Introduction to MVC. least common denominator of all non-identical-zero minors of all order of G(s). Example: The minor of order 2: 1 2 ( s 1)
|
|
- Kristian Webb
- 6 years ago
- Views:
Transcription
1 Introduction to MVC Definition---Proerness and strictly roerness A system G(s) is roer if all its elements { gij ( s)} are roer, and strictly roer if all its elements are strictly roer. Definition---Causal A system G(s) is causal if all its elements are causal, and not causal if all its elements are noncausal. Definition---Poles The eigen values λ i, i=,---,n of the system G(s) are called the ole of the system. The ole olynomial π ( s) n is defined as: π ( s) = ( s λi ), where π ( s) is the least common denominator of all non-identical-zero minors of all order of G(s). Examle: Definition---Zeros ( s )( s + ) 0 ( s ) G( s) = ( s + )( s + )( s ) ( s + )( s + ) ( s )( s + ) ( s )( s + ) The minor of order : ( s ) G = ; G = ; G = ( s + )( s + ) ( s + )( s + ) ( s + )( s + ),,,,,3,3 i= The least common denominator of the minors: π s = s + s + s ( ) ( )( ) ( ) If the rank of G(z) is less than the normal rank, z is a zero of the system. The zero olynomial Z(s) is the greatest common divisor of the numerators of all order-r minors of G(s), where r is the norminal rank of G(s) rovided that these minors have all been adjusted in such a way as to have the ole olynomial π ( s) as their denominator., ( s )( s + ), ( s )( s + ), ( s ) G, ( s) = ; G,3 ( s) = ; G,3 ( s) = π ( s) π ( s) π ( s) So, Z( s) = ( s ) According to this definition, the zero olynomial of a square G(s) is: det{ G( s )} = 0 Notice that, in a MIMO system, there may be no inverse resonse to indicate the
2 resence of RHP-zero. For examle, in the MIMO system of the following: G( s) = ; G(0.5) = 0; σ { G(0.5) } = 0 (0.s + )( s + ) + s G(0.5) = U Σ H V H Transfer functions for Closed loo MIMO systems. Cascade rule. For the cascade interconnection of G ( s) and G ( ) s in the following figure(a), the overall transfer function matrix is G=G G. Feedback rule. With reference to the ositive feedback system in Figure (b): v = ( I L) u = ( I G G ) u 3. Push-through rule. G ( I G G ) = ( I G G ) G G G G G = G ( I G G ) = ( I G G ) G ( I G G ) G ( I G G ) = G ( I G G ) G = G ( I G G )
3 4. MIMO rule. (). Start from the outut, write down the blocks as moving backward by the most direct ath toward the inut (). When exit from a feedback loo, include a term (I-L) - for ositive feedback (or, (I+L) - for negative feedback. Notice that L is the evaluated against signal flow starting at the oint of exit from the loo. Examle: Consider the following block diagram: z = + K( P K) w 5. Consider the closed-loo system: The following relationshis are useful: I L L I L S T ( + ) + ( + ) = + = G( I + KG) = ( I + GK) G ( + ) = ( + ) = ( + ) GK I GK G I KG K I GK GK T = L( I + L) = ( I + L) L = ( I + L ) 3
4 Singular Values and Matrix Norms. Vector Norms A real valued function defined on a vector sace X is said to be a norm on X, if it satisfies the following roerties: () x 0 ; () x = 0 only if x = 0 ; (3) α x = α x, for any scalar α; (4) x + y x + y ; for any x X and y X. The vector -norm of x X articular =,, we have: is defined as: x = x n / i, for. In i= x n n = xi ; x ( x ) i i= i= = ; x = max x. i n i H o&&lder inequality: T x y x y, q + q = Notice that: () x x n x ; () x x n x ; (3) x x n x 4
5 . Matrix norms: A matrix norm is defined as a function valued that satisfies the following: m n () A 0 for all A R with equality only if A=0; m n () α A = α A for all α R, A R ; (3) A + B A + B for all A, B R m n m n Let A = [ a ] R, the vector induced -norms is defined as: ij A = su x 0 Ax x In articular: A A A m = max a (column sum) ; j n i = max ij * ( A A) = λ ; = max aij (row sum) j n n j = Another oular matrix norm is the Frobenius norm, i.e.: A F m n = i= j= a ij The -norms have the following imortant roerty for every m n A R : () AX A x ; =,, ; () A A n A ; F (3) max aij A mn ; { max aij i m j n i m; j n } ; (4) A A m A ; n A A n A (5) m 5
