# Nonlinear Control Lecture 1: Introduction

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Nonlinear Control Lecture 1: Introduction Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 1 1/15

2 Motivation Reference Books Topics Introduction Examples Farzaneh Abdollahi Nonlinear Control Lecture 1 2/15

3 Motivations A system is called linear if its behavior set satisfies linear superposition laws:.i.e. z 1, z 2 B and constant c R z 1 + z 2 B, cz 1 B A nonlinear system is simply a system which is not linear. Powerful tools founded based on superposition principle make analyzing the linear systems simple. All practical systems posses nonlinear dynamics. Sometimes it is possible to describe the operation of physical systems by linear model around its operating points Linearized system can provide us an approximate behavior of the nonlinear system But in analyzing the overall system behavior, often linearized model inadequate or inaccurate. arzaneh Abdollahi Nonlinear Control Lecture 1 3/15

4 Linearization is an approximation in the neighborhood of an operating system it can only predict local behavior of nonlinear system. (No info regarding nonlocal or global behavior of system) Due to richer dynamics of nonlinear systems comparing to the linear ones, there are some essentially nonlinear phenomena that can take place only in presence of nonlinearity Essentially nonlinear phenomena Finite escape time: The state of linear system goes to infinity as t ; nonlinear system s state can go to infinity in finite time. Multiple isolated equilibria: linear system can have only one isolated equilibrium point which attracts the states irrespective on the initial state; nonlinear system can have more than one isolated equilibrium point, the state may converge to each depending on the initial states. Limit cycle: There is no robust oscillation in linear systems. To oscillate there should be a pair of eigenvalues on the imaginary axis which due to presence of perturbations it is almost impossible in practice; For nonlinear systems, there are some oscillations named limit cycle with fixed amplitude and frequency. arzaneh Abdollahi Nonlinear Control Lecture 1 4/15

5 Essentially nonlinear phenomena Subharmonic,harmonic or almost periodic oscillations: A stable linear system under a periodic input output with the same frequency; A nonlinear system under a periodic input can oscillate with submultiple or multiple frequency of input or almost-periodic oscillation. Chaos: A nonlinear system may have a different steady-state behavior which is not equilibrium point, periodic oscillation or almost-periodic oscillation. This chaotic motions exhibit random, despite of deterministic nature of the system. Multiple modes of behavior: A nonlinear system may exhibit multiple modes of behavior based on type of excitation: an unforced system may have one limit cycle. Periodic excitation may exhibit harmonic, subharmonic,or chaotic behavior based on amplitude and frequency of input. if amplitude or frequency is smoothly changed, it may exhibit discontinuous jump of the modes as well. Farzaneh Abdollahi Nonlinear Control Lecture 1 5/15

6 Linear systems: can be described by a set of ordinary differential equations and usually the closed-form expressions for their solutions are derivable. Nonlinear systems: In general this is not possible It is desired to make a prediction of system behavior even in absence of closed-form solution. This type of analysis is called qualitative analysis. Despite of linear systems, no tool or methodology in nonlinear system analysis is universally applicable their analysis requires a wide verity of tools and higher level of mathematic knowledge stability analysis and stabilizablity of such systems and getting familiar with associated control techniques is the basic requirement of graduate studies in control engineering. The aim of this course are developing a basic understanding of nonlinear control system theory and its applications. introducing tools such as Lyapunov s method analyze the system stability Presenting techniques such as feedback linearization to control nonlinear systems. Farzaneh Abdollahi Nonlinear Control Lecture 1 6/15

7 Reference Books Text Book: Nonlinear Systems, H. K. Khalil, 3 rd edition, Prentice-Hall, 2002 Other reference Books: Applied Nonlinear Control, J. J. E. Slotine, and W. Li, Prentice-Hall, 1991 Nonlinear System Analysis, M. Vidyasagar, 2 nd edition, Prentice-Hall, 1993 Nonlinear Control Systems, A. Isidori, 3 rd edition Springer-Verlag, 1995 arzaneh Abdollahi Nonlinear Control Lecture 1 7/15

8 Topics Topic Date Refs Introduction, Phase plane, Analysis Weeks 1,2 Chapters 1-3 Stability Theory Weeks 3-5 Chapters 4,8,9 Input-to-state, I/O stability Weeks 6,7 Chapters 4,5 Passivity Week 8 Chapter 10 Feedback Control Weeks 9,10 Chapter 12 Feedback linearization Weeks 11,12 Chapters 12,13 Sliding Mode Weeks 13,14 Back Stepping Weeks 15,16 Farzaneh Abdollahi Nonlinear Control Lecture 1 8/15

