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


  • 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


    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


    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


    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


    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


    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


    Normalized Transfer Function


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