By K.H. Ng

First of all, the open problem is what is a model? Specifically, what is
a model in control system sense? The word model is generally quite wide
and has many uses. A fashion model is for displaying certain cloths
design, a market model is used to predict the profit and deficit, while a
business model is the strategic planning in an organization.

Model in control system sense is more to a mathematical model. An
extremely simple model is y = 2 * x. Give x to a function and it will
return the double of the original value. A more complicated model is the
Fourier series, where given a variable in function of time, it will
return the amplitudes of the DC gain, sine harmonics and cosine
harmonics.

While in control system, dynamic systems are more of interest where the
ordinary differential equation comes in the game. This is when the
variables and the derivatives comes to form a balanced equation for
example the Newton's Second Law of Motion or the Kirchoff Voltage Law.

A dynamic model gives the output when excited with certain inputs. In
other words, a model gives the output description in function of time
when given a certain inputs, also in function of time. For example, when
a motor is given an voltage for one second, what is the expected angle
it will turn?

There are various method to analyze a model. Or put differently, there
are many methods to work on a certain model. One might go for the most
basic time domain method, which is the convolution of linear time
invariant system. Another simpler method is to apply La Place transform
to a signal and work from there. The most recent and popular approach is
the state space approach.

Now comes to the main part of the article, why it is important? Although
one might not use a model in controlling certain plant, it is important
in the controller design process. For example, when someone wants to
control a motor, they do not just buy a motor and test it. They start
off with the model. And also, plants might not be cheap to start with.

Another example is water level control in a tank. One do not buy a tank
and experiment on it. They would have to come up with a model of the
tank, the pumps, the pipes, etc. After deriving the model, one can
obtain the water level in function of time given the opening of a water
flow valve in function of time.

To have a model is important to validate a controller design. Let's give
an example of PID controller. To determine the effectiveness of the
controller in controlling water level, it is important to have the model
at hand for simulation purposes.

A model can be linear or non linear. But most plants in the real world
is a non linear one. Even Ohm's Law is not linear if the voltage
variation is too large. Also given an example in a motor. Motors have a
certain dead zone where small voltage will not move the motor. The usual
cause is friction. Motors also have saturation where the current cannot
gets too high or it will burn out the motor.

Therefore, it is a challenge to come up with the nonlinear model.
Modeling a non linear plant is never an easy task. It takes experience
and a lot of computation to come up with the perfect model. There are
also many method emerging in modeling. Some are fuzzy modeling, neural
nets modeling, statistical modeling, etc.

In the end of the day, model is just a part of control system design and
analysis. But understanding the plant to control is very important for
understanding of the whole control system.

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