6 Lemma : Let x n F and y m F. () Suose n m. x = y iff.., n m * U F s t x = Uy and U U = I ; () Suose n = m. * x y x y. Moreover, equality holds iff x = α y for some α F or y = 0; n m (3) x y iff F with s. t. x = y. Furthermore, x y iff < ; (4) Ux = x for any aroriate dimensioned unitary matrix U. Lemma : Let A and B be any aroriate dimensioned matrices. () ρ( A) A, ( =,,, F) ; () AB A B ; A A ; (3) UAV = A, and UAV = A, for any unitary matrices U and V ; (4) AB A B, and AB B A. F F F F F F Lemma 3: Let A be a block artitioned matrix with: A A A A A A A A A A q q = = Aij m m mq Then, for any induced matrix -norm, A A A A q A A A q Am A m A mq Further, the equality holds if the F-norm is used. 6
7 7
8 Singular Value Decomosition Consider a fixed frequency ω where G( jω) is a l m comlex matrix. Denote G( jω) as G for simlicity. Any matrix G may be decomosed into G Where, Σ is an l m matrix with k = min{ l, m} non-negative singular values, σ i. arranged in descending order along its main diagonal; the other entries are H = UΣ V. zero. The singular values are the ositive square roots of the eigenvalues of H G G, and H G H σ ( G) = λ ( G G) i y = G d ( ) i is the conjugate transose of G. That is: {,, L, } i i α i i i n i i = = ; Let u be one of the eigenvector of G. d = v v = v + v + v y = G v = Gv = λ v y d i i i i y = λ v α λi vi i v λ i On the other hand, it can be shown that the extreme values of Gd / d are / H λ i G G, which is known as the singular value of G, in the directions of eigenvaectors of H G G. y y y Since, σ = min max = σ d d d As a result, σ λi σ The singular values can be considered as the extreme gains of the MIMO 8
9 system, which are local maximal values with resect to the direction of inuts. For examle, consider the gain matrix at a secific frequency: 5 4 G = 3 The gains with resect to the direction of d are given in the following figure: Tyical singular values with resect to frequency are as shown in the following figure: Theorem : Let m n A F. There exist unitary matrices: U = [ u, u, L, u ] F ; V = [ v, v, L, v ] F m m n n m n such that: where, * Σ 0 A = UΣV, Σ = 0 0 σ 0 L 0 0 σ 0 L Σ = ; σ σ L σ 0 ; = min, M M O M 0 0 L σ 9 { m n}
10 [Proof] Let σ = A, and assume m n. Then, from the definition of A, i.e.: Az A = su ; ( =,, ) Az = A z for some z z In other words, there exists a vector z F n such that Az = σ z = σ z By the Lemma ( m n * x y iff there is a matrix U F such that x Uy and U U I = = = ) there is a matrix ( σ ) Az = U% z = σuz %. % m n such that U F % % = I and * U U Let: x z z n = F and Uz y = % F Uz % m We have: Az σuz % σ Uz % σ Uz % Uz % Ax = = = = = = y z Az Az σuz % Uz % σ Let [, ] m m n n U = y U F ; V = [ x, V ] F be unitary. Thus, A = U AV = [ y, U ] [ Ax, AV ] * * * * * * y Ax y AV σ y y y AV = * * = * * U Ax U AV σu y U AV * σ w = 0 B Since, 0 * * A = σ + w w A σ + w w M 0 0
11 and σ = A = A, we conclude that w = 0. An obvious induction argument gives: * U AV = Σ. The σ i is the i-th eigen value of A, and u and v are i-th left singular vector and j-th right i j singular vectors, resectively. It is obvious to see: Av = σ u and A u = σ v * i i i i i i or, A Av = σ v and AA u = σ u * * i i i i i i Hence, σ i is an eigen value of * * AA or A A, u i is an eigen vector of * AA, and is an eigen vector of singular values are often used: * A A. The following notations for σ ( A) = σ ( A) = σ = max Ax max x = σ ( A) = σ ( A) = σ = min Ax min x = Lemma : Suose A and D are square matrices. () σ ( A + ) σ ( A) σ ( A); () σ ( A ) σ ( A) σ ( ); (3) σ ( A ) = if A is invertible; σ ( A)
12
13 Lemma 3: () σ ( A) λ σ ( A) () λ A (3) Let σ ( A) κ ( A) = = condition number of A, and Ax = b. If σ ( A) 3
14 δ x δ b A( x + δ x) = ( b + δ b), then: = κ ( A) x b (4) Let A A + δ A and x = x + δ x = b. Then: δ x x κ ( A) δ A κ ( A) A Thus, when κ ( A) is large and matrix A is almost singular, a very small change of A will make the ossible range of δ x large. 4