9 At this course we consider dynamical systems modeled by a finite number of coupled first-order ordinary differential equations: ẋ = f (t, x, u) (1) where x = [x 1,..., x n ] T : state vector, u = [u 1,..., u p ] T : input vector, and f (.) = [f 1 (.),..., f n (.)] T : a vector of nonlinear functions. Euq. (1) is called state equation. Another equation named output equation: y = h(t, x, u) (2) where y = [y 1,..., y q ] T : output vector. Equ (2) is employed for particular interest in analysis such as variables which can be measured physically variables which are required to behave in a desirable manner Equs (1) and (2) together are called state-space model. arzaneh Abdollahi Nonlinear Control Lecture 1 9/15

10 Most of our analysis are dealing with unforced state equations where u does not present explicitly in Equ (1): ẋ = f (t, x) In unforced state equations, input to the system is NOT necessarily zero. Input can be a function of time: u = γ(t), a feedback function of state: u = γ(x), or both u = γ(t, x) where is substituted in Equ (1). Autonomous or Time-invariant Systems: ẋ = f (x) (3) function of f does not explicitly depend on t. Autonomous systems are invariant to shift in time origin, i.e. changing t to τ = t a does not change f. The system which is not autonomous is called nonautonomous or time-varying. arzaneh Abdollahi Nonlinear Control Lecture 1 10/15

11 Equilibrium Point x = x x in state space is equilibrium point if whenever the state starts at x, it will remain at x for all future time. for autonomous systems (3), the equilibrium points are the real roots of equation: f (x) = 0. Equilibrium point can be Isolated: There are no other equilibrium points in its vicinity. a continuum of equilibrium points arzaneh Abdollahi Nonlinear Control Lecture 1 11/15

12 Pendulum Employing Newton s second law of motion, equation of pendulum motion is: ml θ = mg sin θ kl θ l: length of pendulum rod; m: mass of pendulum bob; k: coefficient of friction; θ: angle subtended by rod and vertical axis To obtain state space model, let x 1 = θ, x 2 = θ: ẋ 1 = x 2 ẋ 2 = g l sinx 1 k m x 2 arzaneh Abdollahi Nonlinear Control Lecture 1 12/15

13 Pendulum To find equilibrium point: ẋ 1 = ẋ 2 = 0 0 = x 2 0 = g l sinx 1 k m x 2 The Equilibrium points are at (nπ, 0) for n = 0, ±1, ±2,... Pendulum has two equilibrium points: (0, 0) and (π, 0), Other equilibrium points are repetitions of these two which correspond to number of pendulum full swings before it rests Physically we can see that the pendulum rests at (0, 0), but hardly maintain rest at (π, 0) Farzaneh Abdollahi Nonlinear Control Lecture 1 13/15

14 Tunnel Diode Circuit The tunnel diode is characterized by i R = h(v R ) The energy-storing elements are C and L which assumed are linear and time-invariant i C = C dv C dt, v L = L di L dt. Employing Kirchhoff s current law: i C + i R i L = 0 Employing Kirchhoff s voltage law: v C E + Ri L + v L = 0 for state space model, let x 1 = v C, x 2 = i L and u = E as a constant input: ẋ 1 = 1 C [ h(x 1) + x 2 ] ẋ 2 = 1 L [ x 1 Rx 2 + u] Farzaneh Abdollahi Nonlinear Control Lecture 1 14/15

15 Tunnel Diode Circuit To find equilibrium point: ẋ 1 = ẋ 2 = 0 0 = 1 C [ h(x 1) + x 2 ] 0 = 1 L [ x 1 Rx 2 + u] Equilibrium points depends on E and R x 2 = h(x 1 ) = E R 1 R x 1 For certain E and R, it may have 3 equilibrium points (Q 1, Q 2, Q 3 ). if E, and same R only Q3 exists. if E, and same R only Q1 exists. Farzaneh Abdollahi Nonlinear Control Lecture 1 15/15

### Nonlinear Control Lecture 2:Phase Plane Analysis

Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53

### 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

### Using Lyapunov Theory I

Lecture 33 Stability Analysis of Nonlinear Systems Using Lyapunov heory I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline Motivation Definitions

