Process Identification

What is Process Identification ?

Construction of a process model empirically, strictly from
process data

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

__Advantages to Step Response Identification__

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

__Disadvantages to Step Response Identification__

- 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

__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, t_{1},
t_{2}, 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, t_{1},
t_{2}, 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 n^{th} moment is given by

It follows that

K = m_{0} and t = m_{1}/m_{0}

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, m_{j
}

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 n^{th} 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