State Feedback. State Feedback Block Diagram. State Feedback. Closed-Loop State-Space Equations
|
|
|
- Ezra Lewis
- 7 years ago
- Views:
Transcription
1 State Feedback M. S. Fadali Professor of Electrical Engineering University of Nevada, Reno State Feedback The control is a function of the state vector. State Feedback Law: Linear State Feedback Law: state gain matrix (select for good transient response). Affine State Feedback Law: reference input 1 2 State Feedback Block Diagram v(k) u(k) x(k+1) x(k) + B + T C y(k) Closed-Loop State-Space Equations State-Space Equations for LTI System Affine State Feedback Law A K 3 4
2 Output Feedback Output measurements y must be used to obtain the control u x( k 1) v(k) Ax( k) B K ycx( k) A BK C x( k) Bv( k) u(k) + B + y x(k+1) T A x(k) u( k) K v( k) C = K A y(k) y y y y( k) v( k) Cx( k) v( k) A BK y C Control Systems Regulator The control law drives the system to the zero state (zero output, ) or to a specified constant state (constant output, ). Design: Select to provide i. A good transient zero-input response. ii. Good disturbance rejection. Tracker The control law makes the system output track a specified time function. K 5 6 Pole Placement (Assignment, Allocation) Choose the gain matrix or to assign the system eigenvalues to an arbitrary set Easier with state feedback but output feedback is more realistic. Poles can be arbitrarily placed if and only if the system is controllable. Pole placement is easier if the system is given in controllable form. Stability Closed-loop stability: asymptotically stable Select so that the eigenvalues of the closedloop state matrix are inside the unit circle (open LHP for continuous-time systems). 7 8 Lim Lim
3 Theorem 9.1: State Feedback There exists a state feedback gain matrix that arbitrarily places the eigenvalues of the closedloop matrix if and only if the pair is controllable. Furthermore, if the pair is stabilizable, then the controllable modes can all be arbitrarily assigned. Sufficiency Proof (if) Assume w. l. o. g. that the system is in controllable form (any controllable system can be transformed to controllable form) Find the matrix that arbitrarily places the eigenvalues of the closed-loop system with open-loop pair in controllable form (shown later) Proof of Necessity Controllability Invariance: Uncontrollable modes are invariant under state feedback. Let be an uncontrollable mode then Then is an eigenvalue and is a left eigenvector of the closed-loop system for any state feedback gain matrix. Eigenvalues cannot be arbitrarily assigned for uncontrollable. 11 Remarks on Theorem For any controllable single-input pair, and any set of real or complex conjugate values, there exists a unique 1 by vector such that the eigenvalues of the closed-loop matrix are placed at 2. For controllable -input systems, an matrix exists but is not unique. 3. For stabilizable systems, the controllable modes can be arbitrarily selected but the uncontrollable modes are invariant. 12
4 Selection of Eigenvalues Use the time response criteria based on second order approximation. Inequality constraints on design criteria. Poles at, for CT and at ±, for DT systems. For higher order systems, place additional poles at the origin for DT systems (far in the LHP for CT) or to cancel with system zeros. Procedure 9.1: Pole Placement by Equating Coefficients 1. Evaluate the desired characteristic polynomial from the specified eigenvalues using the expression 2. Evaluate the closed-loop characteristic polynomial using the expression 3. Equate the coefficients of the two polynomials to obtain equations to be solved for the entries of the matrix Equating Coefficients Example C The system is controllable The c.l. eigenvalues can be chosen arbitrarily 15 Numerical Values Desired eigenvalues:, Feedback Gain: 16
5 Limitations of Equating Coefficients Difficult to obtain closed-loop characteristic equation in terms of the gains. Equating coefficients may result in complex equations. Suitable for second order systems but difficult in general. What guarantees that the eigenvalues can be arbitrarily assigned? 17 Phase Variable (Controllable) Form, Gain vector in terms of coefficients vectors. 18 Controller (Controllable) Form, Gain vector in terms of coefficients vectors. 19 Procedure: Either Controllable Form 1. Obtain the open-loop characteristic polynomial and the vector or. 2. Obtain the desired closed-loop characteristic equation and vector or. 3. Obtain the gain vector,or (same values: in ascending order, and in descending order). Note: Identical procedure for continuous-time systems but poles selected in LHP (change ) 20
