NECESSARY AND SUFFICIENT CONDITIONS FOR STATE-SPACE NETWORK REALIZATION. Philip E. Paré Masters Thesis Defense
|
|
- Ashley Haynes
- 5 years ago
- Views:
Transcription
1 NECESSARY AND SUFFICIENT CONDITIONS FOR STATE-SPACE NETWORK REALIZATION Philip E. Paré Masters Thesis Defense
2
3 INTRODUCTION
4 MATHEMATICAL MODELING
5 DYNAMIC MODELS A dynamic model has memory
6 DYNAMIC MODELS A dynamic model has memory There are different ways to model dynamic systems Black Box Representation u u x y y u 2 u 2 x 2 y 2 y 2 u 3 u 3 x 3 y 3 y 3
7 DYNAMIC MODELS A dynamic model has memory There are different ways to model dynamic systems State Machine Representation u x y x 4 u 2 x 2 y 2 x 5 x 6 x 7 u 3 x 3 y 3 x 8
8 STATE SPACE MODELS OF LINEAR TIME INVARIANT (LTI) SYSTEMS u x y x 4 u 2 x 2 y 2 x 5 x 6 x 7 u 3 x 3 y 3 x 8 x Ax Bu y Cx Du
9 TRANSFER FUNCTION REPRESENTATION OF LTI SYSTEMS u u x y y u 2 u 2 x 2 y 2 y 2 u 3 u 3 x 3 y 3 y 3 G(s) C(sI A) B D
10 MANY TO ONE RELATIONSHIP D D CX C XB B XAX A ˆ ˆ ˆ ˆ D B A si C D B A si C s G ˆ ˆ ˆ) ˆ( ) ( ) (
11 NETWORK REALIZATION Assume input-output data is produced by the system (A, B, C, D) then what must be known about this system to recover the whole thing from input-output data
12 NETWORK REALIZATION Assume input-output data is produced by the system (A, B, C, D) then what must be known about this system to recover the whole thing from input-output data
13 PROBLEM FORMULATION
14 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices.
15 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices.
16 PARAMETERIZATION The parametrization of the system matrices, is a continuously differentiable function P α : Ω R q R N where q is the number of unknown parameters and N = n n + m + p + mp, the number of parameters in the system matrices. For a β = P α R P R N N mp N mp N nm n nm n n n n D C B A 2 ) (, ) (, ) (, ) (
17 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices
18 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x
19 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x
20 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x I( )
21 IDENTITY PARAMETERIZATION The identity parameterization I θ : R N R N maps θ to itself and is a vectorization of the system matrices x y u x x I( )
22 RESTRICTION OF THE IDENTITY Assuming elements of (A, B, C, D) are known can restrict the identity parameterization
23 RESTRICTION OF THE IDENTITY Assuming elements of (A, B, C, D) are known can restrict the identity parameterization Let θ R R N encode the known information and the indicator function Θ R be a binary vector indicating the known elements and let the affine restriction of the identity parameterization be represented by I θr
24 CONSISTENCY Consider a system (A, B, C, D) and let θ be the vectorization of the system matrices (A, B, C, D). An affine restriction of the identity parametrization I θr is consistent with θ if θ R(I θr )
25 CONSISTENCY Consider a system (A, B, C, D) and let θ be the vectorization of the system matrices (A, B, C, D). An affine restriction of the identity parametrization I θr is consistent with θ if θ R(I θr )
26 GLOBAL IDENTIFIABILITY Let P α : Ω R q R N be a parameterization of system matrices (A, B, C, D). The parameterization P(α) is globally identifiable from the transfer function G(s) if for all α, α 2 Ω if G s = C α (si A α ) B α = C α 2 (si A α 2 ) B α 2 then α = α 2
