This document summarizes a lab where students characterized the performance of PID and bang-bang control schemes on a motor/flywheel system using LabVIEW. Students implemented both PID and bang-bang control and analyzed the performance of each by varying control parameters. PID control had smoother operation than bang-bang and was better at compensating for rise time, stability, and steady-state error. Bang-bang control resulted in more erratic behavior and was unable to eliminate steady-state error. Students also tuned the PID controller manually and using the Ziegler-Nichols method to minimize error.
This document summarizes a student lab experiment calculating control gains for a Furuta pendulum system using the linear quadratic regulator (LQR) method. Students used Matlab code to determine gain values based on the pendulum's state-space model. Different weightings were tested to balance control effort and cost. Testing on the physical pendulum found the lowest errors occurred with a weighting (R) of 7.5, indicating the best balance of control and cost for stabilizing the pendulum.
This document describes a lab experiment where students modeled a motor/flywheel system using LabVIEW. They collected data for sinusoidal and square voltage waveforms and compared the experimental model to a theoretical model based on motor specifications. Key aspects of the comparison included transfer functions, step responses, and Bode plots. Students determined parameter values, created VIs to collect experimental data, and analyzed results to compare experimental and theoretical models.
This document describes the design and implementation of a controller for an inverted pendulum on a cart system. It provides the nonlinear and linearized models of the system and designs a PID controller using root locus analysis. Simulation results show the uncompensated system is unstable but the controlled system with PID controller and pre-compensator meets design specifications with less than 0.2 seconds settling time and 8% overshoot for a unit step input.
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
1) An LQR controller with feedforward control and steady state error tracking was designed and simulated to control an inverted pendulum system.
2) The LQR controller stabilized the unstable system and achieved good performance for the pendulum angle and cart position with minimal overshoot and steady state error.
3) Simulation results demonstrated the robustness of the designed controller under system uncertainties, showing improved performance over existing H-infinity control methods.
2_DOF_Inverted_Pendulum_Laboratory_SessionPeixi Gong
This document provides an introduction and overview of a lab session on controlling a 2-DOF inverted pendulum system. It describes the equipment, typical steps in the control project including modeling and controller design. It also presents the nonlinear and linearized mathematical models of the system and exercises for students to analyze stability, observability and derive the state space models.
Iaetsd modelling and controller design of cart inverted pendulum system using...Iaetsd Iaetsd
This document presents a model reference adaptive control (MRAC) scheme for stabilizing a cart-inverted pendulum system. The cart-inverted pendulum is a highly nonlinear and unstable system that is challenging to control. The proposed controller uses Lyapunov stability theory to design an MRAC controller. Simulation results show the controller is able to balance the inverted pendulum in the unstable upright position and regulate the cart position, demonstrating the effectiveness of the proposed MRAC control approach.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
This document summarizes a student lab experiment calculating control gains for a Furuta pendulum system using the linear quadratic regulator (LQR) method. Students used Matlab code to determine gain values based on the pendulum's state-space model. Different weightings were tested to balance control effort and cost. Testing on the physical pendulum found the lowest errors occurred with a weighting (R) of 7.5, indicating the best balance of control and cost for stabilizing the pendulum.
This document describes a lab experiment where students modeled a motor/flywheel system using LabVIEW. They collected data for sinusoidal and square voltage waveforms and compared the experimental model to a theoretical model based on motor specifications. Key aspects of the comparison included transfer functions, step responses, and Bode plots. Students determined parameter values, created VIs to collect experimental data, and analyzed results to compare experimental and theoretical models.
This document describes the design and implementation of a controller for an inverted pendulum on a cart system. It provides the nonlinear and linearized models of the system and designs a PID controller using root locus analysis. Simulation results show the uncompensated system is unstable but the controlled system with PID controller and pre-compensator meets design specifications with less than 0.2 seconds settling time and 8% overshoot for a unit step input.
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
1) An LQR controller with feedforward control and steady state error tracking was designed and simulated to control an inverted pendulum system.
2) The LQR controller stabilized the unstable system and achieved good performance for the pendulum angle and cart position with minimal overshoot and steady state error.
3) Simulation results demonstrated the robustness of the designed controller under system uncertainties, showing improved performance over existing H-infinity control methods.
2_DOF_Inverted_Pendulum_Laboratory_SessionPeixi Gong
This document provides an introduction and overview of a lab session on controlling a 2-DOF inverted pendulum system. It describes the equipment, typical steps in the control project including modeling and controller design. It also presents the nonlinear and linearized mathematical models of the system and exercises for students to analyze stability, observability and derive the state space models.
Iaetsd modelling and controller design of cart inverted pendulum system using...Iaetsd Iaetsd
This document presents a model reference adaptive control (MRAC) scheme for stabilizing a cart-inverted pendulum system. The cart-inverted pendulum is a highly nonlinear and unstable system that is challenging to control. The proposed controller uses Lyapunov stability theory to design an MRAC controller. Simulation results show the controller is able to balance the inverted pendulum in the unstable upright position and regulate the cart position, demonstrating the effectiveness of the proposed MRAC control approach.