### BIBO STABILITY AND ASYMPTOTIC STABILITY

BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to

### 3 Stability and Lyapunov Functions

CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such

### 7 Pendulum. Part II: More complicated situations

MATH 35, by T. Lakoba, University of Vermont 60 7 Pendulum. Part II: More complicated situations In this Lecture, we will pursue two main goals. First, we will take a glimpse at a method of Classical Mechanics

### Lyapunov Stability Theory

Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous

### Feedback Linearization

Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)

### Vibrations Qualifying Exam Study Material

Vibrations Qualifying Exam Study Material The candidate is expected to have a thorough understanding of engineering vibrations topics. These topics are listed below for clarification. Not all instructors

### Nonlinear Control Lecture 4: Stability Analysis I

Nonlinear Control Lecture 4: Stability Analysis I Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 Farzaneh Abdollahi Nonlinear Control Lecture 4 1/70

### 2 Lyapunov Stability. x(0) x 0 < δ x(t) x 0 < ɛ

1 2 Lyapunov Stability Whereas I/O stability is concerned with the effect of inputs on outputs, Lyapunov stability deals with unforced systems: ẋ = f(x, t) (1) where x R n, t R +, and f : R n R + R n.

### 1 Lyapunov theory of stability

M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability

### Chaotic motion. Phys 750 Lecture 9

Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to

### A GLOBAL STABILIZATION STRATEGY FOR AN INVERTED PENDULUM. B. Srinivasan, P. Huguenin, K. Guemghar, and D. Bonvin

Copyright IFAC 15th Triennial World Congress, Barcelona, Spain A GLOBAL STABILIZATION STRATEGY FOR AN INVERTED PENDULUM B. Srinivasan, P. Huguenin, K. Guemghar, and D. Bonvin Labaratoire d Automatique,

### A simple electronic circuit to demonstrate bifurcation and chaos

A simple electronic circuit to demonstrate bifurcation and chaos P R Hobson and A N Lansbury Brunel University, Middlesex Chaos has generated much interest recently, and many of the important features

### ENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations

ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential

### Applications of Second-Order Differential Equations

Applications of Second-Order Differential Equations ymy/013 Building Intuition Even though there are an infinite number of differential equations, they all share common characteristics that allow intuition

### Lectures on Periodic Orbits

Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete

### 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

### Oscillations in Damped Driven Pendulum: A Chaotic System

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 10, October 2015, PP 14-27 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org Oscillations

### Nonlinear Control Theory. Lecture 9

Nonlinear Control Theory Lecture 9 Periodic Perturbations Averaging Singular Perturbations Khalil Chapter (9, 10) 10.3-10.6, 11 Today: Two Time-scales Averaging ẋ = ǫf(t,x,ǫ) The state x moves slowly compared

### OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL

International Journal in Foundations of Computer Science & Technology (IJFCST),Vol., No., March 01 OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL Sundarapandian Vaidyanathan

### 4 Second-Order Systems

4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization

### Analog Signals and Systems and their properties

Analog Signals and Systems and their properties Main Course Objective: Recall course objectives Understand the fundamentals of systems/signals interaction (know how systems can transform or filter signals)

### PH 120 Project # 2: Pendulum and chaos

PH 120 Project # 2: Pendulum and chaos Due: Friday, January 16, 2004 In PH109, you studied a simple pendulum, which is an effectively massless rod of length l that is fixed at one end with a small mass

### Lab 1: Simple Pendulum 1. The Pendulum. Laboratory 1, Physics 15c Due Friday, February 16, in front of Sci Cen 301

Lab 1: Simple Pendulum 1 The Pendulum Laboratory 1, Physics 15c Due Friday, February 16, in front of Sci Cen 301 Physics 15c; REV 0 1 January 31, 2007 1 Introduction Most oscillating systems behave like

### CONTROLLING CHAOTIC DYNAMICS USING BACKSTEPPING DESIGN WITH APPLICATION TO THE LORENZ SYSTEM AND CHUA S CIRCUIT

Letters International Journal of Bifurcation and Chaos, Vol. 9, No. 7 (1999) 1425 1434 c World Scientific Publishing Company CONTROLLING CHAOTIC DYNAMICS USING BACKSTEPPING DESIGN WITH APPLICATION TO THE

### Modeling and Experimentation: Mass-Spring-Damper System Dynamics

Modeling and Experimentation: Mass-Spring-Damper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to

