Creating and Manipulating Models

Specifying Input/Output Delays

Using the ioDelay property, you can specify frequency-domain models with independent delays in each entry of the transfer function. In continuous time, such models have a transfer function of the form

where the's are rational functions of , and is the time delay between input and output . See Specifying Delays in Discrete-Time Models for details on the discrete-time counterpart. We collectively refer to the scalars as the I/O delays.

The syntax to create above is

H = tf(num,den,'ioDelay',Tau)

or

H = zpk(z,p,k,'ioDelay',Tau)

where

• num, den (respectively, z, p, k) specify the rational part of the transfer function
• Tau is the matrix of time delays for each I/O pair. That is, Tau(i,j) specifies the I/O delay in seconds. Note that Tau and should have the same row and column dimensions.

You can also use the ioDelay property in conjunction with state-space models, as in

sys = ss(A,B,C,D,'ioDelay',Tau)

This creates the LTI model with the following transfer function.

Here is the entry of

Note    State-space models with I/O delays have only a frequency-domain interpretation. They cannot, in general, be described by state-space equations with delayed inputs and outputs.

Distillation Column Example

This example is adapted from [2] and illustrates the use of I/O delays in process modeling. The process of interest is the distillation column depicted by the figure below. This column is used to separate a mix of methanol and water (the feed) into bottom products (mostly water) and a methanol-saturated distillate.

Figure 1-3: Distillation Column

Schematically, the distillation process functions as follows:

• Steam flows into the reboiler and vaporizes the bottom liquid. This vapor is reinjected into the column and mixes with the feed
• Methanol, being more volatile than water, tends to concentrate in the vapor moving upward. Meanwhile, water tends to flow downward and accumulate as the bottom liquid
• The vapor exiting at the top of the column is condensed by a flow of cooling water. Part of this condensed vapor is extracted as the distillate, and the rest of the condensate (the reflux) is sent back to the column.
• Part of the bottom liquid is collected from the reboiler as bottom products (waste).

The regulated output variables are:

• Percentage of methanol in the distillate
• Percentage of methanol in the bottom products.

The goal is to maximize by adjusting the reflux flow rate and the steam flow rate in the reboiler.

To obtain a linearized model around the steady-state operating conditions, the transient responses to pulses in steam and reflux flow are fitted by first-order plus delay models. The resulting transfer function model is

Note the different time delays for each input/output pair.

You can specify this MIMO transfer function by typing

H = tf({12.8 -18.9;6.6 -19.4},...
        {[16.7 1] [21 1];[10.9 1] [14.4 1]},...
        'iodelay',[1 3;7 3],...
        'inputname',{'R' , 'S'},...
        'outputname',{'Xd' , 'Xb'})

The resulting TF model is displayed as

Transfer function from input "R" to output...
                     12.8
 Xd:  exp(-1*s) * ----------
                  16.7 s + 1
 
                     6.6
 Xb:  exp(-7*s) * ----------
                  10.9 s + 1
 
Transfer function from input "S" to output...
                      -18.9
 Xd:  exp(-3*s) * --------
                  21 s + 1
 
                        -19.4
 Xb:  exp(-3*s) * ----------
                  14.4 s + 1

Supported Functionality

Specifying Delays on the Inputs or Outputs