Nonlinear System Theory
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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. -
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. -
Coping with the Bounds: a Neo-Clausewitzean Primer / Thomas J
About the CCRP The Command and Control Research Program (CCRP) has the mission of improving DoD’s understanding of the national security implications of the Information Age. Focusing upon improving both the state of the art and the state of the practice of command and control, the CCRP helps DoD take full advantage of the opportunities afforded by emerging technologies. The CCRP pursues a broad program of research and analysis in information superiority, information operations, command and control theory, and associated operational concepts that enable us to leverage shared awareness to improve the effectiveness and efficiency of assigned missions. An important aspect of the CCRP program is its ability to serve as a bridge between the operational, technical, analytical, and educational communities. The CCRP provides leadership for the command and control research community by: • articulating critical research issues; • working to strengthen command and control research infrastructure; • sponsoring a series of workshops and symposia; • serving as a clearing house for command and control related research funding; and • disseminating outreach initiatives that include the CCRP Publication Series. This is a continuation in the series of publications produced by the Center for Advanced Concepts and Technology (ACT), which was created as a “skunk works” with funding provided by the CCRP under the auspices of the Assistant Secretary of Defense (NII). This program has demonstrated the importance of having a research program focused on the national security implications of the Information Age. It develops the theoretical foundations to provide DoD with information superiority and highlights the importance of active outreach and dissemination initiatives designed to acquaint senior military personnel and civilians with these emerging issues. -
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. -
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. -
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. -
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 -
Control Theory
Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. -
Observation of Nonlinear Verticum-Type Systems Applied to Ecological Monitoring
2nd Reading June 4, 2012 13:53 WSPC S1793-5245 242-IJB 1250051 International Journal of Biomathematics Vol. 5, No. 6 (November 2012) 1250051 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S1793524512500519 OBSERVATION OF NONLINEAR VERTICUM-TYPE SYSTEMS APPLIED TO ECOLOGICAL MONITORING S. MOLNAR´ Institute of Mathematics and Informatics Szent Istv´an University P´ater K. u. 1., H-2103 Godollo, Hungary [email protected] M. GAMEZ´ ∗ and I. LOPEZ´ † Department of Statistics and Applied Mathematics University of Almer´ıa La Ca˜nada de San Urbano 04120 Almer´ıa, Spain ∗[email protected] †[email protected] Received 4 November 2011 Accepted 13 January 2012 Published 7 June 2012 In this paper the concept of a nonlinear verticum-type observation system is introduced. These systems are composed from several “subsystems” connected sequentially in a particular way: a part of the state variables of each “subsystem” also appears in the next “subsystem” as an “exogenous variable” which can also be interpreted as a con- trol generated by an “exosystem”. Therefore, these “subsystems” are not observation systems, but formally can be considered as control-observation systems. The problem of observability of such systems can be reduced to rank conditions on the “subsystems”. Indeed, under the condition of Lyapunov stability of an equilibrium of the “large”, verticum-type system, it is shown that the Kalman rank condition on the linearization of the “subsystems” implies the observability of the original, nonlinear verticum-type system. For an illustration of the above linearization result, a stage-structured fishery model with reserve area is considered. -
Nonlinear System Stability and Behavioral Analysis for Effective
S S symmetry Article Nonlinear System Stability and Behavioral Analysis for Effective Implementation of Artificial Lower Limb Susmita Das 1 , Dalia Nandi 2, Biswarup Neogi 3 and Biswajit Sarkar 4,* 1 Narula Institute of Technology, Kolkata 700109, India; [email protected] 2 Indian Institute of Information Technology Kalyani, Kalyani 741235, India; [email protected] 3 JIS College of Engineering., Kalyani 741235, India; [email protected] 4 Department of Industrial Engineering, Yonsei University, Seoul 03722, Korea * Correspondence: [email protected] or [email protected] Received: 8 September 2020; Accepted: 14 October 2020; Published: 19 October 2020 Abstract: System performance and efficiency depends on the stability criteria. The lower limb prosthetic model design requires some prerequisites such as hardware design functionality and compatibility of the building block materials. Effective implementation of mathematical model simulation symmetry towards the achievement of hardware design is the focus of the present work. Different postures of lower limb have been considered in this paper to be analyzed for artificial system design of lower limb movement. The generated polynomial equations of the sitting and standing positions of the normal limb are represented with overall system transfer function. The behavioral analysis of the lower limb model shows the nonlinear nature. The Euler-Lagrange method is utilized to describe the nonlinearity in the field of forward dynamics of the artificial system. The stability factor through phase portrait analysis is checked with respect to nonlinear system characteristics of the lower limb. The asymptotic stability has been achieved utilizing the most applicable Lyapunov method for nonlinear systems. -
Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems
UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for. -
Systems Theory
1 Systems Theory BRUCE D. FRIEDMAN AND KAREN NEUMAN ALLEN iopsychosocial assessment and the develop - nature of the clinical enterprise, others have chal - Bment of appropriate intervention strategies for lenged the suitability of systems theory as an orga - a particular client require consideration of the indi - nizing framework for clinical practice (Fook, Ryan, vidual in relation to a larger social context. To & Hawkins, 1997; Wakefield, 1996a, 1996b). accomplish this, we use principles and concepts The term system emerged from Émile Durkheim’s derived from systems theory. Systems theory is a early study of social systems (Robbins, Chatterjee, way of elaborating increasingly complex systems & Canda, 2006), as well as from the work of across a continuum that encompasses the person-in- Talcott Parsons. However, within social work, sys - environment (Anderson, Carter, & Lowe, 1999). tems thinking has been more heavily influenced by Systems theory also enables us to understand the the work of the biologist Ludwig von Bertalanffy components and dynamics of client systems in order and later adaptations by the social psychologist Uri to interpret problems and develop balanced inter - Bronfenbrenner, who examined human biological vention strategies, with the goal of enhancing the systems within an ecological environment. With “goodness of fit” between individuals and their its roots in von Bertalanffy’s systems theory and environments. Systems theory does not specify par - Bronfenbrenner’s ecological environment, the ticular theoretical frameworks for understanding ecosys tems perspective provides a framework that problems, and it does not direct the social worker to permits users to draw on theories from different dis - specific intervention strategies.