Control Systems. Time response
|
|
- Arthur Lloyd
- 5 years ago
- Views:
Transcription
1 Control Systems Time response L. Lanari
2 outline zero-state solution matrix exponential total response (sum of zero-state and zero-input responses) Dirac impulse impulse response change of coordinates (state) Lanari: CS - Time response 2
3 system x() = x u(t) input ẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t)+Du(t) x(t) state y(t) output Linear Time Invariant (LTI) dynamical system (Continuous Time) x IR n u IR m y IR p Lanari: CS - Time response 3
4 system representation ẋ(t) =Ax(t)+Bu(t) x() = x implicit representation v = F m v() = v example x(t) =... v(t) =v + 1 m F ( )d explicit representation (solution) we want to study the solution of the set of differential equations in order to have a qualitative knowledge of the system motion Lanari: CS - Time response 4
5 solution: zero-input response zero-input response (i.e. with u(t) = ) ẋ(t) =Ax(t)+Bu(t) scalar case ẋ = ax x() = x x(t) =e at x matrix case ẋ = Ax x() = x x(t) =e At x dimensions n 1 n 1 n n definition e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! check ẋ = d dt e At x = = Ax Lanari: CS - Time response 5
6 matrix exponential definition e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! properties e At = I t= e At1 e At 2 = e A(t 1+t 2 ) e A 1t e A2t = e (A 1+A 2 )t e At 1 = e At consistency composition = () A1A2 = A2A1 equality holds iff d dt eat = Ae At Lanari: CS - Time response 6
7 solution: zero-input response the exponential matrix propagates the initial condition into state at time t x e At propagation x(t) the exponential matrix propagates the state t seconds forward in time x( ) x( + t) =e At x( ) Euler approximation exact solution x( + t) x( )+tẋ( )=(I + At)x( ) x( + t) =e At x( )=(I + At + A 2 t 2 /2! +...)x( ) Lanari: CS - Time response 7
8 solution: zero-input response phase plane x = x1 x 2 it is possible to represent the vector field ẋ = Ax ẋ(t) =Ax(t) x(t) vector field ẋ(t) = apple x(t) and build a phase portrait (geometric representation of the trajectories) Lanari: CS - Time response 8
9 solution on phase plane phase portrait Pendulum with no damping (nonlinear differential equation) #(t) solution #(t)+ g` sin #(t) = (#(), #()) = (, ) unstable equilibrium state stable equilibrium state vector field # (#(), #()) = (, ) #(t) #() (#(), #()) = (.45,.8) Lanari: CS - Time response 9
10 solution: total response scalar case ẋ = ax + bu x(t) =e at x + e a(t ) bu( )d (check) Leibniz integral rule d dt f(t, )d = d dt f(t, )d + f(t, ) =t ẋ = ae at x + ae a(t ) bu( )d + bu(t) =ax + bu Lanari: CS - Time response 1
11 solution: total response matrix case x(t) =e At x + e A(t ) Bu( )d zero-input response (ZIR) zero-state response (ZSR) y(t) =Ce At x + Ce A(t ) Bu( )d + Du(t) e At Ce At state-transition matrix output-transition matrix N.B. product is not commutative for matrices Lanari: CS - Time response 11
12 solution: total response total response = zero-input response + zero-state response x() 6= x() = u(t) = u(t) 6= two distinct contributions to the motion of a linear system a non-zero initial condition causes motion as well as a non-zero input alternative names zero-input response = free response/evolution zero-state response = forced response/evolution Lanari: CS - Time response 12
13 superposition principle x(t) =e At x + y(t) =Ce At x + e A(t ) Bu( )d Ce A(t ) Bu( )d + Du(t) if (x a,u a (t)) generates x a (t) and y a (t) if (x b,u b (t)) generates x b (t) and y b (t) then ( x a + x b, u a (t)+ u b (t)) generates x a (t)+ x b (t) only for same linear combination and y a (t)+ y b (t) Lanari: CS - Time response 13
14 Dirac s delta (t ) generalized function (t) = t 6= t Z +1 1 properties ( )d =1 " 1 " t "! approximation f(t) (t ) =f( ) (t ) (as a function of t) u( )Z Z +1 1 f( ) (t )d = f(t) sifting (sampling) property t (as a function of ) Lanari: CS - Time response 14
15 Dirac s delta: application zero-state output response Ce A(t ) Bu( )d + Du(t) = sifting property h i Ce A(t ) B + D (t ) u( )d rewritten as W (t )u( )d with W (t) =Ce At B + D (t) if W (t )u( )d u(t) = (t) W (t ) ( )d = W (t) W(t) defined as the (output) impulse response Lanari: CS - Time response 15
