THE REACTION WHEEL PENDULUM

Size: px
Start display at page:

Download "THE REACTION WHEEL PENDULUM"

Transcription

1 THE REACTION WHEEL PENDULUM By Ana Navarro Yu-Han Sun Final Report for ECE 486, Control Systems, Fall 2013 TA: Dan Soberal 16 December 2013 Thursday 3-6pm

2 Contents 1. Introduction Sensors (Encoders) Actuators The Equilibrium Positions Implementation Mathematical Model Derivation of differential equations from Lagrangian Summary Linearization into state space form Full State Feedback Control with Friction Compensation PI Controller with friction compensation P controller Angular Approximation PI controller Friction compensator Friction compensator implementation Stability Proof Maximum Disturbance Windows Target Implementation Full State Feedback Control with Decoupled Observer Observers Why Observers Decoupling Error Stability Proof Maximum disturbance Windows Target Implementation Conclusion Extra Credit Up and Down Stabilizing Control Swing-Up Control ii

3 1. Introduction The Reaction Wheel Pendulum (RWP) consists of a pendulum with a rotating wheel (rotor). The rotor is actuated by a 24 volt magnet DC motor mounted on the pendulum. This motor can produce a torque on the wheel and causes the wheel to spin. The momentum from the spinning wheel causes the pendulum to move. In this project, we use the motor to control the movement of the pendulum. Figure 1.1 Schematic Diagram of System 1.1 Sensors (Encoders) The RWP is provided with two optical encoders. These encoders are relative as opposed to absolute encoders. They measure relative angle between the fixed and moving rotor. These values are initialized to zero at the start of each run. These two relative encoders are: : The relative angle between the pendulum and fixed base. : relative angle between the pendulum and wheel These encoder angles can be used to define a set of generalized coordinates to represent our system. : The angle of the pendulum measured counterclockwise from the vertical : The angle of the rotor measured counterclockwise from the vertical 1.2 Actuators There is one actuator in our entire system. This is the 24 volt magnet DC motor mounted on the pendulum. 1.3 The Equilibrium Positions The RWP has two equilibrium points: 1. Up: This equilibrium point is unstable because gravity will force the pendulum out of this position and is the point we are interested in designing our system to stay in. This location is at. 2. Down: This position is stable, because the gravity forces the rotor and pendulum to stay in this position. We denote this location as. 1

4 Figure 1.2 Unstable and Stable Equilibrium Points 1.4 Implementation In order to control the encoders, it was made a model with Windows Target. The model was made basis on the file c6xlib. This system has two encoder values, we only used one Encoder block for both data channels. To access both, we used the Demux block at the Encoder block output. The upper output of the Demux is channel 0 and the bottom channel is channel 1. Figure 1.3 Encoder Implementation 2

5 2 Mathematical Model We already know that the RWP uses relative encoders that measure relative angles and these can be used to create generalized equations about our system models. We will be using the Lagrangian method which allows us to deal with scalar rather than vectors which is a simple and convenient method of deriving equations of motion in electromechanical systems. 2.1 Derivation of differential equations from Lagrangian The RWP has two degrees of freedom, being the angles of the pendulum and of the rotor. In order to define the mathematical model, the next variables were defined: mp mr m Jp Jr lp lr l k i mass of the pendulum and motor housing/stator mass of the rotor combined mass of rotor and pendulum moment of inertia of the pendulum about its center of mass moment of inertia of the rotor about its center of mass distance from pivot to the center of mass of the pendulum distance from pivot to the center of mass of the rotor distance from pivot to the center of mass of pendulum and rotor torque constant of the motor input current to motor First, we define a set of generalized coordinates q1 and q2 as follows to represent our 2 degrees of freedom (DOF) system. Next, we compute the Kinetic Energy and Potential Energy equations in terms of these generalized coordinates. Figure 2.1 Schematic maximum and minimum position of the pendulum Potential and Kinetic Energies of Pendulum and Rotor 3

6 Potential Energy (P) Kinetic Energy(P) Potential Energy(R) Kinetic Energy(R) The Rotor potential energy is always zero because the rotor height is less than the pendulum length the height of the rotor is zero in every moment of the movement. The Lagrangian is defined as the difference between the kinetic and potential energies. The lagrangian is a function of the generalized coordinates and their derivatives: Thus, the Lagrangian equations of the RWP system are: And the derivatives respect to the generalized coordinates qr and qp, which were defined previously are: To define the torque of the pendulum and rotor, the next relation was used: Now, integrating everything in the form: generalized torque in the kth direction. where τk is the 4

7 The Lagrange equations of the system are: Lagrange Equations of Pendulum and Rotor Lagrange Equations Pendulum Lagrange Equations Rotor After that, we integrated the friction to our system. The mass on the pendulum is large enough that the friction on the pendulum link can be ignored. However, there is a significant amount of friction on the rotor link. Because of the rotor is attached directly to the motor we modeled the friction as follow: The motor current i is generated by a pulse width modulation system, which is controlled from the computer. Due to current feedback, the current is proportional to the control command u from the computer. The control variable used in the computer is scaled so that 10 units correspond to maximum current. Therefore: We assume the friction is a function of the rotor speed F(ωr), and the resulting equations are: Or we can write them in the next form: where: Summary 2.2 Linearization into state space form The Reaction Wheel Pendulum (RWP) has equations of motion, ignoring friction, given by 5

8 With this equations and two more variables that we defined as: Our system is nonlinear, thus we linearize it about the equilibrium point that we chose:, in other words, we linearized about the inverted position. With this equilibrium point we defined the next equations: And we linearized the sinusoidal function: Our final equations after the linearization are: With this equations we got the next matrix: 6

