Spring semester 2006 ESE 601: Hybrid Systems Review material on continuous systems I.

Slides:



Advertisements
Similar presentations
Signals and Systems – Chapter 2
Advertisements

ECE Department, University of Illinois ECE 552 Numerical Circuit Analysis I. Hajj Spring 2015 Lecture One INTRODUCTION Copyright © I. Hajj 2012 All rights.
Lecture 3: Signals & Systems Concepts
State-variable method, standard-form state equations Seo Yeon Youn [Mathematical Circuit Theory and Analysis]
ECE 8443 – Pattern Recognition ECE 3163 – Signals and Systems Objectives: Review Resources: Wiki: State Variables YMZ: State Variable Technique Wiki: Controllability.
Lect.2 Modeling in The Frequency Domain Basil Hamed
NUU meiling CHENModern control systems1 Lecture 01 --Introduction 1.1 Brief History 1.2 Steps to study a control system 1.3 System classification 1.4 System.
Spring semester 2006 ESE 601: Hybrid Systems Review materials on continuous systems II.
Lecture 321 Linearity. Lecture 322 Introduction Linearity is a mathematical property of circuits that makes very powerful analysis techniques possible:
TIME 2014 Technology in Mathematics Education July 1 st - 5 th 2014, Krems, Austria.
Meiling chensignals & systems1 Lecture #2 Introduction to Systems.
Feb 23, 2007 EEE393 Basic Electrical Engineering K.A.Peker Signals and Systems Introduction EEE393 Basic Electrical Engineering.
Leo Lam © Signals and Systems EE235 Leo Lam © Today’s menu Exponential response of LTI system LCCDE Midterm Tuesday next week.
Leo Lam © Signals and Systems EE235 Lecture 18.
EE3010 SaS, L7 1/19 Lecture 7: Linear Systems and Convolution Specific objectives for today: We’re looking at continuous time signals and systems Understand.
Lec 3. System Modeling Transfer Function Model
Signal and Systems Prof. H. Sameti
1 Chapter 1 Fundamental Concepts. 2 signalpattern of variation of a physical quantity,A signal is a pattern of variation of a physical quantity, often.
EEE 301 Signal Processing and Linear Systems Dr
Lecture 14 Introduction to dynamic systems Energy storage Basic time-varying signals Related educational materials: –Chapter 6.1, 6.2.
Dr. Tamer Samy Gaafar.   Teaching Assistant:- Eng. Hamdy Soltan.
Fundamentals of Electric Circuits Chapter 16 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Lecture 19 Review: First order circuit step response Steady-state response and DC gain Step response examples Related educational modules: –Section
Motivation Thus far we have dealt primarily with the input/output characteristics of linear systems. State variable, or state space, representations describe.
President UniversityErwin SitompulModern Control 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1 Modern Control
Signals And Systems Chapter 2 Signals and systems analysis in time domain.
Leo Lam © Signals and Systems EE235. Leo Lam © Today’s menu Yesterday: Exponentials Today: Linear, Constant-Coefficient Differential.
1 Alexander-Sadiku Fundamentals of Electric Circuits Chapter 16 Applications of the Laplace Transform Copyright © The McGraw-Hill Companies, Inc. Permission.
Signals and Systems 1 Lecture 7 Dr. Ali. A. Jalali September 4, 2002
Nonlinear Control Systems ECSE 6420 Spring 2009 Lecture 1: 12 January 2009.
EE102 – SYSTEMS & SIGNALS Fall Quarter, Instructor: Fernando Paganini.
Lecture 7: State-Space Modeling 1.Introduction to state-space modeling Definitions How it relates to other modeling formalisms 2.State-space examples 3.Transforming.
4. Introduction to Signal and Systems
(Part one: Continuous)
Textbook and Syllabus Textbook: Syllabus:
Solutions Q1: a) False. The Fourier transform is unique. If two signals have the same Fourier transform, then there is a confusion when trying.
Modeling & Simulation of Dynamic Systems (MSDS)
Signals and Systems Lecture #6 EE3010_Lecture6Al-Dhaifallah_Term3321.
DEPARTMENT OF MECHANICAL TECHNOLOGY VI -SEMESTER AUTOMATIC CONTROL 1 CHAPTER NO.6 State space representation of Continuous Time systems 1 Teaching Innovation.
Modeling interactions 1. Pendulum m – mass R – rod length x – angle of elevation Small angles x.
1 Chapter 3 State Variable Models The State Variables of a Dynamic System The State Differential Equation Signal-Flow Graph State Variables The Transfer.
Modern Control TheoryLecture 1, written by Xiaodong Zhao1 Modern Control Theory Syllabus Course Coordinator: Dr. Xiaodong Zhao – Room: Building Lab 2,
Higher Order Circuits – How To Obtain State Equations? Consider a circuit with capacitor, inductors, n-terminal resistors and independent sources. Aim.
Stability and instability in nonlinear dynamical systems
Textbook and Syllabus Textbook: Syllabus:
Automatic Control Theory
MESB374 System Modeling and Analysis
What Have We Learned In This Lecture?
Chapter 16 Applications of the Laplace Transform
What is System? Systems process input signals to produce output signals A system is combination of elements that manipulates one or more signals to accomplish.
EE 309 Signal and Linear System Analysis
State Space Representation
Fundamentals of Electric Circuits Chapter 16
Autonomous Cyber-Physical Systems: Dynamical Systems
Lecture 19 Review: Steady-state response and DC gain
Signals and Systems Using MATLAB Luis F. Chaparro
Digital Control Systems (DCS)
Chapter 1 Fundamental Concepts
Chapter 3 Convolution Representation
Chapter 2 Systems Defined by Differential or Difference Equations
Comparison Functions Islamic University of Gaza Faculty of Engineering
Signals and Systems EE235 Leo Lam ©
Signals & Systems (CNET - 221) Chapter-2 Introduction to Systems
Signals & Systems (CNET - 221) Chapter-2
Lecture 1: Signals & Systems Concepts
Performance evaluation of manufacturing systems
Signals and Systems EE235 Lecture 31 Leo Lam ©
Lecture 3: Signals & Systems Concepts
Chapter 3 Modeling in the Time Domain
SIGNALS & SYSTEMS (ENT 281)
Presentation transcript:

