DOCSLIB.ORG

Linear Control Theory 2001

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Control Theory 2001 References Wonham Linear Multivariable Control – A Geometric Approach, Summer School 3rd edition, Springer Verlag, 1985. on Time Delay Equations and Control Theory Dobbiaco, June 25–29 2001 Basile and Marro Controlled and Conditioned Invariants in Linear Sys- tem Theory, Prentice Hall, 1992 Trentelman, Stoorvogel and Hautus Control Theory for Linear Systems, Springer Verlag, Linear Control Theory 2001 Giovanni MARRO∗, Domenico PRATTICHIZZO‡ Early References Basile and Marro Controlled and Conditioned Invariant Subspaces in Linear System Theory, Journal of Optimization The- ∗DEIS, University of Bologna, Italy ory and Applications, vol. 3, n. 5, 1969. ‡DII, University of Siena, Italy Wonham and Morse Decoupling and Pole Assignment in Linear Multivari- able Systems: a Geometric Approach, SIAM Journal on Control, vol. 8, n. 1, 1970. d Introduction to Control Problems u Σ y Consider the following figure that includes a controlled system (plant) Σ and a controller Σr, with a feedback part Σ and a feedforward part Σ . c f Σr rp e d1 d2 Fig. 1.2. A reduced block diagram. + rp r e u y1 Σf + Σc + Σ _ y2 In the above figure d := {d1,d2}, y := {y1,y2,d1}. Σr All the symbols in the figure denote signals,repre- sentable by real vectors varying in time. The plant Σ is given and the controller Σr is to be Fig. 1.1. A general block diagram for regulation. designed to (possibly) maintain e(·)=0. Both the plant and the controller are assumed to be linear (zero state and superposition property). • rp previewed reference The blocks represent oriented systems (inputs, out- • r reference puts), that are assumed to be causal. • y controlled output 1 In the classical control theory both continuous-time • y2 informative output systems and discrete-time systems are considered. • e error variable • u manipulated input • d1 non-measurable disturbance • d2 measurable disturbance 0 t 0 k 1 2 cr An example r va ω + e PI M PI controller _ K amplifier motor T + ω cr + 1 + r e T z va _ vc Fig. 1.4. The simplified block diagram. Σ w e tachometer e u Σ y Fig. 1.3. The velocity control of a dc motor. Σr The PI controlled yields steady-state control with no Fig. 1.5. The reduced block diagram. error. This property is robust against parameter variations, In Fig. 1.5 w accounts for both the reference and provided asymptotic stability of the loop is achieved. the disturbance. The control purpose is to achieve a This is due to the presence of an internal model of “minimal” error e intheresponsetow. the exosystem that reproduces a constant input sig- If w is assumed to be generated by an exosystem Σe nal (an integrator). like in the previous example, the internal model en- Thus, a step signal r of any value is reproduced with sures zero stedy-state error. no steady-state error and the disturbance cr is steady- This approach can easily be extended to the multi- state rejected. This is called a type 1 controller. variable case with geometric techniques. Similarly, a double integrator reproduces with no Modern approaches consider, besides the internal steady-state error any linear combination of a step model, the minimization of a norm (H2 or H∞)of and a ramp and rejects disturbances of the same type the transfer function from w to e to guarantee a sat- This is a type 2 controller. isfactory transient. 3 4 A more complex example Mathematical Models reduction controller gear Let us consider the velocity control of a motor shown rp in Fig. 1.3 and its reduced block diagram (Fig. 1.5): + delay r e Σ w e motor e u Σ y v0 gage transducer d Σr Fig. 1.6. Rolling mill control. Mathematical model of Σ: dia This example fits the general control scheme given in va(t)=Ra ia(t)+La (t)+vc(t) (1.1) Fig. 1.1. dt dω The gage control has an inherent transportation de- cm(t)=Bω(t)+J (t)+cr(t) (1.2) dt lay. If the aim of the control is to have given amounts of material (in meters) at a specified thickness, it is In (1.1) va is the