DOCSLIB.ORG

Automated Compact Dynamical Modeling: an Enabling Tool for Analog Designers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Automated Compact Dynamical Modeling: an Enabling Tool for Analog Designers Automated compact dynamical modeling: An enabling tool for analog designers The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Bond, Bradley N., and Luca Daniel. “Automated Compact Dynamical Modeling.” ACM Press, 2010. 415. Web. 26 June 2012. © 2010 Association for Computing Machinery (ACM) As Published http://dx.doi.org/ 10.1145/1837274.1837379 Publisher Association for Computing Machinery (ACM) Version Final published version Citable link http://hdl.handle.net/1721.1/71214 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. 26.1 Automated Compact Dynamical Modeling: An Enabling Tool for Analog Designers Bradley N. Bond Luca Daniel Massachusetts Institute of Technology Massachusetts Institute of Technology [email protected] [email protected] ABSTRACT semidefinite matrices. For linear systems with such specific prop- In this paper we summarize recent developments in compact dy- erties, a body of methods exist (e.g. [1, 2]) that can generate au- namical modeling for both linear and nonlinear systems arising tomatically compact stable and passive dynamical models suitable in analog applications. These techniques include methods based for time domain circuit/gate level simulations. On the contrary, on the projection framework, rational fitting of frequency response when working with analog applications, one needs to handle non- samples, and nonlinear system identification from time domain data. linear circuit blocks or microelectromechanical (MEMS) devices. By combining traditional projection and fitting methods with re- Even when considering only the linear part of the analog systems cently developed convex optimization techniques, it is possible to (e.g. RF inductors, passive power combiners, transmission lines) obtain guaranteed stable and passive parameterized models that are the constraint of “constant semidefinite matrices” is always inevitably usable in time domain simulators and may serve as a valuable tool violated. This is because the electromagnetic field solvers used to for analog designers in both top-down and bottom-up design flows. analyze analog structures inevitably introduce numerical discretiza- tion errors at high frequencies, and because they are forced by the large size of the problems to use advanced formulations and ap- Categories and Subject Descriptors: J.6 [Computer Aided En- proximation techniques such as substrate Green functions and fast gineering]: Computer-aided design (CAD), G.1.6 [Optimization]: matrix-vector products. In many cases no internal description of Convex programming, I.6.0 [Simulation and Modeling]: General the systems is even available, and the only information provided by General Terms: Algorithms the field solvers, or by measured data, is a set of frequency response Keywords: Model reduction, Analog design, Semidefinite pro- values (if the system is linear), or a set of time domain input-output gramming, Parameterized modeling, Compact modeling signals (if the system is nonlinear). Another important difference between digital and analog sys- 1. INTRODUCTION tems is that in digital applications higher level system simulators (e.g. timing analyzers) or top-down design methodologies do not As feature sizes keep decreasing and operating frequencies keep require dynamical state-space models for each gate, but rather just increasing, the performance of many complex analog components a small set of simple equations or tabulated values relating per- and systems is becoming crucially dependent on what the design- formance (or behavioral) parameters. However, in order to enable ers identify as “non-idealities” or “second order effects”. These are system level simulation and fast tradeoff explorations for analog phenomena such as non-linearities, process variations, skin effects, applications, leading to robust designs insensitive to process varia- proximity effects, and substrate and magnetic coupling. Analog tions and to all high frequency second order effects, it is essential to designers would therefore greatly benefit from tools that generate have the ability to instantiate instantaneously parameterized com- automatically compact dynamical models which capture quantita- pact dynamical state-space models of circuit blocks. tively all such second order effects. Digital designers have indeed To better highlight the differences between digital and analog been already benefitting for decades from model order reduction applications, we would like to emphasize that the classical name, tools to speed-up circuit simulations within the context of a bottom- “model order reduction”, used in the digital community, is com- up library characterization step for timing analysis. However, sev- pletely inadequate to describe the variety of techniques that are be- eral significant differences between analog and digital applications ing developed to handle analog systems. We propose instead the