Represent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T
|
|
- Janis Weaver
- 6 years ago
- Views:
Transcription
1 Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented by the symbol solution p q P R of 9 ;, that map their input p q P R to the summation blocks, represented by the symbol output pq ; and, that map their input p q P R to the gain memoryless systems, represented by the symbol, that map their input p q P R to the output pq pq for some P R. m θ l g From Newton s law: Figure. Inverted pendulum : θ sin θ 9 θ ` T where T denotes a torque applied at the base, and is the gravitational acceleration. Exercise (Local linearization around equilibria). Consider the inverted pendulum in Figure and assume the input and output to the system are the signals and dened as T satpq θ where sat denotes the unit-slope saturation function that truncates at ` and. (a) Linearize this system around the equilibrium point for which θ. (b) Linearize this system around the equilibrium point for which θ π (assume that the pendulum is free to rotate all the way to this conguration without hitting the table). Linearize this system around the equilibrium point for which θ π 4. Does such an equilibrium point always exist? (d) Assume that { and {4. Compute the torque T pq needed for the pendulum to fall from θpq with constant velocity 9 Linearize the system around this trajectory. Exercise 3 (Local linearization around a trajectory). A single-wheel cart (unicycle) moving on the plane with linear velocity and angular velocity ω can be modeled by the nonlinear system 9 cos θ 9 sin θ 9 θ ω () where p q denote the Cartesian coordinates of the wheel and θ its orientation. Regard this as a system with input ω P R.
2 (a) Construct a state-space model for this system with state fl cos θ `p q sin θ sin θ `p q cos θfl 3 θ and output P R. (b) Compute a local linearization for this system around the equilibrium point eq, eq. Show that ωpq pq, pq sin, pq cos, is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system. Exercise 4 (Feedback linearization controller). Consider the inverted pendulum in Figure. (a) Assume that you can directly control the system in torque, i.e., that the control input is T. Design a feedback linearization controller to drive the pendulum to the upright position. Use the following values for the parameters: m, kg, Nm s, and 98 ms. Verify the performance of your system in the presence of measurement noise using Simulink. Attention! Writing the system in the carefully chosen coordinates 3 is crucial to getting an LTI linearization. If one tried to linearize this system in the original coordinates θ with dynamics given by (), one would get an LTV system. (b) Assume now that the pendulum is mounted on a cart and that you can control the cart s jerk, which is the derivative of its acceleration. In this case, T cos θ 9 Design a feedback linearization controller for the new system. What happens around θ π{? Note that, unfortunately, the pendulum needs to pass by one of these points for a swing-up, i.e., the motion from θ π (pendulum down) to θ (pendulum upright).
3 Exercise 5 (Observable canonical form). Given a transfer function Ĝpq, let pā B C Dq be a realization for its transpose Ḡpq Ĝpq. Show that pa B C Dq, where A Ā, B C, C B, and D D is a realization for Ĝpq. Note that if the realization pā B C Dq for Ḡpq is in controllable canonical form, then the realization pa B C Dq for Ĝpq so obtained is in observable canonical form. Exercise 6 (SISO realizations). This exercise aims at proving the theorem in Section Use the construction outlined in Section 4.3. to arrive at results consistent with the theorem in Section (a) Compute the controllable canonical form realization for the transfer function ˆpq ` α ` α ` `α ` α (b) For the realization in (a), compute the transfer function from the input to the new output, where is the th element of the state. Hint: You can compute pi Aq using the technique used in class for MIMO systems, or you may simply invert pi Aq using the adjoint formula for matrix inversion: M det M padj Mq adj M rcof Ms where cof M denotes the th cofactor of M. In this problem you actually need only to compute a single entry of pi Aq. Compute the controllable canonical form realization for the transfer function ˆpq β ` β ` `β ` β ` α ` α ` `α ` α () (d) Compute the observable canonical form realization for the transfer function in equation (). Hint: See Exercise 5. Exercise 7 (Equivalent realizations). Consider the following two systems: 9 fl ` fl 9 fl ` fl (a) Are these systems zero-state equivalent? (b) Are they algebraically equivalent?