applied voltage, Ra and La the arma- necessary to have a preview of these amounts, that is ture resistance and inductance, ia and vc the armature taken into account with the delay. current and counter emf, while in (1.2) cm is the mo- Of course, this preview can be used with negligible tor torque, B, J,andω the viscous friction coefficient, error if the cilinder rotation is feedback controlled by the moment of inertia, and the angular velocity of the measuring the amount of material with a type 2 con- shaft, and cr the externally applied load torque. troller. Mathematical model of Σr: Thus, robustness is achieved with feedback and makes dz 1 feedforward (preview control) possible. (t)= e(t) (1.3) There are cases in which preaction (action in advance) dt T on the controlled system significantly improves track- va(t)=Ke(t)+z(t) (1.4) ing of a reference signal. The block diagram shown where z denotes the output of the integrator in the in Fig. 1.1 also accounts for these cases. PI controller. 5 6 Their state space representation is The overall system (controlled system and controller) can be represented with a unique mathematical model ofthesametype: x˙(t)=Ax(t)+B1 u(t)+B2 d(t) (1.5) y(t)=Cx(t)+D1 u(t)+D2 d(t) xˆ˙(t)=Aˆxˆ(t)+Bˆ1 u(t)+Bˆ2 d(t) (1.7) T yˆ(t)=Cˆ xˆ(t)+Dˆ u(t)+Dˆ d(t) where for Σ, x := [ia ω] , u := va, d := cr, y := ω and 1 2 −R /L −k /L T A = a a 1 a where for x := [ia ωz] , u := r, d := cr y := ω and k2/J −B/J − − 1/La 0 Ra/La (k1 + K)/La 1/La B1 = B2 = 0 −1/J Aˆ = k2/J −B/J 0 0 −1/T 0 C = 01 D =0 D =0 1 2 K/La 0 while for Σr, xr := z, ur := e, yr := va and Bˆ1 = 0 Bˆ2 = −1/J 1 0 Ar =0 Br =1/T Cr =1 Dr = K Cˆ = 010 Dˆ1 =0 Dˆ2 =0 Mathematical model of Σe: The regulator design problem is: determine T and K dr dc such that the system (1.7) is internally stable,i.e.the =0 r =0 (1.6) dt dt eigenvalues of Aˆ have stricly negative real parts and this property is maintained in presence of admissible This corresponds to an autonomous system (without parameter variations. T input) having xe = y := [rcr] and 00 10 A = C = e 00 e 01 7 8 If only its behavior with respect to step inputs must State Space Models be considered, the overall system in Fig. 1.3 can be represented as the autonomous system Continuous-time systems: x˙(t)=Ax(t)+Bu(t) (1.9) xˆ˙(t)=Aˆxˆ(t) y(t)=Cx(t)+Du(t) (1.8) yˆ(t)=Cˆ xˆ(t) with the state x ∈X = Rn,theinput u ∈U= Rp,the output y ∈Y= Rq and A, BC, D real matrices of suit- T able dimensions. The system will be referred to as the where for x := [ia ωzrcr] , y := ω and quadruple (A, B, C, D)orthetriple (A, B, C)ifD =0. Most of the theory will be derived referring to triples since extension to quadruples is straightforward. −Ra/La −(k1 + K)/La 1/La K/La 0 − − k2/J B/J 001/J Discrete-time systems: ˆ = 0 −1 01 0 A /T 00000 x(k+1) = A x(k)+B u(k) 00000 d d (1.10) y(k)=Cd x(k)+Dd u(k) ˆ C = 01000 Recall that a continuous-time system is internally asymptotically stable iff all the eigenvalues of A be- The regulator design problem is: determine T and K long to C− (the open left half plane of the complex such that the autonomous system (A,ˆ Cˆ)isexternally plane) and a discrete-time system is internally asymp- stable, i.e., limt→∞ y(t) = 0 for any initial state and totically stable iff all the eigenvalues of Ad belong to this property is maintained in presence of admissible C (the open unit disk of the complex plane). parameter variations. In the discrete-time case a significant linear model is also the FIR (Finite Impulse Response) system, defined by the finite convolution sum N − y(k)= l=0 W (l) u(k l) (1.11) where W (k)(k =0,...,N)isaq × p real matrix, re- ferred to as the gain of the FIR system, while N is called the window of the FIR system. 