ME8281-Advanced Control Systems Design
|
|
- Roderick George
- 5 years ago
- Views:
Transcription
1 ME Advanced Control Systems Design Spring 2016 Perry Y. Li Department of Mechanical Engineering University of Minnesota Spring 2016
2 Lecture 4 - Outline 1 Homework 1 to be posted by tonight 2 Transition matrix for periodic A(t) = A(t + T ). 3 and for constant A Matrix exponential: expm(a (t t 0 )) Laplace transform Eigen decomposition 4 Decomposition into system modes Algebraic and geometric meaning of eigen-decomposition Time varying eigen values, time-invariant eigen vectors Jordan form 5 Zero-initial state response (response to input) 6 Discrete time response
3 Periodic A(t) = A(t + T ) 1 Homework problem - Hint. 2 For a periodic system with period T Why? 3 Floquet theory where 0 τ 1, τ 0 < T Φ(t + T, t 0 + T ) = Φ(t, t 0 ) Φ(t 1, t 0 ) = Φ(τ 1, 0)Φ(T, 0) k Φ(T, τ 0 ) 4 Hence Φ(t, t 0 ) for all (t, t 0 ) can be characterized quite easily by knowing Φ(, ) over a a finite range of (t, t 0 ).
4 Constant A case 1 Matrix exponential method (Matlab» expm(a*t)) 2 Laplace transform method 3 Eigen decomposition method
5 Laplace transform method
6 Eigen decomposition Av i = λ i v i 1 (λ i, v i ) - a pair of eigen values and eigen vector 2 If v i, i = 1,... n are independent (i.e. A is semi-simple), let T = [v 1, v 2,..., v n ]. 3 Show that AT = T Λ A = T ΛT 1 exp(a(t t 0 )) = T exp(λ(t t 0 ))T 1 4 Note: exp(λ(t t 0 )) is diagonal. 5 Similarly for other matrix functions: for semi-simple M = T ΛT 1, sin(m) = Tsin(Λ)T 1
7 Some terminology A matrix A R n n is: Simple: if A has n distinct eigen values λ i λ j, i j. This guarantees that A has n independent eign-vectors. Semi-simple: if A has n independent eigen vectors (but does not necessarily have n distinct eigen values.) Example: A = Identity. Jordan form: if A has repeated eigen values and does not have independent eigen vectors. Example ( ) 2 1 A = 0 2
8 Modal Decomposition ẋ = Ax + Bu If A = T ΛT 1 where Λ = diag(λ 1,..., λ n ),... Coordinate transformation: Let z be such that: x = Tz ẋ = T ż = T Λz + Bu ż = Λz + T 1 Bu ż i = λ i z i + B i u Note z i are decoupled!! Solve problem by: 1 z(t 0 ) = T 1 x(t 0 ); 2 Solve scalar eqns: ż i = λ i z i + B i u for i = 1,..., n. 3 x(t) = Tz(t).
9 Dyadic Expansion A = where n v i w i λ i = D i λ i i=1 v i = i-th column of T (right eigen vector) w i = i-th row of T 1 (left eigen vecotr) Then n Φ(t, 0) = D i exp(λ i t) i=1
10 Phase portrait Plot the flow" (state trajectory) {x(t) t t 0 } for various initial states x(t 0 ). Plot flow in z(t) coordinates first, and then transform back to x(t) = T x(t). Several types - characteristics the same under coordinate transformation: Nodes (real, same signs) Saddle (real, different signs) Focus (imagery) What happens when the eigen-vectors become very close to each other... Jordan form (repeated eigen values but with 1 eigen vector)
11 General form of decomposition - 1 Repeated eigen-values λ 1 = λ 2 = λ 3 = 1 with only one eigen vector v 1 : Decomposed into Jordan block: A = TJT 1 λ i 1 0 J = 0 λ i λ i T = ( ) v 1, v 2, v 3 Defining equations: 0 = (λi A)v 1 v 1 = (λi A)v 2 (λi A) 2 v 2 = 0 v 2 = (λi A)v 3 (λi A) 3 v 3 = 0, etc. Hence, solve for v 1, v 2, v 3 successively by finding the increasing null spaces of (λi A) k.