4 Exercise 8 (State transition matrix). Consider the system 9 ` P R P R (a) Compute its state transition matrix (b) Compute the system output to the constant input for an arbitrary initial condition pq pq pq. Exercise 9 (Matrix powers and exponential). Compute A and A for the following matrices A fl A fl A 3 3 3fl (3) 3 Exercise (Jordan normal forms). Compute the Jordan normal form of the A matrix for the system represented by the following block diagram: u s ` ω y s s s ` ω y + y + y 3 Figure. Block interconnection for Exercise.
5 Exercise (Stability margin). Consider the continuous-time LTI system 9 A P R and suppose that there exists a positive constant µ and positive-denite matrices PQ P R for the Lyapunov equation A P ` PA ` µp Q (4) Show that all eigenvalues of A have real parts less than µ. A matrix A with this property is said to be asymptotically stable with stability margin µ. Hint: Start by showing that all eigenvalues of A have real parts less than µ if and only if all eigenvalues of A ` µi have real parts less than (i.e., A ` µi is a stability matrix). Exercise (Stability of nonlinear systems). Investigate whether or not the solutions to the following nonlinear systems converge to the given equilibrium point when they start sufciently close to it. (a) The state-space system with equilibrium point. (b) The second-order system 9 ` p ` q 9 ` p ` q : ` pq 9 ` with equilibrium point 9. Determine for which values of pq we can guarantee convergence to the origin based on the local linearization. This equation is called the Lienard equation and can be used to model several mechanical systems, depending on the choice of the function p q. Exercise 3. Consider the system 9 fl ` fl ` (a) Compute the system s transfer function. (b) Is the matrix A asymptotically stable, marginally stable, or unstable? Is this system BIBO stable?
6 Exercise 4 (A-invariance and controllability). Consider the LTI systems 9 A ` B { ` A ` B P R P R (AB-LTI) Prove the following two statements: (a) The controllable subspace C of the system (AB-LTI) is A-invariant. (b) The controllable subspace C of the system (AB-LTI) contains Im B. Exercise 5 (Satellite). The equations of motion of a satellite, linearized around a steady-state solution, are given by 9 A ` B, where and denote the perturbations in the radius and the radial velocity, respectively, 3 and 4 denote the perturbations in the angle and the angular velocity, and 3ω ω A ω fl B The input vector consists of a radial thruster and a tangential thruster. (a) Show that the system is controllable from. (b) Can the system still be controlled if the radial thruster fails? What if the tangential thruster fails? Exercise 6 (Controllable canonical form). Consider a system in controllable canonical form α I ˆ α I ˆ α I ˆ α I ˆ I ˆ ˆ ˆ ˆ A ˆ I ˆ ˆ ˆ fl ˆ ˆ I ˆ ˆ ˆ I ˆ ˆ B. ˆ fl ˆ ˆ C N N N N ˆ Show that such a system is always controllable. fl
7 Exercise 7 (Eigenvalue assignment). Consider the SISO LTI system in controllable canonical form 9 A ` B P R P R (AB-DLTI) where α α α α A fl ˆ B. fl ˆ (a) Compute the characteristic polynomial of the closed-loop system for K K Hint: Compute the determinant of pi A ` BKq by doing a Laplacian expansion along the rst line of this matrix. (b) Suppose you are given complex numbers λ, λ,...,λ as desired locations for the closed-loop eigenvalues. Which characteristic polynomial for the closed-loop system would lead to these eigenvalues? Based on the answers to parts (a) and (b), propose a procedure to select K that would result in the desired values for the closed-loop eigenvalues. (d) Suppose that A 3 fl B fl Find a matrix K for which the closed-loop eigenvalues are t u. Exercise 8 (Transformation to controllable canonical form). Consider the following third-order SISO LTI system 9 A ` B P R 3 P R (AB-CLTI) Assume that the characteristic polynomial of A is given by detpi Aq 3 ` α ` α ` α 3 and consider the 3 ˆ 3 matrix T α α α fl (5) where is the system s controllability matrix. (a) Show that the following equality holds: B T fl
