Nonlinear Analysis and Control of Aeroelastic Systems

Total Page:16

File Type:pdf, Size:1020Kb

Nonlinear Analysis and Control of Aeroelastic Systems Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Aerospace Engineering Mayuresh J. Patil, Chair Craig A. Woolsey Cornel Sultan Michael K. Philen May 3, 2016 Blacksburg, Virginia Keywords: Aeroelasticity, Control, Limit Cycle Oscillations, Optimization, Subcritical, Supercritical Copyright 2016, Himanshu Shukla Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla ABSTRACT Presence of nonlinearities may lead to limit cycle oscillations (LCOs) in aeroelastic systems. LCOs can result in fatigue in wings leading to catastrophic failures. Existence of LCOs for velocities less than the linear flutter velocity has been observed during flight and wind tunnel tests, making such subcritical behavior highly undesirable. The objective of this dissertation is to investigate the existence of subcritical LCOs in aeroelastic systems and develop state feedback controllers to suppress them. The research results are demonstrated on a two degree of freedom airfoil section model with stiffness nonlinearity. Three different approaches are developed and discussed. The first approach uses a feedback linearization controller employing the aeroelastic modal coordinates. The use of modal coor- dinates results in a system which is linearly decoupled making it possible to avoid cancellation of any linear terms when compared to existing feedback linearization controllers which use the physical coordinates. The state and control costs of the developed controller are com- pared to the costs of the traditional feedback linearization controllers. Second approach involves the use of nonlinear normal modes (NNMs) as a tool to predict LCO amplitudes of the aeroelastic system. NNM dynamics along with harmonic balance method are used to generate analytical estimates of LCO amplitude and its sensitivities with respect to the introduced control parameters. A multiobjective optimization problem is solved to gener- ate optimal control parameters which minimize the LCO amplitude and the control cost. The third approach uses a nonlinear state feedback control input obtained as the solution of a multiobjective optimization problem which minimizes the difference between the LCO commencement velocity and the linear flutter velocity. The estimates of LCO commence- ment velocity and its sensitivities are obtained using numerical continuation methods and harmonic balance methods. It is shown that the developed optimal controller eliminates any existing subcritical LCOs by converting them to supercritical LCOs. Nonlinear Analysis and Control of Aeroelastic Systems Himanshu Shukla GENERAL AUDIENCE ABSTRACT Aeroelasticity deals with the problem of fluid-structure interaction in aerial systems. The desire for high speed travel and increased maneuverability has led to the development of complex designs, use of advanced materials and integration of control systems in aerial systems. All these factors cause the system to exhibit diverse response behavior which cannot be captured by linear models in general. The need to analyze and predict the behavior of such systems for design optimization and control design resulted in the use of nonlinear models to represent such systems mathematically. Flutter is a catastrophic phenomenon where the system absorbs energy from the surrounding air and leads to oscillations with increasing amplitudes. Presence of nonlinearities results in limited amplitude flutter or limit cycle oscillations (LCO). The objective of this dissertation is to analyze the effects of stiffness nonlinearities on the response of aeroelastic systems and development of nonlinear controllers to suppress LCOs. A two degree of freedom airfoil section model with quasi- steady aerodynamics is used for this purpose. Three different approaches are discussed and developed. The first approach focuses on development of a feedback linearization controller highlighting the advantages of using modal coordinates in the control design process. The developed controller is optimal for small disturbances and guarantees stability of response for large disturbances. The second and third approaches develop analysis tools to capture response behavior of aeroelastic systems. Using the developed framework analytical estimates of LCO amplitude, LCO commencement velocity and their sensitivities to control parameters are derived. These are used to solve an optimization problem to generate strictly nonlinear feedback control laws to control the LCO amplitudes and eliminate the subcritical LCOs. Attribution Mayuresh J. Patil is co-author for the manuscripts of Chapters 3, 4 and 5. His contributions include strategic advice and review of the articles for technical accuracy, completeness and grammatical correctness. iv Dedicated to my parents, sister and brother Acknowledgements First and foremost, I would like to express my appreciation and gratitude to my advisor, Dr. Mayuresh Patil for all the support and guidance and being a great mentor for me. I want to thank you for encouraging me with research and helping me improve as a researcher. Thanks for guiding me in right direction and meeting with me at even short notices to answer my queries patiently however simple they were. I would also like to thank Dr. Craig Woolsey, Dr. Cornel Sultan and Dr. Michael Philen for serving on my advisory committee. The discussions, comments and brilliant ideas have helped me a lot during the period of my PhD work. I am very thankful to Dr. Mazen