### Modelling Research Group

Modelling Research Group The Importance of Noise in Dynamical Systems E. Staunton, P.T. Piiroinen November 20, 2015 Eoghan Staunton Modelling Research Group 2015 1 / 12 Introduction Historically mathematicians

### Dynamics of a mass-spring-pendulum system with vastly different frequencies

Dynamics of a mass-spring-pendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate

### Phase Model for the relaxed van der Pol oscillator and its application to synchronization analysis

Phase Model for the relaxed van der Pol oscillator and its application to synchronization analysis Mimila Prost O. Collado J. Automatic Control Department, CINVESTAV IPN, A.P. 4 74 Mexico, D.F. ( e mail:

### Computational Intelligence Lecture 3: Simple Neural Networks for Pattern Classification

Computational Intelligence Lecture 3: Simple Neural Networks for Pattern Classification Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 arzaneh Abdollahi

### CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Lyapunov Stability - Hassan K. Khalil

LYAPUNO STABILITY Hassan K. Khalil Department of Electrical and Computer Enigneering, Michigan State University, USA. Keywords: Asymptotic stability, Autonomous systems, Exponential stability, Global asymptotic

### Topic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis

Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of

### Chaos and R-L diode Circuit

Chaos and R-L diode Circuit Rabia Aslam Chaudary Roll no: 2012-10-0011 LUMS School of Science and Engineering Thursday, December 20, 2010 1 Abstract In this experiment, we will use an R-L diode circuit

### Waves & Oscillations

Physics 42200 Waves & Oscillations Lecture 21 Review Spring 2013 Semester Matthew Jones Midterm Exam: Date: Wednesday, March 6 th Time: 8:00 10:00 pm Room: PHYS 203 Material: French, chapters 1-8 Review

### Introduction to Dynamical Systems Basic Concepts of Dynamics

Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic

### Nonlinear Control of Electrodynamic Tether in Equatorial or Somewhat Inclined Orbits

Proceedings of the 15th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece T-5 Nonlinear Control of Electrodynamic Tether in Equatorial or Somewhat Inclined Martin B. Larsen

### 6.2 Brief review of fundamental concepts about chaotic systems

6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification

### Exponential Controller for Robot Manipulators

Exponential Controller for Robot Manipulators Fernando Reyes Benemérita Universidad Autónoma de Puebla Grupo de Robótica de la Facultad de Ciencias de la Electrónica Apartado Postal 542, Puebla 7200, México

### Control Systems I. Lecture 2: Modeling. Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch Emilio Frazzoli

Control Systems I Lecture 2: Modeling Suggested Readings: Åström & Murray Ch. 2-3, Guzzella Ch. 2-3 Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 29, 2017 E. Frazzoli

### AP Physics. Harmonic Motion. Multiple Choice. Test E

AP Physics Harmonic Motion Multiple Choice Test E A 0.10-Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.

### Exact Linearization Of Stochastic Dynamical Systems By State Space Coordinate Transformation And Feedback I g-linearization

Applied Mathematics E-Notes, 3(2003), 99-106 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Exact Linearization Of Stochastic Dynamical Systems By State Space Coordinate

### Nonlinear Control Lecture 9: Feedback Linearization

Nonlinear Control Lecture 9: Feedback Linearization Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 9 1/75

### Lecture 5: Lyapunov Functions and Storage Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 5: Lyapunov Functions and Storage

### Linearization problem. The simplest example

Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and

### Response of A Hard Duffing Oscillator to Harmonic Excitation An Overview

INDIN INSTITUTE OF TECHNOLOGY, KHRGPUR 710, DECEMBER 8-0, 00 1 Response of Hard Duffing Oscillator to Harmonic Excitation n Overview.K. Mallik Department of Mechanical Engineering Indian Institute of Technology

### Analysis of periodic solutions in tapping-mode AFM: An IQC approach

Analysis of periodic solutions in tapping-mode AFM: An IQC approach A. Sebastian, M. V. Salapaka Department of Electrical and Computer Engineering Iowa State University Ames, IA 5004 USA Abstract The feedback

### Examination paper for TMA4195 Mathematical Modeling

Department of Mathematical Sciences Examination paper for TMA4195 Mathematical Modeling Academic contact during examination: Elena Celledoni Phone: 48238584, 73593541 Examination date: 11th of December

### Computational Physics (6810): Session 8

Computational Physics (6810): Session 8 Dick Furnstahl Nuclear Theory Group OSU Physics Department February 24, 2014 Differential equation solving Session 7 Preview Session 8 Stuff Solving differential