15 Alications of SVD 5
16 6
17 7
18 8
19 Problem of small singular values: Very small singular values in a multivariable system are analogus to very small gains in a conventional siso system. It requires very large controller gains and results in excessively large controller actions. The tyical resence of constraints in the maniulated variable and noise in the sensor makes it difficult even for siso system. In the context of mimo system, the additional comlications resented by hidden loos, interactions make the roblem even more severe. A general rule of thumb to measure the small singular value is the magnitude of the noise in the signal. Singular values are equal or less than the magnitude of sensor noise should be assumed degenerate. Problem of large singular values: Large singular values are not as serious a roblem as small values. It requires very small controller gains. This results in small controller oututs, which can be easily become lost in the resolution of the maniulator. The symtomatic behaviors are cyclic resonses, which never settling down to a reasonable steady state. A general rule of thumb concerning large singular values is that all singular values that are equal to or greater than the recirocal of the valve resolution should be avoided. Determining Good Sensor Locations Consider, for examle, the ethonal-water distillation column as shown in Figure 3.Assume that the first level objective is to control two column temeratures by maniulating D and Q. The basic concern is to determine which combination out of 5 ossible combinations of sensor locations from the oint of view of column control. 9
20 The 50 gain matrix is as shown in figure 4. 0
21
22 From the SVD result in Figure 5 and is lotted on Figure 6. The largest elements in U and U occurs on stage 8 and 3. On the other hand, if we use abs(u )-abs(u ) as criterion, as shown on Figure 7, the largest differences suggests that stage 8 and stage 3 are good choices.
23 3
24 Selection of Proer Maniulated variables For examle, in the design of control of a control for a distillation column, four maniulated variables are tyically be considered: D distillate flow rate L reflux flow rate B bottoms flow rate Q steam rate to the reboiler Two out of the four have to control levels (i.e. accumulator and column base). Thus, only the remaining two can be maniulated for the comositions. To choose two out of the four, SVD rovides a straightforward method to comare the steady state behavior of various first level control strategies. In this examle, DQ is the best first-level control scheme. It has a much better condition number than the others. The second singular value of this DQ scheme is much stronger than for the other schemes. 4
25 Alication to Feedback Proerties 5
26 6
27 Use of the minimum singular value of the lant: The minimum singular value of the lant evaluated as a function of frequency is a useful measure for evaluating the feasibility of achieving accetable control. In general, we want σ as large as ossible. Singular values for erformance: In general, it is reasonable to require that the gain e( ω) / r( ω) remains small for any direction of r( ω ), including the worst-case direction which gives a gin of σ ( S( jω )). Let / w ( jω ) reresent the maximum allowable magnitude of e( ω) / r( ω) at each frequency, This results in the following erformance requirement: σ ( S( jω)) <, ω σ { ws( jω) } <, ω ws( jω < w ( jω) Tyical weight function is given asl s / M + ωb w ( s) = s + ω A which means / b w equals A at low frequencies, and equals to M at high frequencies, and the asymtote crosses at frequency). ω b (the bandwidth 7
Numerical Linear Algebra
Numerical Linear Algebra Numerous alications in statistics, articularly in the fitting of linear models. Notation and conventions: Elements of a matrix A are denoted by a ij, where i indexes the rows and
More informationLecture 7 (Weeks 13-14)
Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.