12 General form of decomposition - 2 In general: A = ( T 1 ) J T 2 T 3 0 J 2 0 ( T 1 T 2 ) 1 T J 3 where J i is a Jordan block which can have length of 1 Also, the blocks may have the same eigenvalues Unions T i for the eigen value is the eigen-subspace for that eigen value.
13 Zero-initial state transition (effect of input) Recall taht: s(t, t 0, x 0, u) = Φ(t, t 0 )x 0 + s(t, t 0, 0 x, u) we focus now on s(t, t 0, 0 x, u).
14 Heuristic guess - 1 Decompose inputs into piecewise continuous parts {u i : R R m } for i =, 2, 1, 1, 0, 1,, { u(t 0 + h i) t 0 + h i t < t 0 + h (i + 1) u i (t) = 0 otherwise where h > 0 is a small positive number: Intuitively we can see that as h 0, u(t) = i= u i (t) as h 0. Let ū(t) = i= u i(t). By linearity of the transition map, s(t, t 0, 0, ū) = i s(t, t 0, 0, u i ).
15 Heuristic guess - 2. Response to u i ( ) Step 1: t 0 t < t 0 + h i. Since u(τ) = 0, τ [t 0, t 0 + h i) and x(t 0 ) = 0, x(t) = 0 for t 0 t < t 0 + h i Step 2: t [t 0 + h i, t 0 + h(i + 1)). Input is active: x(t) x(t 0 + h i) + [A(t 0 + h i)x(t 0 + h i) +B(t 0 + h i)u(t 0 + h i)] T = [B(t 0 + h i)u(t 0 + h i)] T where T = t (t 0 + h i). Step 3: t t 0 + h (i + 1). Input is no longer active, u i (t) = 0. So the state is again given by the zero-input transition map: Φ (t, t 0 + h (i + 1)) B(t 0 + i h)u(t 0 + h i) }{{} x(t 0 +(i+1) h)
16 Heuristic guess -3 Since Φ(t, t 0 ) is continuous, if we make the approximation Φ(t, t 0 + (h + 1)i) Φ(t, t 0 + h i) we only introduce second order error in h. Hence, s(t, t 0, x 0, u i ) Φ(t, t 0 + h i)b(t 0 + h i)u(t 0 + h i). The total zero-state state transition due to the input u( ) is therefore given by: s(t, t 0, 0, u) (t t 0 )/h i=0 Φ (t, t 0 + h i) B(t 0 + h i)u(t 0 + h i) As h 0, the sum becomes an integral so that: s(t, t 0, 0, u) = t t 0 Φ(t, τ)b(τ)u(τ)dτ. (4)
17 Formal proof Show z(t) = t t 0 Φ(t, τ)b(τ)u(τ)dτ satisfies ż = A(t)z + B(t)u(t) and z(t 0 ) = 0 will do.
18 Discrete time system transition map x(k + 1) = A(k)x(k) + B(k)u(k) Existence and uniqueness of solution only in the forward time direction (unless A( ) is invertible) This leads to linearity in (x 0, u( )): s(k 1, k 0, αx a + βx b, αu a ( ) + βu b ( )) =αs(k 1, k 0, x a, u a ( )) + βs(k 1, k 0, x b, u b ( )) Decomposition into zero-input and zero-initial-state transitions: s(k 1, k 0, x 0, u( )) = s(k 1, k 0, x 0, 0 u ) + s(k }{{} 1, k 0, 0 x, u( )) }{{} zero-input zero-initial-state Linearity of zero-input transition map: s(k 1, k 0, x 0, 0 u ) = Φ(k 1, k 0 )x 0
19 Discrete time transition matrix Matrix difference equations: Φ(k + 1, k 0 ) = A(k)Φ(k, k 0 ) Φ(k 1, k 1) = Φ(k 1, k)a(k 1) Φ(k 1, k 0 ) = A(k 1 1)A(k 1 2)... A(k 0 ) Semi-group property: k 0 k 1 k 2 : Φ(k 2, k 0 ) = Φ(k 2, k 1 )Φ(k 1, k 0 ) Inveritibility of Φ(k 1, k 0 )??? Only if A(k) invertible for all k [k 0, k 1 1].
