Disturbance Decoupling Problem
|
|
- Kathryn Woods
- 6 years ago
- Views:
Transcription
1 DISC Systems and Control Theory of Nonlinear Systems, 21 1 Disturbance Decoupling Problem Lecture 4 Nonlinear Dynamical Control Systems, Chapter 7 The disturbance decoupling problem is a typical example of a structural control problem which can be elegantly solved by geometric methods
2 DISC Systems and Control Theory of Nonlinear Systems, 21 2 Consider the nonlinear system affected by disturbances d ẋ = f(x) + g(x)u + e(x)d, u R m, d R l, y = h(x), y R p, on the state space manifold X Theorem 1 Suppose the system is analytic The disturbance d does not influence the output y, irrespectively of the initial condition x() = x and input u, if and only if there exists a distribution D on X such that
3 DISC Systems and Control Theory of Nonlinear Systems, 21 3 (i) (ii) (iii) D is invariant for ẋ = f(x) + g(x)u D ker dh e j D, j = 1,, l Recall that a distribution D is invariant for a vector field f if [f, X] D, X D or in shorthand notation [f, D] D D is invariant for ẋ = f(x) + g(x)u if [f, D] D, [g j, D] D, j = 1,, m
4 DISC Systems and Control Theory of Nonlinear Systems, 21 4 Necessity Take D = ker do where O is the observation space Sufficiency Assume that D is constant-dimensional By Frobenius theorem we can find local coordinates for X (x 1,, x k, x k+1,, x n ) = (x 1, x 2 ) such that D = span{ 1, 1,, 1 }
5 DISC Systems and Control Theory of Nonlinear Systems, 21 5 In such coordinates (i) implies that ẋ = f(x) + g(x)u takes the form Furthermore by (ii) we have ẋ 1 = f 1 (x 1, x 2 ) + g 1 (x 1, x 2 )u ẋ 2 = f 2 (x 2 ) + g 2 (x 2 )u L x i h(x) =, i = 1,, k, implying that h only depends on x 2, that is h(x 2 ) Finally by (iii) the disturbance vectorfields e j are in D
6 DISC Systems and Control Theory of Nonlinear Systems, 21 6 It follows that the system has the form ẋ1 = f1 (x 1, x 2 ) + g1 (x 1, x 2 ) ẋ 2 f 2 (x 2 ) g 2 (x 2 ) u + e1 (x 1, x 2 ) d y = h(x 2 ) and thus d only affects the x 1 -dynamics, and therefore does not affect y
7 DISC Systems and Control Theory of Nonlinear Systems, 21 7 Next thing: find conditions such that a distribution D satisfying conditions (ii) and (iii) can be rendered invariant by static state feedback Definition 2 A distribution D on X is called controlled invariant for the system ẋ = f(x) + g(x)u if there exists a regular state feedback u = α(x) + β(x)v, detβ(x), x X, such that, denoting the closed-loop system by ẋ = f(x) + m j=1 g j (x)v j f(x) = f(x) + g(x)α(x), g(x) = [g(x)β(x)]
8 DISC Systems and Control Theory of Nonlinear Systems, 21 8 we have [ f, D] D, [ g j, D] D, j = 1,, m Key observation: By the property [X γ, Y ] = γ[x, Y ] X L Y γ, this implies [f, D] D + G, [g j, D] D + G, j = 1,, m where G = span{g 1,, g m }
9 DISC Systems and Control Theory of Nonlinear Systems, 21 9 Theorem 3 Consider the system ẋ = f(x) + g(x)u on X, and let D be a constant dimensional and involutive distribution Suppose that (i) dim[d(x) + G(x)] = constant (ii) [f, D] D + G (iii) [g j, D] D + G, j = 1,, m Then for every point x X there exists a neighborhood V of x and a regular feedback such that [ f, D] D, [ g j, D] D, j = 1,, m The distribution D is therefore called locally controlled invariant
10 DISC Systems and Control Theory of Nonlinear Systems, 21 1 Proof Take local coordinates x = (x 1,, x k, x k+1,, x n ) = (x 1, x 2 ) around x such that D = span{ x 1,, } =: span{ x k x 1 } Write correspondingly f = f1 f 2, g = g1 g 2 By (i) the (n k, m)-matrix g 2 (x) has constant rank, say l Without loss of generality we may assume that the first l rows of g 2 (x) are independent
