**Chapter 7.1 Introduction
**It’s the simplest dynamical unit except

**Proportional.**

**Fig.**

**7-1**

**Inertial**unit parameters:

–

**K**-steady state gain

–

**T**– time constant

We will do some experiments with different

**T**time constants.

We will do some experiments with different input signals

**x(t)**. The output signals

**y(t)**will be observed on the bargraf or on the oscilloscope.

**Chapter 7.2 T=5 sec, step x(t) from the virtual potentiometer, y(t) on the bargraf**

Call Desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/01_inercyjny_suwak_bargraf.zcos

** Fig.7-2
**Remember that parameter

**K=1**is a gain in the steady state!

Push **“Start”** (as you did for **proportional **before)

** Fig.7-3
**You see virtual

**potentiometer**window (TKsource), bargraf window (BARXY) ont the

**block diagram**window. You don’t see

**Scilab**window because it’s covered by the

**diagram**window. Move the windows, windows that you see block diagarm with the digital meters.

Move the slider from

**0**to

**1**in a flash so you give the step signal

**x(t)**and observe the

**bargraf**! The response isn’t up to date as in

**proportional**unit now! Wait when will be the steady state. When

**x(t)=y(t)=1.000**. The digital meters will be usefull here!

The maximal velocity

**y(t)**signal is when started. The

**y(t)**velocity decreases (velocity, not

**y(t)**value!) then. The velocity is

**0**when steady state.

“Swing” the slide several times. You observe the smoothened effect of the inertial unit. Click the red button “finished the experiment”. The button colour changed to grey again. The experiment is finished now.

It should be opened the 3 windows only before we start the new experiment:

–

**Scilab**

–

**Xcos with the block diagra**–>”01_proporcjonalny_bargraf…”

-“naked”

**Xcos**“Untitled”

Close all the other windows.

Do it always (as on the

**and the**

**Fig.**7-4)**Scilab**will work perfectly.

**Fig.** 7-4

What about new time constant **T=10 sec**? You will teach to change **Go(s) **parameters in **Xcos **by the way.

**Chapter 7.3 T=10 sec, step x(t) from the virtual potentiometer, y(t) on the bargraf**

We change the parameters in existing block diagram and don’t call a new one.

Move the mouse on transmision **G(s) **change parameters **1+5*s** –>**1+10*s**.

** Fig.7-5
**Parameters change

Click “start” when accepted

**The**

**Fig.**7-6**T=10sec**is doubled. The object

**G(s)**will be “doukled lazy” too. I hope that you see it

Close the

**Xcos**but don’t save the parameters change! The “old”

**T=5sec**parameter would be when you call this

**G(s)**again.

**Chapter 7.4 T=5 sec, steps “+1 -1” x(t) from the virtual potentiometer, y(t) on the oscilloscope.
**We test the respons

**y(t)**for to consecutive

**“+1,-1”**steps input signal

**x(t)**

Call Desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/02_inercyjny_suwak_oscyloskop.zcos

**Block “0” draws**

**Fig.**7-7**y=0**axis

Click “start”

**Fig.**7-8

All is clear I hope.

Experiment ends in 60 seconds. Remember that **3** windows should be opened, before you you “start” new experiment in existing block diagram . There are **SCILAB**, “naked” untitled **XCOS** window and existing block diagram **XCOS** window.

We are ready to new experiment in existing block diagram now.

**Chapter 7.5 T=5 sec, random x(t) from the virtual potentiometer, y(t) on the oscilloscope.**

Click “start” i and “swing” the potentiometer slider.

** Fig.7-9
**You observe the smoothened effect of the inertial unit.

**Chapter 7.6 T=5 sec, x(t) from the step genearator, y(t) on the oscilloscope.
**Step generator enables more precisely

**G(s)**parameter testing.

Call desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/03_inercyjny_skok_oscyloskop.zcos

**Fig.**7-10

Click “start”

**Fig.**7-11

Transient performance shows “who i who” in the transfer function **G(s)**.

For example here:

**1** in the **G(s)** numerator–> gain **K=1** in steady state.

**5** in the **G(s) **denominator–>time constant **T=5 sec**.

The **y(t) **velocity at the **x(t) **start is maximal. Imagine that this velocity is constant all the time. When does **y(t) **attain state **y(t****)=1**? After **5 sec** of course. This time is called contant time **T=5sec**. of the** inertial unit**.

**Chapter 7.7 Two different inertial units comparison.**

Call Desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/04_porownanie_2_inercyjnych.zcos

**Fig.**7-12

The step pulse **x(t) **given for two inputs simultaneously. Can you predict the **y1(t) **and **y2(t)** signals?

