 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

• Problem Definition
• Model Formulation

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

• Inspecting Step Response data
• Proposing candidate models
• Obtaining a theoretical response

Parameter Estimation

• Obtaining values for the model parameters

Model Validation

• Plotting the experimental response
• Plotting the theoretical response
• Comparing the responses

• Theoretical step response functions are easy to derive.
• Experiments are easy to implement

• Similar responses are possible from different models
• Steps may not be sufficiently small to avoid non-linear behavior. Theoretical Step-Response Expressions

First-order-plus-time-delay Time response Parameters : K, t, a

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

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

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   Go to next lecture 