More informationChater Matrix Norms and Singular Value Decomosition Introduction In this lecture, we introduce the notion of a norm for matrices The singular value de
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A Dahleh George Verghese Deartment of Electrical Engineering and Comuter Science Massachuasetts Institute of Technology c Chater Matrix Norms
More informationPositive decomposition of transfer functions with multiple poles
Positive decomosition of transfer functions with multile oles Béla Nagy 1, Máté Matolcsi 2, and Márta Szilvási 1 Deartment of Analysis, Technical University of Budaest (BME), H-1111, Budaest, Egry J. u.
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationy p 2 p 1 y p Flexible System p 2 y p2 p1 y p u, y 1 -u, y 2 Component Breakdown z 1 w 1 1 y p 1 P 1 P 2 w 2 z 2
ROBUSTNESS OF FLEXIBLE SYSTEMS WITH COMPONENT-LEVEL UNCERTAINTIES Peiman G. Maghami Λ NASA Langley Research Center, Hamton, VA 368 Robustness of flexible systems in the resence of model uncertainties at
More informationFeedback-Based Iterative Learning Control for MIMO LTI Systems
International Journal of Control, Feedback-Based Automation, Iterative and Systems, Learning vol. Control 6, no., for. MIMO 69-77, LTI Systems Aril 8 69 Feedback-Based Iterative Learning Control for MIMO
More informationLecture 4: Analysis of MIMO Systems
Lecture 4: Analysis of MIMO Systems Norms The concept of norm will be extremely useful for evaluating signals and systems quantitatively during this course In the following, we will present vector norms
More informationQuantitative estimates of propagation of chaos for stochastic systems with W 1, kernels
oname manuscrit o. will be inserted by the editor) Quantitative estimates of roagation of chaos for stochastic systems with W, kernels Pierre-Emmanuel Jabin Zhenfu Wang Received: date / Acceted: date Abstract
More informationPrincipal Components Analysis and Unsupervised Hebbian Learning
Princial Comonents Analysis and Unsuervised Hebbian Learning Robert Jacobs Deartment of Brain & Cognitive Sciences University of Rochester Rochester, NY 1467, USA August 8, 008 Reference: Much of the material
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Numerical Linear Algebra Background Cho-Jui Hsieh UC Davis May 15, 2018 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationEL2520 Control Theory and Practice
So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal
More informationOn the Toppling of a Sand Pile
Discrete Mathematics and Theoretical Comuter Science Proceedings AA (DM-CCG), 2001, 275 286 On the Toling of a Sand Pile Jean-Christohe Novelli 1 and Dominique Rossin 2 1 CNRS, LIFL, Bâtiment M3, Université
More informationSTABILITY ANALYSIS TOOL FOR TUNING UNCONSTRAINED DECENTRALIZED MODEL PREDICTIVE CONTROLLERS
STABILITY ANALYSIS TOOL FOR TUNING UNCONSTRAINED DECENTRALIZED MODEL PREDICTIVE CONTROLLERS Massimo Vaccarini Sauro Longhi M. Reza Katebi D.I.I.G.A., Università Politecnica delle Marche, Ancona, Italy
More informationBackground Mathematics (2/2) 1. David Barber
Background Mathematics (2/2) 1 David Barber University College London Modified by Samson Cheung (sccheung@ieee.org) 1 These slides accompany the book Bayesian Reasoning and Machine Learning. The book and
More informationControllability and Resiliency Analysis in Heat Exchanger Networks
609 A ublication of CHEMICAL ENGINEERING RANSACIONS VOL. 6, 07 Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš Coyright 07, AIDIC Servizi S.r.l. ISBN 978-88-95608-5-8;
More informationUse of Transformations and the Repeated Statement in PROC GLM in SAS Ed Stanek
Use of Transformations and the Reeated Statement in PROC GLM in SAS Ed Stanek Introduction We describe how the Reeated Statement in PROC GLM in SAS transforms the data to rovide tests of hyotheses of interest.