20 Discrete time - zero-initial state response x(k) = A(k 1)x(k 1) + B(k 1)u(k 1). = A(k 1)A(k 2)x(k 2) + A(k 1)B(k 2)u(k 2) + B(k 1)u(k 1) = A(k 1)A(k 2)... A(k 0 )x(k 0 ) + k 1 i=k 0 Π k 1 j=i+1 A(j)B(i)u(i) Thus, since x(k 0 ) = 0 for the the zero-initial state response: s(k, k 0, 0 x, u) = = k 1 i=k 0 Π k 1 j=i+1 A(j)B(i)u(i) k 1 i=k 0 Φ(k, i + 1)B(i)u(i)
21 Summary - Lecture 4 Transition matrix for periodic A(t) = A(t + T ) & constant A Eigen decomposition modal decomposition For constant A matrix, eigen decomposition provides a decoupling of the system into simpler systems Geometric meaning of eigen value and eigen vectors Generalized decomposition (allowing for Jordan blocks). Eigen vectors become eigen subspaces. Response due to inputs (zero-initial state response) is a convolution Discrete time systems response similar to LDS except: uniqueness is guaranteed only for forward time (unless A(k) are inveritible).
6.241 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 informationModal Decomposition and the Time-Domain Response of Linear Systems 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the Time-Domain Response of Linear Systems 1 In a previous handout
More informationECEN 605 LINEAR SYSTEMS. Lecture 8 Invariant Subspaces 1/26
1/26 ECEN 605 LINEAR SYSTEMS Lecture 8 Invariant Subspaces Subspaces Let ẋ(t) = A x(t) + B u(t) y(t) = C x(t) (1a) (1b) denote a dynamic system where X, U and Y denote n, r and m dimensional vector spaces,
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 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 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 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 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 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 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 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 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 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 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. 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 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 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 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 informationLecture Notes 6: Dynamic Equations Part C: Linear Difference Equation Systems
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 45 Lecture Notes 6: Dynamic Equations Part C: Linear Difference Equation Systems Peter J. Hammond latest revision 2017 September
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
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 informationDynamic interpretation of eigenvectors
EE263 Autumn 2015 S. Boyd and S. Lall Dynamic interpretation of eigenvectors invariant sets complex eigenvectors & invariant planes left eigenvectors modal form discrete-time stability 1 Dynamic interpretation
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 informationPositive Stabilization of Infinite-Dimensional Linear Systems
Positive Stabilization of Infinite-Dimensional Linear Systems Joseph Winkin Namur Center of Complex Systems (NaXys) and Department of Mathematics, University of Namur, Belgium Joint work with Bouchra Abouzaid
More informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 4 Apr. 4, 2018 1 / 40 Recap Vector space, linear space, linear vector space Subspace Linearly independence and dependence Dimension, Basis, Change of Basis 2 / 40
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
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 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 Exercises
9. 8.03 Linear Algebra Exercises 9A. Matrix Multiplication, Rank, Echelon Form 9A-. Which of the following matrices is in row-echelon form? 2 0 0 5 0 (i) (ii) (iii) (iv) 0 0 0 (v) [ 0 ] 0 0 0 0 0 0 0 9A-2.