27 NETWORK REALIZATION PROBLEM Consider a system (A, B, C, D), with A, B controllable and A, C observable and suppose G s = C(sI A) B + D is given. Find an affine restriction of the identity parameterization, consistent with A, B, C, D, that is globally identifiable from G(s)
28 SOLUTION TO NETWORK REALIZATION PROBLEM
29 SUFFICIENT CONDITION AX ˆ XB CX ˆ XA Bˆ C
30 SUFFICIENT CONDITION C CX B XB XA AX ˆ ˆ ˆ ) ( ˆ) ( ˆ ˆ C vec B vec x C I I B A A n n T T x b A
31 ANOTHER SUFFICIENT CONDITION X Aˆ AX BX ˆ B CX Cˆ
32 ANOTHER SUFFICIENT CONDITION C CX B BX AX A X ˆ ˆ ˆ ˆ) ( ) ( ˆ ˆ C vec B vec x C I I B A A n n T T x b A
33 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. = k, and
34 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. A = k, and N(A) N(A T ) x R(A T ) b R(A)
35 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. A = k, and N(A) N(A T ) R(A T ) R(A) x b
36 COMBINING SUFFICIENT CONDITIONS
37 COMBINING SUFFICIENT CONDITIONS Let x = A b, x = A b, dim N A dim N A = l. = k, and Also let N A = span{x n,, x nk } and N A = span{x n,, x n l }, where x ni = vec(x ni ).
38 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).
39 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).
40 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).
41 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).
42 MAIN RESULT Given a proper G(s), with two minimal realizations, S = (A, B, C, D) and S = (A, B, C, D) Suppose part of S is known, specified by the entries in θ R R N and the indicator function Θ R, characterizing an affine restriction of the identity parameterization I θr Suppose all of S is known Note that I θr and S are compatible giving A, b, A, b, X, X ni, X, and X nj for i =,, k and j =,, l Then I θr is globally identifiable from G s if and only if X + w X n + + w k X nk = (X + w X n + + w l X nl ) has a unique solution (w,, w k, w,, w l ).
43 PHARMACOKINETICS EXAMPLE
44 PHARMACOKINETICS EXAMPLE
45 PHARMACOKINETICS EXAMPLE x y k k k 2 x k k 2 x b u
46 PHARMACOKINETICS EXAMPLE x y u b x k k k k k x 2 2 x y u x x S
47 PHARMACOKINETICS EXAMPLE x y u x x x y u x x S S
48 PHARMACOKINETICS EXAMPLE Assume C = [ I ] is known So for I C=[ I ] we see that k = l = 2
49 PHARMACOKINETICS EXAMPLE Assume C = [ I ] is known So for I C=[ I ] we see that k = l = w w w w
50 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2
51 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2 X
52 PHARMACOKINETICS EXAMPLE Assume C = [ I ] and A :, = [ ] T are known So for I C,A :, we see that k = and l = 2 X x y x.227 x.25 u
53 THANK YOU
54 THANK YOU Questions?
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science : MULTIVARIABLE CONTROL SYSTEMS by A.
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Q-Parameterization 1 This lecture introduces the so-called
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe ECE133A (Winter 2018) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More information1 Controllability and Observability
1 Controllability and Observability 1.1 Linear Time-Invariant (LTI) Systems State-space: Dimensions: Notation Transfer function: ẋ = Ax+Bu, x() = x, y = Cx+Du. x R n, u R m, y R p. Note that H(s) is always
More informationThe norms can also be characterized in terms of Riccati inequalities.
