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
Definite 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 Definite 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, Fig. 16-2 shows a 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 secdefinite 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)

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