16 impulse response for any input u(t), the zero-state output response is a convolution integral of the impulse response W(t) with the input u(t) output ZSR = W (t )u( )d example: mass + friction, measuring velocity C = 1 v = µ m v + 1 m F W (t) =Ce At B = 1 m e µ m t 1.8 zero-state response to sin(5t) y(t) u(t).6 for u(t) =sin!t output y(t) = = 1 m W (t )u( )d = 1 2 +! 2 h µ m sin!t! cos!t +!e µ m ti time µ m Lanari: CS - Time response
17 Dirac s delta: application zero-state response (state) H(t )u( )d if u(t) = (t) H(t ) ( )d H( ) (t )d = H(t) sifting property t = =! = t = t! = H(t) state impulse response e At B one interpretation H(t) =e At B e At x x = B the impulse is transferring formally looks like with the state instantaneously from to B (ZIR) as if it were starting from x = B Lanari: CS - Time response 17
18 impulse response impact hammer Impulse Response Amplitude Time (sec) impulse response (here not experimental) Hubble Space Telescope Lanari: CS - Time response 18
19 solution: total response more compact notation x(t) = (t)x + y(t) = (t)x + H(t )u( )d W (t )u( )d state (t) =e At H(t) =e At B output (t) =Ce At W (t) =Ce At B + D (t) transition matrix impulse response Dirac impulse Lanari: CS - Time response 19
20 change of coordinates (A, B, C, D) ( A, e B, e C, e D) e z = Tx det(t ) 6= ẋ = Ax + Bu ż = Az e + Bu e y = Cx + Du same system y = Cz e + Du e (proof) input u & output y do not change, only state is chosen differently ea = TAT 1 e B = TB e C = CT 1 e D = D equivalent system representation Lanari: CS - Time response 2
21 impulse response 1 experiment 1 response ẋ = Ax + Bu y = Cx + Du same system (different representation) ż = e Az + e Bu y = e Cz + e Du W (t) =Ce At B + D (t) f W (t) = e Ce e At e B + e D (t) W (t) = f W (t) (proof) i.e. independent from the chosen set of coordinates (state) Lanari: CS - Time response 21
22 general solution (recap) x(t) = (t)x + y(t) = (t)x + H(t )u( )d W (t )u( )d (t) =e At H(t) =e At B (t) =Ce At W (t) =Ce At B + D (t) the matrix exponential appears everywhere do we need to compute the exponential using its definition? e At = I + At + A 2 t2 2! + = 1 X k= A k tk k! Lanari: CS - Time response 22
23 matrix exponential SIAM REVIEW c 23 Society for Industrial and Applied Mathematics Vol. 45, No. 1, pp. 3 Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later Cleve Moler Charles Van Loan there are many different ways to compute the matrix exponential... Abstract. In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others, but that none are completely satisfactory. Most of this paper was originally published in An update, with a separate bibliography, describes a few recent developments. Key words. matrix, exponential, roundoff error, truncation error, condition AMS subject classifications. 15A15, 65F15, 65F3, 65L99 PII. S Introduction. Mathematical models of many physical, biological, and economic processes involve systems of linear, constant coefficient ordinary differential equations ẋ(t) =Ax(t). Here A is a given, fixed, real or complex n-by-n matrix. A solution vector x(t) is sought which satisfies an initial condition x() = x. In control theory, A is known as the state companion matrix and x(t) isthesystem response. In principle, the solution is given by x(t) =e ta x where e ta can be formally defined by the convergent power series e ta = I + ta + t2 A ! Published electronically February 3, 23. A portion of this paper originally appeared in SIAM Review, Volume2,Number4,1978,pages The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA (moler@mathworks.com). Department of Computer Science, Cornell University, 413 Upson Hall, Ithaca, NY (cv@cs.cornell.edu). 1 Lanari: CS - Time response 23
24 matrix exponential we will use changes of coordinates (state) z = T x ẋ = Ax + Bu y = Cx + Du e At = T 1 e At e T easier to compute if 9 T : z = Tx det(t ) 6= then ż = Az e + Bu e y = Cz e + Du e with e e At easier to compute e At e = e TAT 1t = Te At T 1 (proof) Lanari: CS - Time response 24
25 since it will be used frequently, we prove that if A = T 1 AT e then e At = T 1 e At e T from ea = TAT 1 we obtain A = T 1 AT e e At = e T 1 e AT t = Lanari: CS - Time response 25 = 1X k= = T 1 " 1 X 1X k= t k k! T 1 ( e A) k T k= = T 1 e e At T t k k! ( e A) k t k k! (T 1 e AT ) k # T
26 vocabulary English phase plane ZIR/ZSR vector field (state) impulse response convolution integral transition matrix sampling property superposition principle Italiano piano delle fasi evoluzione libera/forzata campo vettoriale risposta impulsiva (nello stato) integrale di convoluzione matrice di transizione proprietà di campionamento principio di sovrapposizione Lanari: CS - Time response 26
