Continuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון
|
|
- Lynn Hubbard
- 5 years ago
- Views:
Transcription
1 Continuous Dynamics Solving LTI state-space equations גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון
2 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 state-space description is obtained by introducing n latent variables x t R n called state variables In order to obtain a description of the form: 1. An equation for updating x (usually also takes u into account) 2. An equation for mapping values of x to values of y (sometimes also influenced by u)
3 3 Linear Time Invariant State Space Models For continuous time systems: dx/dt = fax x t t +, ubu(t) t, t y(t) = Cx g xt t +, udu(t) t, t For discrete time systems: x[k x[t + 1] 1] = fax[t] d x k +, ubu[t] k, k y[k] y[t] = gcx[t] d x[k], + u[k], Du[t] k
4 4 Example: One-Dimensional Movement of an Object The Newton's laws of motion for an object moving horizontally on a plane and attached to a wall with a spring: where mξ (t) = u(t) k 1 ξ (t) k 2 ξ(t) ξ(t) is position; ξ (t) is velocity; ξ (t) is acceleration F(t) is an applied force k 1 is the viscous friction coefficient k 2 is the spring constant m is the mass of the object
5 5 Example: One-Dimensional Movement of an Object m ξ (t) = u(t) k 1 ξ (t) k 2 ξ(t) x 1 t x 2 t = 0 1 k 2 /m k 1 /m x 1 t x 2 t + 0 1/m u(t) y(t) = 1 0 x 1 t x 2 t u x = Ax + Bu y = Cx + Du y x 1 t = ξ(t) represents the position of the object x 2 (t) = x 1(t) is the velocity of the object x 2(t) = x 1 t is the acceleration of the object The input u t is the force applied to the object The output y(t) is the position of the object
6 6 Practice Exercise Prove that the explicit solution to discrete-time linear state equation is: x[t + 1] = Ax[t] + Bu[t] y[t] = Cx[t] + Du[t] t 1 x t = A t x t 0 + A t 1 τ Bu[τ] τ=0
7 7 The Matrix Exponential A key quantity in determining solutions to continuous-time LTI state equations is the matrix exponential defined as e At = i=0 1 i! Ai t i Practice exercise: 1. Prove that deat dt = Ae At = e At A 2. Does this mean that x t = e At x 0 is a solution to the differential equation x t = Ax t, x 0 = x 0?
8 8 Example: A is nilpotent Let A = It is easy to check that A 2 = 0. Thus, e At = i=0 1 i! Ai t i = I + At = 6t + 1 9t 4t 1 6t
9 9 Example: A is diagnosable Let A = The eigenvectors of A are 1 2 and 1 1. The eigenvalues of A are 2 and 1. Therefore A = T 1 DT where T = and D = Thus, e At = T 1 e Dt T = e 2t ( e t ) e 2t ( +e t ) 2e 2t ( 1 + e t ) e 2t 2 + e t
10 10 Practice Exercise Apply Leibnitz s rule: to prove that is a solution to the linear state equation x t = Ax t + Bu t
11 11 Output Solution Notice that the state and the output solutions consist of two terms each: An initial condition response due to x 0 And a forced response which depends on the input u(t) over the interval [t 0, t] Practice Exercise: Prove that the system is LTI if x 0 = 0
12 12 Linearization Although almost every real system includes nonlinear features, many systems can be reasonably described, at least within certain operating ranges, by linear models
13 13 Linearization
14 14 Linearization in Two Dimensions
15 15 Linearization in Two Dimensions The function at the right hand side of x = f x represents a surface So the local linear representation should geometrically be a plane This is obtained by the Jacobian matrix of the functional form f(x) at an equilibrium point: If the state space is two dimensional, given by equations of the form x = f 1 x, y y = f 2 (x, y) then the local linearization at an equilibrium point δx δy = f 1/ x f 1 / y f 2 / x f 2 / y where δx = x x and δy = y y The matrix containing the partial derivatives is called the Jacobian matrix The partial derivatives are calculated at the equilibrium point δx δy
16 16 Linearization with MATLAB Modern computational packages include special commands to compute linearized models around a user defined (pre-computed) operating point. In the case of MATLAB-Simulink,the appropriate commands are: linmod (for continuous time systems) and dlinmod (for discrete time and hybrid systems).
