Process Identification

What is Process Identification ?

Construction of a process model empirically, strictly from process data

  • Identification
  • On-Line
  • Off-Line
  • Models
  • Continuous
  • Discrete
  • Principles of Empirical Modeling

    Data Files

    Inputs

  • Step
  • Impulse
  • Pulse
  • Sinusoids
  • White Noise
  • Pseudo-random binary sequences

  • Candidate Process Models

    First order plus time delay


    Second order plus time delay


    Single zero, two poles plus time delay


    Step Response Identification

    Model Formulation

    Parameter Estimation

     

    Model Validation

    Advantages to Step Response Identification

     

    Disadvantages to Step Response Identification

    Theoretical Step-Response Expressions

    First-order-plus-time-delay

    Time response


    Parameters : K, t, a

    Steady State Gain, K

    The ultimate value of the response affords an estimate of the steady state gain


    Step Response Analysis

    Time constant , t, and time delay, a

    The response can be re-written as


    and represented in a linear form for simpler evaluation of the parameters


    Example

    Identification of an Industrial Distillation Column

    The temperature on tray #16 of the column is regulated by the underflow reflux t0 the tray below.

    Steady state values of Temperature and Reflux rate are

    T = 380 °F and R = 25,000 #/hr


    More Step-Response Stuff

    Second-order-plus-time-delay

    Time response


    Parameters : K, t1, t2, a


    Steady State Gain

    As earlier, the ultimate value of the response affords an estimate of the steady state gain



    Even More Step-Response Stuff

    (2,1)-system-plus time-delay


    Time response


    Parameters : K, t1, t2, x, a

    Here we have a much more complex system that is actually nonlinear with as many as 5 parameters to determine. This is where we learn to use the tools we discussed on non-linear estimation.

    Impulse Response Identification

    Comments

    The impulse response provides the direct identification of a process and does not require the estimation procedures described above.

    An Example

    Consider a first-order process


    whose temporal response is


    Note the following




    The ratio of the two integrals yields the time constant, t .

    Moment Analysis

    The first integral is the zeroth moment of the response function.

    The second integral is the first moment of the response function

    where the nth moment is given by


    It follows that

    K = m0 and t = m1/m0

     


    Experimental Evaluation of Moments

    To evaluate a zeroth moment


    By Simpson's Rule

     

    To evaluate a first moment

    By Simpson's Rule


    For this the estimate of the time constant is


    General Theory of Moment Analysis

    of Impulse Response

    The Laplace transform of the Impulse Response is the Transfer Function


    We can decompose the transform if we note that

    so that

    The result is


    and is related to the set of moments, mj


    The process gain is

    Normalized Responses

    We can define a normalized transfer function as


    The moments of the normalized response are


    which provide a Laurent's expansion for G*


    In simpler form, the Laurent's expansion is


    A general nth transfer function, G(s), is


    Its normalized cousin is

    Transfer Functions and Moments

    The two relations for G* lead to the following equation

     


    Cross-multiplying and equating tems of like order, we obtain




    Identification of Simple Model Forms

    Procedure

    Determine experimentally the impulse response of the system

    Calculate the Normalized Moments

    Postulate an approximate transfer function

    Estimate the parameters using the moments

    Compare the inverse transform with the experiment

    First Order Model


    The relation with moments is


    (2,1) Model


    The relations with the moments are

    More Impulse Modeling

    First-Order-plus-Time-Delay

    Normalized Transfer Function




    Then

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