Control Systems. Time response. L. Lanari
Control Systems Time response L. Lanari outline zero-state solution matrix exponential total response (sum of zero-state and zero-input responses) Dirac impulse impulse response change of coordinates (state)
More informationControl Systems. Dynamic response in the time domain. L. Lanari
Control Systems Dynamic response in the time domain L. Lanari outline A diagonalizable - real eigenvalues (aperiodic natural modes) - complex conjugate eigenvalues (pseudoperiodic natural modes) - phase
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationIntroduction to Modern Control MT 2016
CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear
More informationControl Systems. Internal Stability - LTI systems. L. Lanari
Control Systems Internal Stability - LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable
More informationSolution of Linear State-space Systems
Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. JANUARY 3, 25 Summary. This is an introduction to ordinary differential equations.
More informationECEN 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 informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More informationLinear dynamical systems with inputs & outputs
EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time
More informationObservability and state estimation
EE263 Autumn 2015 S Boyd and S Lall Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time observability
More informationCyber-Physical Systems Modeling and Simulation of Continuous Systems
Cyber-Physical Systems Modeling and Simulation of Continuous Systems Matthias Althoff TU München 29. May 2015 Matthias Althoff Modeling and Simulation of Cont. Systems 29. May 2015 1 / 38 Ordinary Differential
More informationAutonomous system = system without inputs
Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation
More informationLecture 19 Observability and state estimation
EE263 Autumn 2007-08 Stephen Boyd Lecture 19 Observability and state estimation state estimation discrete-time observability observability controllability duality observers for noiseless case continuous-time
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationIdentification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from
More informationACM/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 informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear time-invariant systems Katalin Hangos University of Pannonia Faculty
More informationEE Control Systems LECTURE 9
Updated: Sunday, February, 999 EE - Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of state-space
More informationControl 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)
More informationContinuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון
Continuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון 2 State Space Models For a causal system with m inputs u t R m and p outputs y t R p, an nth-order
More informationControl Systems. Internal Stability - LTI systems. L. Lanari
Control Systems Internal Stability - LTI systems L. Lanari definitions (AS) - A system S is said to be asymptotically stable if its state zeroinput response converges to the origin for any initial condition
More information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More information06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationLecture 2. Linear Systems
Lecture 2. Linear Systems Ivan Papusha CDS270 2: Mathematical Methods in Control and System Engineering April 6, 2015 1 / 31 Logistics hw1 due this Wed, Apr 8 hw due every Wed in class, or my mailbox on
More informationFinal Exam Sample Problems, Math 246, Spring 2018
Final Exam Sample Problems, Math 246, Spring 2018 1) Consider the differential equation dy dt = 9 y2 )y 2. a) Find all of its stationary points and classify their stability. b) Sketch its phase-line portrait
More informationChapter 6: The Laplace Transform. Chih-Wei Liu
Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More informationDifferential Equations and Modeling
Differential Equations and Modeling Preliminary Lecture Notes Adolfo J. Rumbos c Draft date: March 22, 2018 March 22, 2018 2 Contents 1 Preface 5 2 Introduction to Modeling 7 2.1 Constructing Models.........................