Controller design of inverted pendulum using pole placement and lqreSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Simulation of inverted pendulum presentationPourya Parsa
This document presents a simulation of controlling an inverted pendulum. It includes equations of motion to model the system state based on the pendulum angle and position over time. A sliding control method is used to control the pendulum angle and ride height, choosing control inputs to minimize errors between the actual and desired states. Simulation results are shown controlling the pendulum velocity and angle to stabilize the system. An animation demonstrates the full simulated control of the inverted pendulum.
This document describes the modeling and simulation of an inverted pendulum system. It begins with deriving the nonlinear equations of motion for an inverted pendulum mounted on a moving cart. It then linearizes the model around the equilibrium point and simulates both the linear and nonlinear models. Various controller designs are tested, including state feedback, PID control, and using position of the cart and pendulum as feedback. The linear model is shown to approximate the nonlinear model well. Increased mass or length are found to decrease stability. PID control is optimized by tuning gains.
This document discusses model reference adaptive control (MRAC). It provides an overview of the concept, the MIT rule for updating controller parameters, and an example of applying MRAC to control the position of a pendulum. Simulation and experimental results show the controller requires proportional-derivative feedback and tuning to stabilize the unstable pendulum system. More advanced control methods could provide better practical performance than the basic MRAC approach presented.
This project was developed for an Embedded systems class: we implemented a PID controller for a mechanical inverted pendulum. It was very interesting to experiment in practice with a simple control plant.
Consider the following nonlinear system:
dx/dt = f(x) + g(x)u
Where x is an n-dimensional vector and f and g are sufficiently smooth vector fields.
The goal is to design a feedback control law u = α(x) that renders the origin globally asymptotically stable (GAS).
Backstepping provides a systematic approach to solve this problem by considering the system as a cascade of "pseudo" linear systems with intermediate virtual controls.
The procedure recursively constructs stabilizing functions and control laws to backstep through this cascade until the actual control input is determined.
This approach systematically cancels out the nonlinearities in f while preserving the desirable properties introduced by g
1) The document provides an overview of inverted pendulum control, focusing on mobile inverted pendulums.
2) It describes the structure of a mobile inverted pendulum system with a cart and mounted pendulum. Equations of motion are provided.
3) Two common control strategies for inverted pendulums are discussed: PID control and fuzzy logic control. Performance comparisons using simulations show fuzzy logic control provides better response.
Iaetsd position control of servo systems using pidIaetsd Iaetsd
This document discusses using PID controllers tuned with soft computing techniques for position control of servo systems. It analyzes tuning PID controllers for a 3rd order plant model of a servo motor using the Ziegler-Nichols (ZN) method, genetic algorithm (GA), and particle swarm optimization (PSO). The step responses show that PSO provides the best performance with the fastest rise and settling times and lowest overshoot and errors. While ZN is easy to apply, GA and especially PSO give better results for controlling the servo motor's position.
Mathematical model analysis and control algorithms design based on state feed...hunypink
XZ-Ⅱtype rotary inverted pendulum is a typical mechatronic system; it completes real-time motion control using DSP motion controller and motor torque. In this paper, we recognize XZ-Ⅱrotational inverted pendulum and learn system composition, working principle, using method, precautions and software platform. We master how to build mathematical model and state feedback control method (pole assignment algorithm) of the one order rotational inverted pendulum system and finish simulation study of system using Mat lab. In the end we grasp debugging method of the actual system, and finish online control of the one order rotational inverted pendulum system as well.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Electromagnetic Levitation (control project)Salim Al Oufi
This project is about controlling the position of a magnetic ball by using PID controller. The ball position controlled by changing the magnetic field from the solenoid. Sensors give a feedback of the position of the magnetic ball. This is the simplest description for this paper.
This document summarizes a student project on stabilizing and balancing linear and rotary inverted pendulum systems. It discusses the design and implementation of PID controllers to balance an inverted pendulum mounted on a cart (linear system) and a rotary inverted pendulum prototype. Key steps included mathematical modeling, simulation in MATLAB, PID controller tuning, and applying the controller to experimental setups. Results showed the systems could be stabilized using optimized PID and LQR controllers designed via pole placement and minimizing cost functions.
This document discusses mathematical modeling of mechanical systems involving translational and rotational motion. It explains how to form differential equations of motion using Newton's laws and analogies to electrical systems. Models with multiple degrees of freedom are addressed by considering the independent motion of individual points/components and summing the relevant forces for each. Examples of 2 and 3 degree of freedom systems are presented for both translation and rotation.