9 3. Full State Feedback Control with Friction Compensation We want to model the friction in terms of velocity. Because the information provided from the encoders are not velocity, we use an approximation instead. The motion of the wheel can be modeled as: where Jr is moment of inertia, and ωr is the speed of the rotor. There is also friction, due mainly to the motor brushes and represented as a torque τf. To obtain ωr, we take the first derivative of θr. Putting the motion of the wheel equation in terms of friction, we got the following: 3.1 PI Controller with friction compensation P controller In order to develop a PI control with friction compensation, first we design the p controller without the friction compensator using the next block diagram: Figure 3.1 General block diagram of velocity controller Computing the transfer function of this block diagram, we got the following: We know that br = 198 and that the rise time is inversely proportional to the factor σ, which represent the pole real part in the s-plane. Therefore we can conclude: Looking at P control only, for a rise time of 0.2 sec and using the following rise time equation, we can compute the value of Kp: 7

10 We chose a gain Kp of However, P control is not enough to get rid of steady state error. Figure 3.2 Simulink P control block diagram Angular Approximation In order to implement our proportional controller it is necessary to estimate the angular velocity of our RWP. We used a continuous derivative approximation to estimate the angular velocity of the wheel and the pendulum. This approximation consist in to keep the frequency response of the derivative function s similar at low frequencies, but refrain from amplifying the high frequency noise. To accomplish this we place a pole at a sufficiently high frequency, we got the following transfer function: PI controller Because of a P control is not enough to get rid of steady state error, we designed a PI control to meet the specification of no steady state error. First we computed the damping ratio ( ζ ) and frequency (ωn ) based on the specifications: We used ωn = 5.45 and ζ=0.545 to develop our PI controller. The transfer function of our controller G(s) and the transfer function of our system H(s) are: To compute we used the denominator of our system transfer function and the characteristic polynomial for a second order transfer function: 8

11 3.1.4 Friction compensator To characterize friction we used the control effort of our system using a closed-loop system identification. Analyzing the following friction equation, defined before: We see that for non-zero friction, the control effort will be non-zero as well. The value of friction for any velocity is merely the steady state control effort for a setpoint of that velocity. Thus, we defined the following equation: where ucl is the closed-loop controller. In order to find an expression for F(ω), we developed a series of experiments running the motor at various speeds and recording the steady-state control effort for each speed. We did this for negative and positive velocities. After, we plot and fit the data acquired in a plot of ω vs ucl.we got the next results: Figure 3.3 Negative Velocities Plot 9

12 Figure 3.4 Positive Velocities Plot Then we compute the values for +b and +c from the linear equation got from the positive velocities plot, and we computed the values -b and -c from the linear equation got from the negative velocities plot. With this information we wrote one positive and one negative expression for F(ω): Friction compensator implementation With this values of +b,-b,+c and -c we add the friction compensator to our system with the block Coulomb and viscous friction. The final block diagram of our system is the following: Figure 3.5 With and Without Friction Compensation Systems We made a comparison between the results implementing the P controller, PI controller and with or without friction compensator in our system. The results are the following: 10

13 The conclusion of this table is that it is unnecessary to implement the friction compensator with any kind of controller, because the friction compensator can meet the specifications by itself. This is possible because the unique disturbance of our system is the friction and we are eliminating it with the friction compensator. 3.2 Stability Proof Because of we don t care what θr is, and it does not affect any of the other states we created the following system in order to check the stability of our system: Then we computed the equilibrium points of our system solving the following equations: From this equations we concluded that the equilibrium points are: 3.3 Maximum Disturbance Max Two State Feedback Three-State Feedback IC dev *π (δθp) 0.388*π (θp).0368*π (δθp) 0.331*π (θp) 11

14 Pulse dist Windows Target Implementation Figure 3.6 Three-State Feedback Control In Windows Target, x matrix delta_theta_p, theta_p_dot, delta_theta_r and theta_r_dot were multiplied with the respective feedback gain matrix K. Only three gains were applied since we have only a 3-state feedback control. These values were summed, saturated and put through the Motor. In Figure 3.6, we can see the constant pi was subtracted from theta_p to ensure that we are converging to the correct equilibrium point. 12

15 4. Full State Feedback Control with Decoupled Observer 4.1 Observers An observer is a dynamic system whose input includes the control u and the output y and whose output is an estimate of the state vector x. We use observers because not all state variables can be measured and therefore we need an estimate of the state vector for state feedback Why Observers The observer replaces the full state feedback controller we designed previously. The observer will estimate both velocities of the system. And since we re designing a full-order observer, it will also estimate both positions. Observer poles have to be sufficiently farther than the desired closed loop poles. Figure 4.1 Illustration of Pole locations There were originally four states together as shown in this diagram: Figure 4.2 Block Diagram of System with Observer the standard differential equation for an observer is: 13

16 where: The above equation can be manipulated to be in the familiar form of Ax+Bu: The derivation is as follows: In conclusion, our Observer state form can be broken down into A, B, C and D matrices Decoupling We can decouple the Observer because our state-space model has the form of: Matrix A can be sectioned off into blocks of 2x2 matrices where the top right and bottom left blocks are 0. 14

17 such that our new vector Equations are This proves that our subsystems are independent of each other, in other words decoupled. This is because A is in block diagonal form. This property also proves that the eigenvalues of A is the union of the eigenvalues of M and the eigenvalues of N. Because of this, we can create the two blocks indepently in matlab with the place command and keep the same poles. Unfortunately, Matlab doesn t check for independence, so our A-LC matrix is wrong. However, it is sufficient to just zero out the terms instead of calculating the block diagonals and then combining. 4.2 Error Stability Proof 4.3 Maximum disturbance Max Observer 4-state Observer 2-state IC deviation *π (δθp) 0.206*π (θp) *π (δθp) 0.317*π (θp) Pulse Disturbance