Farhood for teaching me a lot about control design. I extend my gratitude to my friends and members of the Aeroelasticity and Flight Dynamics group, Mandar, Eddie, Ryan, Rob, Ben, Patrick, Casey, Chris and Aaron for all the group meetings which taught me a lot about structures. I also want to thank my friends from Nonlinear Systems Lab, Dave, Kyle, Ony, Artur and Ankit for the discussions in Femoyer and NSL and some memorable time. My stay at Blacksburg wouldn’t have been possible without my friends at Blacksburg and I would like to thank Ashok, Gaurav, Riddhika, Sreyoshi, Tamoghna, Vireshwar, Ishan and Dhiraj for all their support and memorable moments during this time. I also want to thank my friends from IIT Kharagpur, Akash, Tanmay, Vimal, Priyesh and Shashank for being just a phone call away and for the few unforgettable trips during this time. vi Finally, I would like to thank my parents, Smt. Geeta Shukla and Shri Ramesh Chandra Shukla, my sister Neetu and my brother Sudhanshu for their love, support and care. This thesis wouldn’t have been possible without your faith in me. vii Contents Contents viii List of Figures xii List of Tables xvii 1 Introduction 1 1.1 AimsandObjectives ............................... 2 1.2 Outline....................................... 4 Bibliography ...................................... 6 2 Literature Review 7 2.1 Aeroelasticity ................................... 7 2.2 Effects of nonlinearities on aeroelastic systems . .... 9 2.3 Predicting response characteristics of aeroelastic systems . ......... 13 2.3.1 Harmonic balance method . 14 2.3.2 Nonlinear normal modes . 15 viii 2.4 Activecontrolofaeroelasticsystems . ... 16 Bibliography ...................................... 23 3 Feedback Linearization based Control of Aeroelastic System represented in Modal Coordinates 32 3.1 Abstract...................................... 32 3.2 Introduction.................................... 33 3.3 Equationsofmotion ............................... 36 3.4 Controlsynthesisformulation . 41 3.4.1 Nonlinear aeroelastic system . 42 3.4.2 Modal transformation to obtain linearly decoupled second order system 42 3.4.3 Feedback Linearization . 46 3.4.4 Stability of the zero dynamics . 50 3.5 Results....................................... 52 3.5.1 Open loop system behavior . 53 3.5.2 Closed loop response - Linear control . 53 3.5.3 Comparison of controller performance . 57 3.6 Conclusion..................................... 63 Bibliography ...................................... 64 4 Controlling Limit Cycle Oscillations Amplitudes in Nonlinear Aeroelastic Systems 68 4.1 Abstract...................................... 68 ix 4.2 Introduction.................................... 69 4.3 Nonlinear Aeroelastic System . 71 4.4 SolutionApproach ................................ 73 4.5 NonlinearNormalModes............................. 75 4.6 Estimation of LCO amplitudes . 79 4.6.1 Harmonic Balance method . 80 4.6.2 Fundamental Harmonic balance for LCO Nonlinear normal mode . 81 4.7 Closedloopsystem ................................ 82 4.7.1 Sensitivity of LCO amplitude to control parameters . 84 4.8 OptimizationFramework............................. 86 4.8.1 Approximatecontrolcost. 86 4.8.2 Optimization Problem . 88 4.9 Results....................................... 89 4.9.1 NNMresponse .............................. 90 4.9.2 Nonlinear Aeroelastic system response . 91 4.10Conclusions .................................... 94 Bibliography ...................................... 96 5 Nonlinear Control Design to Eliminate Subcritical LCOs in Aeroelastic Systems 100 5.1 Abstract...................................... 100 5.2 Introduction.................................... 101 x 5.3 SystemModel................................... 103 5.3.1 Linear Analysis . 105 5.3.2 OpenLoopbehavior ........................... 107 5.4 ProblemStatement ................................ 108 5.4.1 Solution
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]
  • 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]
  • Control Theory for Linear Systems
    Control theory for linear systems Harry L. Trentelman Research Institute of Mathematics and Computer Science University of Groningen P.O. Box 800, 9700 AV Groningen The Netherlands Tel. +31-50-3633998 Fax. +31-50-3633976 E-mail. [email protected] Anton A. Stoorvogel Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472378 Fax. +31-40-2442489 E-mail. [email protected] Malo Hautus Dept. of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. Box 513, 5600 MB Eindhoven The Netherlands Tel. +31-40-2472628 Fax. +31-40-2442489 E-mail. [email protected] May 15, 2002 2 Preface This book originates from several editions of lecture notes that were used as teach- ing material for the course ‘Control Theory for Linear Systems’, given within the framework of the national Dutch graduate school of systems and control, in the pe- riod from 1987 to 1999. The aim of this course is to provide an extensive treatment of the theory of feedback control design for linear, finite-dimensional, time-invariant state space systems with inputs and outputs. One of the important themes of control is the design of controllers that, while achieving an internally stable closed system, make the influence of certain exogenous disturbance inputs on given to-be-controlled output variables as small as possible. In- deed, in the appropriate sense this theme is covered by the classical linear quadratic regulator problem and the linear quadratic Gaussian problem, as well as, more re- cently, by the H2 and H∞ control problems.
