## Saturday, 16 August 2008

### Math As A System Modelling Tool

A huge system consists of smaller sub-systems with functions related to the system operation. Sometime the function of the individual sub-systems are known, but the function of the system as a whole is not clear. We just know that given a certain set of inputs, we will get another set of outputs.

But what are the constraints and limits to the system?

What is the strength or weakness of the system?

These are typical questions that system engineer need to know.
How then does he know?

It is through modelling that we can get the answers to the scope of the system operation, its weakness, strength and limits.

Many techniques address this modelling issue, but the simplest is the Math Model.

Here, mathematical function or expression describes the relationship between the input and output. A complex drawing of sub-systems can be simplified into a single block indicated by the Math Model.

Let an example illustrate the usefulness of Math Model.

A negative feedback audio amplifier consists of a main amplifier (A) in the forward path with a feedback circuit (X) that returns a portion of the output signal for control purpose. The main input signal is fed into a circuit that subtracts the feedback signal from the main input signal. (This is the detail operation of the system involving its sub-systems).

To simplify this model, math is used to summarise the overall function. Math is therefore a system modelling tool, in this instance, to describe relationship between the input and output.

Brief explanation on derivation of math model expression:

Using algebra, we can see that the direct input to the amplifier (A) is V1 - XV2, and the output is V2.

The ratio of the (output of A) / (input of A) is V2 / (V1-XV2) = A.

Re-arranging the above expression, we will get V2/V1 = A / (1 + AX) which is the system gain ratio of the complete negative feedback amplifier.

This final ratio is then the Math Model of diagram 1 and is shown in diagram 2.

Diagram 2

After getting this simplified version, system analysis can be carried out to test the extent of its operation and discover its limits. Without this modelling, the testing will be tedious and time consuming.

From the above example, you can see the usefulness of math and its application as a system modelling tool. From an abstract case, math understanding can be used to solve an real-life situation through this specific application.

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