11 DISC Systems and Control Theory of Nonlinear Systems, Lemma 4 There exists, locally around x, an invertible matrix β(x) such that for some R(x) g 2 (x)β(x) = I l R(x) l (m l) (n k l) (m l) n k m It follows that (unspecified elements denoted by ) g(x) := g(x)β(x) = I l R(x) } k } l } n k l
12 DISC Systems and Control Theory of Nonlinear Systems, Then (iii) yields for i = 1,, k [ g, ] (x) = x i R x i (x) im I k + im I l R(x) which necessarily implies that R x i (x) =, i = 1,, k, and thus that [ g, ] (x) D(x), x i i = 1,, k or equivalently, [ g j, D] D, j = 1,, m Hence β(x) is as required
13 DISC Systems and Control Theory of Nonlinear Systems, In order to construct α(x) write f 2 as f 2 = f21 f 22 Now define α(x) = β(x) f21 (x) l m l
14 DISC Systems and Control Theory of Nonlinear Systems, Then f = f + gα = f 1 f 21 f 22 I l R f21 = f 22 Rf 21 and (ii) implies for i = 1,, k [ f, x i ] = x i (f 22 Rf 21 ) im I k + im I l R showing, as above, that equivalently x i (f 22 Rf 21 ) =, i = 1,, k, or [ f, D] D
15 DISC Systems and Control Theory of Nonlinear Systems, Proof of Lemma Denote the matrix consisting of the first l independent rows of g 2 (x) by g 21 (x) Consider the equation g 21 (x)β(x) [I l l (m l) ] = Clearly in x this has an invertible solution β(x ) since g 21 (x ) has full row rank Then by the implicit function theorem it follows that locally around x there exists an invertible solution β(x) of g 2 (x)β(x) = I l R(x) l (m l) Since the rank of g 2 (x) is l, also the rank of g 2 (x)β(x) is l, and thus necessarily the unspecified elements have to be zero
16 DISC Systems and Control Theory of Nonlinear Systems, Example 5 Consider a linear system ẋ = Ax + Bu and let D be the distribution corresponding to a linear subspace V R n, ie, if V = span{e 1,, e k }, then D = span{ x 1,, x k } Then (ii) amounts to AV V + imb Furthermore, (iii) is automatically satisfied since [b j, x i ] = for the constant columns b j of B
17 DISC Systems and Control Theory of Nonlinear Systems, Theorem 6 Suppose there exists an involutive and constant dimensional distribution D on X such that (i) [f, D] D + G, [g j, D] D + G (ii) dim(d + G) is constant (iii) D kerdh (iv) e j D, j = 1,, l Then around each x X there exists a regular state feedback u = α(x) + β(x)v such that the closed-loop system is disturbance decoupled
18 DISC Systems and Control Theory of Nonlinear Systems, Example 7 Consider the cart with fixed rear axis d dt x 1 x 2 ϕ θ = cos(ϕ + θ) sin(ϕ + θ) sinθ u u 2 with u 1 the driving input, and u 2 the steering input
19 DISC Systems and Control Theory of Nonlinear Systems, Suppose there is additionally a disturbance vector field e = ( 1 1) T (corresponding to sideways slipping of the rear axis) The distribution D spanned by e, ie, D = span{ 1 1 } is involutive and constant dimensional, and satisfies the conditions Define x 3 := ϕ + θ, x 4 := θ Then the full dynamics is d dt x 1 x 2 x 3 x 4 = cosx 3 sin x 3 sin x 4 u u d
20 DISC Systems and Control Theory of Nonlinear Systems, 21 2 while D = span{ } 1 Thus modulo permutations we have obtained the required coordinates, with g 2 (x) = cosx 3 sinx 3 sinx 4 1
21 DISC Systems and Control Theory of Nonlinear Systems, Consider now an arbitrary point x = (x 1, x 2, x 3, x 4) Suppose cosx 3 Then the first and third row of g 2 (x) are independent, and we construct β(x) such that cosx 3 sinx 3 β(x) = 1 sinx A possible solution is β(x) = 1 cos x 3 sin x 4 cos x 3 1
22 DISC Systems and Control Theory of Nonlinear Systems, If cosx 3 =, then sinx 3, in which case the second and third row of g 2 (x) are independent, and we construct β(x) such that cosx 3 sinx 3 sinx 4 1 β(x) = 1 1 yielding as possible solution β(x) = 1 sin x 3 sin x 4 sin x 3 1