MODELING AND DESIGN OF CRUISE CONTROL SYSTEM WITH FEEDFORWARD FOR ALL TERRIAN...csandit
This paper presents PID controller with feed-forward control. The cruise control system is one
of the most enduringly popular and important models for control system engineering. The
system is widely used because it is very simple to understand and yet the control techniques
cover many important classical and modern design methods. In this paper, the mathematical
modeling for PID with feed-forward controller is proposed for nonlinear model with
disturbance effect. Feed-forward controller is proposed in this study in order to eliminate the
gravitational and wind disturbance effect. Simulation will be carried out . Finally, a C++
program written and feed to the microcontroller type AMR on our robot
This document analyzes the problem of balancing an inverted pendulum, where a steel ball rolls on arched tracks attached to a movable cart. It describes the control objective of keeping the ball balanced on top of the arc while positioning the cart. The key points are:
1) The problem is modeled using basic physical equations accounting for the vertical and horizontal reaction forces on the ball and cart.
2) The equations are nonlinear and coupled, but can be linearized around the origin for control purposes.
3) State feedback control is implemented using linearized model parameters to feed back the four states to the controller.
4) Cascade control divides the problem into inner-loop ball control and outer-loop cart
STABILIZATION AT UPRIGHT EQUILIBRIUM POSITION OF A DOUBLE INVERTED PENDULUM W...ijcsa
A double inverted pendulum plant has been in the domain of control researchers as an established model for studies on stability. The stability of such as a system taking the linearized plant dynamics has yielded satisfactory results by many researchers using classical control techniques. The established model that is analyzed as part of this work was tested under the influence of time delay, where the controller was fine tuned using a BAT algorithm taking into considering the fitness function of square of error. This proposed
method gave results which were better when compared without time delay wherein the calculated values
indicated the issues when incorporating time delay
Troubleshooting and Enhancement of Inverted Pendulum System Controlled by DSP...Thomas Templin
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart-and-pole system. A normal pendulum is always stable since the pendulum hangs downward, whereas the inverted pendulum is inherently unstable and trivially underactuated (because the number of actuators is less than the degrees of freedom). For these reasons, the inverted pendulum has become one of the most important classical problems of control engineering. Since the 1950s, the inverted-pendulum benchmark, especially the cart version, has been used for the teaching and understanding of the use of linear-feedback control theory to stabilize an open-loop unstable system.
The objectives of this project are to:
• Focus on hardware and software troubleshooting and enhancement of an inverted-pendulum system controlled by a DSP28355 microprocessor and CCSv7.1 software.
• Use the swing-up strategy to move the pendulum into the unstable upward position (‘saddle’). The cart/pole system employs linear bearings for back-and-forward motion. The motor shaft has a pinion gear that rides on a track permitting the cart to move in a linear fashion. Both rack and pinion are made of hardened steel and mesh with a tight tolerance. The rack-and-pinion mechanism eliminates undesirable effects found in belt-driven and free-wheel systems, such as slippage or belt stretching, ensuring consistent and continuous traction.
• The motor shaft is coupled to a high-resolution optical encoder that accurately measures the position of the cart. The angle of the pendulum is also measured by an optical encoder, and the system employs an LQR controller to stabilize the pendulum rod at the unstable-equilibrium position.
• Addition of real-time status reporting and visualization of the system.
For the project, the Quanser High Frequency Linear Cart (HFLC) was used. The HFLC system consists of a precisely machined solid aluminum cart driven by a high-power 3-phase brushless DC motor. The cart slides along two high-precision, ground-hardened stainless steel guide rails, allowing for multiple turns and continuous measurement over the entire range of motion.
Our team implemented a control strategy that consists of a linear stabilizing LQR controller, proportional-integral swing-up control, and a supervisory coordinator that determines the control strategy (LQR or swing-up) to be used at any given time. The function of the linear stabilizer is to stabilize the system when it is in the vicinity of the unstable equilibrium. When the pendulum is in its natural state (straight-down stable-equilibrium node), the swing-up controller provides the cart/pendulum system with adequate energy to move the pendulum to the unstable equilibrium inside the “region of attraction” in which the linearized LQR controller is functional.
study of yaw and pitch control in quad copter PranaliPatil76
This document describes an experimental study of yaw and pitch control in quadcopters. It discusses how quadcopters use independent variation of rotor speeds to control pitch, roll, and yaw. Yaw control is achieved by varying the net torque on the quadcopter by increasing the thrust of rotors spinning in one direction compared to the other. Pitch control is achieved by varying the net center of thrust. The document provides equations for calculating torque and describes how adjusting rotor thrusts can cause the quadcopter to yaw or rotate in different directions.
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.
This document provides an overview of different approaches for tuning PID controllers. It first introduces PID controllers and their proportional, integral and derivative terms. It then describes several common methods for tuning PID controllers, including manual tuning on-site, Ziegler-Nichols reaction curve method, Ziegler-Nichols oscillation method, and Cohen-Coon method. These tuning methods are compared based on their performance and applicability to different process control systems.
This document provides an overview of PID controllers, including:
- The three components of a PID controller are proportional, integral, and derivative terms.
- PID controllers are widely used in industrial control systems due to their general applicability even without a mathematical model of the system.
- Ziegler-Nichols tuning rules can be used to experimentally determine initial PID parameters to provide a stable initial response for the system. Fine-tuning is then used to optimize the response.