18 4.4 Windows Target Implementation Figure 4.3 Windows Target simulation with observer In the simulation, we take the decoded relative angles to create θp and θr. Then take the derivative for theta_p_dot and theta_r_dot. These four values are being fed into the observer State System whose A, B, C and D matrices are the observer matrices. The matrix K is from the previous 3-state feedback controller. And the -1 gain makes sure that that K is negative. The results are combined with the friction compensator and then the resulting u re-enters the Motor. 16

19 5. Conclusion 17

20 6. Extra Credit 6.1 Up and Down Stabilizing Control In the previous chapters, when the pendulum fell outside a maximum IC deviation, the pendulum would become unstable and try endlessly to return back to the equilibrium point at theta_p = pi. In this section, we use switching control such that if the pendulum falls outside the theta_p=pi equilibrium point, the pendulum will converge to a second equilibrium point theta_p=0. Figure 6.1 Switching Control The down controller is for the lower equilibrium point and the up controller is for the upper equilibrium point as from the previous sections. The deciding factor uses the function block cosine on theta_p. If the cosine of theta_p is greater than the breakpoint, then we use down control and if it is less than the breakpoint, we use up control. The down controller requires a new set of feedback values that converge to the other equilibrium point. The derivation is as follows. Since the only difference between this controller and the previous controller is that delta_theta_p = theta_p+0 instead of theta_p_pi, then: Where the resulting K values were: 18

21 Figure 6.2 Switch Control for two equilibrium points 6.2 Swing-Up Control Non-linear control can be useful when used with other control algoithms. We use a switch to switch between nonlinear and linear control. The switch is approximately the same as the previous switch control, with the cosine of the angle as a deciding factor. The idea for swing-up control is to give momentum to the system at the correct times and location. Figure 6.3 Illustration of Nonlinear Swingup Control This can be achieved by looking at energy. We can measure kinetic and potential energy and based on this information, determine how much work the motor must do onto the wheel to propel the pendulum upwards. 19

22 In the Windows Target implementation, we can see that if the cosine of theta_p causes the switch control to switch to non-linear control, the new system implements the above equations with a gain of 8.8 (found through trial and error) that will swing the system up. Figure 6.4 Switch Control for Up Swing 20

Mechatronic System Case Study: Rotary Inverted Pendulum Dynamic System Investigation

Mechatronic System Case Study: Rotary Inverted Pendulum Dynamic System Investigation Mechatronic System Case Study: Rotary Inverted Pendulum Dynamic System Investigation Dr. Kevin Craig Greenheck Chair in Engineering Design & Professor of Mechanical Engineering Marquette University K.

More information

Automatic Control Systems. -Lecture Note 15-

Automatic Control Systems. -Lecture Note 15- -Lecture Note 15- Modeling of Physical Systems 5 1/52 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Cost-effective ii)

More information

Example: Modeling DC Motor Position Physical Setup System Equations Design Requirements MATLAB Representation and Open-Loop Response

Example: Modeling DC Motor Position Physical Setup System Equations Design Requirements MATLAB Representation and Open-Loop Response Page 1 of 5 Example: Modeling DC Motor Position Physical Setup System Equations Design Requirements MATLAB Representation and Open-Loop Response Physical Setup A common actuator in control systems is the

More information

Example: DC Motor Speed Modeling

Example: DC Motor Speed Modeling Page 1 of 5 Example: DC Motor Speed Modeling Physical setup and system equations Design requirements MATLAB representation and open-loop response Physical setup and system equations A common actuator in

More information

Real-Time Implementation of a LQR-Based Controller for the Stabilization of a Double Inverted Pendulum

Real-Time Implementation of a LQR-Based Controller for the Stabilization of a Double Inverted Pendulum Proceedings of the International MultiConference of Engineers and Computer Scientists 017 Vol I,, March 15-17, 017, Hong Kong Real-Time Implementation of a LQR-Based Controller for the Stabilization of

More information

The Control of an Inverted Pendulum

The Control of an Inverted Pendulum The Control of an Inverted Pendulum AAE 364L This experiment is devoted to the inverted pendulum. Clearly, the inverted pendulum will fall without any control. We will design a controller to balance the

More information

Lab 6a: Pole Placement for the Inverted Pendulum

Lab 6a: Pole Placement for the Inverted Pendulum Lab 6a: Pole Placement for the Inverted Pendulum Idiot. Above her head was the only stable place in the cosmos, the only refuge from the damnation of the Panta Rei, and she guessed it was the Pendulum

More information

SRV02-Series Rotary Experiment # 7. Rotary Inverted Pendulum. Student Handout

SRV02-Series Rotary Experiment # 7. Rotary Inverted Pendulum. Student Handout SRV02-Series Rotary Experiment # 7 Rotary Inverted Pendulum Student Handout SRV02-Series Rotary Experiment # 7 Rotary Inverted Pendulum Student Handout 1. Objectives The objective in this experiment is

More information

FEEDBACK CONTROL SYSTEMS

FEEDBACK CONTROL SYSTEMS FEEDBAC CONTROL SYSTEMS. Control System Design. Open and Closed-Loop Control Systems 3. Why Closed-Loop Control? 4. Case Study --- Speed Control of a DC Motor 5. Steady-State Errors in Unity Feedback Control

More information

Lab 6d: Self-Erecting Inverted Pendulum (SEIP)

Lab 6d: Self-Erecting Inverted Pendulum (SEIP) Lab 6d: Self-Erecting Inverted Pendulum (SEIP) Arthur Schopen- Life swings like a pendulum backward and forward between pain and boredom. hauer 1 Objectives The goal of this project is to design a controller

More information

Inverted Pendulum System

Inverted Pendulum System Introduction Inverted Pendulum System This lab experiment consists of two experimental procedures, each with sub parts. Experiment 1 is used to determine the system parameters needed to implement a controller.