17 17 Caution! It is obvious that linearized models are only approximate models Thus these models should be used with appropriate caution (as indeed should all models) In the case of linearized models, the next term in the Taylor s series expansion can often be usefully employed to tell us something about the size of the associated modeling error
18 18 Summary of Linearization Linear models often give deep insights and lead to simple control strategies Linear models can be obtained by linearizing a nonlinear model at an operating point Caution is needed to deal with unavoidable modeling errors
Module 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 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 information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 7: State-space Models Readings: DDV, Chapters 7,8 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 25, 2011 E. Frazzoli
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 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 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 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 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 informationLinear System Theory. Wonhee Kim Lecture 1. March 7, 2018
Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems
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 informationControl Systems Design
ELEC4410 Control Systems Design Lecture 14: Controllability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 14: Controllability p.1/23 Outline
More information16.30 Estimation and Control of Aerospace Systems
16.30 Estimation and Control of Aerospace Systems Topic 5 addendum: Signals and Systems Aeronautics and Astronautics Massachusetts Institute of Technology Fall 2010 (MIT) Topic 5 addendum: Signals, Systems
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 informationLinear Algebra. P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 5243 INTRODUCTION TO CYBER-PHYSICAL SYSTEMS P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015 The objective of this exercise is to assess
More informationReview: 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 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 informationModule 07 Controllability and Controller Design of Dynamical LTI Systems
Module 07 Controllability and Controller Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha October
More informationControl Systems. Time response
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 informationLTI system response. Daniele Carnevale. Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata
LTI system response 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 1 / 15 Laplace
More informationControl Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC4026 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) and Observability Algebraic Tests (Kalman rank condition & Hautus test) A few
More informationEquilibrium points: continuous-time systems
Capitolo 0 INTRODUCTION 81 Equilibrium points: continuous-time systems Let us consider the following continuous-time linear system ẋ(t) Ax(t)+Bu(t) y(t) Cx(t)+Du(t) The equilibrium points x 0 of the system
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 8: Solutions of State-space Models Readings: DDV, Chapters 10, 11, 12 (skip the parts on transform methods) Emilio Frazzoli Aeronautics and Astronautics Massachusetts
More informationSystems and Control Theory Lecture Notes. Laura Giarré
Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 5: State Space Systems State Dimension Infinite-Dimensional systems State-space model (nonlinear) LTI State Space model
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 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 informationControl 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
More informationCALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems
CDS 101 1. For each of the following linear systems, determine whether the origin is asymptotically stable and, if so, plot the step response and frequency response for the system. If there are multiple
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 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 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 informationLinear Systems Theory
ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace
More informationMATH 2250 Final Exam Solutions
MATH 225 Final Exam Solutions Tuesday, April 29, 28, 6: 8:PM Write your name and ID number at the top of this page. Show all your work. You may refer to one double-sided sheet of notes during the exam
More informationModule 08 Observability and State Estimator Design of Dynamical LTI Systems
Module 08 Observability and State Estimator Design of Dynamical LTI Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha November
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More information1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is
ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important
More informationDynamic Modeling. For the mechanical translational system shown in Figure 1, determine a set of first order
QUESTION 1 For the mechanical translational system shown in, determine a set of first order differential equations describing the system dynamics. Identify the state variables and inputs. y(t) x(t) k m
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 informationControl 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 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 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 informationModule 02 CPS Background: Linear Systems Preliminaries
Module 02 CPS Background: Linear Systems Preliminaries Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html August
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 informationTopic # /31 Feedback Control Systems
Topic #7 16.30/31 Feedback Control Systems State-Space Systems What are the basic properties of a state-space model, and how do we analyze these? Time Domain Interpretations System Modes Fall 2010 16.30/31
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 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 informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 19: Stabilization via LMIs Optimization Optimization can be posed in functional form: min x F objective function : inequality constraints
More informationModule 02 Control Systems Preliminaries, Intro to State Space
Module 02 Control Systems Preliminaries, Intro to State Space Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha August 28, 2017 Ahmad
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.4: Dynamic Systems Spring Homework Solutions Exercise 3. a) We are given the single input LTI system: [
More informationFundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad
Fundamentals of Dynamical Systems / Discrete-Time Models Dr. Dylan McNamara people.uncw.edu/ mcnamarad Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian
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 informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationEE451/551: Digital Control. Chapter 7: State Space Representations
EE45/55: Digital Control Chapter 7: State Spae Representations State Variables Definition 7.: The system state is a minimal set of { variables x ( t ), i =, 2,, n needed together with the i input ut, and
More informationTopic # Feedback Control
Topic #7 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop
More informationMATH 251 Examination II July 28, Name: Student Number: Section:
MATH 251 Examination II July 28, 2008 Name: Student Number: Section: This exam has 9 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown.