The document discusses PID controllers, including:
1) PID controllers use proportional, integral and derivative modes to control systems. The proportional mode determines how much correction is made, the integral mode determines how long a correction is applied, and the derivative mode determines how fast a correction is made.
2) Ziegler-Nichols tuning rules provide methods to experimentally determine PID parameters (Kp, Ti, Td) when mathematical models are unknown, including open-loop and closed-loop methods using a plant's step response.
3) An electronic PID controller can be implemented as a circuit using resistors and capacitors to realize the proportional, integral and derivative terms.
Simulation of inverted pendulum presentationPourya Parsa
This document presents a simulation of controlling an inverted pendulum. It includes equations of motion to model the system state based on the pendulum angle and position over time. A sliding control method is used to control the pendulum angle and ride height, choosing control inputs to minimize errors between the actual and desired states. Simulation results are shown controlling the pendulum velocity and angle to stabilize the system. An animation demonstrates the full simulated control of the inverted pendulum.
This document describes the modeling and simulation of an inverted pendulum system. It begins with deriving the nonlinear equations of motion for an inverted pendulum mounted on a moving cart. It then linearizes the model around the equilibrium point and simulates both the linear and nonlinear models. Various controller designs are tested, including state feedback, PID control, and using position of the cart and pendulum as feedback. The linear model is shown to approximate the nonlinear model well. Increased mass or length are found to decrease stability. PID control is optimized by tuning gains.
This document discusses model reference adaptive control (MRAC). It provides an overview of the concept, the MIT rule for updating controller parameters, and an example of applying MRAC to control the position of a pendulum. Simulation and experimental results show the controller requires proportional-derivative feedback and tuning to stabilize the unstable pendulum system. More advanced control methods could provide better practical performance than the basic MRAC approach presented.
This project was developed for an Embedded systems class: we implemented a PID controller for a mechanical inverted pendulum. It was very interesting to experiment in practice with a simple control plant.
Consider the following nonlinear system:
dx/dt = f(x) + g(x)u
Where x is an n-dimensional vector and f and g are sufficiently smooth vector fields.
The goal is to design a feedback control law u = α(x) that renders the origin globally asymptotically stable (GAS).
Backstepping provides a systematic approach to solve this problem by considering the system as a cascade of "pseudo" linear systems with intermediate virtual controls.
The procedure recursively constructs stabilizing functions and control laws to backstep through this cascade until the actual control input is determined.
This approach systematically cancels out the nonlinearities in f while preserving the desirable properties introduced by g
1) The document provides an overview of inverted pendulum control, focusing on mobile inverted pendulums.
2) It describes the structure of a mobile inverted pendulum system with a cart and mounted pendulum. Equations of motion are provided.
3) Two common control strategies for inverted pendulums are discussed: PID control and fuzzy logic control. Performance comparisons using simulations show fuzzy logic control provides better response.
Iaetsd position control of servo systems using pidIaetsd Iaetsd
This document discusses using PID controllers tuned with soft computing techniques for position control of servo systems. It analyzes tuning PID controllers for a 3rd order plant model of a servo motor using the Ziegler-Nichols (ZN) method, genetic algorithm (GA), and particle swarm optimization (PSO). The step responses show that PSO provides the best performance with the fastest rise and settling times and lowest overshoot and errors. While ZN is easy to apply, GA and especially PSO give better results for controlling the servo motor's position.
Mathematical model analysis and control algorithms design based on state feed...hunypink
XZ-Ⅱtype rotary inverted pendulum is a typical mechatronic system; it completes real-time motion control using DSP motion controller and motor torque. In this paper, we recognize XZ-Ⅱrotational inverted pendulum and learn system composition, working principle, using method, precautions and software platform. We master how to build mathematical model and state feedback control method (pole assignment algorithm) of the one order rotational inverted pendulum system and finish simulation study of system using Mat lab. In the end we grasp debugging method of the actual system, and finish online control of the one order rotational inverted pendulum system as well.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Electromagnetic Levitation (control project)Salim Al Oufi
This project is about controlling the position of a magnetic ball by using PID controller. The ball position controlled by changing the magnetic field from the solenoid. Sensors give a feedback of the position of the magnetic ball. This is the simplest description for this paper.
This document summarizes a student project on stabilizing and balancing linear and rotary inverted pendulum systems. It discusses the design and implementation of PID controllers to balance an inverted pendulum mounted on a cart (linear system) and a rotary inverted pendulum prototype. Key steps included mathematical modeling, simulation in MATLAB, PID controller tuning, and applying the controller to experimental setups. Results showed the systems could be stabilized using optimized PID and LQR controllers designed via pole placement and minimizing cost functions.
This document discusses mathematical modeling of mechanical systems involving translational and rotational motion. It explains how to form differential equations of motion using Newton's laws and analogies to electrical systems. Models with multiple degrees of freedom are addressed by considering the independent motion of individual points/components and summing the relevant forces for each. Examples of 2 and 3 degree of freedom systems are presented for both translation and rotation.