23 DISC Systems and Control Theory of Nonlinear Systems, Both feedback expressions are only locally defined In the present case we may also find a globally defined β(x) which solves the disturbance decoupling problem: β(x) = 1 sinx 4 1
24 DISC Systems and Control Theory of Nonlinear Systems, transforming the input vectorfields g 1 and g 2 to g 1 = cos(ϕ + θ) sin(ϕ + θ) sinθ, g 2 = 1 1 Note that the third entry of the new driving input vectorfield g 1 is zero, and thus ϕ + θ = implying that ϕ + θ remains constant Hence the front axis moves in the same direction Furthermore, since θ = sinθ the angle θ converges to zero, and therefore the rear axis tends to a position parallel to the front axis
25 DISC Systems and Control Theory of Nonlinear Systems, Conclusion: solution of the disturbance decoupling problem has been reduced to the search for a distribution D satisfying all the conditions How do we find such a distribution? Answer Compute the maximal distribution satisfying conditions (i)-(iii): Theorem 8 Compute the maximal distribution D satisfying conditions (i)-(iii) Suppose D is constant dimensional and D + G is constant dimensional Then the disturbance decoupling problem is solvable around any x X if and only if e j D, j = 1,, l
26 DISC Systems and Control Theory of Nonlinear Systems, Algorithm for computing D Define the sequence of distributions D µ, µ = 1, 2,, as D 1 = kerdh D µ+1 = kerdh span{x vectorfield [f, X] D µ + G, [g j, X] D µ + G, j = 1,, m} Then (i) D 1 D 2 D 3 (ii) D µ is involutive for µ = 1, 2, (iii) Denote D = lim µ Dµ If D satisfies (i), (iii) then D D (iv) D satisfies (i) and (iii)
27 DISC Systems and Control Theory of Nonlinear Systems, Proof (i) Clearly D 1 D 2 Suppose D µ D µ+1 Then D µ+2 = kerdh {X vectorfield [f, X] D µ+1 + G [g j, X] D µ+1 + G, j = 1,, m} kerdh {X vector field [f, X] D µ + G [g j, X] D µ + G, j = 1,, m} = D µ+1 (ii) D 1 = kerdh is involutive Indeed, let X, Y kerdh, then L [X,Y ] h = L X (L Y h) L Y (L X h) =, and thus [X, Y ] kerdh Now
28 DISC Systems and Control Theory of Nonlinear Systems, suppose D µ is involutive, and let X, Y be in D µ+1, ie [f, X] D µ + G, [g j, X] D µ + G, [f, Y ] D µ + G, [g j, X] D µ + G Then by the Jacobi-identity, it follows that for some Z X, Z Y D µ + G [f, [X, Y ]] = [[f, X], Y ] + [X, [f, Y ]] = = [Z X, Y ] + [X, Z Y ] D µ + G and similarly [g j, [X, Y ]] D µ + G, j = 1,, m
29 DISC Systems and Control Theory of Nonlinear Systems, (iii) Take any D satisfying (i) and (iii), Then D D 1 = kerdh Suppose D D µ, then by (i) [f, D] D + G D µ + G and [g j, D] D + G D µ + G, j = 1,, m, and thus D D µ+1 It follows that D D µ, µ = 1, 2, (iv) By construction D = kerdh {X vector field [f, X] D + G, [g j, X] D + G}, from which everything follows
30 DISC Systems and Control Theory of Nonlinear Systems, 21 3 Special case Consider the single-input single-output system ẋ = f(x) + g(x)u y = h(x) Let ρ be the smallest nonnegative integer such that the function L g L ρ f h(x) is different from the zero-function Assume this function is nowhere equal to zero Then D = ker(spandh, dl f h,,dl ρ f h) Similar expressions will hold for the multi-input multi-output case whenever the input-output decoupling matrix has maximal rank everywhere; cf Chapter 8 and next lecture
Lecture 2: Controllability of nonlinear systems
DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 2: Controllability of nonlinear systems Nonlinear Dynamical Control Systems, Chapter 3 See www.math.rug.nl/ arjan (under teaching) for info
More informationStabilization and Passivity-Based Control
DISC Systems and Control Theory of Nonlinear Systems, 2010 1 Stabilization and Passivity-Based Control Lecture 8 Nonlinear Dynamical Control Systems, Chapter 10, plus handout from R. Sepulchre, Constructive