More information

DC Motor Position: System Modeling

DC Motor Position: System Modeling 1 of 7 01/03/2014 22:07 Tips Effects TIPS ABOUT BASICS INDEX NEXT INTRODUCTION CRUISE CONTROL MOTOR SPEED MOTOR POSITION SUSPENSION INVERTED PENDULUM SYSTEM MODELING ANALYSIS DC Motor Position: System

More information

Stabilization of Motion of the Segway 1

Stabilization of Motion of the Segway 1 Stabilization of Motion of the Segway 1 Houtman P. Siregar, 2 Yuri G. Martynenko 1 Department of Mechatronics Engineering, Indonesia Institute of Technology, Jl. Raya Puspiptek-Serpong, Indonesia 15320,

More information

Lecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization

Lecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization Lecture 9 Nonlinear Control Design Course Outline Eact-linearization Lyapunov-based design Lab Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.] and [Glad-Ljung,ch.17] Lecture

More information

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar

More information

The Control of an Inverted Pendulum

The Control of an Inverted Pendulum The Control of an Inverted Pendulum AAE 364L This experiment is devoted to the inverted pendulum. Clearly, the inverted pendulum will fall without any control. We will design a controller to balance the

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Mechanical Engineering 2.04A Systems and Controls Spring 2013 Problem Set #4 Posted: Thursday, Mar. 7, 13 Due: Thursday, Mar. 14, 13 1. Sketch the Root

More information

Satellite Attitude Control System Design Using Reaction Wheels Bhanu Gouda Brian Fast Dan Simon

Satellite Attitude Control System Design Using Reaction Wheels Bhanu Gouda Brian Fast Dan Simon Satellite Attitude Control System Design Using Reaction Wheels Bhanu Gouda Brian Fast Dan Simon Outline 1. Overview of Attitude Determination and Control system. Problem formulation 3. Control schemes

More information

Inverted Pendulum: State-Space Methods for Controller Design

Inverted Pendulum: State-Space Methods for Controller Design 1 de 12 18/10/2015 22:45 Tips Effects TIPS ABOUT BASICS HARDWARE INDEX NEXT INTRODUCTION CRUISE CONTROL MOTOR SPEED MOTOR POSITION SYSTEM MODELING ANALYSIS Inverted Pendulum: State-Space Methods for Controller

More information

Modeling and control design for a semi-active suspension system with magnetorheological rotary brake

Modeling and control design for a semi-active suspension system with magnetorheological rotary brake Modeling and control design for a semi-active suspension system with magnetorheological rotary brake Geir-Arne Moslått, Erik Myklebust, Palmer Kolberg and Hamid Reza Karimi Department of Engineering, University

More information

Double Inverted Pendulum (DBIP)

Double Inverted Pendulum (DBIP) Linear Motion Servo Plant: IP01_2 Linear Experiment #15: LQR Control Double Inverted Pendulum (DBIP) All of Quanser s systems have an inherent open architecture design. It should be noted that the following

More information

MEM04: Rotary Inverted Pendulum

MEM04: Rotary Inverted Pendulum MEM4: Rotary Inverted Pendulum Interdisciplinary Automatic Controls Laboratory - ME/ECE/CHE 389 April 8, 7 Contents Overview. Configure ELVIS and DC Motor................................ Goals..............................................3

More information

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67 1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure

More information

Lab 3: Quanser Hardware and Proportional Control

Lab 3: Quanser Hardware and Proportional Control Lab 3: Quanser Hardware and Proportional Control The worst wheel of the cart makes the most noise. Benjamin Franklin 1 Objectives The goal of this lab is to: 1. familiarize you with Quanser s QuaRC tools

More information

Rotary Motion Servo Plant: SRV02. Rotary Experiment #11: 1-DOF Torsion. 1-DOF Torsion Position Control using QuaRC. Student Manual

Rotary Motion Servo Plant: SRV02. Rotary Experiment #11: 1-DOF Torsion. 1-DOF Torsion Position Control using QuaRC. Student Manual Rotary Motion Servo Plant: SRV02 Rotary Experiment #11: 1-DOF Torsion 1-DOF Torsion Position Control using QuaRC Student Manual Table of Contents 1. INTRODUCTION...1 2. PREREQUISITES...1 3. OVERVIEW OF

More information

Kinetic Energy of Rolling

Kinetic Energy of Rolling Kinetic Energy of Rolling A solid disk and a hoop (with the same mass and radius) are released from rest and roll down a ramp from a height h. Which one is moving faster at the bottom of the ramp? A. they

More information

Design and Comparison of Different Controllers to Stabilize a Rotary Inverted Pendulum

Design and Comparison of Different Controllers to Stabilize a Rotary Inverted Pendulum ISSN (Online): 347-3878, Impact Factor (5): 3.79 Design and Comparison of Different Controllers to Stabilize a Rotary Inverted Pendulum Kambhampati Tejaswi, Alluri Amarendra, Ganta Ramesh 3 M.Tech, Department