9 10 Transfer Matrix Models Geometric Approach (GA) By taking the Laplace transform of (1.9) or the Z transform of (1.10) we obtain the transfer matrix rep- resentations Geometric Approach: is a control theory for multivari- able linear systems based on: Y (s)=G(s) U(s)with (1.12) G(s)=C (sI − A)−1 B + D • linear transformations and • subspaces ( )= ( ) ( )with Y z Gd z U z (The alternative approach is the transfer function ap- −1 (1.13) Gd(z)=Cd (zI − Ad) Bd + Dd proach) respectively. The geometric approach consists of The H2 norm in the continuous-time case is • an algebraic part (theoretical) ∞ 1/2 1 ∗ • G2 = tr G(jω) G (jω) dω (1.14) an algorithmic part (computational) 2π −∞ ∞ 1/2 Most of the mathematical support is developed in = tr g(t) gT(t) dt (1.15) coordinate-free form, to take advantage of simpler 0 and more elegant results, which facilitate insight into where g(t) denotes the impulse response of the system the actual meaning of statements and procedures; the (the inverse Laplace transform of G(s)), and in the computational aspects are considered independently discrete-time case it is of the theory and handled by means of the standard π 1/2 methods of matrix algebra, once a suitable coordinate 1 jω ∗ jω Gd2 = tr Gd(e ) G (e ) dω (1.16) system is defined.
Load more

Recommended publications
  • 2.161 Signal Processing: Continuous and Discrete Fall 2008
    MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 1 Lecture 1 Reading: • Class handout: The Dirac Delta and Unit-Step Functions 1 Introduction to Signal Processing In this class we will primarily deal with processing time-based functions, but the methods will also be applicable to spatial functions, for example image processing. We will deal with (a) Signal processing of continuous waveforms f(t), using continuous LTI systems (filters). a LTI dy nam ical system input ou t pu t f(t) Continuous Signal y(t) Processor and (b) Discrete-time (digital) signal processing of data sequences {fn} that might be samples of real continuous experimental data, such as recorded through an analog-digital converter (ADC), or implicitly discrete in nature. a LTI dis crete algorithm inp u t se q u e n c e ou t pu t seq u e n c e f Di screte Si gnal y { n } { n } Pr ocessor Some typical applications that we look at will include (a) Data analysis, for example estimation of spectral characteristics, delay estimation in echolocation systems, extraction of signal statistics. (b) Signal enhancement. Suppose a waveform has been contaminated by additive “noise”, for example 60Hz interference from the ac supply in the laboratory. 1copyright c D.Rowell 2008 1–1 inp u t ou t p u t ft + ( ) Fi lte r y( t) + n( t) ad d it ive no is e The task is to design a filter that will minimize the effect of the interference while not destroying information from the experiment.
    [Show full text]
  • Lecture 2 LSI Systems and Convolution in 1D
    Lecture 2 LSI systems and convolution in 1D 2.1 Learning Objectives Understand the concept of a linear system. • Understand the concept of a shift-invariant system. • Recognize that systems that are both linear and shift-invariant can be described • simply as a convolution with a certain “impulse response” function. 2.2 Linear Systems Figure 2.1: Example system H, which takes in an input signal f(x)andproducesan output g(x). AsystemH takes an input signal x and produces an output signal y,whichwecan express as H : f(x) g(x). This very general definition encompasses a vast array of di↵erent possible systems.! A subset of systems of particular relevance to signal processing are linear systems, where if f (x) g (x)andf (x) g (x), then: 1 ! 1 2 ! 2 a f + a f a g + a g 1 · 1 2 · 2 ! 1 · 1 2 · 2 for any input signals f1 and f2 and scalars a1 and a2. In other words, for a linear system, if we feed it a linear combination of inputs, we get the corresponding linear combination of outputs. Question: Why do we care about linear systems? 13 14 LECTURE 2. LSI SYSTEMS AND CONVOLUTION IN 1D Question: Can you think of examples of linear systems? Question: Can you think of examples of non-linear systems? 2.3 Shift-Invariant Systems Another subset of systems we care about are shift-invariant systems, where if f1 g1 and f (x)=f (x x )(ie:f is a shifted version of f ), then: ! 2 1 − 0 2 1 f (x) g (x)=g (x x ) 2 ! 2 1 − 0 for any input signals f1 and shift x0.
