Nothing Special   »   [go: up one dir, main page]
Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    Only $11.99/month after trial. Cancel anytime.

    Systems with Delays: Analysis, Control, and Computations
    Systems with Delays: Analysis, Control, and Computations
    Systems with Delays: Analysis, Control, and Computations
    Ebook201 pages1 hour

    Systems with Delays: Analysis, Control, and Computations

    By A. V. Kim and A. V. Ivanov

    ()

    Read preview

    About this ebook

    The main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays.  Referred to as “systems with delays” in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations.  Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems

    At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician.  there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems. 

    Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research. 

    LanguageEnglish
    PublisherWiley
    Release dateJul 23, 2015
    ISBN9781119117735
    Systems with Delays: Analysis, Control, and Computations

    Related to Systems with Delays

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • Article

      The Mathematics Of Contagion

      Apr 21, 2020

      The Mathematics Of Contagion
      4 min read

    • Article

      How Spooky Science Helps Us Peer Inside The Planets

      Dec 3, 2020

      An assistant professor of computational science at the EPFL research centre in Lausanne, Switzerland, involved in the current research on metallic hydrogen. Could you explain how the machine-learning techniques used in your research work? Why were th

      1 min read

    • Article

      Memristor Setup Could Make Computer Chips More Efficient

      Jul 31, 2018

      A new way of arranging advanced computer components called memristors on a chip could pave the way for their use in general computing. This could cut energy consumption by a factor of 100. Using memristors would improve performance in low power envir

      2 min read

    • Article

      Euclidean And Beyond

      Jul 28, 2020

      Euclidean And Beyond
      1 min read

    • Article

      #43 Enter the Modulation Matrix

      Dec 31, 2019

      In this article we’ll look at the often overlooked modulation matrix. This can be found in most comprehensive synthesisers, both analogue and digital, and allows you to control and automate specific parameters within the synth’s arsenal. I, like man

      5 min read

    • Article

      What Is Cfd

      Jun 21, 2022

      What Is Cfd
      3 min read

    • Article

      It’s About Time… Series

      Jun 1, 2021

      A time series T is a time-ordered sequence of numbers. Time series can come with a time stamp, or they may not contain any time-related information. The time series mentioned in this tutorial don’t contain any date or time information, but they’re st

      1 min read

    • Article

      Kilohearts Snap Heap

      Jan 24, 2024

      Snap Heap (AU, VST, VST3, AAX) from Kilohearts is a fantastic multi-effects host plugin. It normally retails for $29 but this month the extremely kind team at Kilohearts are gifting it to Computer Music readers, and what a great gift! When it comes t

      7 min read

    • Article

      How to Make Sense of Quantum Physics

      Mar 12, 2020

      Quantum mechanics isn’t rocket science. But it’s well on the way to take the place of rocket science as the go-to metaphor for unintelligible math. Quantum mechanics, you have certainly heard, is infamously difficult to understand. It defies intuitio

      11 min read

    • Article

      Electromechanical Keying System

      Nov 4, 2020

      Electromechanical key systems are now very widely accepted in the security landscape and used in a vast array of applications through our region. They provide a hybrid solution that combines many of the benefits of access control with the functionali

      3 min read

    • Article

      Business applications For Quantum computing

      May 1, 2022

      COMPUTERS DO ARITHMETIC. Underlying every amazing application of computers today is math, calculated using binary digits or ‘bits.’ The original computers of the early 1950s could perform about 465 multiplications per second — much faster than the ‘h

      11 min read

    • Article

      BLUE CAT AUDIO Late Replies

      Oct 17, 2019

      It’s very easy to take for granted some of the tools we use as producers and sound engineers. Nowhere is this more true than with delay effects. Rarely seen as the most exciting or glamorous of effects, the delay nonetheless lies at the heart of a nu

      4 min read

    • Article

      The Cm Guide To THE GRID

      Oct 2, 2019

      Bitwig Studio is a DAW that’s always been influenced by modular systems. Version 1 brought nested effects and open-ended controller setup; while v2 took things further with its excellent modulation system, whereby any device could be adorned with ext

      6 min read

    • Article

      IK MULTIMEDIA ARC System 3

      Jul 16, 2020

      A quick question for you: what’s the single most important aspect of your studio, the thing that makes everything else possible? That’s right. Monitoring. Your studio monitoring system is the window through which you hear everything. It’s what all yo

      4 min read

    • Article

      How To Make Sense From And With AI ?

