Target Tracking Using Double Pendulum

Size: px
Start display at page:

Download "Target Tracking Using Double Pendulum"

Transcription

1 Target Tracking Using Double Pendulum Brian Spackman 1, Anusna Chakraborty 1 Department of Electrical and Computer Engineering Utah State University Abstract: This paper deals with the design, implementation and simulation of a linear controller for a double link pendulum which controls its chaotic motion in free fall and guides it to hit a stationary target as well as track a time-varying trajectory. The paper also compares the performance of the linear controller utilizing either a single motor or two motors. The simulation setup considers a double link pendulum in the x-y plane along-with all the targets. The controller design is based on the concept of Linear Quadratic Regulators (LQR) and is tuned to hit 4 different targets during a single run. I. INTRODUCTION The chaotic motion of a double pendulum has been an active area of research for the past few decades. Different models have been designed so that the chaotic motion can be converted into sustained oscillations which can be expressed in terms of simple quadratic equations mathematically. However, the practical applications of the double pendulum have increased extensively in the last decade. Identification that human arm motion is equivalent to that of a controlled double pendulum has been a major breakthrough in the manufacture of prosthetic technology. However since the double pendulum is a highly nonlinear system, the cost of controlling these limbs with sophisticated controllers has increased the cost of these prosthetics dramatically[1]. Specially, the prosthetics of athletics require highly complex control and are very expensive to design and manufacture. Although research is ongoing regarding the design of low cost controllers, no ground-breaking solution has yet been achieved[2]. Another major application of the double pendulum is that of load-lifting cranes. These are used in most fields and control of the load at the end of the second arm is very essential for safety of the payload and ground personnel. Safety is the main reason that these cranes are not automated. This is because to achieve smooth control of the crane with load requires very intricate and complex designing and hence it is not at all cost effective. In this paper, we have shown that the non-linear system can be expressed in terms of linear equations and can also be effectively controlled with the help of linear controllers. The paper also talks about designing the controllers to achieve a critically damped system and attempts to show the balance between the smooth control and the input voltage. The paper has been modelled in the following manner. Section II talks about the equations of motion and also the mathematical calculations required for modelling the system. Section III discusses the simulation set-up and LQR controller. Section IV talks about the results obtained from the experiment and compares the graphical results. Section V summarizes the experimental results and conclusions drawn and section VI discusses the possible future scope of this paper. II. MODELLING A DOUBLE PENDULUM A. SYSTEM DYNAMICS Fig 1. Model of the Double Pendulum The following equations are derived for a system with two motors one motor is at the base of the first arm of the pendulum and the second motor is at the link between the first and the second arm. While designing the double pendulum model we have assumed that

2 both arms of the pendulum are of equal length so that the entire model has a greater reachability [5]. Here: (1) (2) (3) (4) Where (14) (15) (16) (17) (18) The Lagrangian function is defined as: (5) (6) (7) (19) The state space representation of the double link pendulum with a single motor is as follows: The equations of motion are obtained from the following set of equations: (8) (9) B. DETERMINATION OF ANGLE OF IMPACT (20) Where E 1 and E 2 are the motor voltages and θ 1 and θ 2 are the degrees of freedom for the system. For the design with the single motor attached at the base of the first arm of the pendulum equation (9) gets modified as follows: (10) (0,0) (x1,y1) After simplification of equations (8) and (9) we get: (11) (12) The state space representation of the double link pendulum with two motors is as follows [6]: (13) Fig 2. Location of Target with respect to the pendulum Defining the ground location as (0,0) and the target location as (x1,y1), a circle with radius l can be drawn around each location and the intersecting points are the location where the first bob must be located in order for the second bob to reach the target location. For most scenarios within reach, there are two such intersections defined as x and y below. The equations for the two circles are written below. x 2 + y 2 = l 2 (21)