    [Show full text]
  • Introduction to Aircraft Aeroelasticity and Loads
    JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads i JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 ii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Copyright C 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620.
    [Show full text]
  • Linear System Theory
    Chapter 2 Linear System Theory In this course, we will be dealing primarily with linear systems, a special class of sys- tems for which a great deal is known. During the first half of the twentieth century, linear systems were analyzed using frequency domain (e.g., Laplace and z-transform) based approaches in an effort to deal with issues such as noise and bandwidth issues in communication systems. While they provided a great deal of intuition and were suf- ficient to establish some fundamental results, frequency domain approaches presented various drawbacks when control scientists started studying more complicated systems (containing multiple inputs and outputs, nonlinearities, noise, and so forth). Starting in the 1950’s (around the time of the space race), control engineers and scientists started turning to state-space models of control systems in order to address some of these issues. These time-domain approaches are able to effectively represent concepts such as the in- ternal state of the system, and also present a method to introduce optimality conditions into the controller design procedure. We will be using the state-space (or “modern”) approach to control almost exclusively in this course, and the purpose of this chapter is to review some of the essential concepts in this area. 2.1 Discrete-Time Signals Given a field F,asignal is a mapping from a set of numbers to F; in other words, signals are simply functions of the set of numbers that they operate on. More specifically: • A discrete-time signal f is a mapping from the set of integers Z to F, and is denoted by f[k]fork ∈ Z.Eachinstantk is also known as a time-step.
    [Show full text]
  • Digital Image Processing Lectures 3 & 4
    2-D Signals and Systems 2-D Fourier Transform Digital Image Processing Lectures 3 & 4 M.R. Azimi, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Definitions and Extensions: 2-D Signals: A continuous image is represented by a function of two variables e.g. x(u; v) where (u; v) are called spatial coordinates and x is the intensity. A sampled image is represented by x(m; n). If pixel intensity is also quantized (digital images) then each pixel is represented by B bits (typically B = 8 bits/pixel). 2-D Delta Functions: They are separable 2-D functions i.e. 1 (u; v) = (0; 0) Dirac: δ(u; v) = δ(u) δ(v) = 0 Otherwise 1 (m; n) = (0; 0) Kronecker: δ(m; n) = δ(m) δ(n) = 0 Otherwise M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Properties: For 2-D Dirac delta: 1 1- R R x(u0; v0)δ(u − u0; v − v0)du0dv0 = x(u; v) −∞ 2- R R δ(u; v)du dv = 1 8 > 0 − For 2-D Kronecker delta: 1 1 1- x(m; n) = P P x(m0; n0)δ(m − m0; n − n0) m0=−∞ n0=−∞ 1 2- P P δ(m; n) = 1 m;n=−∞ Periodic Signals: Consider an image x(m; n) which satisfies x(m; n + N) = x(m; n) x(m + M; n) = x(m; n) This signal is said to be doubly periodic with horizontal and vertical periods M and N, respectively.