More informationVideo 8.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 8.1 Vijay Kumar 1 Definitions State State equations Equilibrium 2 Stability Stable Unstable Neutrally (Critically) Stable 3 Stability Translate the origin to x e x(t) =0 is stable (Lyapunov stable)
More informationThe Process. 218 Technical Applications of Computers
ble. Right now, we will introduce simulink on transfer functions with the simple transfer function for the differential equation, dx(t)/dt + x(t) = u(t). Our input signal of u(t) will be the step funtion.
More informationDiscrete Dynamics Finite State Machines גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון
Discrete Dynamics Finite State Machines גרא וייס המחלקה למדעי המחשב אוניברסיטת בן-גוריון 2 Recap: Actor Model An actor is a mapping of input signals to output signals S: R R k R R m where k is the number
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 informationStability lectures. Stability of Linear Systems. Stability of Linear Systems. Stability of Continuous Systems. EECE 571M/491M, Spring 2008 Lecture 5
EECE 571M/491M, Spring 2008 Lecture 5 Stability of Continuous Systems http://courses.ece.ubc.ca/491m moishi@ece.ubc.ca Dr. Meeko Oishi Electrical and Computer Engineering University of British Columbia,
More informationModeling 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
More informationLecture 2 and 3: Controllability of DT-LTI systems
1 Lecture 2 and 3: Controllability of DT-LTI systems Spring 2013 - EE 194, Advanced Control (Prof Khan) January 23 (Wed) and 28 (Mon), 2013 I LTI SYSTEMS Recall that continuous-time LTI systems can be
More informationControl Systems Lab - SC4070 Control techniques
Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from
More informationChapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems
Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.
More informationSTATE VARIABLE (SV) SYSTEMS
Copyright F.L. Lewis 999 All rights reserved Updated:Tuesday, August 05, 008 STATE VARIABLE (SV) SYSTEMS A natural description for dynamical systems is the nonlinear state-space or state variable (SV)
More information1 Some Facts on Symmetric Matrices
1 Some Facts on Symmetric Matrices Definition: Matrix A is symmetric if A = A T. Theorem: Any symmetric matrix 1) has only real eigenvalues; 2) is always iagonalizable; 3) has orthogonal eigenvectors.