MODELING AND DESIGN OF CRUISE CONTROL SYSTEM WITH FEEDFORWARD FOR ALL TERRIAN...csandit
This paper presents PID controller with feed-forward control. The cruise control system is one
of the most enduringly popular and important models for control system engineering. The
system is widely used because it is very simple to understand and yet the control techniques
cover many important classical and modern design methods. In this paper, the mathematical
modeling for PID with feed-forward controller is proposed for nonlinear model with
disturbance effect. Feed-forward controller is proposed in this study in order to eliminate the
gravitational and wind disturbance effect. Simulation will be carried out . Finally, a C++
program written and feed to the microcontroller type AMR on our robot
This document analyzes the problem of balancing an inverted pendulum, where a steel ball rolls on arched tracks attached to a movable cart. It describes the control objective of keeping the ball balanced on top of the arc while positioning the cart. The key points are:
1) The problem is modeled using basic physical equations accounting for the vertical and horizontal reaction forces on the ball and cart.
2) The equations are nonlinear and coupled, but can be linearized around the origin for control purposes.
3) State feedback control is implemented using linearized model parameters to feed back the four states to the controller.
4) Cascade control divides the problem into inner-loop ball control and outer-loop cart
STABILIZATION AT UPRIGHT EQUILIBRIUM POSITION OF A DOUBLE INVERTED PENDULUM W...ijcsa
A double inverted pendulum plant has been in the domain of control researchers as an established model for studies on stability. The stability of such as a system taking the linearized plant dynamics has yielded satisfactory results by many researchers using classical control techniques. The established model that is analyzed as part of this work was tested under the influence of time delay, where the controller was fine tuned using a BAT algorithm taking into considering the fitness function of square of error. This proposed
method gave results which were better when compared without time delay wherein the calculated values
indicated the issues when incorporating time delay
Troubleshooting and Enhancement of Inverted Pendulum System Controlled by DSP...Thomas Templin
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart-and-pole system. A normal pendulum is always stable since the pendulum hangs downward, whereas the inverted pendulum is inherently unstable and trivially underactuated (because the number of actuators is less than the degrees of freedom). For these reasons, the inverted pendulum has become one of the most important classical problems of control engineering. Since the 1950s, the inverted-pendulum benchmark, especially the cart version, has been used for the teaching and understanding of the use of linear-feedback control theory to stabilize an open-loop unstable system.
The objectives of this project are to:
• Focus on hardware and software troubleshooting and enhancement of an inverted-pendulum system controlled by a DSP28355 microprocessor and CCSv7.1 software.
• Use the swing-up strategy to move the pendulum into the unstable upward position (‘saddle’). The cart/pole system employs linear bearings for back-and-forward motion. The motor shaft has a pinion gear that rides on a track permitting the cart to move in a linear fashion. Both rack and pinion are made of hardened steel and mesh with a tight tolerance. The rack-and-pinion mechanism eliminates undesirable effects found in belt-driven and free-wheel systems, such as slippage or belt stretching, ensuring consistent and continuous traction.
• The motor shaft is coupled to a high-resolution optical encoder that accurately measures the position of the cart. The angle of the pendulum is also measured by an optical encoder, and the system employs an LQR controller to stabilize the pendulum rod at the unstable-equilibrium position.
• Addition of real-time status reporting and visualization of the system.
For the project, the Quanser High Frequency Linear Cart (HFLC) was used. The HFLC system consists of a precisely machined solid aluminum cart driven by a high-power 3-phase brushless DC motor. The cart slides along two high-precision, ground-hardened stainless steel guide rails, allowing for multiple turns and continuous measurement over the entire range of motion.
Our team implemented a control strategy that consists of a linear stabilizing LQR controller, proportional-integral swing-up control, and a supervisory coordinator that determines the control strategy (LQR or swing-up) to be used at any given time. The function of the linear stabilizer is to stabilize the system when it is in the vicinity of the unstable equilibrium. When the pendulum is in its natural state (straight-down stable-equilibrium node), the swing-up controller provides the cart/pendulum system with adequate energy to move the pendulum to the unstable equilibrium inside the “region of attraction” in which the linearized LQR controller is functional.
study of yaw and pitch control in quad copter PranaliPatil76
This document describes an experimental study of yaw and pitch control in quadcopters. It discusses how quadcopters use independent variation of rotor speeds to control pitch, roll, and yaw. Yaw control is achieved by varying the net torque on the quadcopter by increasing the thrust of rotors spinning in one direction compared to the other. Pitch control is achieved by varying the net center of thrust. The document provides equations for calculating torque and describes how adjusting rotor thrusts can cause the quadcopter to yaw or rotate in different directions.
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.
This document provides an overview of different approaches for tuning PID controllers. It first introduces PID controllers and their proportional, integral and derivative terms. It then describes several common methods for tuning PID controllers, including manual tuning on-site, Ziegler-Nichols reaction curve method, Ziegler-Nichols oscillation method, and Cohen-Coon method. These tuning methods are compared based on their performance and applicability to different process control systems.
This document provides an overview of PID controllers, including:
- The three components of a PID controller are proportional, integral, and derivative terms.