9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements
More informationNew concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse
Lesson 10: Rank-nullity theorem, General solution of Ax = b (A 2 R mm ) New concepts: rank-nullity theorem Inverse matrix Gauss-Jordan algorithm to nd inverse Matrix rank. matrix nullity Denition. The
More informationOptimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications
Optimization problems on the rank and inertia of the Hermitian matrix expression A BX (BX) with applications Yongge Tian China Economics and Management Academy, Central University of Finance and Economics,
More informationMatrices: 2.1 Operations with Matrices
Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,
More informationThis lecture: basis and dimension 4.4. Linear Independence: Suppose that V is a vector space and. r 1 x 1 + r 2 x r k x k = 0
Linear Independence: Suppose that V is a vector space and that x, x 2,, x k belong to V {x, x 2,, x k } are linearly independent if r x + r 2 x 2 + + r k x k = only for r = r 2 = = r k = The vectors x,
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
More informationLinear System Theory
Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability
More informationLinear Matrix Inequality (LMI)
Linear Matrix Inequality (LMI) A linear matrix inequality is an expression of the form where F (x) F 0 + x 1 F 1 + + x m F m > 0 (1) x = (x 1,, x m ) R m, F 0,, F m are real symmetric matrices, and the
More informationControl Systems
6.5 Control Systems Last Time: Introdction Motivation Corse Overview Project Math. Descriptions of Systems ~ Review Classification of Systems Linear Systems LTI Systems The notion of state and state variables
More informationA STRATEGY FOR IDENTIFICATION OF BUILDING STRUCTURES UNDER BASE EXCITATIONS
A STRATEGY FOR IDENTIFICATION OF BUILDING STRUCTURES UNDER BASE EXCITATIONS G. Amato and L. Cavaleri PhD Student, Dipartimento di Ingegneria Strutturale e Geotecnica,University of Palermo, Italy. Professor,
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 4
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 4 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 12, 2012 Andre Tkacenko
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationMatrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.
Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n
More informationLec 6: State Feedback, Controllability, Integral Action
Lec 6: State Feedback, Controllability, Integral Action November 22, 2017 Lund University, Department of Automatic Control Controllability and Observability Example of Kalman decomposition 1 s 1 x 10 x
More informationBALANCING-RELATED MODEL REDUCTION FOR DATA-SPARSE SYSTEMS
BALANCING-RELATED Peter Benner Professur Mathematik in Industrie und Technik Fakultät für Mathematik Technische Universität Chemnitz Computational Methods with Applications Harrachov, 19 25 August 2007
More informationLinear Systems. Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output
Linear Systems Linear systems?!? (Roughly) Systems which obey properties of superposition Input u(t) output Our interest is in dynamic systems Dynamic system means a system with memory of course including
More informationLinear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008
Linear Algebra Chih-Wei Yi Dept. of Computer Science National Chiao Tung University November, 008 Section De nition and Examples Section De nition and Examples Section De nition and Examples De nition
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 information8 A First Glimpse on Design with LMIs
8 A First Glimpse on Design with LMIs 8.1 Conceptual Design Problem Given a linear time invariant system design a linear time invariant controller or filter so as to guarantee some closed loop indices
More informationStability of Parameter Adaptation Algorithms. Big picture
ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about
More informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationENGI Parametric Vector Functions Page 5-01
ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric
More informationFast Convolution; Strassen s Method
Fast Convolution; Strassen s Method 1 Fast Convolution reduction to subquadratic time polynomial evaluation at complex roots of unity interpolation via evaluation at complex roots of unity 2 The Master
More informationIntro. Computer Control Systems: F9
Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationMAE 143B - Homework 8 Solutions
MAE 43B - Homework 8 Solutions P6.4 b) With this system, the root locus simply starts at the pole and ends at the zero. Sketches by hand and matlab are in Figure. In matlab, use zpk to build the system
More information1.6 and 5.3. Curve Fitting One of the broadest applications of linear algebra is to curve fitting, especially in determining unknown coefficients in
16 and 53 Curve Fitting One of the broadest applications of linear algebra is to curve fitting, especially in determining unknown coefficients in functions You should know that, given two points in the
More informationThe Four Fundamental Subspaces
The Four Fundamental Subspaces Introduction Each m n matrix has, associated with it, four subspaces, two in R m and two in R n To understand their relationships is one of the most basic questions in linear
More informationLifted approach to ILC/Repetitive Control
Lifted approach to ILC/Repetitive Control Okko H. Bosgra Maarten Steinbuch TUD Delft Centre for Systems and Control TU/e Control System Technology Dutch Institute of Systems and Control DISC winter semester
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 informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Norms for Signals and Systems
. AERO 632: Design of Advance Flight Control System Norms for Signals and. Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Norms for Signals ...