    [Show full text]
  • CHAPTER TWO - Static Aeroelasticity – Unswept Wing Structural Loads and Performance 21 2.1 Background
    Static aeroelasticity – structural loads and performance CHAPTER TWO - Static Aeroelasticity – Unswept wing structural loads and performance 21 2.1 Background ........................................................................................................................... 21 2.1.2 Scope and purpose ....................................................................................................................... 21 2.1.2 The structures enterprise and its relation to aeroelasticity ............................................................ 22 2.1.3 The evolution of aircraft wing structures-form follows function ................................................ 24 2.2 Analytical modeling............................................................................................................... 30 2.2.1 The typical section, the flying door and Rayleigh-Ritz idealizations ................................................ 31 2.2.2 – Functional diagrams and operators – modeling the aeroelastic feedback process ....................... 33 2.3 Matrix structural analysis – stiffness matrices and strain energy .......................................... 34 2.4 An example - Construction of a structural stiffness matrix – the shear center concept ........ 38 2.5 Subsonic aerodynamics - fundamentals ................................................................................ 40 2.5.1 Reference points – the center of pressure..................................................................................... 44 2.5.2 A different
    [Show full text]
  • Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors Wei-Ping Huang University of Vermont
    University of Vermont ScholarWorks @ UVM Graduate College Dissertations and Theses Dissertations and Theses 2018 Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors Wei-Ping Huang University of Vermont Follow this and additional works at: https://scholarworks.uvm.edu/graddis Part of the Electrical and Electronics Commons Recommended Citation Huang, Wei-Ping, "Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors" (2018). Graduate College Dissertations and Theses. 967. https://scholarworks.uvm.edu/graddis/967 This Thesis is brought to you for free and open access by the Dissertations and Theses at ScholarWorks @ UVM. It has been accepted for inclusion in Graduate College Dissertations and Theses by an authorized administrator of ScholarWorks @ UVM. For more information, please contact [email protected]. Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors A Thesis Presented by Wei-Ping Huang to The Faculty of the Graduate College of The University of Vermont In Partial Fulfillment of the Requirements for the Degree of Master of Science Specializing in Electrical Engineering October, 2018 Defense Date: July 25th, 2018 Thesis Examination Committee: Hamid R. Ossareh, Ph.D., Advisor Eric M. Hernandez, Ph.D., Chairperson Jeff Frolik, Ph.D. Cynthia J. Forehand, Ph.D., Dean of Graduate College Abstract This thesis investigates Quasilinear Control (QLC) of time-delay systems with non- linear actuators and sensors and analyzes the accuracy of stochastic linearization for these systems. QLC leverages the method of stochastic linearization to replace each nonlinearity with an equivalent gain, which is obtained by solving a transcendental equation.
    [Show full text]
  • Applying Classical Control Theory to an Airplane Flap Model on Real Physical Hardware
    Applying Classical Control Theory to an Airplane Flap Model on Real Physical Hardware Edgar Collado Alvarez, BSIE, EIT, Janice Valentin, Jonathan Holguino ME.ME in Design and Controls Polytechnic University of Puerto Rico, [email protected],, [email protected], [email protected] Mentor: Prof. Bernardo Restrepo, Ph.D, PE in Aerospace and Controls Polytechnic University of Puerto Rico Hato Rey Campus, [email protected] Abstract — For the trimester of Winter -14 at Polytechnic University of Puerto Rico, the course Applied Control Systems was offered to students interested in further development and integration in the area of Control Engineering. Students were required to develop a control system utilizing a microprocessor as the controlling hardware. An Arduino Mega board was chosen as the microprocessor for the job. Integration of different mechanical, electrical, and electromechanical parts such as potentiometers, sensors, servo motors, etc., with the microprocessor were the main focus in developing the physical model to be controlled. This paper details the development of one of these control system projects applied to an airplane flap. After physically constructing and integrating additional parts, control was programmed in Simulink environment and further downloads to the microprocessor. MATLAB’s System Identification Tool was used to model the physical system’s transfer function based on input and output data from the physical plant. Tested control types utilized for system performance improvement were proportional (P) and proportional-integrative (PI) control. Key Words — embedded control systems, proportional integrative (PI) control, MATLAB, Simulink, Arduino. I. INTRODUCTION II. FIRST PHASE - POTENTIOMETER AND SERVO MOTOR CONTROL During the master’s course of Mechanical and Aerospace Control System, students were interested in The team started in the development of a potentiometer and developing real-world models with physical components servo motor control circuit.
