Since the feedback control action is not
sufficient to reject periodic load disturbances, an adaptive feedforward
algorithm is designed to estimate the disturbance and generate a
compensation term to attenuate periodic disturbances of known fre-
quency and unknown amplitude. Experiments were conducted on an
industrial grade transmission system to evaluate the control schemes
and compare their performance. Although we have used only belt
compliance as the compliant element in the transmission system,
torsional compliance due to long shafts can also be included and the
analysis conclusions will remain the same.
Acknowledgments
This work was supported by the Web Handling Research Center at
Oklahoma State University and the US National Science Foundation.
Appendix A
A.1. Stability of the System with only Motor Speed Feedback
The following theorem provides the stability of the system with
only motor feedback that is discussed in Section 3.
Theorem 1. The closed-loop system defined by (4) and (1) is stable
and ωm approaches ωrm for all Kpm; Kim 40.
Proof. Substituting (4) into (1) and neglecting backlash effect, we
obtain dynamics of the closed-loop system as
Kpm ωrm Àωmð ÞþKim
Z
ωrm Àωmð Þ
dτ ¼ Jm
€θm þbm
_θm
þR1Kb R1θm ÀGRR2θL
À Á
; ð28aÞ
GRR2Kb R1θm ÀGRR2θL
À Á
¼ JL
€θL þbL
_θL
: ð28bÞ
Differentiate (28) to obtain
ÀKpm _ωm þKim ωrm Àωmð Þ ¼ Jm €ωm þbm _ωm
À Á
þR1Kb R1ωm ÀGRR2ωLð Þ; ð29aÞ
GRR2Kb R1ωm ÀGRR2ωLð Þ ¼ JL €ωL þbL _ωL
À Á
: ð29bÞ
Defining errors, em ¼ ωm Àωrm and eL ¼ ωL ÀðR1=GRR2Þωrm, (29)
may be written as
ÀKpm _em ÀKimem ¼ Jm €em þbm _em þR1Kb R1em ÀGRR2eLð Þ; ð30aÞ
GRR2Kb R1em ÀGRR2eLð Þ ¼ JL
€eL þbL _eL ð30bÞ
Choose
V tð Þ ¼
1
2
Jm
_e2
m þJL
_e2
L þKbðR1em ÀGRR2eLÞ2
þKime2
m
: ð31Þ
Then, the time derivative of V along the trajectories defined by
(30) is obtained to be
dVðtÞ
dt
¼ Àðbm þKpmÞ_e2
m ÀbL _e2
L : ð32Þ
Thus, V(t) is a Lyapunov function and em; eL; _em; _eL AL1 which
implies, from (30), that €em; €eL AL1. From (31) and (32), we con-
clude that because V(t) is bounded from below and is non-
increasing with time, it has a limit, i.e., limt-1VðtÞ ¼ V1. Now
0 20 40 60
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
TorqueInput(lbf−ft)
Control Input without AFF
0 20 40 60
−1.5
−1
−0.5
0
0.5
1
1.5
Time (sec)
TorqueInput(lbf−ft)
Control Input with AFF
Fig. 10. Control input with 0.25 Hz torque disturbance. Top: without AFF. Bottom:
with AFF.
Table 1
Comparison of different control schemes.
Disturbance fre-
quency (Hz)
Standard deviation
Only motor
feedback
Motor þ load
feedback
Motor þ load
feedback þ AFF
0.25 2.09 1.35 0.34
0.15 4.71 3.53 0.87
0.05 3.89 2.47 0.68
P.R. Raul et al. / ISA Transactions 63 (2016) 355–364362
9. from (32), we have
lim
t-1
Z t
0
ðbm þKpmÞ_e2
m þbL _e2
L ¼ V0 ÀV1 o1 ð33Þ
Therefore, _em; _eL AL2 and by Barbalat's Lemma, we have _em-0
and _eL-0. Thus, ωm and ωL tend to become constants as t-1
and from (29), we see that ωm-ωrm and ωL-ðR1=GRR2Þωrm. □
A.2. Speed correction based simultaneous motor and load speed
feedback control scheme
The control scheme that utilizes both motor and load speed
feedback discussed in Section 5 considers the output of the load
speed and motor speed controllers as torque correction. There is
also another control scheme that is employed in practice where
the outer load speed loop provides a speed reference correction to
the inner motor speed which is shown in Fig. 11. In the following
we show that such a control scheme results in an unstable system,
and thus must be avoided. For this analysis, we employ a simple
proportional control action for the load speed controller and a PI
controller for the motor speed loop. The closed-loop transfer
function from ωrL to ωL for this strategy is obtained as
ωLðsÞ
ωrLðsÞ
¼
ðGRR1R2Kb=JmJLÞαmLs
ψmLsðsÞ
ð34Þ
where
αmLsðsÞ ¼ KpmKpLsþKimKpL ð35Þ
ψmLsðsÞ ¼ s4
þe3s3
þe2s2
þe1sþe0;
e3 ¼
bmJL þJmbL þJLKpm
À Á
JmJL
;
e2 ¼
KbJeq þbmbL þbLKpm þJLKim
JmJL
;
e1 ¼
Kbbeq þG2
RR2
2KbKpm þbLKim
JmJL
þ
GRR1R2KbKpmKpL
À Á
JmJL
ð36Þ
e0 ¼
G2
RR2
2KbKim þGRR1R2KbKimKpL
JmJL
:
Singular perturbation analysis results in the following slow and
fast characteristic polynomials:
ψlsðs; εÞ % s2
þδ1sþδ0 ð37aÞ
ψlf ðp; εÞ % p2
Àδ0
1pþδ0
0 ð37bÞ
where
δ1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2Kpm þðG2
RR2
2=R1ÞKpmKpL
G2
RR2
2Jm þR2
1JL
δ0 ¼
G2
RR2
2Kim þðG2
RR2
2=R1ÞKimKpL
G2
RR2
2Jm þR2
1JL
δ0
1 ¼
G2
RR2
2bm þR2
1bL þG2
RR2
2KpLKpm þG2
RR2
2KpmðJL=JmÞ
G2
RR2
2Jm þR2
1JL
ε; ð38Þ
δ0
0 ¼
G2
RR2
2JL þR2
1Jm
JmJL
:
Note that the slow subsystem is stable for all Kpm, Kim, and KpL.
However, the fast subsystem is unstable for all KpL 40. The instability
of the system is also evident from simple root locus analysis of the
closed-loop characteristic with varying KpL, which is shown in Fig. 12.
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