      Sep 25, 2021

      How To Make Sense From And With AI ?
      4 min read

    • Article

      How To Assess An Energy Device

      Jun 4, 2023

      How To Assess An Energy Device
      1 min read

    • Article

      The Cm Guide To THE GRID Part 2

      Nov 1, 2019

      In last month’s , we introduced the basic concepts behind The Grid, Bitwig Studio 3’s open-ended modular instrument and effect building environment. With a whole host of sound sources, filters, effects, modulation devices and helpful utilities on of

      6 min read

    • Article

      Zenith $79

      Mar 24, 2020

      Zenith $79
      4 min read

    Related categories

    Reviews for Systems with Delays

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Systems with Delays - A. V. Kim

      Preface

      At present there are elaborated effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). The progress in this direction results in wide application of these types of equations in practice. Another class of differential equations is represented by delay differential equations (DDE), also called systems with delays, time-delay systems, hereditary systems, functional differential equations.

      Delay differential equations are widely used for describing and mathematical modeling of various processes and systems in different applied problems [3, 5, 1, 27, 32, 33, 34, 40, 50, 62, 63, 183, 91, 107, 108, 111, 127, 183].

      Delay in dynamical systems can have several causes, for example: technological lag, signal transmission and information delay, incubational period (infection diseases), time of mixing reactants (chemical kinetics), time of spreading drugs in a body (pharmaceutical kinetics), latent period (population dynamics), etc.

      Though at present different theoretical aspects of time-delay theory (see, for example, [3, 1, 27, 32, 34, 50, 62, 63, 67, 72, 73, 183, 91, 107, 111, 127] and references therein) are developed with almost the same completeness as the corresponding parts of ODE theory, practical implementation of many methods is very difficult because of infinite dimensional nature of systems with delays.

      Also it is necessary to note that, unlike ODE, even for linear DDE there are no methods of finding solutions in explicit forms, and the absence of generally available general-purpose software packages for simulating such systems cause a big obstacle for analysis and application of time-delay systems.

      In this book we try to fill up this gap.

      The main aim of the book is to present new constructive methods of DDE theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays¹.

      The main outstanding features of this book are the following:

      1. on the basis of i-smooth analysis we give a complete description of the structure and properties of quadratic Lyapunov-Krasovskii functionals²;

      2. we describe a new control design technique for systems with delays, based on an explicit form of solutions of linear quadratic control problems;

      3. we present new numerical algorithms for simulating DDE.

      Acknowledgements

      N.N.Krasovskii, A. B. Lozhnikov, Yu.F.Dolgii, A. I. Korotkii, O. V. Onegova, M. V. Zyryanov, Young Soo Moon, Soo Hee Han.

      Research was supported by the Russian Foundation for Basic Research (projects 08-01-00141, 14-01-00065, 14-01-00477, 13-01-00110), the program Fundamental Sciences for Medicine of the Presidium of the Russian Academy of Sciences, the Ural-Siberia interdisciplinary project.

      ¹ The present volume is devoted to linear time-delay system theory. We plan to prepare a special volume devoted to analysis of nonlinear systems with delays.

      ² Including properties of positiveness, and constructive presentation of the total derivative of functionals with respect to time-delay systems.