More informationChapter 3 Convolution Representation
Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn
More informationIntroduction ODEs and Linear Systems
BENG 221 Mathematical Methods in Bioengineering ODEs and Linear Systems Gert Cauwenberghs Department of Bioengineering UC San Diego 1.1 Course Objectives 1. Acquire methods for quantitative analysis and
More informationLinear Systems. Chapter Basic Definitions
Chapter 5 Linear Systems Few physical elements display truly linear characteristics. For example the relation between force on a spring and displacement of the spring is always nonlinear to some degree.
More informationSolution via Laplace transform and matrix exponential
EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative
More informationChap 4. State-Space Solutions and
Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations
More informationSolving Dynamic Equations: The State Transition Matrix
Overview Solving Dynamic Equations: The State Transition Matrix EGR 326 February 24, 2017 Solutions to coupled dynamic equations Solutions to dynamic circuits from EGR 220 The state transition matrix Discrete
More informationOrdinary differential equations
Class 11 We will address the following topics Convolution of functions Consider the following question: Suppose that u(t) has Laplace transform U(s), v(t) has Laplace transform V(s), what is the inverse
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability
More informationSolutions of Spring 2008 Final Exam
Solutions of Spring 008 Final Exam 1. (a) The isocline for slope 0 is the pair of straight lines y = ±x. The direction field along these lines is flat. The isocline for slope is the hyperbola on the left
More informationUnderstand the existence and uniqueness theorems and what they tell you about solutions to initial value problems.
Review Outline To review for the final, look over the following outline and look at problems from the book and on the old exam s and exam reviews to find problems about each of the following topics.. Basics
More informationLecture 1 From Continuous-Time to Discrete-Time
Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationECE504: Lecture 8. D. Richard Brown III. Worcester Polytechnic Institute. 28-Oct-2008
ECE504: Lecture 8 D. Richard Brown III Worcester Polytechnic Institute 28-Oct-2008 Worcester Polytechnic Institute D. Richard Brown III 28-Oct-2008 1 / 30 Lecture 8 Major Topics ECE504: Lecture 8 We are
More information11.2 Basic First-order System Methods
112 Basic First-order System Methods 797 112 Basic First-order System Methods Solving 2 2 Systems It is shown here that any constant linear system u a b = A u, A = c d can be solved by one of the following
More informationControl Systems I. Lecture 2: Modeling and Linearization. Suggested Readings: Åström & Murray Ch Jacopo Tani
Control Systems I Lecture 2: Modeling and Linearization Suggested Readings: Åström & Murray Ch. 2-3 Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich September 28, 2018 J. Tani, E.
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 4884 NOVEMBER 9, 7 Summary This is an introduction to ordinary differential equations We
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationChapter 1 Fundamental Concepts
Chapter 1 Fundamental Concepts 1 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationCLTI System Response (4A) Young Won Lim 4/11/15
CLTI System Response (4A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More information1 (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 informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationBackground LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More information4 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
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationAnalog 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)
More informationDiscrete-time linear systems
Automatic Control Discrete-time linear systems Prof. Alberto Bemporad University of Trento Academic year 2-2 Prof. Alberto Bemporad (University of Trento) Automatic Control Academic year 2-2 / 34 Introduction
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationDynamical systems: basic concepts
Dynamical systems: basic concepts Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015
More informationSignals and Systems Chapter 2
Signals and Systems Chapter 2 Continuous-Time Systems Prof. Yasser Mostafa Kadah Overview of Chapter 2 Systems and their classification Linear time-invariant systems System Concept Mathematical transformation
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 5: Calculating the Laplace Transform of a Signal Introduction In this Lecture, you will learn: Laplace Transform of Simple
More informationLinear Systems. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output
Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including
More informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationSTATE-SPACE DYNAMIC SYSTEMS (DISCRETE-TIME)
ECE452/552: Multivariable Control Systems I. 3 1 STATE-SPACE DYNAMIC SYSTEMS (DISCRETE-TIME) 3.1: The transform Computer control requires analog-to-digital (A2D) and digital-toanalog (D2A) conversion.
