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Why Time Response Characteristics Derived from Zero State Equation

  1. Mar 29, 2016 #1
    QUESTION:

    1) Why are Time Response Characteristic's Expressions derived from only from Zero State Equations?
    NOTE: Nise Control Systems Engineering 6ed uses step inputs to derive Time Response Characteristics for 1st
    and 2nd order ordinary differential equations.
    -What would the characteristic expressions be for arbitrary inputs?

    2) Why is it not customary to derive general analytic characteristic expressions from Zero Input Equations?
    -Would the characteristic expressions be different?


    3) Why is it not customary to derive general analytic general analytic characteristic expressions from the Total Solution?
    -Would the characteristic expressions be different due to arbitrary inputs and now the inclusion of non-zero
    initial conditions?

    4) Can Impulse Input Functions be considered as initial conditions?

    ASSUMPTIONS:
    -Time Response Characteristics. e.g.: settling time, rise time, percent overshoot, peak time
    -All Time Response Characteristics are derived using a step input
    -Zero State Equation is the solution to a differential equation with zero initial conditions and non-zero inputs
    -Zero Input Equation is the solution to a differential equation with zero inputs and non-zero initial conditions


    (SYSTEM) STABILITY & BOUNDED INPUTS:
    -bounded inputs/forcing functions (i.e. no unbounded inputs, but can have harmonic, periodic, and constant input functions)
    -aperiodic inputs such as ramp, parabola, or impulses are not considered, except of course the constant (step)
    -system is stable (i.e. naturally decays to zero for infinite time)
    -Not considering marginally stable systems: oscillatory nor constant. Therefore our inputs when operating at the system's natural frequency will not cause an unbounded total solution.



    SOURCES:
    -Nise Control Systems Engineering 6ed


    MY ANSWERS:
    1) Yes. I think the typical characteristic expressions would be different for arbitrary inputs.
    If inputs were a constant like a step then I think that it would result in the typical characteristic expressions.
    2) Yes. I think the typical characteristic expressions would be different for non-zero initial conditions and zero inputs.
    3) Yes.
     
    Last edited: Mar 29, 2016
  2. jcsd
  3. Apr 3, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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