6 Example: Phase Variable Form Example: Controller Form Desired eigenvalues: Desired characteristic polynomial: Open-loop characteristic polynomial: Gain vector: Desired eigenvalues: Desired characteristic polynomial: Open-loop characteristic polynomial: Gain vector: Example: Phase Variable Form Example: Controller Form Desired eigenvalues: Desired characteristic polynomial: s 2 +6 s +18 Open-loop characteristic polynomial: s 2 +4 s +3 Gain vector: Desired poles Desired characteristic polynomial: Open-loop characteristic polynomial: Gain vector: Check closed-loop state matrix with MATLAB 23 24
7 MATLAB Commands >> a=[ ; ; ]; >>polya= poly(a) % Characteristic polynomial ans = >> polyd=poly([-10,-3+3*j,-3-j*3]) % Desired polynomial ans = >> polyd-polya; kc=ans(2:4) % For the controller form kc = MATLAB: fliplr needed for phase variable form. 25 Lemma Any pair can be transformed to controllable form if and only if it is controllable. Note: 1. Transform to phase variable form or to controller form. 2. Use the characteristic polynomial to form 3. There exists a nonsingular by matrix such that,where is in controllable form. 26 Proof of Sufficiency Assume the system is controllable, then C is invertible If the system is controllable then CC and C C exist (invertible matrices). C The similarity transformation 27 Proof of Necessity Controllable form is controllable. Similarity transformation does not change controllability. If the system is not controllable we cannot transform it to a controllable form. Transformation CC C C does not exist if the system is not controllable (C is singular). 28
8 State Feedback (Bass-Gura) For any controllable pair exists. For controllable form CC transformation to controllable form CC C C Write the control in terms of (for any ) C C C C Calculate (invert C only). Do not calculate 29 Example Desired eigenvalues: C C C C 30 Example: DT Pole placement for the DT system with Desired eigenvalues: (±)(.) Discretize the system with sampling period Example: DT (cont.) Desired eigenvalues: Characteristic polynomial of DT system C C C 31 32
9 MATLAB Pole Placement >> a=[ ; ; ]; % Earlier Ex. >> b=[0; 0; 1]; >> k=place(a, b, [-10,-3+3*j,-3-j*3]) k = >> eig(a-b*k) % Check closed-loop eigenvalues ans = i i Uncontrollable Modes If some modes are uncontrollable >> b= [1; 3; 1]; >> rank(ctrb(a,b)) % Not even stabilizable! 1 >> k=place(a,b,[-10,-3+3*j,-3-j*3])??? Error using ==> place at 179 Can't place eigenvalues there Ackermann s Formula to place the eigenvalues at C C = row of C 35 Proof For phase var. (controllable) form Cayley-Hamilton (similarity transf. invariant) Subtract then use 36
10 Proof: Elementary Vectors entry only is unity Similarly show: 37 Proof: Controllable Form only entry is unity 38 Proof (Cont.) Similarity transformation Bass-Gura formula and C C 39 Proof (Cont.) Transformation to phase var. form C C C Substitute in the formula derived earlier C C C 40
11 Example: Ackermann Example: Ackermann CT C C Desired poles C C = = , C = Desired poles C As obtained earlier MATLAB: Basic Commands >> pd=poly([-3+j*3,-3-j*3]) % Desired polynomial ans = >> con=ctrb(a,[0;1]) % Controllability matrix con = >> k=[0,1]/con* polyvalm(pd,a) % Gain vector k = Desired eigenvalues: DT Example Discretize the system with sampling period >> poldes=poly(pold) % pold =poles for DT system >> k=[0 1]/cond*polyvalm(poldes,pd.A) k =
12 MATLAB: acker >> A=[0 1;-3-4];B=[0;1]; % Controllable pair >> acker(a,b,[-3+j*3,-3-j*3]) % Gain vector ans = 15 2 Theorem: C.L. Controllability A LTI system with the state feedback is controllable with the external input if and only if is controllable. Closed-loop system controllable controllable Proof of Sufficiency (if): For controllable (reachable), there exists an input such that and for any state. Let =state associated with. Since, define the external input takes the closed-loop system to arbitrary in finite time, i.e. the closed-loop system is controllable. 47 Proof of Necessity (only if): If is not controllable (reachable), then there are some states that are not reachable from the origin regardless of how the input is generated (including the state feedback ). All point in the uncontrollable subspace are not reachable. If is not controllable, the closed-loop system is also not controllable. 48
