Discrete Linear Systems and Z-transform
|
|
- Emil Mathews
- 7 years ago
- Views:
Transcription
1 Discrete Linear Systems and Z-transform Sven Laur University of Tarty 1 Lumped Linear Systems Recall that a lumped system is a system with finite memory that together with future inputs completely determines the output. The finite memory is formally represented as a state x. In most real world examples, the state x corresponds to certain physical properties of the system, like its position in the space, voltage levels etc. A system is time-invariant, if its behaviour does not depend on a time step k. More formally, if we have two runs of the system x[k1 ] = x 0 u[k 1 l] = u 0 [l], l 0 x[k2 ] = x 0 u[k 2 l] = u 0 [l], l 0 then the corresponding output signals y 1 and y 2 are just shifts of each other y 1 [k 1 l] = y 1 [k 2 l], l 0. Recall that the linearity condition is defined as follows. Assume that y 1 and y 2 are the output signals of the systems x[0] = x1 x[0] = x1 and u[k] = u 1 [k], k 0 u[k] = u 2 [k], k 0 Then the output of the following system x[0] = α1 x 1 α 2 x 2 u[k] = α 1 u 1 [k] α 2 u 2 [k], k 0 can be expressed as a linear combination y[k] = α 1 y 1 [k] α 2 y 2 [k]. As a result, we can decompose the run of a system into a sum of zero-state and zero-input responses, as x[0] = x0 u[k], k 0 = x[0] = 0 u[k], l 0 x[0] = x0 u[k] = 0, l 0 Intuitively, the zero-state response y zs describes how the system responds to the control signal and the zero-input response y zi describes how the system behaves without external inference or how it reacts for small disturbances.
2 As we can split the input signal into a sum of step functions 1, if k = m, δ m [k] = 0, if k m, the zero-state response y zs can be computed as a linear combination of zero-state responses to these step functions. For clarity consider SISO system (single input single output system). Then the system determines the correspondence δ m [ ] g m [ ] and zero-state response to any input signal can be computed as a sum y zs [k] = u[m]g m [k] = u[m]g m [k] m=k1 u[m]g m [k] (1) where the first term in the sum quantifies the effect of past and present inputs to the output and the second term quantifies the effect of future inputs to the output. If a system is causal, the effect of future inputs must be zero. For the time-invariant systems, we can go a step further. It is possible though a bit technical to show that the system state x must also depend linearly on initial input x 0 and input signal u. Consequently, a system in a relaxed state x[k] = 0 remains in the relaxed state until the input signal u[k l] = 0. In particular, note that the step function δ m [k] = 0 for k < m and thus the system is still in the zero state at time m. Hence, we can conclude g m [k] = g 0 [k m] := g[k m] and we have to only know how the system reacts to the unit impulse at k = 0. For linear causal systems, the corresponding sum (1) is a convolution y zs [k] = u[m]g[m k]. (2) 2 Convolutions and Z-transform Convolutions as such are difficult to confront in a raw format. Hence, mathematicians use formal methods to circumvent this eminent danger. Discrete convolutions can be treated with formal series. Let ŷ(x) be a product of û(x) = u[k]z k and û(x) = g[k]z k. k=0 Then it is straightforward to verify y[k] = u[k]g[m k] k=0
3 since we must take terms form û(x) and ĝ(x) so that their summary degree is k. This transformation is known as z-transform. For obvious reasons, the z-transform is unique. For each function g[ ], we can compute ĝ[z] and, given an infinite series ĝ[z], we can reconstruct all values of g[ ]. Similarly, it is easy to see that z-transform is linear h[k] = α 1 g 1 [k] α 2 g 2 [k] ĥ[z] = α 1 α 2. 