- PID controllers are widely used in industrial control systems due to their general applicability even without a mathematical model of the system.
- Ziegler-Nichols tuning rules can be used to experimentally determine initial PID parameters to provide a stable initial response for the system. Fine-tuning is then used to optimize the response.
The document discusses PID controllers, including:
1) PID controllers use proportional, integral and derivative modes to control systems. The proportional mode determines how much correction is made, the integral mode determines how long a correction is applied, and the derivative mode determines how fast a correction is made.
2) Ziegler-Nichols tuning rules provide methods to experimentally determine PID parameters (Kp, Ti, Td) when mathematical models are unknown, including open-loop and closed-loop methods using a plant's step response.
3) An electronic PID controller can be implemented as a circuit using resistors and capacitors to realize the proportional, integral and derivative terms.
Okay, let's solve this step-by-step:
* Set point (Io) = 12 rpm
* Range = 15 - 10 = 5 rpm
* Initial controller output = 22%
* KI = -0.15%/s/% error
* Error = Actual - Set point = ?
* Given: Initial output is 22%
* To find: What is the actual speed?
Using the integral control equation:
Iout = Io - KI * ∫edt
22% = 12rpm - 0.15%/s/% * ∫e dt
∫e dt = (22% - 12rpm)/0.15%/s/% = 40%*
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1. Abstract—In this lab, students characterized the performance
of PID and bang-bang control schemes in a motor/flywheel system.
Using LabVIEW, students added onto the LabVIEW file used in
the first lab to accommodate for closed loop PID and bang-bang
control. Using the data collected in LabVIEW, the performance of
both PID and bang-bang control were analyzed. In the next part
of the lab, students were askedto “feel” the PID gains by adjusting
the gains in LabVIEW and feeling the resistance from the
flywheel. Lastly, students were askedto tune theirmotor/flywheel
system to reduce error in the system using the manual tuning
method and the Ziegler-Nichols tuning method.
Index Terms—bang-bang, closed-loop, PID, Ziegler-Nichols
tuning method
I. INTRODUCTION
HIS lab tasked students with characterizing the
performance of both closed-loop PID control and closed-
loop bang-bang control. PID stands for proportional-integral-
derivative and is the most commonly used closed-loop control
system. Bang-bang is a simpler control method that also uses
closed-loop feedback to minimize feedback error in the system.
Students experimented with both control systemschemes using
the motor/flywheel systemand LabVIEW. The motor/flywheel
system is comprised of the motor/flywheel (the part of the
motor that spins), a motor controller (that controls the motors
position), a power amp (that powers the motor), a rotation
sensor(that outputs a voltage value that represents the motor’s
position), a tachometer (that outputs a voltage value that
represents the motor’s angular velocity) and a DAQ (the
interface between the system and the computer with
LabVIEW).
After the DAQ is wired to system, a square wave bang-bang
controller is used to control the system using three different
command efforts (C). After this is done, a square wave PID
controller is used with increasing proportional gain values to
control the system.This is done until the systemgoes unstable.
Next, students determined the “feel” of the PID gains by
adjusting the gains and qualitatively determining the resistance
of the flywheel when opposing motion is applied to the
flywheel.
Lastly, students were asked to tune their systemto minimize
error using a manual tuning method and the Ziegler-Nichols
tuning method.
II. PROCEDURE
PID/bang-bang control
A. PID control
The formula used for a PID controller is as follows:
𝑢(𝑡) = 𝑘 𝑝 𝑒(𝑡) + 𝑘 𝑑 𝑒̇( 𝑡) + 𝑘𝑖 ∫ 𝑒(𝑡)𝑑𝑡
𝑡
𝑡−𝑇
(1)
Where 𝑢(𝑡) is the command signal, 𝑘 𝑝 is the proportional gain,
𝑘 𝑑 is the derivative gain, 𝑘𝑖 is the integral gain and 𝑒(𝑡) is the
error defined as:
𝑒(𝑡) = 𝜃𝑑𝑒𝑠𝑖𝑟𝑒𝑑 (𝑡) − 𝜃𝑎𝑐𝑡𝑢𝑎𝑙 (𝑡) (2)
Where 𝜃𝑑𝑒𝑠𝑖𝑟𝑒𝑑 (𝑡) is the desired motor rotation and 𝜃𝑎𝑐𝑡𝑢𝑎𝑙 (𝑡)
is the actual motor rotation.
B. Bang-bang control
Bang-bang control is a simpler control method that adds
command effort when the error term is positive and subtracts
command effort when the error term is negative.The bang-bang
control law is as follows:
𝑢(𝑡) = {
+𝑐, 𝑒(𝑡) > 0
−𝑐, 𝑒(𝑡) < 0
(3)
Application ofeach control systemto the motor/flywheel system
In order to implement each control method into the
motor/flywheel system, the DAQ must first be wired to each of
the systemcomponents. With LabVIEW, students select which
control method is used by implementing a case structure that
contains both controlschemes.Afterthe desired controlmethod
is selected, the frequency is set to 0.25 Hz and the amplitude is
set to 90 degrees.The first method used is the bang-bang control
method. In order to control the systemwith this method, three
different command effort values (0.05, 0.25 and 1.0) are used
and data is recorded for each.