3 (x x 1 ) 2 + (y y 1 ) 2 = l 2 (22) Setting these equal to each other and solving for x yields the following equation. x = x 1 2 2y 1 y+y 1 2 2x 1 (23) This can be plugged back into the first equation and simplified into a second degree polynomial. (4y x 1 2 )y 2 + ( 4y 1 3 4x 1 2 y 1 )y + (x y x 1 2 y 1 2 4x 1 2 l 2 ) = 0 (24) Where (29) = output energy (30) =input energy (31) This is solved by the quadratic formula. Plugging that solution back into the original equation allows one to solve for x. Both are solved below, representing four locations where the first bob may be located. y = 4y 1(x y 1 2 ) ± 4x 1 4l 2 (x y 1 2 ) (x y 1 2 ) 2 8(x y 1 2 ) x = ± l 2 y 2 (25) Only two of these can be intersect locations. To determine which two of the four sets are intersect locations, they are each used in the distance equation below to determine which points are located at a distance of l away from the target. D 2 = (x x 1 ) 2 + (y y 1 ) 2 (26) Angles θ 1 and θ 2, the angles made by the first and second arm respectively, are determined by trigonometric functions. θ 1 = tan 1 ( y x ) (27) θ 2 = tan 1 ( y 1 y x 1 x ) (28) III. SIMULATION DESIGN A. CONTROLLER DESIGN In this paper the Linear Quadratic Regulator Controller (LQR) has been used to control the arms of the pendulum. The theory of optimal control is concerned with operating a dynamic system at minimum cost [7]. For a continuous linear tie invariant system x = Ax + Bu and y = Cx where u = Fx, the objective is to minimize the cost function which is defined as follows: The LQR controller is desired to reduce both these energies since they both contribute to the cost function. However, decreasing the energy of output requires a large control input and a small control input leads to large outputs. So the positive definite weighting matrices Q and R establishes a trade-off between these conflicting parameters. When we used a single motor in the system, to control the pendulum, we initially checked the controllability of the system to determine whether it is even possible to control the system in the desired manner with a single motor. The controllability of the system is defined as: (32) The B matrix used for this check is that shown in equation (20). The rank of the matrix was 4, meaning the system was controllable with only one motor. For the single motor case, we defined our Q matrix such that we did not have any control over the velocity of the second arm. The second arm was completely dependent on the first arm s velocity to set to the desired angle and hit the target. The beginning guess for weights on each parameter were based on Bryson s rule and then further tuned to match the requirements of the system. For the double motor scenario, our Q matrix had weightage on all the four parameters that is the angles of the first arm and second arm and the velocities of the two arms. B. SIMULATION SET-UP The block diagram of the entire system is as below:

4 Fig 3. Block Diagram of the system The simplified diagram above represents the system operation. Initially we consider that there is some noise in the system. The Plant represents the system. The outputs from the plant, that is the 4 state variables, are passed onto the LQR controller which is in a negative feedback loop with the plant. The controller generates the values for K which are the system gains and we check the outputs, that is the angles of the two arms, and we also record the control input to show the difficulties in balancing the two responses. The Q and R matrices used in the system and the various graphs obtained are all discussed in Section IV of this paper Fig 4. Fluctuation of Θ 1 with change of Targets IV. EXPERIMENTAL RESULTS A. OUTPUT WITH SINGLE MOTOR Our aim was to create a system that was critically damped without any transience in the output. Since our focus was on the output terms rather than the control input we will see in the following graphs that the input is extremely high and often has large transience which is quite undesirable. However, with the single motor scenario, despite repeated tunings we were unable to decrease the oscillation in the system beyond a certain limit. Although the system settles down very quickly, the peaks of the oscillations are quite high and the system was not critically damped with a single motor. CASE 1: The weighting matrices used in Case 1 are: Fig 5. Fluctuation of Θ 2 with change of Targets Fig 6. Fluctuation of Input Voltage E1with change of Targets

5 The huge oscillations of Θ 1 and Θ 2 can be observed in the above graphs although these Q and R matrices are completely tuned. Also the huge oscillations of the input voltage can be seen in figure (5). The oscillations reach up to 300 volts in the beginning which is completely unacceptable. CASE 2: The weighting matrices used in this case are: Fig 9. Fluctuation of Input Voltage E 1with change of Targets Fig 7. The responses obtained are as follows: Fluctuation of Θ 1 with change of Targets Now we put more weight on the control input and hence we can observe from the figure that although the transience is present the peaks are much lower than the previous case. The maximum peak is at 175 volts as opposed to the 300 volts in the previous case. However the main trade off is in the transience of the angles Θ 1 and Θ 2. The oscillations have increased significantly and continue far longer than the previous case. This clearly reflects that the balance between the input control voltage and the output angles is quite subtle. But from this scenario of single motor we concluded that it is not possible to reduce the transients any further than case 1. Although the goal of controlling a non-linear system with a linear controller has been achieved, the system is not damped and hence when practically implemented will cause significant problems to the motor. B. OUTPUTS WITH DOUBLE MOTORS Now we use two motors to control both the arms of the pendulum separately. Under this scenario also we describe two cases with different weights on the control input that will demonstrate quite vividly balance between the deviation from a desired angle and the input voltage required to get there. CASE 1: The weighting matrices used in Case 1 are: Fig 8. Fluctuation of Θ 2 with change of Targets

