**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?

**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.

**Fig. 15-2**

Every 1 year

**Technical University**student knows this formula.

**Chapter 15.3. Second derivative “derivative of the derivative”
Derivative **is the function. So we can obtain derivative of the derivative i.e. s

**econd derivative**.

**Fig**

**. 15-3**

An example

**Fig. 15-4**

We will check it by the

**double differentiating**.

Cal desktop/PID/02_pochodna/01-rozniczkujacy_kwadrat_druga_pochodna.zcos

**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”

**Fig. 15-6**

It’s true!