The next method used is the PID control method using only
a proportional gain term. This is done in a similar fashion to the
bang-bang controllerexcept that instead ofthe systemreceiving
a specific command effort as an input, the system receives a
Lab 2: PID and Bang Bang Control Comparison
Ballingham, Ryland
Section 7042 10/6/16
T
2. <Section 7042_Lab2> Double Click to Edit 2
2
proportional gain value. The proportional gain value is set three
different values (0.0001, 0.0005 and 0.001) and data is recorded
for each of the values. After this is done, the proportional gain
is increased until the system becomes unstable and the
associated gain is recorded.
Determining “feel” of PID gains
The “feel” of the gains is a qualitative assessment ofhow the
system responds when an opposing motion is applied to the
flywheel. In order to perform this task, the magnitude is set to
zero as well as the derivative and integral gains. The
proportional gain is then set to 0.0001 and then the flywheel is
manually rotated by the students and system response is felt.
The proportional gain is then increased by factors of 10, 100
and 1,000 and the system response is felt. This is done in a
similar fashion for the derivative gain. The derivative/integral
gains are done in a similar fashion. For the derivative gain, the
numerical derivative and tachometer input are used to compute
the error and the feel of each is assessed. For the integral gain,
the points to integrate overis assessedat 30 and then at 200 and
then compared.
Tuning the PID controller
A. Manual tuning method
The goal of manual tuning is to reduce rise time, overshoot
and steady-state error. This is done by increasing the
proportional gain until the output starts to oscillate. This gain
value is then divided in half to obtain the quarter amp decay.
Next, the integral gain is increased until the offset is corrected.
Lastly, the derivative gain is increased to reduce overshoot and
excessive response [1].
B. Ziegler-Nichols tuning method
The steps to perform the Ziegler-Nichols tuning method is
described in [2]. Essentially, students manually turn the
flywheel and release it until the systemgoes unstable and then
they slowly decrease the proportional gain value until the
systemhas a stable, oscillatory response.This gain value is 𝐾𝑢.
The associated period of one cycle of this oscillation is 𝑃𝑢. The
following equations showhowthe gains can be calculated from
these parameters.
𝐾𝑝 = 0.60𝐾𝑢 (4)
𝐾𝑖 =
2𝐾𝑝
𝑃𝑢
(5)
𝐾𝑑 =
𝐾𝑝 𝑃𝑢
8
(6)
III. RESULTS
TABLE I
MEAN/STANDARD DEVIATION OF ERROR/COMMAND SIGNALS
Control
scheme
Mean,
error
Standard
deviation,error
Command,
error
Command,
standard.
deviation
Bang-bang
(C=0.05)
3.45 86.83 0.05 0
Bang-bang
(C=0.25)
-1.88 85.89 0.25 0
Bang-bang
(C=1.0)
4.94 73.65 1 0
PID
(Kp=0.0001)
11.84 89.24 0.009 5.11E-05
PID
(Kp=0.0005)
-1.14 56.83 0.018 0.022
PID
(Kp=0.001)
-2.47 59.39 0.033 0.050
Manual
tuning
0.137 52.41 0.0003 0.111
Ziegler-
Nichols
tuning
6.06 59.14 0.004 0.234
TABLE II
ROOT-MEAN SQUARE VALUES
Control
Scheme
Error, rms Command, rms
Bang-bang
(C=0.05)
86.90 0.05
Bang-bang
(C=0.25)
85.91 0.25
Bang-bang
(C=1)
73.82 1.0
PID
(Kp=0.0001)
90.02 0.009
PID
(Kp=0.0005)
56.84 0.028
PID
(Kp=0.001)
59.44 0.059
Manual-
tuning
52.41 0.111
Ziegler-
Nichols
tuning
59.46 0.145
TABLE III
RISE-TIME/OVERSHOOT VALUES
Control
Scheme
Rise-time (s) Overshoot (deg)
Bang-bang
(C=0.05)
0.281 156.78
Bang-bang
(C=0.25)
0.279 162.76
Bang-bang
(C=1)
0.262 167.89
PID
(Kp=0.0001)
N/A N/A
PID
(Kp=0.0005)
0.406 -11.26
PID
(Kp=0.001)
0.537 11.07
Manual-
tuning
0.348 5.09
Ziegler-
Nichols
tuning
0.378 8.11
3. <Section 7042_Lab2> Double Click to Edit 3
3
TABLE IV
TUNING VALUES
Tuningmethod 𝐾𝑝 𝐾𝑑 𝐾𝑖
Manual tuning 0.002 0.0002 0
Ziegler-
Nichols
0.00235 0.0125 0.000110
Fig. 1. Motor rotation/command signal vs. time for C=0.05
Fig. 2. Motor rotation/command signal vs. time for C=0.25
Fig. 3. Motor rotation/command signal vs. time for C=1.0
Fig. 4. Motor rotation/command signal vs. time for Kp=0.0001 (No visual
system response)
Fig. 5. Motor rotation/command signal vs. time for Kp=0.0005
Fig. 6. Motor rotation/command signal vs. time for Kp=0.001
4. <Section 7042_Lab2> Double Click to Edit 4
4
Fig. 7. Motor rotation/command signal vs. time manual tuning
IV. DISCUSSION
A. Root Mean Square of mean/standard deviation for
error/command signals
Table I shows all the values for mean and standard
deviations of the error and command signals. From table II, it
can be seen that with increasing command effort for the bang-
bang controller, the root mean square of the error signal
decreases. For the PID controller, with increasing
proportional gain, the root mean square of the error signal
decreases up until a proportional gain value of 0.005 and then
it increased. The formulas used to find these values can be
found in the appendix section
B. Bang-bang vs. PID comparison
The bang-bang and PID controllers performed much
differently. From figs 1-3, it is clear that the bang-bang
controller is much more erratic in behavior as it always has
large oscillations around the desired command signal. Because
of this, bang-bang controllers can’t compensate for steady-state
error in an efficient fashion. For bang-bang controllers,
overshoot is always going to be a problem since the controller
is either on or off. There is no condition for a bang-bang
controller in which the error is zero.