More information

ME 3210 Mechatronics II Laboratory Lab 4: DC Motor Characteristics

ME 3210 Mechatronics II Laboratory Lab 4: DC Motor Characteristics ME 3210 Mechatronics II Laboratory Lab 4: DC Motor Characteristics Introduction Often, due to budget constraints or convenience, engineers must use whatever tools are available to create new or improved

More information

Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control. DC Motor Control Trainer (DCMCT) Student Manual

Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control. DC Motor Control Trainer (DCMCT) Student Manual Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #02: DC Motor Position Control DC Motor Control Trainer (DCMCT) Student Manual Table of Contents 1 Laboratory Objectives1 2 References1 3 DCMCT Plant

More information

Laboratory Exercise 1 DC servo

Laboratory Exercise 1 DC servo Laboratory Exercise DC servo Per-Olof Källén ø 0,8 POWER SAT. OVL.RESET POS.RESET Moment Reference ø 0,5 ø 0,5 ø 0,5 ø 0,65 ø 0,65 Int ø 0,8 ø 0,8 Σ k Js + d ø 0,8 s ø 0 8 Off Off ø 0,8 Ext. Int. + x0,

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

More information

MEAM 510 Fall 2011 Bruce D. Kothmann

MEAM 510 Fall 2011 Bruce D. Kothmann Balancing g Robot Control MEAM 510 Fall 2011 Bruce D. Kothmann Agenda Bruce s Controls Resume Simple Mechanics (Statics & Dynamics) of the Balancing Robot Basic Ideas About Feedback & Stability Effects

More information

DC-motor PID control

DC-motor PID control DC-motor PID control This version: November 1, 2017 REGLERTEKNIK Name: P-number: AUTOMATIC LINKÖPING CONTROL Date: Passed: Chapter 1 Introduction The purpose of this lab is to give an introduction to

More information

Manufacturing Equipment Control

Manufacturing Equipment Control QUESTION 1 An electric drive spindle has the following parameters: J m = 2 1 3 kg m 2, R a = 8 Ω, K t =.5 N m/a, K v =.5 V/(rad/s), K a = 2, J s = 4 1 2 kg m 2, and K s =.3. Ignore electrical dynamics

More information

Lab 5a: Pole Placement for the Inverted Pendulum

Lab 5a: Pole Placement for the Inverted Pendulum Lab 5a: Pole Placement for the Inverted Pendulum November 1, 2011 1 Purpose The objective of this lab is to achieve simultaneous control of both the angular position of the pendulum and horizontal position

More information

Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise

Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise Mechatronics Modeling and Analysis of Dynamic Systems Case-Study Exercise Goal: This exercise is designed to take a real-world problem and apply the modeling and analysis concepts discussed in class. As

More information

ENGG4420 LECTURE 7. CHAPTER 1 BY RADU MURESAN Page 1. September :29 PM

ENGG4420 LECTURE 7. CHAPTER 1 BY RADU MURESAN Page 1. September :29 PM CHAPTER 1 BY RADU MURESAN Page 1 ENGG4420 LECTURE 7 September 21 10 2:29 PM MODELS OF ELECTRIC CIRCUITS Electric circuits contain sources of electric voltage and current and other electronic elements such

More information

The control of a gantry

The control of a gantry The control of a gantry AAE 364L In this experiment we will design a controller for a gantry or crane. Without a controller the pendulum of crane will swing for a long time. The idea is to use control

More information

Introduction to Control (034040) lecture no. 2

Introduction to Control (034040) lecture no. 2 Introduction to Control (034040) lecture no. 2 Leonid Mirkin Faculty of Mechanical Engineering Technion IIT Setup: Abstract control problem to begin with y P(s) u where P is a plant u is a control signal

More information

Department of Mechanical Engineering

Department of Mechanical Engineering Department of Mechanical Engineering 2.010 CONTROL SYSTEMS PRINCIPLES Laboratory 2: Characterization of the Electro-Mechanical Plant Introduction: It is important (for future lab sessions) that we have

More information

FUZZY LOGIC CONTROL Vs. CONVENTIONAL PID CONTROL OF AN INVERTED PENDULUM ROBOT

FUZZY LOGIC CONTROL Vs. CONVENTIONAL PID CONTROL OF AN INVERTED PENDULUM ROBOT http:// FUZZY LOGIC CONTROL Vs. CONVENTIONAL PID CONTROL OF AN INVERTED PENDULUM ROBOT 1 Ms.Mukesh Beniwal, 2 Mr. Davender Kumar 1 M.Tech Student, 2 Asst.Prof, Department of Electronics and Communication

More information

Automatic Control II Computer exercise 3. LQG Design

Automatic Control II Computer exercise 3. LQG Design Uppsala University Information Technology Systems and Control HN,FS,KN 2000-10 Last revised by HR August 16, 2017 Automatic Control II Computer exercise 3 LQG Design Preparations: Read Chapters 5 and 9

More information

MEAM 510 Fall 2012 Bruce D. Kothmann

MEAM 510 Fall 2012 Bruce D. Kothmann Balancing g Robot Control MEAM 510 Fall 2012 Bruce D. Kothmann Agenda Bruce s Controls Resume Simple Mechanics (Statics & Dynamics) of the Balancing Robot Basic Ideas About Feedback & Stability Effects

More information

Dynamic Modeling of Rotary Double Inverted Pendulum Using Classical Mechanics

Dynamic Modeling of Rotary Double Inverted Pendulum Using Classical Mechanics ISBN 978-93-84468-- Proceedings of 5 International Conference on Future Computational echnologies (ICFC'5) Singapore, March 9-3, 5, pp. 96-3 Dynamic Modeling of Rotary Double Inverted Pendulum Using Classical