More informationFeedback Linearization
Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)
More informationControllability, Observability & Local Decompositions
ontrollability, Observability & Local Decompositions Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Lie Bracket Distributions ontrollability ontrollability Distributions
More informationIntroduction to Geometric Control
Introduction to Geometric Control Relative Degree Consider the square (no of inputs = no of outputs = p) affine control system ẋ = f(x) + g(x)u = f(x) + [ g (x),, g p (x) ] u () h (x) y = h(x) = (2) h
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 7. Feedback Linearization IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs1/ 1 1 Feedback Linearization Given a nonlinear
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationImplicit Functions, Curves and Surfaces
Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationOutput Feedback and State Feedback. EL2620 Nonlinear Control. Nonlinear Observers. Nonlinear Controllers. ẋ = f(x,u), y = h(x)
Output Feedback and State Feedback EL2620 Nonlinear Control Lecture 10 Exact feedback linearization Input-output linearization Lyapunov-based control design methods ẋ = f(x,u) y = h(x) Output feedback:
More informationStatic Problem Set 2 Solutions
Static Problem Set Solutions Jonathan Kreamer July, 0 Question (i) Let g, h be two concave functions. Is f = g + h a concave function? Prove it. Yes. Proof: Consider any two points x, x and α [0, ]. Let
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More informationMathematical Systems Theory: Advanced Course Exercise Session 5. 1 Accessibility of a nonlinear system
Mathematical Systems Theory: dvanced Course Exercise Session 5 1 ccessibility of a nonlinear system Consider an affine nonlinear control system: [ ẋ = f(x)+g(x)u, x() = x, G(x) = g 1 (x) g m (x) ], where
More informationCSL361 Problem set 4: Basic linear algebra
CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Input-Output and Input-State Linearization Zero Dynamics of Nonlinear Systems Hanz Richter Mechanical Engineering Department Cleveland State University
More informationMathematics for Control Theory
Mathematics for Control Theory Geometric Concepts in Control Involutivity and Frobenius Theorem Exact Linearization Hanz Richter Mechanical Engineering Department Cleveland State University Reading materials
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationNonlinear Control Lecture 9: Feedback Linearization
Nonlinear Control Lecture 9: Feedback Linearization Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Nonlinear Control Lecture 9 1/75
More informationThe disturbance decoupling problem (DDP)
CHAPTER 3 The disturbance decoupling problem (DDP) Consider the system 3.1. Geometric formulation { ẋ = Ax + Bu + Ew y = Cx. Problem 3.1 (Disturbance decoupling). Find a state feedback u = Fx+ v such that
More informationCHAPTER 1. Introduction
CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric
More informationOverview of Fourier Series (Sect. 6.2). Origins of the Fourier Series.
Overview of Fourier Series (Sect. 6.2. Origins of the Fourier Series. Periodic functions. Orthogonality of Sines and Cosines. Main result on Fourier Series. Origins of the Fourier Series. Summary: Daniel
More informationControl design with guaranteed ultimate bound for feedback linearizable systems
Control design with guaranteed ultimate bound for feedback linearizable systems Ernesto Kofman, Fernando Fontenla Hernan Haimovich María M. Seron CONICET; Depto. de Electrónica, Fac. de Cs. Exactas, Ing.
More informationMATH 304 Linear Algebra Lecture 15: Linear transformations (continued). Range and kernel. Matrix transformations.