More informationHARMONIC EXTENSION ON NETWORKS
HARMONIC EXTENSION ON NETWORKS MING X. LI Abstract. We study the imlication of geometric roerties of the grah of a network in the extendibility of all γ-harmonic germs at an interior node. We rove that
More informationFeedback-error control
Chater 4 Feedback-error control 4.1 Introduction This chater exlains the feedback-error (FBE) control scheme originally described by Kawato [, 87, 8]. FBE is a widely used neural network based controller
More informationSolved Problems. (a) (b) (c) Figure P4.1 Simple Classification Problems First we draw a line between each set of dark and light data points.
Solved Problems Solved Problems P Solve the three simle classification roblems shown in Figure P by drawing a decision boundary Find weight and bias values that result in single-neuron ercetrons with the
More informationPROFIT MAXIMIZATION. π = p y Σ n i=1 w i x i (2)
PROFIT MAXIMIZATION DEFINITION OF A NEOCLASSICAL FIRM A neoclassical firm is an organization that controls the transformation of inuts (resources it owns or urchases into oututs or roducts (valued roducts
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 5: Numerical Linear Algebra Cho-Jui Hsieh UC Davis April 20, 2017 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationThe Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001
The Hasse Minkowski Theorem Lee Dicker University of Minnesota, REU Summer 2001 The Hasse-Minkowski Theorem rovides a characterization of the rational quadratic forms. What follows is a roof of the Hasse-Minkowski
More informationParticipation Factors. However, it does not give the influence of each state on the mode.
Particiation Factors he mode shae, as indicated by the right eigenvector, gives the relative hase of each state in a articular mode. However, it does not give the influence of each state on the mode. We
More informationConvex Analysis and Economic Theory Winter 2018
Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is
More informationRadial Basis Function Networks: Algorithms
Radial Basis Function Networks: Algorithms Introduction to Neural Networks : Lecture 13 John A. Bullinaria, 2004 1. The RBF Maing 2. The RBF Network Architecture 3. Comutational Power of RBF Networks 4.
More informationCHAPTER 2: SMOOTH MAPS. 1. Introduction In this chapter we introduce smooth maps between manifolds, and some important
CHAPTER 2: SMOOTH MAPS DAVID GLICKENSTEIN 1. Introduction In this chater we introduce smooth mas between manifolds, and some imortant concets. De nition 1. A function f : M! R k is a smooth function if
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationExtension of Minimax to Infinite Matrices
Extension of Minimax to Infinite Matrices Chris Calabro June 21, 2004 Abstract Von Neumann s minimax theorem is tyically alied to a finite ayoff matrix A R m n. Here we show that (i) if m, n are both inite,
More informationOn The Circulant Matrices with Ducci Sequences and Fibonacci Numbers
Filomat 3:15 (018), 5501 5508 htts://doi.org/10.98/fil1815501s Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat On The Circulant Matrices
More informationLecture 8: Linear Algebra Background
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 8: Linear Algebra Background Lecturer: Shayan Oveis Gharan 2/1/2017 Scribe: Swati Padmanabhan Disclaimer: These notes have not been subjected
More information3.4 Design Methods for Fractional Delay Allpass Filters
Chater 3. Fractional Delay Filters 15 3.4 Design Methods for Fractional Delay Allass Filters Above we have studied the design of FIR filters for fractional delay aroximation. ow we show how recursive or
More informationME 375 System Modeling and Analysis. Homework 11 Solution. Out: 18 November 2011 Due: 30 November 2011 = + +
Out: 8 November Due: 3 November Problem : You are given the following system: Gs () =. s + s+ a) Using Lalace and Inverse Lalace, calculate the unit ste resonse of this system (assume zero initial conditions).
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
More informationEstimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationRadar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.
Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Deartment, Middle East Technical University Armature controlled dc motor Outside θ D outut Inside θ r inut r θ m Gearbox Control Transmitter θ D
More informationLINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL
LINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL Mohammad Bozorg Deatment of Mechanical Engineering University of Yazd P. O. Box 89195-741 Yazd Iran Fax: +98-351-750110
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationDimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
More informationMultivariable Generalized Predictive Scheme for Gas Turbine Control in Combined Cycle Power Plant
Multivariable Generalized Predictive Scheme for Gas urbine Control in Combined Cycle Power Plant L.X.Niu and X.J.Liu Deartment of Automation North China Electric Power University Beiing, China, 006 e-mail
More informationA Bound on the Error of Cross Validation Using the Approximation and Estimation Rates, with Consequences for the Training-Test Split
A Bound on the Error of Cross Validation Using the Aroximation and Estimation Rates, with Consequences for the Training-Test Slit Michael Kearns AT&T Bell Laboratories Murray Hill, NJ 7974 mkearns@research.att.com
More information8 Extremal Values & Higher Derivatives
8 Extremal Values & Higher Derivatives Not covered in 2016\17 81 Higher Partial Derivatives For functions of one variable there is a well-known test for the nature of a critical point given by the sign
More informationSingular Value Decomposition Analysis
Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control
More informationKalman Decomposition B 2. z = T 1 x, where C = ( C. z + u (7) T 1, and. where B = T, and
Kalman Decomposition Controllable / uncontrollable decomposition Suppose that the controllability matrix C R n n of a system has rank n 1
More informationIntroduction to Group Theory Note 1
Introduction to Grou Theory Note July 7, 009 Contents INTRODUCTION. Examles OF Symmetry Grous in Physics................................. ELEMENT OF GROUP THEORY. De nition of Grou................................................
More informationThe Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More informationNumerical Linear Algebra Homework Assignment - Week 2
Numerical Linear Algebra Homework Assignment - Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationAveraging sums of powers of integers and Faulhaber polynomials
Annales Mathematicae et Informaticae 42 (20. 0 htt://ami.ektf.hu Averaging sums of owers of integers and Faulhaber olynomials José Luis Cereceda a a Distrito Telefónica Madrid Sain jl.cereceda@movistar.es
More informationRobust Performance Design of PID Controllers with Inverse Multiplicative Uncertainty
American Control Conference on O'Farrell Street San Francisco CA USA June 9 - July Robust Performance Design of PID Controllers with Inverse Multilicative Uncertainty Tooran Emami John M Watkins Senior
More informationComputational math: Assignment 1
Computational math: Assignment 1 Thanks Ting Gao for her Latex file 11 Let B be a 4 4 matrix to which we apply the following operations: 1double column 1, halve row 3, 3add row 3 to row 1, 4interchange
More informationHidden Predictors: A Factor Analysis Primer
Hidden Predictors: A Factor Analysis Primer Ryan C Sanchez Western Washington University Factor Analysis is a owerful statistical method in the modern research sychologist s toolbag When used roerly, factor
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationState Estimation with ARMarkov Models
Deartment of Mechanical and Aerosace Engineering Technical Reort No. 3046, October 1998. Princeton University, Princeton, NJ. State Estimation with ARMarkov Models Ryoung K. Lim 1 Columbia University,
More information216 S. Chandrasearan and I.C.F. Isen Our results dier from those of Sun [14] in two asects: we assume that comuted eigenvalues or singular values are
Numer. Math. 68: 215{223 (1994) Numerische Mathemati c Sringer-Verlag 1994 Electronic Edition Bacward errors for eigenvalue and singular value decomositions? S. Chandrasearan??, I.C.F. Isen??? Deartment
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal
More information2-D Analysis for Iterative Learning Controller for Discrete-Time Systems With Variable Initial Conditions Yong FANG 1, and Tommy W. S.