6 The controllability of the system was checked again and the controllability matrix had a full rank of 4. The response graphs are below: Fig 12. Targets Fluctuation of Input Voltage E1 and E2 with change of Figures (9) and (10) show the fluctuations of Θ 1 and Θ 2. In this case we find that except for very small oscillations in Θ 1 the system is critically damped with no unwanted transience. This gives complete smooth control on the two arms. However, the control input oscillations for both E 1 and E 2 are quite high. This is because we have put negligible weightage on the control input. CASE II: In this case we will put quite a high weight on the control input. The result will be decreased peaks of oscillations in the control input voltages. But oscillations in the output angles increase considerably. Fig 10. Fluctuation of Θ 1 with change of Targets Fig 13. Fluctuation of Θ 1 with change of Targets Fig 11. Fluctuation of Θ 2 with change of Targets

7 Fig 14. Fluctuation of Θ 2 with change of Targets In this trajectory the red dots represent the trajectory and the black dots represents the path followed by the pendulum in tracking the trajectory. We see that the pendulum follows the path quite efficiently except at the point where the trajectory makes a sharp turn. V. CONCLUSION In this paper we have used a linear (LQR) controller and successfully controlled the chaotic motion of a double pendulum which is highly non-linear. From the experimental results it is quite clear that the use of two motors is much more effective in controlling the motion and reaching the target than a single motor. The trade-off between the input energy and the output energy is always present for an LQR controller and the weights of the Q and R matrices will vary depending on the requirement. In our case, we desired a critically damped system with minimum transience in the output and the LQR controller performs very well in that respect. For the moving trajectory we find that the pendulum tracks the moving trajectory with a small amount of time lapse. Except at sharp edges and corners where the pendulum has to adjust it s angle, overall the pendulum tracks the time-varying trajectory quite well. VI. FUTURE SCOPE Fig 15. Targets Fluctuation of Input Voltage E1 and E2 with change of C. TIME VARYING TRAJECTORY In this case we will look at how well the double pendulum tracks a time varying trajectory. In order to have an autonomous crane which is cost effective and very efficient in lifting loads and following fixed trajectories, we introduced the idea of using a linear controller to control the system dynamics. So, as a continuation of this work, a considerable load may be attached at the end of the second arm of the pendulum and the equations of motion have to be modified to take that load into account. The control of that model will give an exact idea of the practical implementation of an autonomous crane with load. Another aspect that can be worked upon is the implementation of a smart controller which can calculate by itself the angle by which it has to turn the least during tracking. In that way the time lapse can be reduced by a big amount. This can be achieved by calculating the difference between the angle the pendulum has to move and the angle that the pendulum is currently at in both directions and whichever angle is smaller will be the more efficient choice for the pendulum. Fig 16. The Pendulum tracking a Time Varying trajectory

8 VII. REFERENCES [1] Wentink, EC; Koopman, HFJM; Stamigioli, S; et al. Variable Stiffness Actuated Prosthetic Knee to Restore Knee Buckling Stance: A Modeling Study. Medical Engineering & Physics, Volume 35, Issue 6, Pages Jun [2] Yoyo Au. Double Inverted Pendulum Control. [3] Troy et al, Chaos in Double Pendulum, Chaos_in_a_double_pendulum-AJP.pdf [4] Google.com [5] Google.com [6] Google.com [7] Linear Systems Theory, Joao. P. Hespanha, Lecture 10: Preview of Optimal Control

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

ELEC4631 s Lecture 2: Dynamic Control Systems 7 March Overview of dynamic control systems