More information5 More on Linear Algebra
14.102, Math for Economists Fall 2004 Lecture Notes, 9/23/2004 These notes are primarily based on those written by George Marios Angeletos for the Harvard Math Camp in 1999 and 2000, and updated by Stavros
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationEigenvalues, Eigenvectors and the Jordan Form
EE/ME 701: Advanced Linear Systems Eigenvalues, Eigenvectors and the Jordan Form Contents 1 Introduction 3 1.1 Review of basic facts about eigenvectors and eigenvalues..... 3 1.1.1 Looking at eigenvalues
More informationAN ITERATION. In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b
AN ITERATION In part as motivation, we consider an iteration method for solving a system of linear equations which has the form x Ax = b In this, A is an n n matrix and b R n.systemsof this form arise
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 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 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 informationChap. 3. Controlled Systems, Controllability
Chap. 3. Controlled Systems, Controllability 1. Controllability of Linear Systems 1.1. Kalman s Criterion Consider the linear system ẋ = Ax + Bu where x R n : state vector and u R m : input vector. A :
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationDiagonalization. P. Danziger. u B = A 1. B u S.
7., 8., 8.2 Diagonalization P. Danziger Change of Basis Given a basis of R n, B {v,..., v n }, we have seen that the matrix whose columns consist of these vectors can be thought of as a change of basis
More informationModule 06 Stability of Dynamical Systems
Module 06 Stability of Dynamical Systems Ahmad F. Taha EE 5143: Linear Systems and Control Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ataha October 10, 2017 Ahmad F. Taha Module 06
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationExtensions and applications of LQ
Extensions and applications of LQ 1 Discrete time systems 2 Assigning closed loop pole location 3 Frequency shaping LQ Regulator for Discrete Time Systems Consider the discrete time system: x(k + 1) =
More informationEigenvalues, Eigenvectors. Eigenvalues and eigenvector will be fundamentally related to the nature of the solutions of state space systems.
Chapter 3 Linear Algebra In this Chapter we provide a review of some basic concepts from Linear Algebra which will be required in order to compute solutions of LTI systems in state space form, discuss
More informationEE363 homework 8 solutions
EE363 Prof. S. Boyd EE363 homework 8 solutions 1. Lyapunov condition for passivity. The system described by ẋ = f(x, u), y = g(x), x() =, with u(t), y(t) R m, is said to be passive if t u(τ) T y(τ) dτ
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 4 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University April 15, 2018 Linear Algebra (MTH 464)
More informationMATH 221, Spring Homework 10 Solutions
MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationMATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018
Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry
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 informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 1. Linear systems Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Linear systems 1.1 Differential Equations 1.2 Linear flows 1.3 Linear maps
More informationCAAM 335 Matrix Analysis
CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points
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 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 information0.1 Rational Canonical Forms
We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it s best
More information= A(λ, t)x. In this chapter we will focus on the case that the matrix A does not depend on time (so that the ODE is autonomous):
Chapter 2 Linear autonomous ODEs 2 Linearity Linear ODEs form an important class of ODEs They are characterized by the fact that the vector field f : R m R p R R m is linear at constant value of the parameters
More informationObservability. Dynamic Systems. Lecture 2 Observability. Observability, continuous time: Observability, discrete time: = h (2) (x, u, u)
Observability Dynamic Systems Lecture 2 Observability Continuous time model: Discrete time model: ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)) Reglerteknik, ISY,
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 informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
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 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 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 informationStabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints
Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints Joseph Winkin Namur Center of Complex Systems (naxys) and Dept. of Mathematics, University of Namur, Belgium
More informationLecture 4 Continuous time linear quadratic regulator
EE363 Winter 2008-09 Lecture 4 Continuous time linear quadratic regulator continuous-time LQR problem dynamic programming solution Hamiltonian system and two point boundary value problem infinite horizon
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
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 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 informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationDefinition of Dynamic System
Capitolo 0. INTRODUCTION 1.1 Definition of Dynamic System There are various types of dynamic systems: continuous-time, discrete-time, linear, non-linear, lumped systems, distributed systems, finite states,
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationThere are six more problems on the next two pages
Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with
More informationLinear Systems Notes for CDS-140a
Linear Systems Notes for CDS-140a October 27, 2008 1 Linear Systems Before beginning our study of linear dynamical systems, it only seems fair to ask the question why study linear systems? One might hope
More informationWe have already seen that the main problem of linear algebra is solving systems of linear equations.