More informationSymmetries 2 - Rotations in Space
Symmetries 2 - Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More information6x 2 8x + 5 ) = 12x 8
Example. If f(x) = x 3 4x + 5x + 1, then f (x) = 6x 8x + 5 Observation: f (x) is also a differentiable function... d dx ( f (x) ) = d dx ( 6x 8x + 5 ) = 1x 8 The derivative of f (x) is called the second
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 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 informationLecture Note 1: Background
ECE5463: Introduction to Robotics Lecture Note 1: Background Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 1 (ECE5463 Sp18)
More informationLinear Algebra Review (Course Notes for Math 308H - Spring 2016)
Linear Algebra Review (Course Notes for Math 308H - Spring 2016) Dr. Michael S. Pilant February 12, 2016 1 Background: We begin with one of the most fundamental notions in R 2, distance. Letting (x 1,
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 informationDepartment of Electrical and Computer Engineering ECED4601 Digital Control System Lab3 Digital State Space Model
Department of Electrical and Computer Engineering ECED46 Digital Control System Lab3 Digital State Space Model Objectives. To learn some MATLAB commands that deals with the discrete time systems.. To give
More informationGrammians. Matthew M. Peet. Lecture 20: Grammians. Illinois Institute of Technology
Grammians Matthew M. Peet Illinois Institute of Technology Lecture 2: Grammians Lyapunov Equations Proposition 1. Suppose A is Hurwitz and Q is a square matrix. Then X = e AT s Qe As ds is the unique solution
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 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 informationPerspective. ECE 3640 Lecture 11 State-Space Analysis. To learn about state-space analysis for continuous and discrete-time. Objective: systems
ECE 3640 Lecture State-Space Analysis Objective: systems To learn about state-space analysis for continuous and discrete-time Perspective Transfer functions provide only an input/output perspective of
More informationLecture 4 and 5 Controllability and Observability: Kalman decompositions
1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More information1. Type your solutions. This homework is mainly a programming assignment.
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 5243 INTRODUCTION TO CYBER-PHYSICAL SYSTEMS H O M E W O R K S # 6 + 7 Ahmad F. Taha October 22, 2015 READ Homework Instructions: 1. Type your solutions. This homework
More informationECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04-Nov-2008
ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 04-Nov-2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are
More informationME Fall 2001, Fall 2002, Spring I/O Stability. Preliminaries: Vector and function norms
I/O Stability Preliminaries: Vector and function norms 1. Sup norms are used for vectors for simplicity: x = max i x i. Other norms are also okay 2. Induced matrix norms: let A R n n, (i stands for induced)
More informationSolving Dynamic Equations: The State Transition Matrix II
Reading the Text Solving Dynamic Equations: The State Transition Matrix II EGR 326 February 27, 2017 Just a reminder to read the text Read through longer passages, to see what is connected to class topics.
More information7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
More information9. Introduction and Chapter Objectives
Real Analog - Circuits 1 Chapter 9: Introduction to State Variable Models 9. Introduction and Chapter Objectives In our analysis approach of dynamic systems so far, we have defined variables which describe
More informationHomogeneous and particular LTI solutions
Homogeneous and particular LTI solutions 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 informationFull State Feedback for State Space Approach
Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial
More informationEE 341 Homework Chapter 2
EE 341 Homework Chapter 2 2.1 The electrical circuit shown in Fig. P2.1 consists of two resistors R1 and R2 and a capacitor C. Determine the differential equation relating the input voltage v(t) to the
More informationA nemlineáris rendszer- és irányításelmélet alapjai Relative degree and Zero dynamics (Lie-derivatives)
A nemlineáris rendszer- és irányításelmélet alapjai Relative degree and Zero dynamics (Lie-derivatives) Hangos Katalin BME Analízis Tanszék Rendszer- és Irányításelméleti Kutató Laboratórium MTA Számítástechnikai
More informationSimple Harmonic Motion
Simple Harmonic Motion (FIZ 101E - Summer 2018) July 29, 2018 Contents 1 Introduction 2 2 The Spring-Mass System 2 3 The Energy in SHM 5 4 The Simple Pendulum 6 5 The Physical Pendulum 8 6 The Damped Oscillations
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 4: LMIs for State-Space Internal Stability Solving the Equations Find the output given the input State-Space:
More informationIntegral action in state feedback control
Automatic Control 1 in state feedback control Prof. Alberto Bemporad University of Trento Academic year 21-211 Prof. Alberto Bemporad (University of Trento) Automatic Control 1 Academic year 21-211 1 /
More informationECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77
1/77 ECEN 605 LINEAR SYSTEMS Lecture 7 Solution of State Equations Solution of State Space Equations Recall from the previous Lecture note, for a system: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t),
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 informationInteractions. Yuan Gao. Spring Applied Mathematics University of Washington
Interactions Yuan Gao Applied Mathematics University of Washington yuangao@uw.edu Spring 2015 1 / 27 Nonlinear System Consider the following coupled ODEs: dx = f (x, y). dt dy = g(x, y). dt In general,
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More information