More information

ECE 5670/6670 Lab 8. Torque Curves of Induction Motors. Objectives

ECE 5670/6670 Lab 8. Torque Curves of Induction Motors. Objectives ECE 5670/6670 Lab 8 Torque Curves of Induction Motors Objectives The objective of the lab is to measure the torque curves of induction motors. Acceleration experiments are used to reconstruct approximately

More information

Control of Electromechanical Systems

Control of Electromechanical Systems Control of Electromechanical Systems November 3, 27 Exercise Consider the feedback control scheme of the motor speed ω in Fig., where the torque actuation includes a time constant τ A =. s and a disturbance

More information

EE 422G - Signals and Systems Laboratory

EE 422G - Signals and Systems Laboratory EE 4G - Signals and Systems Laboratory Lab 9 PID Control Kevin D. Donohue Department of Electrical and Computer Engineering University of Kentucky Lexington, KY 40506 April, 04 Objectives: Identify the

More information

Digital Control Semester Project

Digital Control Semester Project Digital Control Semester Project Part I: Transform-Based Design 1 Introduction For this project you will be designing a digital controller for a system which consists of a DC motor driving a shaft with

More information

University of Utah Electrical & Computer Engineering Department ECE 3510 Lab 9 Inverted Pendulum

University of Utah Electrical & Computer Engineering Department ECE 3510 Lab 9 Inverted Pendulum University of Utah Electrical & Computer Engineering Department ECE 3510 Lab 9 Inverted Pendulum p1 ECE 3510 Lab 9, Inverted Pendulum M. Bodson, A. Stolp, 4/2/13 rev, 4/9/13 Objectives The objective of

More information

Lecture 9 Nonlinear Control Design

Lecture 9 Nonlinear Control Design Lecture 9 Nonlinear Control Design Exact-linearization Lyapunov-based design Lab 2 Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.2] and [Glad-Ljung,ch.17] Course Outline

More information

FAULT DETECTION for SPACECRAFT ATTITUDE CONTROL SYSTEM. M. Amin Vahid D. Mechanical Engineering Department Concordia University December 19 th, 2010

FAULT DETECTION for SPACECRAFT ATTITUDE CONTROL SYSTEM. M. Amin Vahid D. Mechanical Engineering Department Concordia University December 19 th, 2010 FAULT DETECTION for SPACECRAFT ATTITUDE CONTROL SYSTEM M. Amin Vahid D. Mechanical Engineering Department Concordia University December 19 th, 2010 Attitude control : the exercise of control over the orientation

More information

System Parameters and Frequency Response MAE 433 Spring 2012 Lab 2

System Parameters and Frequency Response MAE 433 Spring 2012 Lab 2 System Parameters and Frequency Response MAE 433 Spring 2012 Lab 2 Prof. Rowley, Prof. Littman AIs: Brandt Belson, Jonathan Tu Technical staff: Jonathan Prévost Princeton University Feb. 21-24, 2012 1

More information

Line following of a mobile robot

Line following of a mobile robot Line following of a mobile robot May 18, 004 1 In brief... The project is about controlling a differential steering mobile robot so that it follows a specified track. Steering is achieved by setting different

More information

Lab 11: Rotational Dynamics

Lab 11: Rotational Dynamics Lab 11: Rotational Dynamics Objectives: To understand the relationship between net torque and angular acceleration. To understand the concept of the moment of inertia. To understand the concept of angular

More information

Linear Motion Servo Plant: IP02. Linear Experiment #4: Pole Placement. Single Pendulum Gantry (SPG) Student Handout

Linear Motion Servo Plant: IP02. Linear Experiment #4: Pole Placement. Single Pendulum Gantry (SPG) Student Handout Linear Motion Servo Plant: IP0 Linear Experiment #4: Pole Placement Single Pendulum Gantry (SPG) Student Handout Table of Contents 1. Objectives...1. Prerequisites...1 3. References... 4. Experimental

More information

Position Control Experiment MAE171a

Position Control Experiment MAE171a Position Control Experiment MAE171a January 11, 014 Prof. R.A. de Callafon, Dept. of MAE, UCSD TAs: Jeff Narkis, email: jnarkis@ucsd.edu Gil Collins, email: gwcollin@ucsd.edu Contents 1 Aim and Procedure

More information

Linear Experiment #11: LQR Control. Linear Flexible Joint Cart Plus Single Inverted Pendulum (LFJC+SIP) Student Handout

Linear Experiment #11: LQR Control. Linear Flexible Joint Cart Plus Single Inverted Pendulum (LFJC+SIP) Student Handout Linear Motion Servo Plants: IP01 or IP02 Linear Experiment #11: LQR Control Linear Flexible Joint Cart Plus Single Inverted Pendulum (LFJC+SIP) Student Handout Table of Contents 1. Objectives...1 2. Prerequisites...2

More information

UNIVERSITY OF WASHINGTON Department of Aeronautics and Astronautics

UNIVERSITY OF WASHINGTON Department of Aeronautics and Astronautics UNIVERSITY OF WASHINGTON Department of Aeronautics and Astronautics Modeling and Control of a Flexishaft System March 19, 2003 Christopher Lum Travis Reisner Amanda Stephens Brian Hass AA/EE-448 Controls

More information

Rotary Inverted Pendulum

Rotary Inverted Pendulum Rotary Inverted Pendulum Eric Liu 1 Aug 2013 1 1 State Space Derivations 1.1 Electromechanical Derivation Consider the given diagram. We note that the voltage across the motor can be described by: e b

More information

Mechatronics Engineering. Li Wen

Mechatronics Engineering. Li Wen Mechatronics Engineering Li Wen Bio-inspired robot-dc motor drive Unstable system Mirko Kovac,EPFL Modeling and simulation of the control system Problems 1. Why we establish mathematical model of the control