13 Transmission Zero Solve for and equations in unknowns Can determine and direction of. for..... Transmission Zeros :, 49 Transmission Zero Condition A complex number is a transmission zero if it satisfies Rewrite as two equations Matrix is not full rank if the two equations hold. 50 Theorem: Transmission Zeros The transmission zeros of a LTI system are invariant under the state feedback Proof Let be a transmission zero of the LTI system. Then 51 Closed-Loop System Zero transmission zero of the open-loop system is also a transmission zero for the closed-loop system. Cannot change zeros with state feedback 52
14 Loss of Observability State feedback can place a pole at the location of a zero: pole-zero cancellation. If the original system is controllable, the closed-loop system is controllable. Pole-zero cancellation: closed-loop system can have unobservable modes even if the original system is completely observable. 53
Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems
Linear-Quadratic Optimal Controller 10.3 Optimal Linear Control Systems In Chapters 8 and 9 of this book we have designed dynamic controllers such that the closed-loop systems display the desired transient
CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
Similarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor
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
Linear Control Systems
Chapter 3 Linear Control Systems Topics : 1. Controllability 2. Observability 3. Linear Feedback 4. Realization Theory Copyright c Claudiu C. Remsing, 26. All rights reserved. 7 C.C. Remsing 71 Intuitively,
Controllability and Observability
Controllability and Observability Controllability and observability represent two major concepts of modern control system theory These concepts were introduced by R Kalman in 1960 They can be roughly defined
3.1 State Space Models
31 State Space Models In this section we study state space models of continuous-time linear systems The corresponding results for discrete-time systems, obtained via duality with the continuous-time models,
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
Orthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
by the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
Solving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
Control Systems with Actuator Saturation
Control Systems with Actuator Saturation Analysis and Design Tingshu Hu Zongli Lin With 67 Figures Birkhauser Boston Basel Berlin Preface xiii 1 Introduction 1 1.1 Linear Systems with Actuator Saturation
Recall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
Chapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
University of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
Stability. Chapter 4. Topics : 1. Basic Concepts. 2. Algebraic Criteria for Linear Systems. 3. Lyapunov Theory with Applications to Linear Systems
Chapter 4 Stability Topics : 1. Basic Concepts 2. Algebraic Criteria for Linear Systems 3. Lyapunov Theory with Applications to Linear Systems 4. Stability and Control Copyright c Claudiu C. Remsing, 2006.
Eigenvalues and Eigenvectors
Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution
Applied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
Numerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
LOOP TRANSFER RECOVERY FOR SAMPLED-DATA SYSTEMS 1
LOOP TRANSFER RECOVERY FOR SAMPLED-DATA SYSTEMS 1 Henrik Niemann Jakob Stoustrup Mike Lind Rank Bahram Shafai Dept. of Automation, Technical University of Denmark, Building 326, DK-2800 Lyngby, Denmark
Using the Theory of Reals in. Analyzing Continuous and Hybrid Systems
Using the Theory of Reals in Analyzing Continuous and Hybrid Systems Ashish Tiwari Computer Science Laboratory (CSL) SRI International (SRI) Menlo Park, CA 94025 Email: ashish.tiwari@sri.com Ashish Tiwari
Lecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
Lecture 13 Linear quadratic Lyapunov theory
EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time
1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
LS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
Understanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
Real-Time Systems Versus Cyber-Physical Systems: Where is the Difference?