### Anti-synchronization of a new hyperchaotic system via small-gain theorem

Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised

### Generalized projective synchronization between two chaotic gyros with nonlinear damping

Generalized projective synchronization between two chaotic gyros with nonlinear damping Min Fu-Hong( ) Department of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China

### Oscillations. Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance

Oscillations Simple Harmonic Motion (SHM) Position, Velocity, Acceleration SHM Forces SHM Energy Period of oscillation Damping and Resonance 1 Revision problem Please try problem #31 on page 480 A pendulum

### Vibrations and Waves Physics Year 1. Handout 1: Course Details

Vibrations and Waves Jan-Feb 2011 Handout 1: Course Details Office Hours Vibrations and Waves Physics Year 1 Handout 1: Course Details Dr Carl Paterson (Blackett 621, carl.paterson@imperial.ac.uk Office

### M2A2 Problem Sheet 3 - Hamiltonian Mechanics

MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,

### AP Physics B Syllabus

AP Physics B Syllabus Course Overview Advanced Placement Physics B is a rigorous course designed to be the equivalent of a college introductory Physics course. The focus is to provide students with a broad

### 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

### Simple and Physical Pendulums Challenge Problem Solutions

Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional

### Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series

Fourier series Electrical Circuits Lecture - Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform - Fourier

### Healy/DiMurro. Vibrations 2016

Name Vibrations 2016 Healy/DiMurro 1. In the diagram below, an ideal pendulum released from point A swings freely through point B. 4. As the pendulum swings freely from A to B as shown in the diagram to

### 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

### Unforced Mechanical Vibrations

Unforced Mechanical Vibrations Today we begin to consider applications of second order ordinary differential equations. 1. Spring-Mass Systems 2. Unforced Systems: Damped Motion 1 Spring-Mass Systems We

### Nonlinear Control Lecture # 36 Tracking & Regulation. Nonlinear Control

Nonlinear Control Lecture # 36 Tracking & Regulation Normal form: η = f 0 (η,ξ) ξ i = ξ i+1, for 1 i ρ 1 ξ ρ = a(η,ξ)+b(η,ξ)u y = ξ 1 η D η R n ρ, ξ = col(ξ 1,...,ξ ρ ) D ξ R ρ Tracking Problem: Design

### Physics 2001/2051 The Compound Pendulum Experiment 4 and Helical Springs

PY001/051 Compound Pendulum and Helical Springs Experiment 4 Physics 001/051 The Compound Pendulum Experiment 4 and Helical Springs Prelab 1 Read the following background/setup and ensure you are familiar

### E209A: Analysis and Control of Nonlinear Systems Problem Set 6 Solutions

E9A: Analysis and Control of Nonlinear Systems Problem Set 6 Solutions Michael Vitus Gabe Hoffmann Stanford University Winter 7 Problem 1 The governing equations are: ẋ 1 = x 1 + x 1 x ẋ = x + x 3 Using

### Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx

Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull

### Modeling and Experimentation: Compound Pendulum

Modeling and Experimentation: Compound Pendulum Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin Fall 2014 Overview This lab focuses on developing a mathematical

### Some Dynamical Behaviors In Lorenz Model

International Journal Of Computational Engineering Research (ijceronline.com) Vol. Issue. 7 Some Dynamical Behaviors In Lorenz Model Dr. Nabajyoti Das Assistant Professor, Department of Mathematics, Jawaharlal

### Reverse Order Swing-up Control of Serial Double Inverted Pendulums

Reverse Order Swing-up Control of Serial Double Inverted Pendulums T.Henmi, M.Deng, A.Inoue, N.Ueki and Y.Hirashima Okayama University, 3-1-1, Tsushima-Naka, Okayama, Japan inoue@suri.sys.okayama-u.ac.jp

### 2:2:1 Resonance in the Quasiperiodic Mathieu Equation

Nonlinear Dynamics 31: 367 374, 003. 003 Kluwer Academic Publishers. Printed in the Netherlands. ::1 Resonance in the Quasiperiodic Mathieu Equation RICHARD RAND Department of Theoretical and Applied Mechanics,

### APPPHYS217 Tuesday 25 May 2010

APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag

### Feedback stabilisation with positive control of dissipative compartmental systems

Feedback stabilisation with positive control of dissipative compartmental systems G. Bastin and A. Provost Centre for Systems Engineering and Applied Mechanics (CESAME Université Catholique de Louvain