ELEC4631 s Lecture 2: Dynamic Control Systems 7 March Overview of dynamic control systems ELEC4631 s Lecture 2: Dynamic Control Systems 7 March 2011 Overview of dynamic control systems Goals of Controller design Autonomous dynamic systems Linear Multi-input multi-output (MIMO) systems Bat flight

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

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

EE C128 / ME C134 Feedback Control Systems

EE C128 / ME C134 Feedback Control Systems EE C128 / ME C134 Feedback Control Systems Lecture Additional Material Introduction to Model Predictive Control Maximilian Balandat Department of Electrical Engineering & Computer Science University of

More information

Polynomial Functions and Their Graphs

Polynomial Functions and Their Graphs Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,, a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) a

More information

Digital Pendulum Control Experiments

Digital Pendulum Control Experiments EE-341L CONTROL SYSTEMS LAB 2013 Digital Pendulum Control Experiments Ahmed Zia Sheikh 2010030 M. Salman Khalid 2010235 Suleman Belal Kazi 2010341 TABLE OF CONTENTS ABSTRACT...2 PENDULUM OVERVIEW...3 EXERCISE

More information

State Feedback MAE 433 Spring 2012 Lab 7

State Feedback MAE 433 Spring 2012 Lab 7 State Feedback MAE 433 Spring 1 Lab 7 Prof. C. Rowley and M. Littman AIs: Brandt Belson, onathan Tu Princeton University April 4-7, 1 1 Overview This lab addresses the control of an inverted pendulum balanced

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

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

The output voltage is given by,

The output voltage is given by, 71 The output voltage is given by, = (3.1) The inductor and capacitor values of the Boost converter are derived by having the same assumption as that of the Buck converter. Now the critical value of the

More information

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc. Chapter 14 Oscillations 14-1 Oscillations of a Spring If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The

More information

Comparison of LQR and PD controller for stabilizing Double Inverted Pendulum System

Comparison of LQR and PD controller for stabilizing Double Inverted Pendulum System International Journal of Engineering Research and Development ISSN: 78-67X, Volume 1, Issue 1 (July 1), PP. 69-74 www.ijerd.com Comparison of LQR and PD controller for stabilizing Double Inverted Pendulum

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

Chapter 14 Oscillations

Chapter 14 Oscillations Chapter 14 Oscillations If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic. The mass and spring system is a

More information

pg B7. A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is dra

pg B7. A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is dra pg 165 A 0.20 kg object moves along a straight line. The net force acting on the object varies with the object's displacement as shown in the graph above. The object starts from rest at displacement x

More information

1 (30 pts) Dominant Pole

1 (30 pts) Dominant Pole EECS C8/ME C34 Fall Problem Set 9 Solutions (3 pts) Dominant Pole For the following transfer function: Y (s) U(s) = (s + )(s + ) a) Give state space description of the system in parallel form (ẋ = Ax +

More information

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators.

EECS C128/ ME C134 Final Wed. Dec. 15, am. Closed book. Two pages of formula sheets. No calculators. Name: SID: EECS C28/ ME C34 Final Wed. Dec. 5, 2 8- am Closed book. Two pages of formula sheets. No calculators. There are 8 problems worth points total. Problem Points Score 2 2 6 3 4 4 5 6 6 7 8 2 Total

More information

ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016

ACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016 ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe

More information

Control Systems. Design of State Feedback Control.

Control Systems. Design of State Feedback Control. Control Systems Design of State Feedback Control chibum@seoultech.ac.kr Outline Design of State feedback control Dominant pole design Symmetric root locus (linear quadratic regulation) 2 Selection of closed-loop

More information

Introduction to Root Locus. What is root locus?

Introduction to Root Locus. What is root locus? Introduction to Root Locus What is root locus? A graphical representation of the closed loop poles as a system parameter (Gain K) is varied Method of analysis and design for stability and transient response

More information

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.

