## Preliminary Automatics Course

**Chapter 16. Integration
**

**Chapter 16.1 Introduction**

There are

**undefinite integrals**and

**definite integrals**

**Chapter 16.2 Undefinite integral F(t) of the function f(t)
**Undefinite integral

**F(t)**is often named an integral

**F(t)**.

**Fig. 16-1**

The derivative of the antiderivative function

**F(t)**is this same function

**f(t)**

and vice versa- the antiderivative of the function

**f**

**(t)**is a function

**F**

**(t)**.

**Fig. 16-2**

Integration and differentiation are the inverse operations.

**Chapter 16.3 Definite integral of the function x(t)**

**Chapter 16.3.1 Introduction
**The control theory has more a brush with the

**definite integral**than with the

**undefinite integral**. The latter enables an easy

**definite integral**calculation.

**Chapter 16.3.2 Definite integral of the function x(t) as a surface area under the function x(t)
**

**Fig. 16-3**

**D****efinite integral from t1 up to t2 **is a surface area **S** uder a function **x(t), **so it’s a particular number e.g. **S=27.13**. Something begins in time **t1=0 **in automatic control theory usually, so we assume that **x(t)****=0 **for** t<0**.

**Fig. 16-4
**This is

**Fig. 16-3**version when

**t1=0**and

**t2=t**. Please note that

**D**

**efinite integral**is a

**function y(t)**now, not a concrete number

**S**!

And most important now.

The

**definite integral=Surface Area**calculus method.

**Fig. 16-5**

We wil test this theory .

**Chapter 16.3.3 Definite integral of the function x(t) as step function x(t)=1**

Why the step **x(t)=1**? Because this function is as simple as possible and its area **S **calculus method is easy.

The **definite integral** of the function **x(t) **is an **integral unit** output **y(t)**.

Call Desktop/PID/03_calka/01_calka_ze_skoku_jednostkowego.zcos

**Fig. 16-6**

Click “Start”

**Fig. 16-7**

The **x(t)=1** and it’s easy to calculate **y(t) **as a area surface from **t1=0 **up to** t2=t **under the **x(t)=1. **This area is **y(t)=1*t
**

**Fig. 16-8**

There is some analogy to the

**Fig. 16-2**, but not at all. Why?

–

**x(t)=0**for

**t<0**and

**x(t)=1**for

**t>0**and not for all

**t**as in classical

**mathematical analysis**

– there is a

**definite integral,**shows a

**Fig. 16-2****definite integral**

**Chapter 16.3.4 Definite integral of the function x(t) as ramp type function x(t)=0.2*t**

The ramp type **x(t) **and it’s easy to calculate **y(t) **as a area triangel surface from **t1=0 **up to** t2=t **under the **x(t)=0.2*t**.

Call Desktop/PID/03_calka/02_calka_z_pily.zcos

**Fig. 16-9**

Input is a ramp type **x(t)=0.2*t**.

Click “Start”

**Fig. 16-10
**We will count

**definite integral**of the

**x(t)**as a triangel surface. We don’t use

**mathematical analysis**! This

**definite integral**is an

**integral unit**output

**y(t)**too! Test that (for example for

**t=8 sec**)

**definite integral**of the

**x(t)**is the same as parabola

**y(t)**!

**Chapter 16.3.4 Definite integral of the function x(t) as “potentiometer slider swinging” **

Call Desktop/PID/03_calka/03_calka_suwak_oscyloskop.zcos

**FIg. 16-11
**Function

**x(t)**is hand operated now. Use the digital meter to set

**x(t)=0**.

Click “Start”

**Fig. 16-12**

Set gently

**x(t)**:

– x(t)=+0.025

– x(t)=+0.05

– x(t)=+0.025

– x(t)=0

– x(t)=-0.025

– x(t)=-0.05

– x(t)=-0.025

– x(t)=0

– ….itd

You note:

– big

**+x(t)**–>+big speed of the

**y(t)**

–

**big**

**-x(t)**–>

**–**big speed of the

**y(t)**

–

**x(t)**=0–>

**y(t)**steady

**Chapter 16.4 Conclusions**

**Fig. 16-13**

Integrating unit symbols

**Fig. 16-14
**Output

**y(t)**is a

**definite integral**of the input

**x(t)**

**)**

Input

**x(t)**is

**a**

**derivative**of the output

**y(t)**