Notes on Eigenvalues and Eigenvectors by Arunas Rudvalis Definition 1: Given a linear transformation T : R n R n a non-zero vector v in R n is called an eigenvector of T if Tv = λv for some real number
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 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 informationME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
More informationData Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples
Data Analysis and Manifold Learning Lecture 2: Properties of Symmetric Matrices and Examples Radu Horaud INRIA Grenoble Rhone-Alpes, France Radu.Horaud@inrialpes.fr http://perception.inrialpes.fr/ Outline
More informationMIT Final Exam Solutions, Spring 2017
MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationEcon Slides from Lecture 7
Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
More informationThe Jordan Normal Form and its Applications
The and its Applications Jeremy IMPACT Brigham Young University A square matrix A is a linear operator on {R, C} n. A is diagonalizable if and only if it has n linearly independent eigenvectors. What happens
More informationTheorem 1. ẋ = Ax is globally exponentially stable (GES) iff A is Hurwitz (i.e., max(re(σ(a))) < 0).
Linear Systems Notes Lecture Proposition. A M n (R) is positive definite iff all nested minors are greater than or equal to zero. n Proof. ( ): Positive definite iff λ i >. Let det(a) = λj and H = {x D
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
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 informationLinear algebra II Tutorial solutions #1 A = x 1
Linear algebra II Tutorial solutions #. Find the eigenvalues and the eigenvectors of the matrix [ ] 5 2 A =. 4 3 Since tra = 8 and deta = 5 8 = 7, the characteristic polynomial is f(λ) = λ 2 (tra)λ+deta
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
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 informationExamples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.
The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section
More informationPOLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19
POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order
More informationDiscrete-Time State-Space Equations. M. Sami Fadali Professor of Electrical Engineering UNR
Discrete-Time State-Space Equations M. Sami Fadali Professor of Electrical Engineering UNR 1 Outline Discrete-time (DT) state equation from solution of continuous-time state equation. Expressions in terms
More informationJordan Canonical Form Homework Solutions
Jordan Canonical Form Homework Solutions For each of the following, put the matrix in Jordan canonical form and find the matrix S such that S AS = J. [ ]. A = A λi = λ λ = ( λ) = λ λ = λ =, Since we have
More informationJordan normal form notes (version date: 11/21/07)
Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationEE5120 Linear Algebra: Tutorial 6, July-Dec Covers sec 4.2, 5.1, 5.2 of GS
EE0 Linear Algebra: Tutorial 6, July-Dec 07-8 Covers sec 4.,.,. of GS. State True or False with proper explanation: (a) All vectors are eigenvectors of the Identity matrix. (b) Any matrix can be diagonalized.
More informationLecture Note 12: The Eigenvalue Problem
MATH 5330: Computational Methods of Linear Algebra Lecture Note 12: The Eigenvalue Problem 1 Theoretical Background Xianyi Zeng Department of Mathematical Sciences, UTEP The eigenvalue problem is a classical
More informationEE363 homework 2 solutions
EE363 Prof. S. Boyd EE363 homework 2 solutions. Derivative of matrix inverse. Suppose that X : R R n n, and that X(t is invertible. Show that ( d d dt X(t = X(t dt X(t X(t. Hint: differentiate X(tX(t =
More informationLinear Algebra II Lecture 13
Linear Algebra II Lecture 13 Xi Chen 1 1 University of Alberta November 14, 2014 Outline 1 2 If v is an eigenvector of T : V V corresponding to λ, then v is an eigenvector of T m corresponding to λ m since
More informationState will have dimension 5. One possible choice is given by y and its derivatives up to y (4)
A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To
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 informationEigenvalues and Eigenvectors
/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix
More information