8 (b) Show that the following equality holds: α α α 3 AT T fl Hint: Compute separately the left- and right-hand side of the equation above and then show that the two matrices are equal with the help of the Cayley-Hamilton theorem. Show that if the system (AB-CLTI) is controllable, then T is a nonsingular matrix. (d) Combining parts (a), you showed that, if the system (AB-CLTI) is controllable, then the matrix T given by equation (5) can be viewed as a similarity transformation that transforms the system into the controllable canonical form α T α α 3 AT fl T B Use this to nd the similarity transformation that transforms the following pair into the controllable canonical form A fl B fl 4 3 Hint: You may use the MATLAB R functions poly(a) to compute the characteristic polynomial of A and ctrb(a,b) to compute the controllability matrix of the pair (A,B). fl
9 Exercise 9 (Diagonal Systems). Consider the following system 9 fl 3 where,, and 3 are unknown scalars. (a) Provide an example of values for,, and 3 for which the system is not observable. (b) Provide an example of values for,, and 3 for which the system is observable. Provide a necessary and sufcient condition on the so that the system is observable. Hint: Use the eigenvector test. Make sure that you provide a condition that when true the system is guaranteed to be observable, but when false the system is guaranteed to not be observable. (d) Generalize the previous result for an arbitrary system with a single output and diagonal matrix A. Exercise (Diagonal Systems). Consider the system 9 fl 3 where,, and 3 are unknown scalars. (a) Provide a necessary and sufcient condition on the so that the system is detectable. (b) Generalize the previous result for an arbitrary system with a single output and diagonal matrix A. Exercise (Repeated eigenvalues). Consider the SISO LTI system 9{` A ` B C ` D P R P R (a) Assume that A is a diagonal matrix and B, C are column/row vectors with entries and, respectively. Write the controllability and observability matrices for this system. (b) Show that if A is a diagonal matrix with repeated eigenvalues, then the pair pa Bq cannot be controllable and the pair pa Cq cannot be observable. Given a SISO transfer function T pq, can you nd a minimal realization for T pq for which the matrix A is diagonalizable with repeated eigenvalues? Justify your answer. (d) Given a SISO transfer function T pq, can you nd a minimal realization for T pq for which the matrix A is not diagonalizable with repeated eigenvalues? Justify your answer. Hint: An example sufces to justify the answer yes in or (d).
Contents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31
Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems 5 1.1 State-Space Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization
More informationJoão P. Hespanha. January 16, 2009
LINEAR SYSTEMS THEORY João P. Hespanha January 16, 2009 Disclaimer: This is a draft and probably contains a few typos. Comments and information about typos are welcome. Please contact the author at hespanha@ece.ucsb.edu.