More information

1 x(k +1)=(Φ LH) x(k) = T 1 x 2 (k) x1 (0) 1 T x 2(0) T x 1 (0) x 2 (0) x(1) = x(2) = x(3) =

1 x(k +1)=(Φ LH) x(k) = T 1 x 2 (k) x1 (0) 1 T x 2(0) T x 1 (0) x 2 (0) x(1) = x(2) = x(3) = 567 This is often referred to as Þnite settling time or deadbeat design because the dynamics will settle in a Þnite number of sample periods. This estimator always drives the error to zero in time 2T or

More information

kx m x B N 1 C L, M Mg θ

kx m x B N 1 C L, M Mg θ .004 MODELING DYNAMICS AND CONTROL II Spring 00 Solutions to Problem Set No. 7 Problem 1. Pendulum mounted on elastic support. This problem is an execise in the application of momentum principles. Two

More information

QNET Experiment #04: Inverted Pendulum Control. Rotary Pendulum (ROTPEN) Inverted Pendulum Trainer. Instructor Manual

QNET Experiment #04: Inverted Pendulum Control. Rotary Pendulum (ROTPEN) Inverted Pendulum Trainer. Instructor Manual Quanser NI-ELVIS Trainer (QNET) Series: QNET Experiment #04: Inverted Pendulum Control Rotary Pendulum (ROTPEN) Inverted Pendulum Trainer Instructor Manual Table of Contents 1 Laboratory Objectives1 2

More information

Stepping Motors. Chapter 11 L E L F L D

Stepping Motors. Chapter 11 L E L F L D Chapter 11 Stepping Motors In the synchronous motor, the combination of sinusoidally distributed windings and sinusoidally time varying current produces a smoothly rotating magnetic field. We can eliminate

More information

Acceleration Feedback

Acceleration Feedback Acceleration Feedback Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic

More information

Introduction to Feedback Control

Introduction to Feedback Control Introduction to Feedback Control Control System Design Why Control? Open-Loop vs Closed-Loop (Feedback) Why Use Feedback Control? Closed-Loop Control System Structure Elements of a Feedback Control System

More information

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30

Lecture 12. Upcoming labs: Final Exam on 12/21/2015 (Monday)10:30-12:30 289 Upcoming labs: Lecture 12 Lab 20: Internal model control (finish up) Lab 22: Force or Torque control experiments [Integrative] (2-3 sessions) Final Exam on 12/21/2015 (Monday)10:30-12:30 Today: Recap

More information

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory

Bangladesh University of Engineering and Technology. EEE 402: Control System I Laboratory Bangladesh University of Engineering and Technology Electrical and Electronic Engineering Department EEE 402: Control System I Laboratory Experiment No. 4 a) Effect of input waveform, loop gain, and system

More information

State Feedback Controller for Position Control of a Flexible Link

State Feedback Controller for Position Control of a Flexible Link Laboratory 12 Control Systems Laboratory ECE3557 Laboratory 12 State Feedback Controller for Position Control of a Flexible Link 12.1 Objective The objective of this laboratory is to design a full state

More information

(Refer Slide Time: 00:01:30 min)

(Refer Slide Time: 00:01:30 min) Control Engineering Prof. M. Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 3 Introduction to Control Problem (Contd.) Well friends, I have been giving you various

More information

Lab 3: Model based Position Control of a Cart

Lab 3: Model based Position Control of a Cart I. Objective Lab 3: Model based Position Control of a Cart The goal of this lab is to help understand the methodology to design a controller using the given plant dynamics. Specifically, we would do position

More information

Lecture Module 5: Introduction to Attitude Stabilization and Control

Lecture Module 5: Introduction to Attitude Stabilization and Control 1 Lecture Module 5: Introduction to Attitude Stabilization and Control Lectures 1-3 Stability is referred to as a system s behaviour to external/internal disturbances (small) in/from equilibrium states.

More information

Video 5.1 Vijay Kumar and Ani Hsieh

Video 5.1 Vijay Kumar and Ani Hsieh Video 5.1 Vijay Kumar and Ani Hsieh Robo3x-1.1 1 The Purpose of Control Input/Stimulus/ Disturbance System or Plant Output/ Response Understand the Black Box Evaluate the Performance Change the Behavior

More information

YTÜ Mechanical Engineering Department

YTÜ Mechanical Engineering Department YTÜ Mechanical Engineering Department Lecture of Special Laboratory of Machine Theory, System Dynamics and Control Division Coupled Tank 1 Level Control with using Feedforward PI Controller Lab Date: Lab

More information

Lecture 6: Control Problems and Solutions. CS 344R: Robotics Benjamin Kuipers

Lecture 6: Control Problems and Solutions. CS 344R: Robotics Benjamin Kuipers Lecture 6: Control Problems and Solutions CS 344R: Robotics Benjamin Kuipers But First, Assignment 1: Followers A follower is a control law where the robot moves forward while keeping some error term small.

