Essentials of Mechatronics

Chapter 6 - Essential Control Theory

Control theory is traditionally taught from the point of view of the frequency
response, with great emphasis on the manipulation of transfer functions.
Instead we will start with the state space approach, based on differential equations
that you can identify from the “real” system.

6.1   STATE VARIABLES

The relationship between state variables and initial conditions has already
been mentioned, but let us try to make the concept as clear as possible.

A cup of coffee has just been prepared. It is rather too hot at the moment,
at 80°C. If left for some hours, it would cool down to room temperature at
20°C, but just how fast would it cool, and when would it be at 60°C?

The cup remains full for now, so just one variable interests us: T, the temperature
of the coffee. It is a reasonable assumption that the rate of fall in
temperature is proportional to the temperature above ambient. So we see that

 

If we can determine the value of the constant k, perhaps by a simple experiment,
then the equation can be solved for any particular initial temperature—although
we’ll look at the form of the solution later.

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