MATH 304 Linear Algebra Lecture 15: Linear transformations (continued). Range and kernel. Matrix transformations. Linear mapping = linear transformation = linear function Definition. Given vector spaces
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationNPTEL Online Course: Control Engineering
NPTEL Online Course: Control Engineering Ramkrishna Pasumarthy Assignment-11 : s 1. Consider a system described by state space model [ ] [ 0 1 1 x + u 5 1 2] y = [ 1 2 ] x What is the transfer function
More informationMathematical Economics: Lecture 2
Mathematical Economics: Lecture 2 Yu Ren WISE, Xiamen University September 25, 2012 Outline 1 Number Line The number line, origin (Figure 2.1 Page 11) Number Line Interval (a, b) = {x R 1 : a < x < b}
More informationSpan and Linear Independence
Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous
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 informationA Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction
A Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction R. W. Brockett* and Hongyi Li* Engineering and Applied Sciences Harvard University Cambridge, MA 38, USA {brockett, hongyi}@hrl.harvard.edu
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationSolutions. Problems of Chapter 1
Solutions Problems of Chapter 1 1.1 A real square matrix A IR n n is invertible if and only if its determinant is non zero. The determinant of A is a polynomial in the entries a ij of A, whose set of zeros
More informationSpring 2015 Sample Final Exam
Math 1151 Spring 2015 Sample Final Exam Final Exam on 4/30/14 Name (Print): Time Limit on Final: 105 Minutes Go on carmen.osu.edu to see where your final exam will be. NOTE: This exam is much longer than
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationPattern generation, topology, and non-holonomic systems
Systems & Control Letters ( www.elsevier.com/locate/sysconle Pattern generation, topology, and non-holonomic systems Abdol-Reza Mansouri Division of Engineering and Applied Sciences, Harvard University,
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationMATH 532: Linear Algebra
MATH 532: Linear Algebra Chapter 5: Norms, Inner Products and Orthogonality Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Spring 2015 fasshauer@iit.edu MATH 532 1 Outline
More informationOHSX XM511 Linear Algebra: Multiple Choice Exercises for Chapter 2
OHSX XM5 Linear Algebra: Multiple Choice Exercises for Chapter. In the following, a set is given together with operations of addition and scalar multiplication. Which is not a vector space under the given
More informationLinear Algebra Lecture Notes
Linear Algebra Lecture Notes Lecturers: Inna Capdeboscq and Damiano Testa Warwick, January 2017 Contents 1 Number Systems and Fields 3 1.1 Axioms for number systems............................ 3 2 Vector
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationDisturbance Decoupling and Unknown-Input Observation Problems
Disturbance Decoupling and Unknown-Input Observation Problems Lorenzo Ntogramatzidis Curtin University of Technology, Australia MTNS - July 5-9, 2010 L. Ntogramatzidis (Curtin University) MTNS - July 5-9,
More informationNonlinear Control Theory. Lecture 9
Nonlinear Control Theory Lecture 9 Periodic Perturbations Averaging Singular Perturbations Khalil Chapter (9, 10) 10.3-10.6, 11 Today: Two Time-scales Averaging ẋ = ǫf(t,x,ǫ) The state x moves slowly compared
More informationGeometric Control Theory
1 Geometric Control Theory Lecture notes by Xiaoming Hu and Anders Lindquist in collaboration with Jorge Mari and Janne Sand 2012 Optimization and Systems Theory Royal institute of technology SE-100 44
More informationNOTES ON MULTIVARIABLE CALCULUS: DIFFERENTIAL CALCULUS
NOTES ON MULTIVARIABLE CALCULUS: DIFFERENTIAL CALCULUS SAMEER CHAVAN Abstract. This is the first part of Notes on Multivariable Calculus based on the classical texts [6] and [5]. We present here the geometric
More informationUNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test
UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following
More informationReview of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE)
Review of Multi-Calculus (Study Guide for Spivak s CHPTER ONE TO THREE) This material is for June 9 to 16 (Monday to Monday) Chapter I: Functions on R n Dot product and norm for vectors in R n : Let X
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationAnalysis II: The Implicit and Inverse Function Theorems
Analysis II: The Implicit and Inverse Function Theorems Jesse Ratzkin November 17, 2009 Let f : R n R m be C 1. When is the zero set Z = {x R n : f(x) = 0} the graph of another function? When is Z nicely
More informationConsider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity
1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m
More informationMATH 167: APPLIED LINEAR ALGEBRA Chapter 2
MATH 167: APPLIED LINEAR ALGEBRA Chapter 2 Jesús De Loera, UC Davis February 1, 2012 General Linear Systems of Equations (2.2). Given a system of m equations and n unknowns. Now m n is OK! Apply elementary
More informationBasic Theory of Linear Differential Equations
Basic Theory of Linear Differential Equations Picard-Lindelöf Existence-Uniqueness Vector nth Order Theorem Second Order Linear Theorem Higher Order Linear Theorem Homogeneous Structure Recipe for Constant-Coefficient
More informationEigenvalues and Eigenvectors
LECTURE 3 Eigenvalues and Eigenvectors Definition 3.. Let A be an n n matrix. The eigenvalue-eigenvector problem for A is the problem of finding numbers λ and vectors v R 3 such that Av = λv. If λ, v are
More informationEquivalence of dynamical systems by bisimulation
Equivalence of dynamical systems by bisimulation Arjan van der Schaft Department of Applied Mathematics, University of Twente P.O. Box 217, 75 AE Enschede, The Netherlands Phone +31-53-4893449, Fax +31-53-48938
More informationPartial Derivatives October 2013
Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a
More informationMATH 409 Advanced Calculus I Lecture 11: More on continuous functions.