More information

DEVELOPMENT OF DIRECT TORQUE CONTROL MODELWITH USING SVI FOR THREE PHASE INDUCTION MOTOR

DEVELOPMENT OF DIRECT TORQUE CONTROL MODELWITH USING SVI FOR THREE PHASE INDUCTION MOTOR DEVELOPMENT OF DIRECT TORQUE CONTROL MODELWITH USING SVI FOR THREE PHASE INDUCTION MOTOR MUKESH KUMAR ARYA * Electrical Engg. Department, Madhav Institute of Technology & Science, Gwalior, Gwalior, 474005,

More information

Linear State Feedback Controller Design

Linear State Feedback Controller Design Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University

More information

YTÜ Mechanical Engineering Department

YTÜ Mechanical Engineering Department YTÜ Mechanical Engineering Department Lecture of Special Laboratory of Machine Theory, System Dynamics and Control Division Coupled Tank 1 Level Control with using Feedforward PI Controller Lab Report

More information

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint

Laboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control

More information

Control System Design

Control System Design ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science

More information

Embedded Control: Applications and Theory

Embedded Control: Applications and Theory Embedded Control: Applications and Theory IEEE Rock River Valley Section Ramavarapu RS Sreenivas UIUC 30 September 2010 Ramavarapu RS Sreenivas (UIUC) Embedded Control: Applications and Theory 30 September

More information

Rotational Kinetic Energy

Rotational Kinetic Energy Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body

More information

Two-Mass, Three-Spring Dynamic System Investigation Case Study

Two-Mass, Three-Spring Dynamic System Investigation Case Study Two-ass, Three-Spring Dynamic System Investigation Case Study easurements, Calculations, anufacturer's Specifications odel Parameter Identification Which Parameters to Identify? What Tests to Perform?

More information

1 Introduction. 2 Process description

1 Introduction. 2 Process description 1 Introduction This document describes the backround and theory of the Rotary Inverted Pendulum laboratory excercise. The purpose of this laboratory excercise is to familiarize the participants with state

More information

Coupled Drive Apparatus Modelling and Simulation

Coupled Drive Apparatus Modelling and Simulation University of Ljubljana Faculty of Electrical Engineering Victor Centellas Gil Coupled Drive Apparatus Modelling and Simulation Diploma thesis Menthor: prof. dr. Maja Atanasijević-Kunc Ljubljana, 2015

More information

Understanding Precession

Understanding Precession University of Rochester PHY35 Term Paper Understanding Precession Author: Peter Heuer Professor: Dr. Douglas Cline December 1th 01 1 Introduction Figure 1: Bicycle wheel gyroscope demonstration used at

More information

Matlab-Based Tools for Analysis and Control of Inverted Pendula Systems

Matlab-Based Tools for Analysis and Control of Inverted Pendula Systems Matlab-Based Tools for Analysis and Control of Inverted Pendula Systems Slávka Jadlovská, Ján Sarnovský Dept. of Cybernetics and Artificial Intelligence, FEI TU of Košice, Slovak Republic sjadlovska@gmail.com,

More information

Trajectory-tracking control of a planar 3-RRR parallel manipulator

Trajectory-tracking control of a planar 3-RRR parallel manipulator Trajectory-tracking control of a planar 3-RRR parallel manipulator Chaman Nasa and Sandipan Bandyopadhyay Department of Engineering Design Indian Institute of Technology Madras Chennai, India Abstract

More information

SRV02-Series Rotary Experiment # 1. Position Control. Student Handout

SRV02-Series Rotary Experiment # 1. Position Control. Student Handout SRV02-Series Rotary Experiment # 1 Position Control Student Handout SRV02-Series Rotary Experiment # 1 Position Control Student Handout 1. Objectives The objective in this experiment is to introduce the

More information

Final Examination Thursday May Please initial the statement below to show that you have read it

Final Examination Thursday May Please initial the statement below to show that you have read it EN40: Dynamics and Vibrations Final Examination Thursday May 0 010 Division of Engineering rown University NME: General Instructions No collaboration of any kind is permitted on this examination. You may

More information

Represent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T

Represent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented

More information

Physics 351, Spring 2015, Homework #6. Due at start of class, Friday, February 27, 2015

Physics 351, Spring 2015, Homework #6. Due at start of class, Friday, February 27, 2015 Physics 351, Spring 2015, Homework #6. Due at start of class, Friday, February 27, 2015 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page

More information

System simulation using Matlab, state plane plots

System simulation using Matlab, state plane plots System simulation using Matlab, state plane plots his lab is mainly concerned with making state plane (also referred to as phase plane ) plots for various linear and nonlinear systems with two states he

More information

Advanced Control Theory

Advanced Control Theory State Feedback Control Design chibum@seoultech.ac.kr Outline State feedback control design Benefits of CCF 2 Conceptual steps in controller design We begin by considering the regulation problem the task

More information

D(s) G(s) A control system design definition

D(s) G(s) A control system design definition R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure

More information

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University

ECEn 483 / ME 431 Case Studies. Randal W. Beard Brigham Young University ECEn 483 / ME 431 Case Studies Randal W. Beard Brigham Young University Updated: December 2, 2014 ii Contents 1 Single Link Robot Arm 1 2 Pendulum on a Cart 9 3 Satellite Attitude Control 17 4 UUV Roll

More information

Passivity-based Control of Euler-Lagrange Systems

Passivity-based Control of Euler-Lagrange Systems Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramfrez Passivity-based Control of Euler-Lagrange Systems Mechanical, Electrical and Electromechanical Applications Springer Contents

More information

Positioning Servo Design Example

Positioning Servo Design Example Positioning Servo Design Example 1 Goal. The goal in this design example is to design a control system that will be used in a pick-and-place robot to move the link of a robot between two positions. Usually

More information

Multivariable Control Laboratory experiment 2 The Quadruple Tank 1

Multivariable Control Laboratory experiment 2 The Quadruple Tank 1 Multivariable Control Laboratory experiment 2 The Quadruple Tank 1 Department of Automatic Control Lund Institute of Technology 1. Introduction The aim of this laboratory exercise is to study some different

More information

2.004 Dynamics and Control II Spring 2008

2.004 Dynamics and Control II Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute

More information