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 informationCayley-Hamilton Theorem
Cayley-Hamilton Theorem Massoud Malek In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n Let A be an n n matrix Although det (λ I n A
More informationChapter 30 Minimality and Stability of Interconnected Systems 30.1 Introduction: Relating I/O and State-Space Properties We have already seen in Chapt
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
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3. 8. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid
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 informationAMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A =
AMS1 HW Solutions All credit is given for effort. (- pts for any missing sections) Problem 1 ( pts) Consider the following matrix 1 1 9 a. Calculate the eigenvalues of A. Eigenvalues are 1 1.1, 9.81,.1
More informationDefinition (T -invariant subspace) Example. Example
Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin
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 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 informationEC Control Engineering Quiz II IIT Madras
EC34 - Control Engineering Quiz II IIT Madras Linear algebra Find the eigenvalues and eigenvectors of A, A, A and A + 4I Find the eigenvalues and eigenvectors of the following matrices: (a) cos θ sin θ
More informationLecture 11: Diagonalization
Lecture 11: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 11 1 / 11 Definition The n n matrix A is diagonalizableif there exits nonsingular matrix P d 1 0 0. such that P 1 AP = D, where
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More informationLab 6a: Pole Placement for the Inverted Pendulum
Lab 6a: Pole Placement for the Inverted Pendulum Idiot. Above her head was the only stable place in the cosmos, the only refuge from the damnation of the Panta Rei, and she guessed it was the Pendulum
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
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 informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
More informationExercise Set 7.2. Skills
Orthogonally diagonalizable matrix Spectral decomposition (or eigenvalue decomposition) Schur decomposition Subdiagonal Upper Hessenburg form Upper Hessenburg decomposition Skills Be able to recognize
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
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 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 informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
More informationDiagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Diagonalization MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Motivation Today we consider two fundamental questions: Given an n n matrix A, does there exist a basis
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationTheory and Problems of Signals and Systems
SCHAUM'S OUTLINES OF Theory and Problems of Signals and Systems HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University
More informationa 11 a 12 a 11 a 12 a 13 a 21 a 22 a 23 . a 31 a 32 a 33 a 12 a 21 a 23 a 31 a = = = = 12
24 8 Matrices Determinant of 2 2 matrix Given a 2 2 matrix [ ] a a A = 2 a 2 a 22 the real number a a 22 a 2 a 2 is determinant and denoted by det(a) = a a 2 a 2 a 22 Example 8 Find determinant of 2 2
More informationAUTOMATIC CONTROL. Andrea M. Zanchettin, PhD Spring Semester, Introduction to Automatic Control & Linear systems (time domain)
1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Introduction to Automatic Control & Linear systems (time domain) 2 What is automatic control? From Wikipedia Control theory is an interdisciplinary
More informationMEM 355 Performance Enhancement of Dynamical Systems MIMO Introduction
MEM 355 Performance Enhancement of Dynamical Systems MIMO Introduction Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 11/2/214 Outline Solving State Equations Variation
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
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 informationAnalysis of Discrete-Time Systems
TU Berlin Discrete-Time Control Systems 1 Analysis of Discrete-Time Systems Overview Stability Sensitivity and Robustness Controllability, Reachability, Observability, and Detectabiliy TU Berlin Discrete-Time
More informationRobust Control 2 Controllability, Observability & Transfer Functions
Robust Control 2 Controllability, Observability & Transfer Functions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /26/24 Outline Reachable Controllability Distinguishable
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More informationAutomatic Control Systems theory overview (discrete time systems)
Automatic Control Systems theory overview (discrete time systems) Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Motivations
More informationLecture 9 Nonlinear Control Design
Lecture 9 Nonlinear Control Design Exact-linearization Lyapunov-based design Lab 2 Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.2] and [Glad-Ljung,ch.17] Course Outline
More 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 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 informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More informationControl of Mobile Robots
Control of Mobile Robots Regulation and trajectory tracking Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Organization and
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 informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationEE221A Linear System Theory Final Exam
EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,
More information18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationLinear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
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 informationAnalysis of Discrete-Time Systems
TU Berlin Discrete-Time Control Systems TU Berlin Discrete-Time Control Systems 2 Stability Definitions We define stability first with respect to changes in the initial conditions Analysis of Discrete-Time
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 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 / 54 Outline 1 G. Ducard c 2 / 54 Outline 1 G. Ducard
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationSYSTEMTEORI - ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation:
SYSTEMTEORI - ÖVNING 1 GIANANTONIO BORTOLIN AND RYOZO NAGAMUNE In this exercise, we will learn how to solve the following linear differential equation: 01 ẋt Atxt, xt 0 x 0, xt R n, At R n n The equation
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 informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationDIAGONALIZATION BY SIMILARITY TRANSFORMATIONS
DIAGONALIZATION BY SIMILARITY TRANSFORMATIONS The correct choice of a coordinate system (or basis) often can simplify the form of an equation or the analysis of a particular problem. For example, consider
More informationLecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization
Lecture 9 Nonlinear Control Design Course Outline Eact-linearization Lyapunov-based design Lab Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.] and [Glad-Ljung,ch.17] Lecture
More informationName: Final Exam MATH 3320
Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following
More informationEigenvalues and Eigenvectors
November 3, 2016 1 Definition () The (complex) number λ is called an eigenvalue of the n n matrix A provided there exists a nonzero (complex) vector v such that Av = λv, in which case the vector v is called
More informationLINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS
LINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationUniversity of Toronto Department of Electrical and Computer Engineering ECE410F Control Systems Problem Set #3 Solutions = Q o = CA.