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
More informationAssessment Exemplars: Polynomials, Radical and Rational Functions & Equations
Class: Date: Assessment Exemplars: Polynomials, Radical and Rational Functions & Equations 1 Express the following polynomial function in factored form: P( x) = 10x 3 + x 2 52x + 20 2 SE: Express the following
More informationLECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK)
LECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK) In this lecture, F is a fixed field. One can assume F = R or C. 1. More about the spanning set 1.1. Let S = { v 1, v n } be n vectors in V, we have defined
More informationMath 31CH - Spring Final Exam
Math 3H - Spring 24 - Final Exam Problem. The parabolic cylinder y = x 2 (aligned along the z-axis) is cut by the planes y =, z = and z = y. Find the volume of the solid thus obtained. Solution:We calculate
More informationNPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-mechanical System Module 4- Lecture 31. Observer Design
Observer Design Dr. Bishakh Bhattacharya h Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD This Lecture Contains Full State Feedback Control
More informationOn some interpolation problems
On some interpolation problems A. Gombani Gy. Michaletzky LADSEB-CNR Eötvös Loránd University Corso Stati Uniti 4 H-1111 Pázmány Péter sétány 1/C, 35127 Padova, Italy Computer and Automation Institute
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
More informationAnalysis of Systems with State-Dependent Delay
Analysis of Systems with State-Dependent Delay Matthew M. Peet Arizona State University Tempe, AZ USA American Institute of Aeronautics and Astronautics Guidance, Navigation and Control Conference Boston,
More informationLecture 1. Finite difference and finite element methods. Partial differential equations (PDEs) Solving the heat equation numerically
Finite difference and finite element methods Lecture 1 Scope of the course Analysis and implementation of numerical methods for pricing options. Models: Black-Scholes, stochastic volatility, exponential
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationEfficient robust optimization for robust control with constraints Paul Goulart, Eric Kerrigan and Danny Ralph
Efficient robust optimization for robust control with constraints p. 1 Efficient robust optimization for robust control with constraints Paul Goulart, Eric Kerrigan and Danny Ralph Efficient robust optimization
More informationParametric Model Order Reduction for Linear Control Systems
Parametric Model Order Reduction for Linear Control Systems Peter Benner HRZZ Project Control of Dynamical Systems (ConDys) second project meeting Zagreb, 2 3 November 2017 Outline 1. Introduction 2. PMOR
More informationCANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM
CANONICAL LOSSLESS STATE-SPACE SYSTEMS: STAIRCASE FORMS AND THE SCHUR ALGORITHM Ralf L.M. Peeters Bernard Hanzon Martine Olivi Dept. Mathematics, Universiteit Maastricht, P.O. Box 616, 6200 MD Maastricht,
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture 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
More information7 : APPENDIX. Vectors and Matrices
7 : APPENDIX Vectors and Matrices An n-tuple vector x is defined as an ordered set of n numbers. Usually we write these numbers x 1,...,x n in a column in the order indicated by their subscripts. The transpose
More informationIntroduction to Nonlinear Control Lecture # 4 Passivity
p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive
More informationDSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 1
DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 1 A schematic for a small laboratory electromechanical shaker is shown below, along with a bond graph that can be used for initial modeling studies. Our intent
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 19: Stabilization via LMIs Optimization Optimization can be posed in functional form: min x F objective function : inequality constraints
More informationAn LMI Approach to the Control of a Compact Disc Player. Marco Dettori SC Solutions Inc. Santa Clara, California
An LMI Approach to the Control of a Compact Disc Player Marco Dettori SC Solutions Inc. Santa Clara, California IEEE SCV Control Systems Society Santa Clara University March 15, 2001 Overview of my Ph.D.