      Chapter 1

      Linear time-delay systems

      1.1 Introduction

      1.1.1 Linear systems with delays

      In this book we consider methods of analysis, control and computer simulation of linear systems with delays

      (1.1)

      equation

      where A(t), Aτ(t) are n × n matrices with piece-wise continuous elements, G(t, s) is n × n matrix with piece-wise continuous elements on R × [−τ, 0], u is a given n-dimensional vector-function, τ(t) : R → [−τ, 0] is a continuous function, τ is a positive constant.

      Much attention will be paid to the special class of linear time-invariant systems

      (1.2)

      equation

      where A, Aτ are n × n constant matrices, G(s) is n × n matrix with piece-wise continuous elements on [−τ, 0], τ is a positive constant¹.

      Usually we will consider u as the vector of control parameters. There are two possible variants:

      1) u = u(t) is the function of time t;

      2) u depend on the current and previous state of the system, for example,

      (1.3)

      equation

      Consider some models of control systems with delays.

      1.1.2 Wind tunnel model

      A linearized model of the high-speed closed-air unit wind tunnel is [134, 135]

      (1.4)

      equation

      with

      .

      The state variable x1, x2, x3 represent deviations from a chosen operating point (equilibrium point) of the following quantities: x1 = Mach number, x2 = actuator position guide vane angle in a driving fan, x3 = actuator rate. The delay represents the time of the transport between the fan and the test section.

      The system can be written in matrix form

      (1.5)

      equation

      where

      equation

      1.1.3 Combustion stability in liquid propellant rocket motors

      A linearized version of the feed system and combustion chamber equations, assuming nonsteady flow, is given by²

      (1.6)

      equation

      Here

      ϕ(t) = fractional variation of pressure in the combustion chamber,

      t is the unit of time normalized with gas residence time,

      θg, in steady operation,

      = value of time lag in steady operation,

      = pressure in combustion chamber in steady operation,

      = const for some number γ,

      μ(t) = fractional variation of injection and burning rate,

      ψ(t) = relative variation of p1,

      p1 = instantaneous pressure at that place in the feeding line where the capacitance representing the elasticity is located,

      ξ = fractional length for the constant pressure supply,

      J = inertial parameter of the line,

      P = pressure drop parameter,

      μ1(t) = fractional variation of instantaneous mass flow upstream of the capacitance,

      Δp = injector pressure drop in steady operation,

      p0 = regulated gas pressure for constant pressure supply,

      E = elasticity parameter of the line.

      For our purpose we have taken

      equation

      to be a control variable and guided by [36] have adopted the following representative numerical values:

      γ = 0.8, ξ = 0.5, δ = 1, J = 2, P = 1, E = 1.

      This gives, for x(t) = (φ(t), μ1(t), μ(t), ψ(t))′,

      (1.7)

      equation

      where

      equation

      The system (1.7) has two roots with positive real part: λ1,2 = 0.11255 ± 1.52015 i.

      1.2 Conditional representation of differential equations

      1.2.1 Conditional representation of ODE and PDE

      Let us remember that for ODE

      (1.8) equation

      the conditional representation is

      (1.9) equation

      i.e. the argument t is not pointed out in state variable x(t).

      The conditional representation of the partial differential equation

      equation

      is

      (1.10) equation

      i.e. the arguments t and x are not pointed out in the function y(t, x).

      Thus in order to obtain the conditional representation of an ODE it is necessary to make in this equation the following substitutions

      (1.11) equation

      Example 1.1. The linear control ODE

      equation

      can be written in the conditional form as

      equation

      note, we omit variable t only in the state variable x(t) but not in the coefficients a(t) and u(t). One can omit t also in the control variable u(t), in this case the conditional representation will be

      equation

      Remark 1.1 It is necessary to emphasize, conditional representation is very useful for describing local properties of differential equations, for application of geometrical language and methods.

      1.2.2 Conditional representation of DDE

      Let us introduce the conditional representation of systems with delays (1.1). First of all it necessary to note, differential equations with time lags differ from ODE by presence (involving) point x(t − τ) and/or segment x(t + s),

      Enjoying the preview?
      Page 1 of 1