Controller Design and System Modelling

Modelling a Process

Introduction

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.

Simple Black-Box Models

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.

Typical Responses

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

Observations of typical `processes', from aircraft to papermills, suggest that there are three main types of behaviour which may be seen in an output y.

Instantaneous response

The first typical response is called the instantaneous response. In this case y also responds in a step, but in general of different size to that in u (in any case y will normally have different dimensions to u) as shown below

The simplest mathematical relationship is of the form:

Classical control theory assumes that behaviour can be represented by linear equations like the above, and so this is the only type of equation required to represent this type of behaviour.

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

Lagging response

Here y starts to change the moment that y changes, but the full extent of the response `lags' behind the disturbance. After a while, y will have responded fully.

The simplest mathematical form which provides this behaviour is an ordinary differential equation with time t as the independent variable, having the form:

Here

is as before the gain, and

is called the Time Constant of the equation, system or model. Because it is described by a single first order o.d.e. this is called a First Order model, system, lag or response. The interpretation of these parameters is described below.

Delayed response

When u changes, no immediate change in y is observed. However after a time T, y responds completely to the change in u as in the instantaneous response case.

Mathematically this is represented by a difference equation:

This does not have simple analytical properties, but is easily understood by chemical engineers as corresponding to a plug flow or pipeline system with residence time T. It is also referred to as a time delay or pure time delay system.

Representing Complex Responses

Complex systems may be reasonably well approximated by combinations of the above three elements.

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.

Theoretical Response

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.

It is possible to solve this equation analytically to obtain the expression

Here

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

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

This equation can be now be used directly to calculate the new value of the output variable if the change in u, the gain and time constant are all known. Otherwise it is necessary to estimate values for the gain and time constant as shown below.

Anaysis of Reponse

It will be shown later in the section on tuning controllers that it is useful to be able to look at the open loop response of a process and try and estimate the values of the gain and time constant. Below are notes on how to do this and then you can try it for yourself in the exercises associated with this part of the module.

Estimating the Gain

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 u and hence leads to a quicker response.

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 y.

After the system has settled down following the step disturbance

Or, as shown in the graph below

From this we can see that it is a simple calculation to evaluate the gain of the process given the change in u and y.

Estimating the Time Constant

is the time constant for the process. This is related to the speed of response of the system. The diagram below shows a graphical method of evaluating its value.

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.

This is shown mathematically below

The following points should be noted about the time constant

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.

Changing the Gain and Time Constant

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.