More informationAffine transformations
Reading Required: Affine transformations Brian Curless CSEP 557 Fall 2016 Angel 3.1, 3.7-3.11 Further reading: Angel, the rest of Chapter 3 Fole, et al, Chapter 5.1-5.5. David F. Rogers and J. Alan Adams,
More informationy= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.
. (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the
More informationEEE582 Homework Problems
EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use
More informationState Feedback and State Estimators Linear System Theory and Design, Chapter 8.
1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 2 Homework 7: State Estimators (a) For the same system as discussed in previous slides, design another closed-loop state estimator,
More informationTransfer function and linearization
Transfer function and linearization Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Corso di Controlli Automatici, A.A. 24-25 Testo del corso:
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationj=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p.
LINEAR ALGEBRA Fall 203 The final exam Almost all of the problems solved Exercise Let (V, ) be a normed vector space. Prove x y x y for all x, y V. Everybody knows how to do this! Exercise 2 If V is a
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 informationKRONECKER PRODUCT AND LINEAR MATRIX EQUATIONS
Proceedings of the Second International Conference on Nonlinear Systems (Bulletin of the Marathwada Mathematical Society Vol 8, No 2, December 27, Pages 78 9) KRONECKER PRODUCT AND LINEAR MATRIX EQUATIONS
More informationCDS Solutions to the Midterm Exam
CDS 22 - Solutions to the Midterm Exam Instructor: Danielle C. Tarraf November 6, 27 Problem (a) Recall that the H norm of a transfer function is time-delay invariant. Hence: ( ) Ĝ(s) = s + a = sup /2
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
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 informationWeek 9-10: Recurrence Relations and Generating Functions
Week 9-10: Recurrence Relations and Generating Functions April 3, 2017 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationRepresenting Structure in Linear Interconnected Dynamical Systems
49th IEEE Conference on Decision and Control December 5-7, 200 Hilton Atlanta Hotel, Atlanta, GA, USA Representing Structure in Linear Interconnected Dynamical Systems E. Yeung, J. Gonçalves, H. Sandberg,
More informationCategorical techniques for NC geometry and gravity
Categorical techniques for NC geometry and gravity Towards homotopical algebraic quantum field theory lexander Schenkel lexander Schenkel School of Mathematical Sciences, University of Nottingham School
More informationZeros and zero dynamics
CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)
More informationIntroduction to Modern Control MT 2016
CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear
More informationENGG5781 Matrix Analysis and Computations Lecture 9: Kronecker Product
ENGG5781 Matrix Analysis and Computations Lecture 9: Kronecker Product Wing-Kin (Ken) Ma 2017 2018 Term 2 Department of Electronic Engineering The Chinese University of Hong Kong Kronecker product and
More information7.4: Integration of rational functions
A rational function is a function of the form: f (x) = P(x) Q(x), where P(x) and Q(x) are polynomials in x. P(x) = a n x n + a n 1 x n 1 + + a 0. Q(x) = b m x m + b m 1 x m 1 + + b 0. How to express a
More informationSubrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING
Subrings and Ideals Chapter 2 2.1 INTRODUCTION In this chapter, we discuss, subrings, sub fields. Ideals and quotient ring. We begin our study by defining a subring. If (R, +, ) is a ring and S is a non-empty
More informationOn Input Design for System Identification
On Input Design for System Identification Input Design Using Markov Chains CHIARA BRIGHENTI Masters Degree Project Stockholm, Sweden March 2009 XR-EE-RT 2009:002 Abstract When system identification methods
More informationEE263: Introduction to Linear Dynamical Systems Review Session 2
EE263: Introduction to Linear Dynamical Systems Review Session 2 Basic concepts from linear algebra nullspace range rank and conservation of dimension EE263 RS2 1 Prerequisites We assume that you are familiar
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2017) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal
More informationTopic # Feedback Control
Topic #7 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For
More informationSignal Structure for a Class of Nonlinear Dynamic Systems
Brigham Young University BYU ScholarsArchive All Theses and Dissertations 2018-05-01 Signal Structure for a Class of Nonlinear Dynamic Systems Meilan Jin Brigham Young University Follow this and additional
More informationINVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN. Leonid Lyubchyk
CINVESTAV Department of Automatic Control November 3, 20 INVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN Leonid Lyubchyk National Technical University of Ukraine Kharkov
More informationEE221A Linear System Theory Final Exam
EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this
More informationMath 515 Fall, 2008 Homework 2, due Friday, September 26.