limit severely the usability of the existing model order reduction name “automated compact dynamical modeling”. Here the term techniques. dynamical indicates state space models, emphasizing the fact that The first important difference is that, within the digital applica- for system level analog simulations it is not sufficient to generate tions, the systems to be “reduced” are mainly, if not exclusively, simple static behavioral models relating performance parameters RLC interconnect networks generated by parasitic extractors (e.g. of a circuit block as in digital applications. The term compact as resistance, capacitance and inductance extractors). Such systems opposed to “reduced order”, emphasizes the fact that model com- are linear, and are always described by constant, structured, and plexity is not strictly tied to the model order (i.e. the number of equations) as is the case in the digital interconnect linear system world. The complexity of the generated system should instead be measured in terms of how efficiently the generated model can be employed in a time domain simulator. When working with analog nonlinear systems one quickly realizes that small model orders do Permission to make digital or hard copies of all or part of this work for not necessarily imply fast simulations. Finally, although in some personal or classroom use is granted without fee provided that copies are cases one might want to “reduce” a large model already given in not made or distributed for profit or commercial advantage and that dynamical state space form, in all the other cases typical of the copies bear this notice and the full citation on the first page. To copy analog world, one is only given frequency domain data, or input- otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. DAC'10, June 13-18, 2010, Anaheim, California, USA Copyright 2010 ACM 978-1-4503-0002-5 /10/06...$10.00 415 26.1 output time domain data. Therefore a compact dynamical model For weakly nonlinear systems it is common to use a polynomial might need to be “‘identified” or “fitted” as opposed to “reduced”. expansion for the approximation. For example, for a simplified As a matter of fact, in order to address the many new challenging nonlinear system x˙ = f(x)+Bu, the vector field f(x) is approx- analog constraints, during the last few years the automated compact imated as dynamical modeling community has been forced to move beyond ˜ f(x) ≈ f(x)=A0 + A1x + A2(x ⊗ x)+.... traditional projection-based model order reduction approaches, em- (5) bracing for instance recent developments in convex optimization. k th where Ak is an N × N matrix containing the k derivatives of In some cases, some of the more traditional projection appraoches f(x). The new vector field f˜(x) can be easily projected given the or unconstrained fitting techniques have been adapted to the analog Kronecker product property (V xˆ) ⊗ (V xˆ)=(V ⊗ V )(ˆx ⊗ xˆ), world by simply adding a perturbation step that enforces proper- resulting in the reduced vector field ties such as stability and passivity as a post-processing operation. ˆ T ˜ ˆ ˆ ˆ In other cases, completely new techniques have been, and are still f(ˆx)=U f(V xˆ)=A0 + A1xˆ + A2(ˆx ⊗ xˆ)+... currently being invented. ˆ T ˆ T ˆ T The remainder of this paper is organized as follows. Section 2 where A0 = U A0, A1 = U A1V and A2 = U A(V ⊗ V ). presents projection methods for stable and passive modeling of Analysis of polynomial reduced models can be further simplified linear and nonlinear systems described in state-space form. Sec- using ideas from Volterra theory, as was done in [4, 5]. tion 3 surveys methods for compact modeling of linear systems An alternative approach, better suited for highly nonlinear sys- using rational approximation techniques for transfer matrix fitting, tems, consists of interpolating between a collection of local models combined with optimization-based stability and passivity enforce- ˜ ˜ ment. Section 4 summarizes recent developments in stable dynam- f(x) ≈ f(x)= wi(x)fi(x), (6) ical model fitting techniques using optimization to enforce stability i in nonlinear dynamical systems. Lastly, Section 5 discusses what where w(x) are weighting functions. The local models are chosen further steps need to be taken, both by model reduction researchers ˜ so that each individual model fi(x) can be reduced using projection and by the rest of the EDA community, to help compact dynamical modeling become a mainstream tool supporting analog design. ˆ T ˜ T ˜ f(ˆx)=U f(V xˆ)= wi(ˆx)U fi(V xˆ), (7) i T ˜ ˜ 2. PROJECTION-BASED METHODS such that U fi(V xˆ) has low complexity. For instance, fi could be ˜ polynomial functions, or even linear fi(x)=Aix + ki,inwhich T ˜ T T 2.1 The Traditional Projection Framework case U fi(V xˆ)=U AiV xˆ + U ki. The key idea behind the Most of the traditional model order reduction techniques can be Trajectory PieceWise Linear (TPWL) method [6] and related tech- interpreted within a projection framework.
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]
  • 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]