3 Transfer Function A transfer function ĝ[z] of a linear system is just a z-transform of impulse response g[ ]. In particular, the convolution relation (2) becomes a product ŷ zs [z] = ĝ[z]û[z]. (3) For causal and lumped systems, it is possible to show that a transfer function must be a rational function ĝ[z] = N[z] D[z] where N[z] and D[z] are polynomials and deg N[z] deg D[z]. The stability and controllability of a linear system is greatly influenced by zeroes and poles of transfer function. A pole of ĝ[z] is a point z such that ĝ[z] = D[z] = 0 and a zero of ĝ[z] is a point z such that ĝ[z] = 0 N[z] = 0. Matlab function tf2zp computes zeros and poles of a rational function. 4 State Space Description and Transfer Function Normally, it is almost impossible to derive transfer function from the definition, since the system will have an infinite response to a unit impulse. Similarly, it is non-trivial task to device a linear system that implements the transfer function ĝ[z] by trying to device a system that has corresponding impulse response or impulse response matrix. Hence, one needs more clever conversion algorithms. Although a description of a lumped linear system can in principle be arbitrarily complex, all of these can be converted into the following canonical form x[k 1] = A[k]x[k] B[k]u[k] (4) y[k] = C[k]x[k] D[k]u[k]
4 where A[ ], B[ ], C[ ] and D[ ] are time-varying matrices with an appropriate shape. Moreover, any system that can be described through the equations (4) is linear. A system is time-invariant iff all matrices are constant in time. Now given a state space description (4) of a time-invariant system, it is possible to compute the transfer matrix Ĝ[z] as follows Ĝ[z] = C(zI A) 1 B D. This formula reveals an obvious fact that we can divide any linear system into a stateless subsystem y1 [k] = Du[k] and a remaining sub-system with inertia x[k 1] = Ax[k] Bu[k] y 2 [k] = Cx[k] that are run in parallel. Indeed, if y[k] = y 1 [k] y 2 [k], then the corresponding transfer function is Ĝ1[z] Ĝ2[z] = C(zI A) 1 B D. For any proper transfer matrix it is also possible to construct a corresponding canonical state space equation. For clarity, we present the corresponding formula for the SISO system. Let the transfer function be ĝ[z] = β 1 z n 1 β 2 z n 2 β n z n α 1 z n 1 α 2 z n 2 α n γ then the corresponding matrices that determine state space equation are α 1 α 2 α n 1 α n D = B = (5) C = [ ] β 1 β 2... β n 1 β n D = [ γ ] In Matlab these transformations are implemented as tf2ss and ss2tf. There are many other Matlab functions that can be used to analyse and optimise the realisations of a transfer function. However, all these functions are require the purchase of Control System Toolbox. 5 Analyse and Synthesis of Block Diagrams It is often convenient to decompose a linear system into small sub-systems that are interconnected. Moreover, all discrete systems can be built form addition,
5 ĝ[z] = α ĝ[z] = α ĝ[z] = ĝ[z] = ĝ1[z] 1 Fig.1. Transfer function of parallel, sequential and loop-back constructions multiplication, duplication and delay modules. Hence, it practical to know how to compute a transfer function from a block diagram and vice versa. The corresponding computation rule for simple configurations are given in Figure 1. One possible configuration how to reduce construction of MIMO (multiple input multiple output) systems to SISO systems is given in Figure 2. Most of these formulae follow directly from the definitions, whereas the loopback construction is a bit tricky to analyse. As usual, let u and y denote input and output signals. Additionally, let v be the feedback signal. Then the corresponding z-transforms must satisfy the following constraints ŷ zs = (û ˆv), (6) ˆv = ŷ zs. (7) Now iterative substitution of the equation (7) into the equation (6) yields ŷ zs = (1 ĝ2 2 [z] )û = ĝ1[z] which justifies the formula for the loop-back construction. 1 û, u 1 ĝ 11[z] y 1 u 2 ĝ 21[z] ĝ 12[z] u 3 ĝ 22[z] ĝ 13[z] ĝ 23[z] y 2 Fig.2. The synthesis of a MIMO system with a transfer matrix Ĝ[z] = (ĝij[z]) can be reduced to the design of individual transfer functions ĝ ij[z].
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
More information3.1 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,
More informationG(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).
Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,
More informationFormulations of Model Predictive Control. Dipartimento di Elettronica e Informazione
Formulations of Model Predictive Control Riccardo Scattolini Riccardo Scattolini Dipartimento di Elettronica e Informazione Impulse and step response models 2 At the beginning of the 80, the early formulations
More informationThe continuous and discrete Fourier transforms
FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More information1.4 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
More informationControl System Definition
Control System Definition A control system consist of subsytems and processes (or plants) assembled for the purpose of controlling the outputs of the process. For example, a furnace produces heat as a
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationMATH10212 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
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationCSE 135: Introduction to Theory of Computation Decidability and Recognizability
CSE 135: Introduction to Theory of Computation Decidability and Recognizability Sungjin Im University of California, Merced 04-28, 30-2014 High-Level Descriptions of Computation Instead of giving a Turing
More informationMATH 551 - APPLIED MATRIX THEORY
MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationSGN-1158 Introduction to Signal Processing Test. Solutions
SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:
More information(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,
More informationLecture 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
More informationLS.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
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationCONTROLLABILITY. 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
More informationNOTES 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
More informationInteractive 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,
More informationSolution 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
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationFormal Languages and Automata Theory - Regular Expressions and Finite Automata -
Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
More information3.Basic Gate Combinations
3.Basic Gate Combinations 3.1 TTL NAND Gate In logic circuits transistors play the role of switches. For those in the TTL gate the conducting state (on) occurs when the baseemmiter signal is high, and
More informationLecture 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
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationTHE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationGeneric Polynomials of Degree Three
Generic Polynomials of Degree Three Benjamin C. Wallace April 2012 1 Introduction In the nineteenth century, the mathematician Évariste Galois discovered an elegant solution to the fundamental problem
More informationFinal Year Project Progress Report. Frequency-Domain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones
Final Year Project Progress Report Frequency-Domain Adaptive Filtering Myles Friel 01510401 Supervisor: Dr.Edward Jones Abstract The Final Year Project is an important part of the final year of the Electronic
More informationFFT Algorithms. Chapter 6. Contents 6.1
Chapter 6 FFT Algorithms Contents Efficient computation of the DFT............................................ 6.2 Applications of FFT................................................... 6.6 Computing DFT
More informationDesign of FIR Filters
Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as
More informationSystems 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
More informationDIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION
DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION Introduction The outputs from sensors and communications receivers are analogue signals that have continuously varying amplitudes. In many systems
More informationRecall 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
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationUnderstanding 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
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationVector 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
More informationThe Calculation of G rms
The Calculation of G rms QualMark Corp. Neill Doertenbach The metric of G rms is typically used to specify and compare the energy in repetitive shock vibration systems. However, the method of arriving
More informationUniversity 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
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationTTT4120 Digital Signal Processing Suggested Solution to Exam Fall 2008
Norwegian University of Science and Technology Department of Electronics and Telecommunications TTT40 Digital Signal Processing Suggested Solution to Exam Fall 008 Problem (a) The input and the input-output
More informationPositive Feedback and Oscillators
Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active
More informationLecture 20: Transmission (ABCD) Matrix.
Whites, EE 48/58 Lecture 0 Page of 7 Lecture 0: Transmission (ABC) Matrix. Concerning the equivalent port representations of networks we ve seen in this course:. Z parameters are useful for series connected
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationSection 3. Sensor to ADC Design Example
Section 3 Sensor to ADC Design Example 3-1 This section describes the design of a sensor to ADC system. The sensor measures temperature, and the measurement is interfaced into an ADC selected by the systems
More informationDigital to Analog Converter. Raghu Tumati
Digital to Analog Converter Raghu Tumati May 11, 2006 Contents 1) Introduction............................... 3 2) DAC types................................... 4 3) DAC Presented.............................
More informationConvex Programming Tools for Disjunctive Programs
Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More information160 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
More informationConstrained optimization.
ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values
More informationMeasurement of Capacitance
Measurement of Capacitance Pre-Lab Questions Page Name: Class: Roster Number: Instructor:. A capacitor is used to store. 2. What is the SI unit for capacitance? 3. A capacitor basically consists of two
More informationTrend and Seasonal Components
Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing
More informationReal-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
More information5.3 Improper Integrals Involving Rational and Exponential Functions
Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a
More informationTEACHING AUTOMATIC CONTROL IN NON-SPECIALIST ENGINEERING SCHOOLS
TEACHING AUTOMATIC CONTROL IN NON-SPECIALIST ENGINEERING SCHOOLS J.A.Somolinos 1, R. Morales 2, T.Leo 1, D.Díaz 1 and M.C. Rodríguez 1 1 E.T.S. Ingenieros Navales. Universidad Politécnica de Madrid. Arco
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued
More informationSeries and Parallel Resistive Circuits
Series and Parallel Resistive Circuits The configuration of circuit elements clearly affects the behaviour of a circuit. Resistors connected in series or in parallel are very common in a circuit and act
More information7 Gaussian Elimination and LU Factorization
7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method
More informationMathematics 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
More informationCircuits 1 M H Miller
Introduction to Graph Theory Introduction These notes are primarily a digression to provide general background remarks. The subject is an efficient procedure for the determination of voltages and currents
More informationAnalysis of Algorithms I: Binary Search Trees
Analysis of Algorithms I: Binary Search Trees Xi Chen Columbia University Hash table: A data structure that maintains a subset of keys from a universe set U = {0, 1,..., p 1} and supports all three dictionary
More informationMethods 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,
More informationα = 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
More informationJim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 009-10 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationMixedÀ¾ нOptimization Problem via Lagrange Multiplier Theory
MixedÀ¾ нOptimization Problem via Lagrange Multiplier Theory Jun WuÝ, Sheng ChenÞand Jian ChuÝ ÝNational Laboratory of Industrial Control Technology Institute of Advanced Process Control Zhejiang University,
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
More informationMATH 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
More informationLecture 2: Universality
CS 710: Complexity Theory 1/21/2010 Lecture 2: Universality Instructor: Dieter van Melkebeek Scribe: Tyson Williams In this lecture, we introduce the notion of a universal machine, develop efficient universal
More informationStress Recovery 28 1
. 8 Stress Recovery 8 Chapter 8: STRESS RECOVERY 8 TABLE OF CONTENTS Page 8.. Introduction 8 8.. Calculation of Element Strains and Stresses 8 8.. Direct Stress Evaluation at Nodes 8 8.. Extrapolation
More information8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes
Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationCHAPTER 2 Estimating Probabilities
CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationdspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor
dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor Jaswandi Sawant, Divyesh Ginoya Department of Instrumentation and control, College of Engineering, Pune. ABSTRACT This
More informationMEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationDesign of Efficient Digital Interpolation Filters for Integer Upsampling. Daniel B. Turek
Design of Efficient Digital Interpolation Filters for Integer Upsampling by Daniel B. Turek Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationBasic Op Amp Circuits
Basic Op Amp ircuits Manuel Toledo INEL 5205 Instrumentation August 3, 2008 Introduction The operational amplifier (op amp or OA for short) is perhaps the most important building block for the design of
More informationIN current film media, the increase in areal density has
IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 1, JANUARY 2008 193 A New Read Channel Model for Patterned Media Storage Seyhan Karakulak, Paul H. Siegel, Fellow, IEEE, Jack K. Wolf, Life Fellow, IEEE, and
More informationFast analytical techniques for electrical and electronic circuits. Jet Propulsion Laboratory California Institute of Technology
Fast analytical techniques for electrical and electronic circuits Vatché Vorpérian Jet Propulsion Laboratory California Institute of Technology PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
More informationFACTORING SPARSE POLYNOMIALS
FACTORING SPARSE POLYNOMIALS Theorem 1 (Schinzel): Let r be a positive integer, and fix non-zero integers a 0,..., a r. Let F (x 1,..., x r ) = a r x r + + a 1 x 1 + a 0. Then there exist finite sets S
More informationAuto-Tuning Using Fourier Coefficients
Auto-Tuning Using Fourier Coefficients Math 56 Tom Whalen May 20, 2013 The Fourier transform is an integral part of signal processing of any kind. To be able to analyze an input signal as a superposition
More informationProduct Mix as a Framing Exercise: The Role of Cost Allocation. Anil Arya The Ohio State University. Jonathan Glover Carnegie Mellon University
Product Mix as a Framing Exercise: The Role of Cost Allocation Anil Arya The Ohio State University Jonathan Glover Carnegie Mellon University Richard Young The Ohio State University December 1999 Product
More informationLecture 7 Circuit analysis via Laplace transform
S. Boyd EE12 Lecture 7 Circuit analysis via Laplace transform analysis of general LRC circuits impedance and admittance descriptions natural and forced response circuit analysis with impedances natural
More informationFrom Workflow Design Patterns to Logical Specifications
AUTOMATYKA/ AUTOMATICS 2013 Vol. 17 No. 1 http://dx.doi.org/10.7494/automat.2013.17.1.59 Rados³aw Klimek* From Workflow Design Patterns to Logical Specifications 1. Introduction Formal methods in software
More information