The PID controller has much a smoother operation than the
bang-bang controller as seen in figs 4-7. The PID controller is
better at compensating for rise-time, stability and steady-state
error as it has proportional gain term that is used to manipulate
rise-time, a derivative gain term for adjusting the stability of the
system and an integral gain term that focuses on reducing
steady-state error.
C. Control effort magnitude effect
When the control effort magnitude is increased for the
bang-bang controller, the rise-time decreases and the percent
overshoot increases. With large effort values,the systemgoes
unstable.
D. Effect of proportional gain value on system performance
The larger the proportional gain value, the faster the rise-
time and the more likely that overshoot will occur. If the gain
value is increased to a value too large, the system will go
unstable. This systemwent unstable at 𝐾𝑝 = 0.015.
E. “Feel” of PID gains
The higher the 𝐾𝑝 value, the more the flywheel resist manual
movements in a linear fashion (the more force used to turn to
the flywheel, the more it resists the motion). This is because the
proportional gain is proportional to position. The higher 𝐾𝑑
value, the more the flywheel fights against manual hand
movements in an incremental fashion (The faster the flywheel
is rotated, the more it resists the motion). This is because the
derivative gain is proportional to the velocity of the flywheel.
For 𝐾𝑖, there was no visual response until 𝐾𝑖 was increased to
1. From there 𝐾𝑖 was gradually decreased until the flywheel
could be manually turned without risk of injury. The system
response seemed very slow. It doesn’t resist manual hand turns
initially, but after some time the flywheel resistance gradually
increases with time. This because the error term is being
integrated over a long time interval thus the errors of position
are becoming greater and greater.
F. Tuning method Comparison
The manual tuning method had betterresults and was easier
to implement. The manual tuning method was less intricate
and allowed for the system to be tuned in a more visual
manner. The Ziegler-Nichols method was much more time
consuming as well. Table IV shows the results forboth tuning
methods.
V. CONCLUSION
A. Bang-bang vs. PID
PID is a better way to control the motor flywheel systemdue
to its less erratic behavior. Bang-bang control seems to be much
too crude to have effective closed-loop control for the
motor/flywheel system. Since PID control has gains that can be
adjusted, it is much easier to fine tune a system properly to
increase rise time, yet decrease overshoot and steady-state
errors.
B. Manual tuning vs. Ziegler-Nichols tuning
In the lab, manual tuning yielded better results than Ziegler-
Nichols tuning. This is likely because there is less error
propagation using manual tuning as it is a less arbitrary tuning
method and it is more simple to implement.
C. Ways to improve lab
Unfortunately, while performing the lab the motor that was
being used kept overheating.This was frustrating as the there
was a large amount of down time waiting for the motor to cool
off. If the lab had a better way of keeping the motors cool it
would be highly beneficial to lab efficiency.
APPENDIX
.
𝐸𝑟𝑟𝑜𝑟𝑅𝑀 𝑆 = √ 𝑀𝑒𝑎𝑛 𝐸𝑟𝑟𝑜𝑟2 + 𝑆𝑇𝐷 𝐸𝑟𝑟𝑜𝑟2 (7)
𝐶𝑜𝑚𝑚𝑎𝑛𝑑 𝑅𝑀𝑆
= √ 𝑀𝑒𝑎𝑛 𝐶𝑜𝑚𝑚𝑎𝑛𝑑 2 + 𝑆𝑇𝐷 𝐶𝑜𝑚𝑚𝑎𝑛𝑑2
(8)
REFERENCES
[1] Wikipedia, “PID Controller”, 2016 [Online].
Available: https://en.wikipedia.org/wiki/PID_controller. Accessed: Oct
7th
, 2016
[2] LabAssignment 2, S. Banks, 2016