** Fig.7-13
**Have you predicted. I propose to repeat the chapter if the answer is “no”.

**Chapter 7**.**8 T=1 sek, Dirac pulse x(t), y(t) oscilloscope**Dirac pulse

**δ(t)**is a funny type.

– the pulse

**δ(t)**time is infinitely short

-the pulse amplitude

**δ(t)**is infinitely high

-but the pulse

**δ(t)**energy (

**δ(t)**area) is

**=1**–>normal value

Its mechanical interpretation is a hammer hit.

The ideal

**Dirac**pulse

**δ(t)**doesn’t exist in the real world. There is always finitely short

**δ(t)**pulse time and big but finitely

**δ(t)**amplitude.

Call desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/05_inercyjny_dirac_oscyloskop.zcos

**Fig.**7-14

Dirac pulse area=**1**

Click “start”

** Fig.7-15
**The output

**y(t)**is rising very fast during

**3..3.01 sec**. But not immediately! The final state

**y(t**

**)=0**. This final state

**y(t**

**)=0**for input

**x(t)=δ(t)**dirac type, is typical for all so-called dynamic

**static units**. We will discuss them later.

**Chapter 7**.**9**** T=1 sek, “better” Dirac pulse x(t), y(t) oscilloscope**The previous Dirac pulse wasn’t ideal of course. But we try do it “better”. I propose pulse with amplitude

**100**and time

**0.01 sec**.

Note that it’s

**energy-area =100*0.01=1**!

Call Desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/06_inercyjny_idealniejszy_dirac_oscyloskop.zcos

The diagram is the same as

**. The pulse parameters are different only:**

**Fig.**7-16– amplitude=100

– time=0.01 sec.

Click “start”

**It looks like ideal Dirac pulse. But it’s oscilloscope limitation!**

**Fig.**7-16**Chapter 7.10 **** Why do we need Dirac pulses?**

Their execution is difficult. It requires big power in the short time.

Let’ go to the **Chapter 18** for a moment.

We give the **Dirac **pulse **δ(t)** to the **G(s) **object.

The **Laplace transform **of this **y(t)** is **Y(s)**=**G(s)*δ(s)=G(s) **because **Laplace transform **Dirac pulse **δ(s)=1 !**

**Chapter 7.11 **** What to do when G1(s) is as undermentioned?**

Call Desktop/PID/01_podstawowe_człony_dynamiczne/02_człon_inercyjny/07_rozne_postacie_tych_samych_czlonow.zcos

** Fig.7-17
**We suppose that the

**G1(s)**inertial type. But what are

**K**and

**T**? We don’t see it for a first look. So make the

**G1(s)**denominator as

**1+…**(not as previous

**7+21*s**)–> divide denominator and numerator by

**7**. You have

**G2(s)=G1(s)**. You see the

**K=2**and

**T=3 sec**. here. Check it.

Click “start”

**You see that**

**Fig.**7-18**G1(s)=G2(s)**

**G2(s)**form shows:

–

**K=2**

–

**T=3 sek**

**Chapter 7.11 **** Typical inertial units**

**RC circuit **

** Fig.7-19
R = 100 kΩ** and

**C =10 µF**–>

**T**= R*C = 100 000 Ω * 10*0,000001F =

**1 sec**.

**The direct-current motor
**Input

**x(t) –**voltage step

Output

**y(t) –**rotational speed

The rotational speed

**at**start is

**0**. It rises as in

**inertial units**and attain maximum value. The

**K**and

**T**parameter depend on the mechanical and electrical

**qualities**. This is simplified motor model. The real is more complicated of course.

**The bathtub with the taken out plug
**Input

**x(t) –**you open immediately the valve –>

**Q(t)**input flow step

Output

**h**

**(t) –**the water level.

The

**h(t)**level speed is maximal at the start. The

**h(t)**level rises with maximal initial speed. This speed declines because the output flow through the open hole rises when

**h**

**(t)**rise. The maximum level

**h**

**(t)**is when

**input flow=output**

**flow**.

This is smplified “bathtub with the taken out plug”

**model**. More–> see

**Chapter 17**.

**Chapter. 7.12 Summary
**The

**proportional**unit is the

**first**approximation of the every* dynamic units.

The

**second**and more accurate

**approximation**approximation of the every* dynamic units is the

**inertial**unit. When you give

**x(t)**input step type signal, then the

**response**

**y(t)**will reach the steady level

**y**after some steady-state time

**t**.

We assume that it’s a

**inertia**

**l**unit with parameters:

–

**K=y**because

**x(t)=1**

–

**T**is

**4…**

**5**times smaller than steady-state time

**t.**

This

**G(s)**

**approximation**is very primitive of course, but much better than

**proportional**unit

**approximation.**

*almost –> doesn’t concern

*

**integral**unit for example.

Excellent post. I’m facing a few of these issues as well..