Chapter 15.1 Introduction
If you know all  the “A BIT OF MATHS” subjects then go to the chapter 19.
The transfer function G(s) was more often than not used word in this course. But I haven’ t defined it. My main goal was to associate the input x(t) (step, ramp or dirac type) with the output y(t) and with the G(s) parameters as K, T, Ti
There is the time for some mathematical principles now.
Do you connote G1(s), G2(s) and G3(s) behaviours with their parameters? What does cause one small single letter s in the transfer function denominator or nominator?

15-1a
Fig. 15-1
When the answer is no–> repeat please these chapters.

Chapter 15.2 Function derivative
Function derivative<==>Function speed
Study the Fig.9-7 chapter 9 once more. You will know why.
see Fig.9-12  chapter 9.  The x(t)=t^2  is transformed in its derivative =2t here.

15-2a
Fig. 15-2
Every 1 year Technical University student knows this formula.

Chapter 15.3. Second derivetive “derivative of the derivative”
Derivative
is the function. So we can obtain derivative of the derivative i.e. second derivative.
15-3
Fig. 15-3
An example
15-4a
Fig. 15-4
We will check it by the double differentiating.
Cal desktop/PID/02_pochodna/01-rozniczkujacy_kwadrat_druga_pochodna.zcos
15-5a
Fig. 15-5
The input x(t)  is a quadratic function of the time t.  The derivative x'(t) (strictly first derivative) is after the first differentiating unit. The second derivative is after the second differentiating unit.
Click”Start”
15-6a
Fig. 15-6
It’s true!

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