There are two ways of approaching the problem of obtaining a
mathematical representation or **model** of a chemical process,
or indeed anything.

- Create a
**fundamental**or**mechanistic**model based on knowledge of the physics and chemistry of the system to be modelled. This can be quite a hard thing to do, indeed nearly all of a chemical engineering degree course might be regarded as being about the creation of such models! The advantage of such a model is that it is basically `right' (provided of course the model builder's knowledge of physics and chemistry is right and is applied correctly). Such a model should be robust in that it can be applied again under conditions of operation different from those for which it was first constructed. If the process being modelled is modified, then analogous modifications to the model will enable it to continue to be used.We will describe briefly some rules for constructing this type of model which help to ensure that if the modeller's understanding of the problem is correct then a correct model will be obtained.

- Choose a mathematical form which is convenient (e.g. it is simple or
easy to manipulate) and which represents fairly well the observed
behaviour of
the system being modelled. Fit numerical parameters to the mathematical form.
This is a so-called `black box' or `input-output' model, which seeks only to reproduce the behaviour of the system's output in response to changes in its setpoint or inputs. The mathematical form chosen may bear no relation to the form of the equations which truly describe the system. As a result, such models must be used with the greatest care under conditions in the least bit different from those at which the original parameters were determined.

The advantage of such 'arbitrary' models is that they can be developed with little or no knowledge of the system to be represented, and hence complicated systems can be modelled quickly.

Input-output models form the basis of most classical process control theory. They are usually subdivided according to whether they have one or more than one input and/or output. We will consider initially only single input, single output (SISO) models, although some ideas associated with multiple input-output models will be touched on elsewhere in the course.

The basic SISO model can be thought of as relating an output y to an input u. In general both of these quanties will change with time, the model must represent how y responds to changes in its input or inputs.

Suppose an input u is given a **step change** at some time, as shown in the
figure.

In the above equation
is called the **Gain** of the process or
model.

Models of such systems can be assembled as networks of the elements as shown below.

Analytical and numerical techniques are available to work with models constructed in this way.

Classical control theory constructs all its models from sets of linear ordinary differential equations. (The instantaneous response is the limiting case of the the o.d.e. where is zero, and the plug flow delay, like the plug flow reactor, is the limit of an infinite number of first order lags.)

There is no good physical reason why a real process should be well represented by such a set of equations, except that in the limit of infinitesimally small changes, all nonlinear equations approximate to linear ones.

However, the theoretical advantage of linear representation is
twofold. Firstly, the whole system may be represented by o.d.e.s,
whereas if there were any nonlinear algebraic equations a mixed
set of differential-algebraic equations would be required.
Further, a system of linear differential equations always has
an analytical solution, but more particularly, is amenable to
various other types of analysis which cannot be performed on nonlinear
equations. The *tuning* methods for controllers described later
make use of this type of analysis to obtain generalised equations
for suitable controller settings in terms of parameters of a process
model written in terms of the above three types of behaviour. This is
not possible for nonlinear systems.

It should be stressed that if we wish to simulate the behaviour of a process, which requires only the solution of the relevant equations, and not their analysis, then there is no particular point in approximating it with this type of simplified approximate model. A `real' model should be constructed, as discussed later, and solved.

Let us look again at the differential equation which describes first order behaviour.

is the value of*yo*at*y*= 0*t*- is the size of the step change in
at*u*= 0*t*

Note that a graph of this equation gives the response curve shown above under the section on the lag response.

The first thing to consider is **What is the Change in y**

is known as the gain. It tells us how much
the output variable will change per unit change in the input variable.
A large gain implies a large change in ** y** for a given change in

To calculate its value we have to consider the system
going from one steady state value to another. Thus we can see what
effect a change in ** u** has on the value of

After the system has settled down following the step disturbance

- The first stage is to draw the initial slope
- Then the final steady state value is drawn
- The time at which these two lines intercept is the value of the time constant

Note that this is also the time taken for the output value to travel 63% of the distance to its new value.

- y(t) reaches 63.2% of its final value in one time constant.
- The smaller the time constant the steeper (quicker) the response.
- After 3 to 4 time constants the system is essentially at its new steady state.

Finally, how does the response change when
and
are altered but the
change in ** u** stays the same?

The diagram below shows that changing alters the slope of the initial slope and changing alters the final steady state.