Real-Time Systems Versus Cyber-Physical Systems: Where is the Difference? Samarjit Chakraborty www.rcs.ei.tum.de TU Munich, Germany Joint work with Dip Goswami*, Reinhard Schneider #, Alejandro Masrur
Methods for Finding Bases
Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
Factorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
Inner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
Notes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
Systems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
NOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
3. Let A and B be two n n orthogonal matrices. Then prove that AB and BA are both orthogonal matrices. Prove a similar result for unitary matrices.
Exercise 1 1. Let A be an n n orthogonal matrix. Then prove that (a) the rows of A form an orthonormal basis of R n. (b) the columns of A form an orthonormal basis of R n. (c) for any two vectors x,y R
MAT 242 Test 2 SOLUTIONS, FORM T
MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these
LINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
Linear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
LINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2-106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 2-06 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are right-hand-sides b for which A x = b has no solution (a) What
2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
Question 2: How do you solve a matrix equation using the matrix inverse?
Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients
Section 6.1 - Inner Products and Norms
Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
Inner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column
1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
Vector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
Interactive applications to explore the parametric space of multivariable controllers
Milano (Italy) August 28 - September 2, 211 Interactive applications to explore the parametric space of multivariable controllers Yves Piguet Roland Longchamp Calerga Sàrl, Av. de la Chablière 35, 14 Lausanne,
Similar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every
Modélisation et résolutions numérique et symbolique
Modélisation et résolutions numérique et symbolique via les logiciels Maple et Matlab Jeremy Berthomieu Mohab Safey El Din Stef Graillat Mohab.Safey@lip6.fr Outline Previous course: partial review of what
MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =
MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the
CONTROLLER DESIGN USING POLYNOMIAL MATRIX DESCRIPTION
CONTROLLER DESIGN USING POLYNOMIAL MATRIX DESCRIPTION Didier Henrion, Centre National de la Recherche Scientifique, France Michael Šebek, Center for Applied Cybernetics, Faculty of Electrical Eng., Czech
Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability. p. 1/?
Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability p. 1/? p. 2/? Definition: A p p proper rational transfer function matrix G(s) is positive
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
T ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
MATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
Zeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
Solution of Linear Systems
Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
Matrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.
Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems
Numerical Methods I Solving Linear Systems: Sparse Matrices, Iterative Methods and Non-Square Systems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420-001,
α = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1
19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point
Chapter 20. Vector Spaces and Bases
Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit
BX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
Chapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
Brief Introduction to Vectors and Matrices
CHAPTER 1 Brief Introduction to Vectors and Matrices In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valued
1.4 Fast Fourier Transform (FFT) Algorithm
74 CHAPTER AALYSIS OF DISCRETE-TIME LIEAR TIME-IVARIAT SYSTEMS 4 Fast Fourier Transform (FFT Algorithm Fast Fourier Transform, or FFT, is any algorithm for computing the -point DFT with a computational
Understanding and Applying Kalman Filtering
Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding
(Refer Slide Time: 01:11-01:27)
Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,
Linear Algebra and TI 89
Linear Algebra and TI 89 Abdul Hassen and Jay Schiffman This short manual is a quick guide to the use of TI89 for Linear Algebra. We do this in two sections. In the first section, we will go over the editing
is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
ROUTH S STABILITY CRITERION
ECE 680 Modern Automatic Control Routh s Stability Criterion June 13, 2007 1 ROUTH S STABILITY CRITERION Consider a closed-loop transfer function H(s) = b 0s m + b 1 s m 1 + + b m 1 s + b m a 0 s n + s
A note on companion matrices
Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod
Subspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
Linear Codes. Chapter 3. 3.1 Basics
Chapter 3 Linear Codes In order to define codes that we can encode and decode efficiently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length
1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
the points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
8 Square matrices continued: Determinants
8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You
Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI
Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:
DRAFT. Further mathematics. GCE AS and A level subject content
Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed
Lecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
Typical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
State Feedback Stabilization with Guaranteed Transient Bounds
State Feedback Stabilization with Guaranteed Transient Bounds D. Hinrichsen, E. Plischke, F. Wirth Zentrum für Technomathematik Universität Bremen 28334 Bremen, Germany Abstract We analyze under which