Math 515 Fall, 2008 Homework 2, due Friday, September 26 In this assignment you will write efficient MATLAB codes to solve least squares problems involving block structured matrices known as Kronecker
More informationWeak Monotonicity of Interval Matrices
Electronic Journal of Linear Algebra Volume 25 Volume 25 (2012) Article 9 2012 Weak Monotonicity of Interval Matrices Agarwal N. Sushama K. Premakumari K. C. Sivakumar kcskumar@iitm.ac.in Follow this and
More informationA = u + V. u + (0) = u
Recall: Last time we defined an affine subset of R n to be a subset of the form A = u + V = {u + v u R n,v V } where V is a subspace of R n We said that we would use the notation A = {u,v } to indicate
More informationLinear Algebra 1 Exam 2 Solutions 7/14/3
Linear Algebra 1 Exam Solutions 7/14/3 Question 1 The line L has the symmetric equation: x 1 = y + 3 The line M has the parametric equation: = z 4. [x, y, z] = [ 4, 10, 5] + s[10, 7, ]. The line N is perpendicular
More information1 Continuous-time Systems
Observability Completely controllable systems can be restructured by means of state feedback to have many desirable properties. But what if the state is not available for feedback? What if only the output
More informationMathematics 136 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 19 and 21, 2016
Mathematics 36 Calculus 2 Everything You Need Or Want To Know About Partial Fractions (and maybe more!) October 9 and 2, 206 Every rational function (quotient of polynomials) can be written as a polynomial
More informationNonlinear Control. Nonlinear Control Lecture # 6 Passivity and Input-Output Stability
Nonlinear Control Lecture # 6 Passivity and Input-Output Stability Passivity: Memoryless Functions y y y u u u (a) (b) (c) Passive Passive Not passive y = h(t,u), h [0, ] Vector case: y = h(t,u), h T =
More informationMEM-BUS ABSOLUTE ENCODER
MEM-BUS ABSOLUTE ENCODER With PROFINET Interface Application Examples Software version STEP7 CONTENTS Applications with ELAP encoder... 3 Example 1: Angle position measurement on a rotary table with mechanical
More informationECE504: Lecture 9. D. Richard Brown III. Worcester Polytechnic Institute. 04-Nov-2008
ECE504: Lecture 9 D. Richard Brown III Worcester Polytechnic Institute 04-Nov-2008 Worcester Polytechnic Institute D. Richard Brown III 04-Nov-2008 1 / 38 Lecture 9 Major Topics ECE504: Lecture 9 We are
More informationA Benders Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse
A Benders Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse Ted Ralphs 1 Joint work with Menal Güzelsoy 2 and Anahita Hassanzadeh 1 1 COR@L Lab, Department of Industrial
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationMethods in Computer Vision: Introduction to Matrix Lie Groups
Methods in Computer Vision: Introduction to Matrix Lie Groups Oren Freifeld Computer Science, Ben-Gurion University June 14, 2017 June 14, 2017 1 / 46 Definition and Basic Properties Definition (Matrix
More informationModel Order Reduction for Parameter-Varying Systems
Model Order Reduction for Parameter-Varying Systems Xingang Cao Promotors: Wil Schilders Siep Weiland Supervisor: Joseph Maubach Eindhoven University of Technology CASA Day November 1, 216 Xingang Cao
More information