MATH 409 Advanced Calculus I Lecture 11: More on continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if for any ε > 0 there
More informationFact: Every matrix transformation is a linear transformation, and vice versa.
Linear Transformations Definition: A transformation (or mapping) T is linear if: (i) T (u + v) = T (u) + T (v) for all u, v in the domain of T ; (ii) T (cu) = ct (u) for all scalars c and all u in the
More informationMATH 54 QUIZ I, KYLE MILLER MARCH 1, 2016, 40 MINUTES (5 PAGES) Problem Number Total
MATH 54 QUIZ I, KYLE MILLER MARCH, 206, 40 MINUTES (5 PAGES) Problem Number 2 3 4 Total Score YOUR NAME: SOLUTIONS No calculators, no references, no cheat sheets. Answers without justification will receive
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 22, Time Allowed: 150 Minutes Maximum Marks: 30
NATIONAL BOARD FOR HIGHER MATHEMATICS M A and MSc Scholarship Test September 22, 2018 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions that follow INSTRUCTIONS TO CANDIDATES
More informationMA 262, Fall 2017, Final Version 01(Green)
INSTRUCTIONS MA 262, Fall 2017, Final Version 01(Green) (1) Switch off your phone upon entering the exam room. (2) Do not open the exam booklet until you are instructed to do so. (3) Before you open the
More informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
More information1. Introduction. 2. Outlines
1. Introduction Graphs are beneficial because they summarize and display information in a manner that is easy for most people to comprehend. Graphs are used in many academic disciplines, including math,
More informationMath 3C Lecture 20. John Douglas Moore
Math 3C Lecture 20 John Douglas Moore May 18, 2009 TENTATIVE FORMULA I Midterm I: 20% Midterm II: 20% Homework: 10% Quizzes: 10% Final: 40% TENTATIVE FORMULA II Higher of two midterms: 30% Homework: 10%
More informationMath Advanced Calculus II
Math 452 - Advanced Calculus II Manifolds and Lagrange Multipliers In this section, we will investigate the structure of critical points of differentiable functions. In practice, one often is trying to
More informationInstructions. 2. Four possible answers are provided for each question and only one of these is correct.
Instructions 1. This question paper has forty multiple choice questions. 2. Four possible answers are provided for each question and only one of these is correct. 3. Marking scheme: Each correct answer
More informationChapter 3a Topics in differentiation. Problems in differentiation. Problems in differentiation. LC Abueg: mathematical economics
Chapter 3a Topics in differentiation Lectures in Mathematical Economics L Cagandahan Abueg De La Salle University School of Economics Problems in differentiation Problems in differentiation Problem 1.
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More informationPhysics 411 Lecture 7. Tensors. Lecture 7. Physics 411 Classical Mechanics II
Physics 411 Lecture 7 Tensors Lecture 7 Physics 411 Classical Mechanics II September 12th 2007 In Electrodynamics, the implicit law governing the motion of particles is F α = m ẍ α. This is also true,
More information3.2 Frobenius Theorem
62 CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE 3.2 Frobenius Theorem 3.2.1 Distributions Definition 3.2.1 Let M be a n-dimensional manifold. A k-dimensional distribution (or a tangent subbundle) Δ : M Δ
More informationVector Spaces. (1) Every vector space V has a zero vector 0 V
Vector Spaces 1. Vector Spaces A (real) vector space V is a set which has two operations: 1. An association of x, y V to an element x+y V. This operation is called vector addition. 2. The association of
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationMath 61CM - Quick answer key to section problems Fall 2018
Math 6CM - Quick answer key to section problems Fall 08 Cédric De Groote These are NOT complete solutions! You are always expected to write down all the details and justify everything. This document is
More informationDifferentiation. f(x + h) f(x) Lh = L.