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationLinear System Theory
Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability
More information+ i. cos(t) + 2 sin(t) + c 2.
MATH HOMEWORK #7 PART A SOLUTIONS Problem 7.6.. Consider the system x = 5 x. a Express the general solution of the given system of equations in terms of realvalued functions. b Draw a direction field,
More information信號與系統 Signals and Systems
Spring 2010 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb10 Jun10 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan
More informationThe Laplace transform
The Laplace transform Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Laplace transform Differential equations 1
More informationEE263: Introduction to Linear Dynamical Systems Review Session 6
EE263: Introduction to Linear Dynamical Systems Review Session 6 Outline diagonalizability eigen decomposition theorem applications (modal forms, asymptotic growth rate) EE263 RS6 1 Diagonalizability consider
More informationLinearization 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
More informationControl Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationThird In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix
Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.
More informationNonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México
Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October
More informationDifferential Equations 2280 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 2015 at 12:50pm
Differential Equations 228 Sample Midterm Exam 3 with Solutions Exam Date: 24 April 25 at 2:5pm Instructions: This in-class exam is 5 minutes. No calculators, notes, tables or books. No answer check is
More informationLinear Differential Equations. Problems
Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =
More informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
More informationCopyright (c) 2006 Warren Weckesser
2.2. PLANAR LINEAR SYSTEMS 3 2.2. Planar Linear Systems We consider the linear system of two first order differential equations or equivalently, = ax + by (2.7) dy = cx + dy [ d x x = A x, where x =, and
More informationDifferential and Difference LTI systems
Signals and Systems Lecture: 6 Differential and Difference LTI systems Differential and difference linear time-invariant (LTI) systems constitute an extremely important class of systems in engineering.
More informationSource-Free RC Circuit
First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order
More informationME 132, Fall 2017, UC Berkeley, A. Packard 334 # 6 # 7 # 13 # 15 # 14
ME 132, Fall 2017, UC Berkeley, A. Packard 334 30.3 Fall 2017 Final # 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 NAME 20 15 20 15 15 18 15 20 # 9 # 10 # 11 # 12 # 13 # 14 # 15 # 16 18 12 12 15 12 20 18 15 Facts: 1.
More information信號與系統 Signals and Systems
Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan
More informationApplications of CAS to analyze the step response of a system with parameters
Applications of CAS to analyze the step response of a system with parameters Takuya Kitamoto kitamoto@yamaguchi-u.ac.jp Faculty of Education Yamaguchi University 753-8513 Japan Abstract Recently, Computer
More informationQuestion: Total. Points:
MATH 308 May 23, 2011 Final Exam Name: ID: Question: 1 2 3 4 5 6 7 8 9 Total Points: 0 20 20 20 20 20 20 20 20 160 Score: There are 9 problems on 9 pages in this exam (not counting the cover sheet). Make
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More informationLINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding
LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display
More informationFinite Difference and Finite Element Methods
Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element
More informationModule 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control
Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/
More informationAPPM 2360: Midterm exam 1 February 15, 2017
APPM 36: Midterm exam 1 February 15, 17 On the front of your Bluebook write: (1) your name, () your instructor s name, (3) your recitation section number and () a grading table. Text books, class notes,
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationMATH 23 Exam 2 Review Solutions
MATH 23 Exam 2 Review Solutions Problem 1. Use the method of reduction of order to find a second solution of the given differential equation x 2 y (x 0.1875)y = 0, x > 0, y 1 (x) = x 1/4 e 2 x Solution
More information1 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
More informationSolutions to Final Exam Sample Problems, Math 246, Spring 2011
Solutions to Final Exam Sample Problems, Math 246, Spring 2 () Consider the differential equation dy dt = (9 y2 )y 2 (a) Identify its equilibrium (stationary) points and classify their stability (b) Sketch
More informationTransfer function and linearization
Transfer function and linearization Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Corso di Controlli Automatici, A.A. 24-25 Testo del corso:
More informationBIBO 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
More information5. Observer-based Controller Design
EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1
More informationEN Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015
EN530.678 Nonlinear Control and Planning in Robotics Lecture 3: Stability February 4, 2015 Prof: Marin Kobilarov 0.1 Model prerequisites Consider ẋ = f(t, x). We will make the following basic assumptions
More information