University of Toronto Department of Electrical and Computer Engineering ECE41F Control Systems Problem Set #3 Solutions 1. The observability matrix is Q o C CA 5 6 3 34. Since det(q o ), the matrix is
More informationTopic # /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
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 informationk is a product of elementary matrices.
Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation
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 informationMATH 1553, C. JANKOWSKI MIDTERM 3
MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until
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 informationHomework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9
Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year - Is
More informationLecture 7 and 8. Fall EE 105, Feedback Control Systems (Prof. Khan) September 30 and October 05, 2015
1 Lecture 7 and 8 Fall 2015 - EE 105, Feedback Control Systems (Prof Khan) September 30 and October 05, 2015 I CONTROLLABILITY OF AN DT-LTI SYSTEM IN k TIME-STEPS The DT-LTI system is given by the following
More informationMAT2342 : Introduction to Linear Algebra Mike Newman, 5 October assignment 1
[/8 MAT4 : Introduction to Linear Algebra Mike Newman, 5 October 07 assignment You must show your work. You may use software or solvers of some sort to check calculation of eigenvalues, but you should
More informationChapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected
More informationReview of Controllability Results of Dynamical System
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 4 Ver. II (Jul. Aug. 2017), PP 01-05 www.iosrjournals.org Review of Controllability Results of Dynamical System
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationMath 110 (Fall 2018) Midterm II (Monday October 29, 12:10-1:00)
Math 110 (Fall 2018) Midterm II (Monday October 29, 12:10-1:00) Name: SID: Please write clearly and legibly. Justify your answers. Partial credits may be given to Problems 2, 3, 4, and 5. The last sheet
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 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 informationLecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3.12 in Boas)
Lecture 9: Eigenvalues and Eigenvectors in Classical Mechanics (See Section 3 in Boas) As suggested in Lecture 8 the formalism of eigenvalues/eigenvectors has many applications in physics, especially in
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationPractice Final Exam Solutions
MAT 242 CLASS 90205 FALL 206 Practice Final Exam Solutions The final exam will be cumulative However, the following problems are only from the material covered since the second exam For the material prior
More informationState Feedback and State Estimators Linear System Theory and Design, Chapter 8.
1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 2 Homework 7: State Estimators (a) For the same system as discussed in previous slides, design another closed-loop state estimator,
More informationChapter 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
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 9. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -
More informationEigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationJordan Canonical Form
Jordan Canonical Form Massoud Malek Jordan normal form or Jordan canonical form (named in honor of Camille Jordan) shows that by changing the basis, a given square matrix M can be transformed into a certain
More informationANSWERS. E k E 2 E 1 A = B
MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
More information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More informationSYSTEMTEORI - ÖVNING Stability of linear systems Exercise 3.1 (LTI system). Consider the following matrix:
SYSTEMTEORI - ÖVNING 3 1. Stability of linear systems Exercise 3.1 (LTI system. Consider the following matrix: ( A = 2 1 Use the Lyapunov method to determine if A is a stability matrix: a: in continuous
More informationMath 113 Final Exam: Solutions
Math 113 Final Exam: Solutions Thursday, June 11, 2013, 3.30-6.30pm. 1. (25 points total) Let P 2 (R) denote the real vector space of polynomials of degree 2. Consider the following inner product on P
More information