Spring semester 2006 ESE 601: Hybrid Systems Review material on continuous systems I

References Kwakernaak, H. and Sivan, R. “Modern signal and systems”, Prentice Hall, Brogan, W., “Modern control theory”, Prentice Hall Int’l, Textbooks or lecture notes on linear systems or systems theory.

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Physical systems Resistor Inductor Capacitor Damper Mass Spring

Electric circuit V + I(t) 1 0 t V(t) t L L

More electric circuit V I(t) + R L C

A pendulum Mg r

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Linear vs nonlinear Linear systems: if the set of solutions is closed under linear operation, i.e. scaling and addition. All the examples are linear systems, except for the pendulum.

Time invariant vs time varying Time invariant: the set of solutions is closed under time shifting. Time varying: the set of solutions is not closed under time shifting.

Autonomous vs non-autonomous Autonomous systems: given the past of the signals, the future is already fixed. Non-autonomous systems: there is possibility for input, non-determinism.

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Techniques for autonomous systems

Techniques for non-autonomous systems

Example: 1 u(t) t 1 y(t) t

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability

Solution concepts

Example of weak solution

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability

Simulation methods x(t) x[1] x[2] x[3]

Simulation methods

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability

State space representation One of the most important representations of linear time invariant systems.

State space representation

Solution to state space rep. Solution:

Exact discretization of autonomous systems x(t) x[1] x[2] x[3] t

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs Simulation and numerical methods State space representation Stability Reachability Discrete time systems

Stability of LTI systems

Stability of nonlinear systems pp stable

Stability of nonlinear systems p Asymptotically stable

Lyapunov functions

Contents Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Reachability of linear systems