Analysis in R n Math 204, Section 30 Winter Quarter 2008 Paul Sally, e-mail: sally@math.uchicago.edu John Boller, e-mail: boller@math.uchicago.edu website: http://www.math.uchicago.edu/ boller/m203 Differentiation
More informationExample: Limit definition. Geometric meaning. Geometric meaning, y. Notes. Notes. Notes. f (x, y) = x 2 y 3 :
Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationAEA 2003 Extended Solutions
AEA 003 Extended Solutions These extended solutions for Advanced Extension Awards in Mathematics are intended to supplement the original mark schemes, which are available on the Edexcel website. 1. Since
More information1. Let r, s, t, v be the homogeneous relations defined on the set M = {2, 3, 4, 5, 6} by
Seminar 1 1. Which ones of the usual symbols of addition, subtraction, multiplication and division define an operation (composition law) on the numerical sets N, Z, Q, R, C? 2. Let A = {a 1, a 2, a 3 }.
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationExercises * on Linear Algebra
Exercises * on Linear Algebra Laurenz Wiskott Institut für Neuroinformatik Ruhr-Universität Bochum, Germany, EU 4 February 7 Contents Vector spaces 4. Definition...............................................
More informationWinter 2014 Practice Final 3/21/14 Student ID
Math 4C Winter 2014 Practice Final 3/21/14 Name (Print): Student ID This exam contains 5 pages (including this cover page) and 20 problems. Check to see if any pages are missing. Enter all requested information
More informationNotes on multivariable calculus
Notes on multivariable calculus Jonathan Wise February 2, 2010 1 Review of trigonometry Trigonometry is essentially the study of the relationship between polar coordinates and Cartesian coordinates in
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 informationSec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h
1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets
More informationMath 240, 4.3 Linear Independence; Bases A. DeCelles. 1. definitions of linear independence, linear dependence, dependence relation, basis
Math 24 4.3 Linear Independence; Bases A. DeCelles Overview Main ideas:. definitions of linear independence linear dependence dependence relation basis 2. characterization of linearly dependent set using
More informationUNDERGROUND LECTURE NOTES 1: Optimality Conditions for Constrained Optimization Problems
UNDERGROUND LECTURE NOTES 1: Optimality Conditions for Constrained Optimization Problems Robert M. Freund February 2016 c 2016 Massachusetts Institute of Technology. All rights reserved. 1 1 Introduction
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 21: Optimal Output Feedback Control connection is called the (lower) star-product of P and Optimal Output Feedback ansformation (LFT).
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
More informationLecture 13 - Wednesday April 29th
Lecture 13 - Wednesday April 29th jacques@ucsdedu Key words: Systems of equations, Implicit differentiation Know how to do implicit differentiation, how to use implicit and inverse function theorems 131
More informationMath 180, Exam 2, Practice Fall 2009 Problem 1 Solution. f(x) = arcsin(2x + 1) = sin 1 (3x + 1), lnx
Math 80, Exam, Practice Fall 009 Problem Solution. Differentiate the functions: (do not simplify) f(x) = x ln(x + ), f(x) = xe x f(x) = arcsin(x + ) = sin (3x + ), f(x) = e3x lnx Solution: For the first
More informationVector Spaces ปร ภ ม เวกเตอร
Vector Spaces ปร ภ ม เวกเตอร 1 5.1 Real Vector Spaces ปร ภ ม เวกเตอร ของจ านวนจร ง Vector Space Axioms (1/2) Let V be an arbitrary nonempty set of objects on which two operations are defined, addition
More informationSMSTC (2017/18) Geometry and Topology 2.
SMSTC (2017/18) Geometry and Topology 2 Lecture 1: Differentiable Functions and Manifolds in R n Lecturer: Diletta Martinelli (Notes by Bernd Schroers) a wwwsmstcacuk 11 General remarks In this lecture
More informationFunctional Analysis I
Functional Analysis I Course Notes by Stefan Richter Transcribed and Annotated by Gregory Zitelli Polar Decomposition Definition. An operator W B(H) is called a partial isometry if W x = X for all x (ker
More informationMcGill University Math 325A: Differential Equations LECTURE 12: SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (II)
McGill University Math 325A: Differential Equations LECTURE 12: SOLUTIONS FOR EQUATIONS WITH CONSTANTS COEFFICIENTS (II) HIGHER ORDER DIFFERENTIAL EQUATIONS (IV) 1 Introduction (Text: pp. 338-367, Chap.
More informationCDS 101/110a: Lecture 2.1 Dynamic Behavior
CDS 11/11a: Lecture 2.1 Dynamic Behavior Richard M. Murray 6 October 28 Goals: Learn to use phase portraits to visualize behavior of dynamical systems Understand different types of stability for an equilibrium
More informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More information