### 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

### First Order RC and RL Transient Circuits

First Order R and RL Transient ircuits Objectives To introduce the transients phenomena. To analyze step and natural responses of first order R circuits. To analyze step and natural responses of first

### arxiv: v1 [nlin.ao] 19 May 2017

Feature-rich bifurcations in a simple electronic circuit Debdipta Goswami 1, and Subhankar Ray 2, 1 Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA

### Topic 6: Projected Dynamical Systems

Topic 6: Projected Dynamical Systems John F. Smith Memorial Professor and Director Virtual Center for Supernetworks Isenberg School of Management University of Massachusetts Amherst, Massachusetts 01003

### Lecture Notes of EE 714

Lecture Notes of EE 714 Lecture 1 Motivation Systems theory that we have studied so far deals with the notion of specified input and output spaces. But there are systems which do not have a clear demarcation

### Chapter 9 Global Nonlinear Techniques

Chapter 9 Global Nonlinear Techniques Consider nonlinear dynamical system 0 Nullcline X 0 = F (X) = B @ f 1 (X) f 2 (X). f n (X) x j nullcline = fx : f j (X) = 0g equilibrium solutions = intersection of

### Application of Viscous Vortex Domains Method for Solving Flow-Structure Problems

Application of Viscous Vortex Domains Method for Solving Flow-Structure Problems Yaroslav Dynnikov 1, Galina Dynnikova 1 1 Institute of Mechanics of Lomonosov Moscow State University, Michurinskiy pr.

### DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS

DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University

### APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS

APPLICATIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS Second-order linear differential equations have a variety of applications in science and engineering. In this section we explore two of them: the vibration

### Control Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli

Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)

### Chaos suppression of uncertain gyros in a given finite time

Chin. Phys. B Vol. 1, No. 11 1 1155 Chaos suppression of uncertain gyros in a given finite time Mohammad Pourmahmood Aghababa a and Hasan Pourmahmood Aghababa bc a Electrical Engineering Department, Urmia

### Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,

Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you

### The Important State Coordinates of a Nonlinear System

The Important State Coordinates of a Nonlinear System Arthur J. Krener 1 University of California, Davis, CA and Naval Postgraduate School, Monterey, CA ajkrener@ucdavis.edu Summary. We offer an alternative

### Chapter 13 Internal (Lyapunov) Stability 13.1 Introduction We have already seen some examples of both stable and unstable systems. The objective of th

Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

### Stability Analysis for ODEs

Stability Analysis for ODEs Marc R Roussel September 13, 2005 1 Linear stability analysis Equilibria are not always stable Since stable and unstable equilibria play quite different roles in the dynamics

### 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

### Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 9, 2017 11:00AM to 1:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

### Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

### Dynamics of Non-Smooth Systems

Dynamics of Non-Smooth Systems P Chandramouli Indian Institute of Technology Madras October 5, 2013 Mouli, IIT Madras Non-Smooth System Dynamics 1/39 Introduction Filippov Systems Behaviour around hyper-surface

### Oscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums

PHYS 101 Previous Exam Problems CHAPTER 15 Oscillations Simple harmonic motion Mass-spring system Energy in SHM Pendulums 1. The displacement of a particle oscillating along the x axis is given as a function

### Dynamical behaviour of a controlled vibro-impact system

Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and

### CDS Solutions to the Midterm Exam

CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2

### Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters

Vol 16 No 5, May 2007 c 2007 Chin. Phys. Soc. 1009-1963/2007/16(05)/1246-06 Chinese Physics and IOP Publishing Ltd Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with

### Convergence Rate of Nonlinear Switched Systems

Convergence Rate of Nonlinear Switched Systems Philippe JOUAN and Saïd NACIRI arxiv:1511.01737v1 [math.oc] 5 Nov 2015 January 23, 2018 Abstract This paper is concerned with the convergence rate of the

### CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE

CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE PETER ROBICHEAUX Abstract. The goal of this paper is to examine characterizations of linear differential equations. We define the flow of an equation and examine

### Classical Particles Having Complex Energy Exhibit Quantum-Like Behavior

Classical Particles Having Complex Energy Exhibit Quantum-Like Behavior Carl M. Bender Department of Physics Washington University St. Louis MO 63130, USA Email: cmb@wustl.edu 1 Introduction Because the