-D Analysis for Iterative Learning Controller for Discrete-ime Systems With Variable Initial Conditions Yong FANG, and ommy W. S. Chow Abstract In this aer, an iterative learning controller alying to linear
More informationSome results on Lipschitz quasi-arithmetic means
IFSA-EUSFLAT 009 Some results on Lischitz quasi-arithmetic means Gleb Beliakov, Tomasa Calvo, Simon James 3,3 School of Information Technology, Deakin University Burwood Hwy, Burwood, 35, Australia Deartamento
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More informationElements of Asymptotic Theory. James L. Powell Department of Economics University of California, Berkeley
Elements of Asymtotic Theory James L. Powell Deartment of Economics University of California, Berkeley Objectives of Asymtotic Theory While exact results are available for, say, the distribution of the
More informationAn Inverse Problem for Two Spectra of Complex Finite Jacobi Matrices
Coyright 202 Tech Science Press CMES, vol.86, no.4,.30-39, 202 An Inverse Problem for Two Sectra of Comlex Finite Jacobi Matrices Gusein Sh. Guseinov Abstract: This aer deals with the inverse sectral roblem
More informationTRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES
Khayyam J. Math. DOI:10.22034/kjm.2019.84207 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and
More informationV. Practical Optimization
V. Practical Otimization Scaling Practical Otimality Parameterization Otimization Objectives Means Solutions ω c -otimization Observer based design V- Definition of Gain and Frequency Scales G ( s) kg
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationParametrization of All Stabilizing Controllers Subject to Any Structural Constraint
49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Parametrization of All Stabilizing Controllers Subject to Any Structural Constraint Michael C Rotkowitz
More informationNumber Theory Naoki Sato
Number Theory Naoki Sato 0 Preface This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO
More informationLecture 3. Chapter 4: Elements of Linear System Theory. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 3 Chapter 4: Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 3 p. 1/77 3.1 System Descriptions [4.1] Let f(u) be a liner operator, u 1 and u
More informationSolution sheet ξi ξ < ξ i+1 0 otherwise ξ ξ i N i,p 1 (ξ) + where 0 0
Advanced Finite Elements MA5337 - WS7/8 Solution sheet This exercise sheets deals with B-slines and NURBS, which are the basis of isogeometric analysis as they will later relace the olynomial ansatz-functions
More informationGeneral Linear Model Introduction, Classes of Linear models and Estimation
Stat 740 General Linear Model Introduction, Classes of Linear models and Estimation An aim of scientific enquiry: To describe or to discover relationshis among events (variables) in the controlled (laboratory)
More informationGeneration of Linear Models using Simulation Results
4. IMACS-Symosium MATHMOD, Wien, 5..003,. 436-443 Generation of Linear Models using Simulation Results Georg Otte, Sven Reitz, Joachim Haase Fraunhofer Institute for Integrated Circuits, Branch Lab Design
More informationResearch of power plant parameter based on the Principal Component Analysis method
Research of ower lant arameter based on the Princial Comonent Analysis method Yang Yang *a, Di Zhang b a b School of Engineering, Bohai University, Liaoning Jinzhou, 3; Liaoning Datang international Jinzhou
More informationChapter 1 Fundamentals
Chater Fundamentals. Overview of Thermodynamics Industrial Revolution brought in large scale automation of many tedious tasks which were earlier being erformed through manual or animal labour. Inventors
More informationDISCRIMINANTS IN TOWERS
DISCRIMINANTS IN TOWERS JOSEPH RABINOFF Let A be a Dedekind domain with fraction field F, let K/F be a finite searable extension field, and let B be the integral closure of A in K. In this note, we will
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More information3 Properties of Dedekind domains
18.785 Number theory I Fall 2016 Lecture #3 09/15/2016 3 Proerties of Dedekind domains In the revious lecture we defined a Dedekind domain as a noetherian domain A that satisfies either of the following
More informationApplications to stochastic PDE
15 Alications to stochastic PE In this final lecture we resent some alications of the theory develoed in this course to stochastic artial differential equations. We concentrate on two secific examles:
More informationGalois Fields, Linear Feedback Shift Registers and their Applications
Galois Fields, Linear Feedback Shift Registers and their Alications With 85 illustrations as well as numerous tables, diagrams and examles by Ulrich Jetzek ISBN (Book): 978-3-446-45140-7 ISBN (E-Book):