Chapter 14 Oscillations. Copyright 2009 Pearson Education, Inc. Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical

More information

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

Review: control, feedback, etc. Today s topic: state-space models of systems; linearization Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered

More information

Control. CSC752: Autonomous Robotic Systems. Ubbo Visser. March 9, Department of Computer Science University of Miami

Control. CSC752: Autonomous Robotic Systems. Ubbo Visser. March 9, Department of Computer Science University of Miami Control CSC752: Autonomous Robotic Systems Ubbo Visser Department of Computer Science University of Miami March 9, 2017 Outline 1 Control system 2 Controller Images from http://en.wikipedia.org/wiki/feed-forward

More information

DOUBLE ARM JUGGLING SYSTEM Progress Presentation ECSE-4962 Control Systems Design

DOUBLE ARM JUGGLING SYSTEM Progress Presentation ECSE-4962 Control Systems Design DOUBLE ARM JUGGLING SYSTEM Progress Presentation ECSE-4962 Control Systems Design Group Members: John Kua Trinell Ball Linda Rivera Introduction Where are we? Bulk of Design and Build Complete Testing

More information

ECE557 Systems Control

ECE557 Systems Control ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,

More information

The basic principle to be used in mechanical systems to derive a mathematical model is Newton s law,

The basic principle to be used in mechanical systems to derive a mathematical model is Newton s law, Chapter. DYNAMIC MODELING Understanding the nature of the process to be controlled is a central issue for a control engineer. Thus the engineer must construct a model of the process with whatever information

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

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

Math Assignment 5

Math Assignment 5 Math 2280 - Assignment 5 Dylan Zwick Fall 2013 Section 3.4-1, 5, 18, 21 Section 3.5-1, 11, 23, 28, 35, 47, 56 Section 3.6-1, 2, 9, 17, 24 1 Section 3.4 - Mechanical Vibrations 3.4.1 - Determine the period

More information

Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems. Process Dynamics and Control Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

More information

EE C128 / ME C134 Final Exam Fall 2014

EE C128 / ME C134 Final Exam Fall 2014 EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket

More information

An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems

An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems Journal of Automation Control Engineering Vol 3 No 2 April 2015 An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems Nguyen Duy Cuong Nguyen Van Lanh Gia Thi Dinh Electronics Faculty

More information

MEMS Gyroscope Control Systems for Direct Angle Measurements

MEMS Gyroscope Control Systems for Direct Angle Measurements MEMS Gyroscope Control Systems for Direct Angle Measurements Chien-Yu Chi Mechanical Engineering National Chiao Tung University Hsin-Chu, Taiwan (R.O.C.) 3 Email: chienyu.me93g@nctu.edu.tw Tsung-Lin Chen

More information

Application of Newton/GMRES Method to Nonlinear Model Predictive Control of Functional Electrical Stimulation

Application of Newton/GMRES Method to Nonlinear Model Predictive Control of Functional Electrical Stimulation Proceedings of the 3 rd International Conference on Control, Dynamic Systems, and Robotics (CDSR 16) Ottawa, Canada May 9 10, 2016 Paper No. 121 DOI: 10.11159/cdsr16.121 Application of Newton/GMRES Method

More information

Table of Laplacetransform

Table of Laplacetransform Appendix Table of Laplacetransform pairs 1(t) f(s) oct), unit impulse at t = 0 a, a constant or step of magnitude a at t = 0 a s t, a ramp function e- at, an exponential function s + a sin wt, a sine fun

More information

TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR

TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR Samo Lasič, Gorazd Planinšič,, Faculty of Mathematics and Physics University of Ljubljana, Slovenija Giacomo Torzo, Department of Physics, University

More information

QFT Framework for Robust Tuning of Power System Stabilizers

QFT Framework for Robust Tuning of Power System Stabilizers 45-E-PSS-75 QFT Framework for Robust Tuning of Power System Stabilizers Seyyed Mohammad Mahdi Alavi, Roozbeh Izadi-Zamanabadi Department of Control Engineering, Aalborg University, Denmark Correspondence

More information

Controlling the Inverted Pendulum

Controlling the Inverted Pendulum Controlling the Inverted Pendulum Steven A. P. Quintero Department of Electrical and Computer Engineering University of California, Santa Barbara Email: squintero@umail.ucsb.edu Abstract The strategies

More information

Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10) Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane

More information

Transitioning to Chaos in a Simple Mechanical Oscillator

Transitioning to Chaos in a Simple Mechanical Oscillator Transitioning to Chaos in a Simple Mechanical Oscillator Hwan Bae Physics Department, The College of Wooster, Wooster, Ohio 69, USA (Dated: May 9, 8) We vary the magnetic damping, driver frequency, and