    [Show full text]
  • From Linear to Nonlinear Control Means: a Practical Progression
    Cleveland State University EngagedScholarship@CSU Electrical Engineering & Computer Science Electrical Engineering & Computer Science Faculty Publications Department 4-2002 From Linear to Nonlinear Control Means: A Practical Progression Zhiqiang Gao Cleveland State University, [email protected] Follow this and additional works at: https://engagedscholarship.csuohio.edu/enece_facpub Part of the Controls and Control Theory Commons How does access to this work benefit ou?y Let us know! Publisher's Statement NOTICE: this is the author’s version of a work that was accepted for publication in ISA Transactions. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in ISA Transactions, 41, 2, (04-01-2002); 10.1016/S0019-0578(07)60077-9 Original Citation Gao, Z. (2002). From linear to nonlinear control means: A practical progression. ISA Transactions, 41(2), 177-189. doi:10.1016/S0019-0578(07)60077-9 Repository Citation Gao, Zhiqiang, "From Linear to Nonlinear Control Means: A Practical Progression" (2002). Electrical Engineering & Computer Science Faculty Publications. 61. https://engagedscholarship.csuohio.edu/enece_facpub/61 This Article is brought to you for free and open access by the Electrical Engineering & Computer Science Department at EngagedScholarship@CSU. It has been accepted for inclusion in Electrical Engineering & Computer Science Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contact [email protected]. From linear to nonlinear control means: A practical progression Zhiqiang Gao* DeP.1l'lll Ie/!/ of EJoclI"icai ami Computer Eng ineering.
    [Show full text]
  • Calculus and Differential Equations II
    Calculus and Differential Equations II MATH 250 B Linear systems of differential equations Linear systems of differential equations Calculus and Differential Equations II Second order autonomous linear systems We are mostly interested with2 × 2 first order autonomous systems of the form x0 = a x + b y y 0 = c x + d y where x and y are functions of t and a, b, c, and d are real constants. Such a system may be re-written in matrix form as d x x a b = M ; M = : dt y y c d The purpose of this section is to classify the dynamics of the solutions of the above system, in terms of the properties of the matrix M. Linear systems of differential equations Calculus and Differential Equations II Existence and uniqueness (general statement) Consider a linear system of the form dY = M(t)Y + F (t); dt where Y and F (t) are n × 1 column vectors, and M(t) is an n × n matrix whose entries may depend on t. Existence and uniqueness theorem: If the entries of the matrix M(t) and of the vector F (t) are continuous on some open interval I containing t0, then the initial value problem dY = M(t)Y + F (t); Y (t ) = Y dt 0 0 has a unique solution on I . In particular, this means that trajectories in the phase space do not cross. Linear systems of differential equations Calculus and Differential Equations II General solution The general solution to Y 0 = M(t)Y + F (t) reads Y (t) = C1 Y1(t) + C2 Y2(t) + ··· + Cn Yn(t) + Yp(t); = U(t) C + Yp(t); where 0 Yp(t) is a particular solution to Y = M(t)Y + F (t).
    [Show full text]
  • The Scientist and Engineer's Guide to Digital Signal Processing Properties of Convolution
    CHAPTER 7 Properties of Convolution A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers. Common Impulse Responses Delta Function The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change. Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: EQUATION 7-1 The delta function is the identity for ( ' convolution. Any signal convolved with x[n] *[n] x[n] a delta function is left unchanged. This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition (a%0 ' a), and one being the identity for multiplication (a×1 ' a).
    [Show full text]
  • Linear Time Invariant Systems
    UNIT III LINEAR TIME INVARIANT CONTINUOUS TIME SYSTEMS CT systems – Linear Time invariant Systems – Basic properties of continuous time systems – Linearity, Causality, Time invariance, Stability – Frequency response of LTI systems – Analysis and characterization of LTI systems using Laplace transform – Computation of impulse response and transfer function using Laplace transform – Differential equation – Impulse response – Convolution integral and Frequency response. System A system may be defined as a set of elements or functional blocks which are connected together and produces an output in response to an input signal. The response or output of the system depends upon transfer function of the system. Mathematically, the functional relationship between input and output may be written as y(t)=f[x(t)] Types of system Like signals, systems may also be of two types as under: 1. Continuous-time system 2. Discrete time system Continuous time System Continuous time system may be defined as those systems in which the associated signals are also continuous. This means that input and output of continuous – time system are both continuous time signals. For example: Audio, video amplifiers, power supplies etc., are continuous time systems. Discrete time systems Discrete time system may be defined as a system in which the associated signals are also discrete time signals. This means that in a discrete time system, the input and output are both discrete time signals. For example, microprocessors, semiconductor memories, shift registers etc., are discrete time signals. LTI system:- Systems are broadly classified as continuous time systems and discrete time systems. Continuous time systems deal with continuous time signals and discrete time systems deal with discrete time system.
    [Show full text]