More informationMultivariable PID Control Design For Wastewater Systems
Proceedings of the 5th Mediterranean Conference on Control & Automation, July 7-9, 7, Athens - Greece 4- Multivariable PID Control Design For Wastewater Systems N A Wahab, M R Katebi and J Balderud Industrial
More informationThe Graph Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule
The Grah Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule STEFAN D. BRUDA Deartment of Comuter Science Bisho s University Lennoxville, Quebec J1M 1Z7 CANADA bruda@cs.ubishos.ca
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationEvaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models
Evaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models Ketan N. Patel, Igor L. Markov and John P. Hayes University of Michigan, Ann Arbor 48109-2122 {knatel,imarkov,jhayes}@eecs.umich.edu
More informationWolfgang POESSNECKER and Ulrich GROSS*
Proceedings of the Asian Thermohysical Proerties onference -4 August, 007, Fukuoka, Jaan Paer No. 0 A QUASI-STEADY YLINDER METHOD FOR THE SIMULTANEOUS DETERMINATION OF HEAT APAITY, THERMAL ONDUTIVITY AND
More informationMULTIVARIATE STATISTICAL PROCESS OF HOTELLING S T CONTROL CHARTS PROCEDURES WITH INDUSTRIAL APPLICATION
Journal of Statistics: Advances in heory and Alications Volume 8, Number, 07, Pages -44 Available at htt://scientificadvances.co.in DOI: htt://dx.doi.org/0.864/jsata_700868 MULIVARIAE SAISICAL PROCESS
More informationLecture Note 7: Iterative methods for solving linear systems. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 7: Iterative methods for solving linear systems Xiaoqun Zhang Shanghai Jiao Tong University Last updated: December 24, 2014 1.1 Review on linear algebra Norms of vectors and matrices vector
More informationActual exergy intake to perform the same task
CHAPER : PRINCIPLES OF ENERGY CONSERVAION INRODUCION Energy conservation rinciles are based on thermodynamics If we look into the simle and most direct statement of the first law of thermodynamics, we
More informationNetwork Configuration Control Via Connectivity Graph Processes
Network Configuration Control Via Connectivity Grah Processes Abubakr Muhammad Deartment of Electrical and Systems Engineering University of Pennsylvania Philadelhia, PA 90 abubakr@seas.uenn.edu Magnus
More informationComputations in Quantum Tensor Networks
Comutations in Quantum Tensor Networks T Huckle a,, K Waldherr a, T Schulte-Herbrüggen b a Technische Universität München, Boltzmannstr 3, 85748 Garching, Germany b Technische Universität München, Lichtenbergstr
More informationMath 751 Lecture Notes Week 3
Math 751 Lecture Notes Week 3 Setember 25, 2014 1 Fundamental grou of a circle Theorem 1. Let φ : Z π 1 (S 1 ) be given by n [ω n ], where ω n : I S 1 R 2 is the loo ω n (s) = (cos(2πns), sin(2πns)). Then
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analysis of Variance and Design of Exeriment-I MODULE II LECTURE -4 GENERAL LINEAR HPOTHESIS AND ANALSIS OF VARIANCE Dr. Shalabh Deartment of Mathematics and Statistics Indian Institute of Technology Kanur
More informationMODEL-BASED MULTIPLE FAULT DETECTION AND ISOLATION FOR NONLINEAR SYSTEMS
MODEL-BASED MULIPLE FAUL DEECION AND ISOLAION FOR NONLINEAR SYSEMS Ivan Castillo, and homas F. Edgar he University of exas at Austin Austin, X 78712 David Hill Chemstations Houston, X 77009 Abstract A
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationALGEBRAIC TOPOLOGY MASTERMATH (FALL 2014) Written exam, 21/01/2015, 3 hours Outline of solutions
ALGERAIC TOPOLOGY MASTERMATH FALL 014) Written exam, 1/01/015, 3 hours Outline of solutions Exercise 1. i) There are various definitions in the literature. ased on the discussion on. 5 of Lecture 3, as
More informationSingular Value Decomposition
Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationAn Estimate For Heilbronn s Exponential Sum
An Estimate For Heilbronn s Exonential Sum D.R. Heath-Brown Magdalen College, Oxford For Heini Halberstam, on his retirement Let be a rime, and set e(x) = ex(2πix). Heilbronn s exonential sum is defined
More informationbe a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u
MATH 434/534 Theoretical Assignment 7 Solution Chapter 7 (71) Let H = I 2uuT Hu = u (ii) Hv = v if = 0 be a Householder matrix Then prove the followings H = I 2 uut Hu = (I 2 uu )u = u 2 uut u = u 2u =
More informationEnumeration of Balanced Symmetric Functions over GF (p)
Enumeration of Balanced Symmetric Functions over GF () Shaojing Fu 1, Chao Li 1 and Longjiang Qu 1 Ping Li 1 Deartment of Mathematics and System Science, Science College of ational University of Defence
More information