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

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

Balancing of an Inverted Pendulum with a SCARA Robot

Balancing of an Inverted Pendulum with a SCARA Robot Balancing of an Inverted Pendulum with a SCARA Robot Bernhard Sprenger, Ladislav Kucera, and Safer Mourad Swiss Federal Institute of Technology Zurich (ETHZ Institute of Robotics 89 Zurich, Switzerland

More information

Topic # Feedback Control

Topic # Feedback Control Topic #5 6.3 Feedback Control State-Space Systems Full-state Feedback Control How do we change the poles of the state-space system? Or,evenifwecanchangethepolelocations. Where do we put the poles? Linear

More information

A Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction

A Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction A Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction R. W. Brockett* and Hongyi Li* Engineering and Applied Sciences Harvard University Cambridge, MA 38, USA {brockett, hongyi}@hrl.harvard.edu

More information

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state

More information

PID Control. Objectives

PID Control. Objectives PID Control Objectives The objective of this lab is to study basic design issues for proportional-integral-derivative control laws. Emphasis is placed on transient responses and steady-state errors. The

More information

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.

Chapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc. Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics

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

Autonomous Mobile Robot Design

Autonomous Mobile Robot Design Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:

More information

Optimal Linear Control of an Energy Harvesting System

Optimal Linear Control of an Energy Harvesting System International Journal of Scientific and Research Publications, Volume 6, Issue 6, June 2016 385 Optimal Linear Control of an Energy Harvesting System Ukoima Kelvin Nkalo Department of Electrical & Electronic

More information

TWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations

TWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1

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

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

Report E-Project Henriette Laabsch Toni Luhdo Steffen Mitzscherling Jens Paasche Thomas Pache

Report E-Project Henriette Laabsch Toni Luhdo Steffen Mitzscherling Jens Paasche Thomas Pache Potsdam, August 006 Report E-Project Henriette Laabsch 7685 Toni Luhdo 7589 Steffen Mitzscherling 7540 Jens Paasche 7575 Thomas Pache 754 Introduction From 7 th February till 3 rd March, we had our laboratory

More information

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10)

Subject: Optimal Control Assignment-1 (Related to Lecture notes 1-10) Subject: Optimal Control Assignment- (Related to Lecture notes -). Design a oil mug, shown in fig., to hold as much oil possible. The height and radius of the mug should not be more than 6cm. The mug must

More information

A plane autonomous system is a pair of simultaneous first-order differential equations,

A plane autonomous system is a pair of simultaneous first-order differential equations, Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium

More information

Pitch Rate CAS Design Project

Pitch Rate CAS Design Project Pitch Rate CAS Design Project Washington University in St. Louis MAE 433 Control Systems Bob Rowe 4.4.7 Design Project Part 2 This is the second part of an ongoing project to design a control and stability

More information

Nonlinear Oscillations and Chaos

Nonlinear Oscillations and Chaos CHAPTER 4 Nonlinear Oscillations and Chaos 4-. l l = l + d s d d l l = l + d m θ m (a) (b) (c) The unetended length of each spring is, as shown in (a). In order to attach the mass m, each spring must be

More information

The content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam.

The content contained in all sections of chapter 6 of the textbook is included on the AP Physics B exam. WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system is always

More information

Vibration Suppression of a 2-Mass Drive System with Multiple Feedbacks

Vibration Suppression of a 2-Mass Drive System with Multiple Feedbacks International Journal of Scientific and Research Publications, Volume 5, Issue 11, November 2015 168 Vibration Suppression of a 2-Mass Drive System with Multiple Feedbacks L. Vidyaratan Meetei, Benjamin

More information

Coordinated Tracking Control of Multiple Laboratory Helicopters: Centralized and De-Centralized Design Approaches

Coordinated Tracking Control of Multiple Laboratory Helicopters: Centralized and De-Centralized Design Approaches Coordinated Tracking Control of Multiple Laboratory Helicopters: Centralized and De-Centralized Design Approaches Hugh H. T. Liu University of Toronto, Toronto, Ontario, M3H 5T6, Canada Sebastian Nowotny

More information

Tropical Polynomials

Tropical Polynomials 1 Tropical Arithmetic Tropical Polynomials Los Angeles Math Circle, May 15, 2016 Bryant Mathews, Azusa Pacific University In tropical arithmetic, we define new addition and multiplication operations on