    [Show full text]
  • Signals and Images Kronecker Delta Function
    10 cm z 5 cm d Bioengineering 280A" Principles of Biomedical Imaging" Assume z=d/2 " 10 cm 10 cm Fall Quarter 2013" X-Rays Lecture 2" z z 5 cm 10 cm d d Assume z=d/2 Assume z=d/2 TT Liu, BE280A, UCSD Fall 2013 TT Liu, BE280A, UCSD Fall 2013 Signals and Images Kronecker Delta Function Discrete-time/space signal /image: continuous valued function with a discrete time/space index, #1 for n = 0 δ[n] = $ denoted as s[n] for 1D, s[m,n] for 2D , etc. %0 otherwise n n δ[n] m € n Continuous-time/space signal /image: continuous 0 valued function with a continuous time/space index, δ[n-2] denoted as s(t) or s(x) for 1D, s(x,y) for 2D, etc. n 0 t y x TT Liu, BE280A, UCSD Fall 2013 x TT Liu, BE280A, UCSD Fall 2013 1 Kronecker Delta Function Discrete Signal Expansion ∞ # 1 for m = 0,n = 0 g[n] = g[k]δ[n − k] δ[m,n] = $ ∑ % 0 otherwise k=−∞ ∞ ∞ g[m,n] = ∑ ∑ g[k,l]δ[m − k,n − l] δ[m,n] δ[m-2,n] k=−∞ l=−∞ € δ[n] g[n] n 0 € -δ[n-1] δ[m,n-2] δ[m-2,n-2] n n 0 1.5δ[n-2] n 0 TT Liu, BE280A, UCSD Fall 2013 TT Liu, BE280A, UCSD Fall 2013 n 2D Signal Image Decomposition 1 0 0 1 c +d a 0 0 b c d 0 0 0 0 + = + a b 0 0 0 0 a b a 0 0 0 0 + +b = 1 0 0 1 c d 0 0 0 0 + c 0 0 d g[m,n] = aδ[m,n]+ bδ[m,n −1]+ cδ[m −1,n]+ dδ[m −1,n −1] 1 1 = ∑∑g[k,l]δ[m − k,n − l] k= 0 l= 0 TT Liu, BE280A, UCSD Fall 2013 TT Liu, BE280A, UCSD Fall 2013 € 2 Dirac Delta Function 1D Dirac Delta Function ∞ δ(x) = 0 when x ≠ 0 and δ(x)dx =1 ∫−∞ Can interpret the integral as a limit of the integral of an ordinary function that is shrinking in width and growing in height, while maintaining a Notation : constant area.
    [Show full text]
  • Feedback Systems: Notes on Linear Systems Theory
    Feedback Systems: Notes on Linear Systems Theory Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT – Fall 2019 October 31, 2019 These notes are a supplement for the second edition of Feedback Systems by Astr¨om˚ and Murray (referred to as FBS2e), focused on providing some additional mathematical background and theory for the study of linear systems. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. You are free to copy and redistribute the material and to remix, transform, and build upon the material for any purpose. 2 Contents 1 Signals and Systems 5 1.1 Linear Spaces and Mappings . 5 1.2 Input/OutputDynamicalSystems . 8 1.3 Linear Systems and Transfer Functions . 12 1.4 SystemNorms ....................................... 13 1.5 Exercises .......................................... 15 2 Linear Input/Output Systems 19 2.1 MatrixExponential..................................... 19 2.2 Convolution Equation . 20 2.3 LinearSystemSubspaces ................................. 21 2.4 Input/outputstability ................................... 24 2.5 Time-VaryingSystems ................................... 25 2.6 Exercises .......................................... 27 3 Reachability and Stabilization 31 3.1 ConceptsandDefinitions ................................. 31 3.2 Reachability for Linear State Space Systems . 32 3.3 SystemNorms ....................................... 36 3.4 Stabilization via Linear Feedback . 37 3.5 Exercises .........................................
    [Show full text]