More information

Learn2Control Laboratory

Learn2Control Laboratory Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should

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

MECH 3140 Final Project

MECH 3140 Final Project MECH 3140 Final Project Final presentation will be held December 7-8. The presentation will be the only deliverable for the final project and should be approximately 20-25 minutes with an additional 10

More information

CHBE320 LECTURE XI CONTROLLER DESIGN AND PID CONTOLLER TUNING. Professor Dae Ryook Yang

CHBE320 LECTURE XI CONTROLLER DESIGN AND PID CONTOLLER TUNING. Professor Dae Ryook Yang CHBE320 LECTURE XI CONTROLLER DESIGN AND PID CONTOLLER TUNING Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 11-1 Road Map of the Lecture XI Controller Design and PID

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

Structural Damage Detection Using Time Windowing Technique from Measured Acceleration during Earthquake

Structural Damage Detection Using Time Windowing Technique from Measured Acceleration during Earthquake Structural Damage Detection Using Time Windowing Technique from Measured Acceleration during Earthquake Seung Keun Park and Hae Sung Lee ABSTRACT This paper presents a system identification (SI) scheme

More information

Empirical Models Interpolation Polynomial Models

Empirical Models Interpolation Polynomial Models Mathematical Modeling Lia Vas Empirical Models Interpolation Polynomial Models Lagrange Polynomial. Recall that two points (x 1, y 1 ) and (x 2, y 2 ) determine a unique line y = ax + b passing them (obtained

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

ME 132, Dynamic Systems and Feedback. Class Notes. Spring Instructor: Prof. A Packard

ME 132, Dynamic Systems and Feedback. Class Notes. Spring Instructor: Prof. A Packard ME 132, Dynamic Systems and Feedback Class Notes by Andrew Packard, Kameshwar Poolla & Roberto Horowitz Spring 2005 Instructor: Prof. A Packard Department of Mechanical Engineering University of California

More information

8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)

8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) 8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)

More information

GAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL

GAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL GAIN SCHEDULING CONTROL WITH MULTI-LOOP PID FOR 2- DOF ARM ROBOT TRAJECTORY CONTROL 1 KHALED M. HELAL, 2 MOSTAFA R.A. ATIA, 3 MOHAMED I. ABU EL-SEBAH 1, 2 Mechanical Engineering Department ARAB ACADEMY

More information

Nonlinear System Analysis

Nonlinear System Analysis Nonlinear System Analysis Lyapunov Based Approach Lecture 4 Module 1 Dr. Laxmidhar Behera Department of Electrical Engineering, Indian Institute of Technology, Kanpur. January 4, 2003 Intelligent Control

More information

LQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin

LQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin LQR, Kalman Filter, and LQG Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin May 2015 Linear Quadratic Regulator (LQR) Consider a linear system

More information

ACTIVE VIBRATION CONTROL PROTOTYPING IN ANSYS: A VERIFICATION EXPERIMENT

ACTIVE VIBRATION CONTROL PROTOTYPING IN ANSYS: A VERIFICATION EXPERIMENT ACTIVE VIBRATION CONTROL PROTOTYPING IN ANSYS: A VERIFICATION EXPERIMENT Ing. Gergely TAKÁCS, PhD.* * Institute of Automation, Measurement and Applied Informatics Faculty of Mechanical Engineering Slovak

More information

LQG/LTR CONTROLLER DESIGN FOR ROTARY INVERTED PENDULUM QUANSER REAL-TIME EXPERIMENT

LQG/LTR CONTROLLER DESIGN FOR ROTARY INVERTED PENDULUM QUANSER REAL-TIME EXPERIMENT LQG/LR CONROLLER DESIGN FOR ROARY INVERED PENDULUM QUANSER REAL-IME EXPERIMEN Cosmin Ionete University of Craiova, Faculty of Automation, Computers and Electronics Department of Automation, e-mail: cosmin@automation.ucv.ro

More information

State space control for the Two degrees of freedom Helicopter

State space control for the Two degrees of freedom Helicopter State space control for the Two degrees of freedom Helicopter AAE364L In this Lab we will use state space methods to design a controller to fly the two degrees of freedom helicopter. 1 The state space

More information

EE221A Linear System Theory Final Exam

EE221A Linear System Theory Final Exam EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,

More information

Proportional plus Integral (PI) Controller

Proportional plus Integral (PI) Controller Proportional plus Integral (PI) Controller 1. A pole is placed at the origin 2. This causes the system type to increase by 1 and as a result the error is reduced to zero. 3. Originally a point A is on

More information

Stable Limit Cycle Generation for Underactuated Mechanical Systems, Application: Inertia Wheel Inverted Pendulum

Stable Limit Cycle Generation for Underactuated Mechanical Systems, Application: Inertia Wheel Inverted Pendulum Stable Limit Cycle Generation for Underactuated Mechanical Systems, Application: Inertia Wheel Inverted Pendulum Sébastien Andary Ahmed Chemori Sébastien Krut LIRMM, Univ. Montpellier - CNRS, 6, rue Ada

More information

Conventional Paper-I-2011 PART-A

Conventional Paper-I-2011 PART-A Conventional Paper-I-0 PART-A.a Give five properties of static magnetic field intensity. What are the different methods by which it can be calculated? Write a Maxwell s equation relating this in integral

More information

SKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.

SKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution. SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or

More information

A conjecture on sustained oscillations for a closed-loop heat equation

A conjecture on sustained oscillations for a closed-loop heat equation A conjecture on sustained oscillations for a closed-loop heat equation C.I. Byrnes, D.S. Gilliam Abstract The conjecture in this paper represents an initial step aimed toward understanding and shaping

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

The Torsion Pendulum (One or two weights)

The Torsion Pendulum (One or two weights) The Torsion Pendulum (One or two weights) Exercises I through V form the one-weight experiment. Exercises VI and VII, completed after Exercises I -V, add one weight more. Preparatory Questions: 1. The

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

Nonlinear dynamics & chaos BECS

Nonlinear dynamics & chaos BECS Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes

More information

One Dimensional Dynamical Systems

One Dimensional Dynamical Systems 16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar

More information

DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. EE 537 Homewors Friedland Text Updated: Wednesday November 8 Some homewor assignments refer to Friedland s text For full credit show all wor. Some problems require hand calculations. In those cases do

More information

Chapter 14: Periodic motion

Chapter 14: Periodic motion Chapter 14: Periodic motion Describing oscillations Simple harmonic motion Energy of simple harmonic motion Applications of simple harmonic motion Simple pendulum & physical pendulum Damped oscillations

More information

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.

Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition

More information

Graphical Analysis and Errors - MBL

Graphical Analysis and Errors - MBL I. Graphical Analysis Graphical Analysis and Errors - MBL Graphs are vital tools for analyzing and displaying data throughout the natural sciences and in a wide variety of other fields. It is imperative

More information

Variable Radius Pulley Design Methodology for Pneumatic Artificial Muscle-based Antagonistic Actuation Systems

Variable Radius Pulley Design Methodology for Pneumatic Artificial Muscle-based Antagonistic Actuation Systems 211 IEEE/RSJ International Conference on Intelligent Robots and Systems September 25-3, 211. San Francisco, CA, USA Variable Radius Pulley Design Methodology for Pneumatic Artificial Muscle-based Antagonistic

More information

Application of Neural Networks for Control of Inverted Pendulum

Application of Neural Networks for Control of Inverted Pendulum Application of Neural Networks for Control of Inverted Pendulum VALERI MLADENOV Department of Theoretical Electrical Engineering Technical University of Sofia Sofia, Kliment Ohridski blvd. 8; BULARIA valerim@tu-sofia.bg

More information

CHAPTER 1. Introduction

CHAPTER 1. Introduction CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric

More information

Linear Control Systems General Informations. Guillaume Drion Academic year

Linear Control Systems General Informations. Guillaume Drion Academic year Linear Control Systems General Informations Guillaume Drion Academic year 2017-2018 1 SYST0003 - General informations Website: http://sites.google.com/site/gdrion25/teaching/syst0003 Contacts: Guillaume

More information

Graphical Analysis and Errors MBL

Graphical Analysis and Errors MBL Graphical Analysis and Errors MBL I Graphical Analysis Graphs are vital tools for analyzing and displaying data Graphs allow us to explore the relationship between two quantities -- an independent variable

More information