**Chapter 28.1 Introduction**

You know the **P **and **PD **controller already.

**Fig. 28-1
**These controllers assure closed loop gain

**Kz**and error gain

**Ke**in steady state. The

**K**is a static gain of all the open loop and it includes

**controller Kp**and

**object Ko**. The gain of the controlled object

**Go(s)**is

**Ko=1**mostly–> see

**Fig. 25-3**in

**chapter**

**25**.

**Conclusion**

All the open loop gain is

**K=Kp**. It’s very convenient solution to addjust controllers parameters. —>

**Chapter 31**.

Big

**Kp**means

**Kz≈1**and

**Ke≈0.**It means that

**y(t)≈**

**x(t)**and control error

**e(t)≈0**in steady state. It’s almost

**ideal situation**! Output signal

**y(t)**tries to follow set point

**x(t)**but the ideal steady state situation

**x(t)=y(t)**will be never.

**The very important message !**

The

**I**control assures

**!**

**Fig. 28-2**

Easier formulas are hard to imagine! It means that the output signal

**y(t)=**

**x(t)**or

**e(t)=0**in steady states. The formulas are the special case of the

**Fig. 28-1**. What’s the steady state of the integral unit

**I**which is a part of the open loop. See

**Fig. 28-4.**This state is

**y(t)=infinity**! –> The open loop gain

**K**is

**infinity**too. Put the

**K=**

**infinity**to the formulas

**Fig. 28-1**. The result is

**Fig. 28-2**

**Chapter 28.2 Integral Unit**

**Chapter 28.2.1 Introduction**

We will test **3** **integral** units with different speed of integration

–**“Slow”**

–**“Middle”**

–**“Fast”
**These units aren’t

**I**controllers because they haven’t subtractor node. This node calculates control error

**e(t)=x(t)-y(t)**

**Chapter 28.2.2 “Slow” I unit Ti=2 sec**

Call PID/14_regulacja_typu_I/01_calkujacy_wolny.zcos

**Fig. 28-3**

**Ti=2 sec**

Click “start”

**Fig. 28-4
**The input

**x(t)=1**is step type and the

**y(t)**speed is steady. Note that

**x(t)=y(t)**after

**Ti=2sec**. Conclusion -The bigger is

**Ti**, the “slower” is the

**I**integration unit. The next units will be faster.

**Chapter 28.2.3 “Middle” I unit Ti=1 sec**

Call PID/14_regulacja_typu_I/02_calkujacy_taki_sobie.zcos

**Fig. 28-5**

**Ti=1 sek**

Click “start”

**Fig. 28-6**

**Ti=1 sec**

The unit is **2 **times faster

**Chapter 28.2.4 “Fast” I unit Ti=0.5 sec
**Call PID/14_regulacja_typu_I/03_calkujacy_szybki.zcos

**Fig. 28-7**

**Ti=0.5 sek**.

Click “start”

**Fig. 28-8**

The unit is

**2**times faster.

**Chapter 28.3 How does** **I control make control error e(t)=0. **

**Chapter 28.3.1 Introduction
**It’s an example that

**I control**(Integration control type) is absolutely accurate–>

**x(t)=y(t)**

**Fig. 28-9**

This servomechanism shows that

**I control**can make error

**e=x-y=0**! Other words. The potentiometer

**B x(t)**voltage will absolutely equal to potentiometer

**A**

**y(t)**voltage in steady state

**x(t)=y(t)=0**. Other words. The initial state is

**x(t)=y(t)=0.**We move the potentiometer slider up to

**+5 V**. The voltage

**+Kp*5V**occures immadietaly because

**y(t)=0V**in this moment. The D.C. motor start rotation and the

**B**potentiometer slider will start with the maximal intial speed. Please analize this process. This

**B**slider ascent speed is decreasing because D.C. motor voltage

**+Kp*[5V-y(t)]**is decreasing (

**y(t)**voltage is growing). When does motor stop? It stops when

**x(t)=y(t)**! Please note that our motor is ideal! Very small voltage, for example

**1µV**even, causes rotation. The gear is ideal too. I hope you haven’t problems with block diagrams

**Fig.28-9b**and

**Fig.28-9c**.

**Ks**and

**Kp**depends on the electrical and mechanical parameters. Note that d.c. motor is integral

**I**type. I will not sink in this subject deeper.

**The main conclusion.**

There is

**K**

**/s*Ti**integral unit (D.C. motor!) in the closed loop and it causes control error

**e(t)=0**. The output signal doesn’t oscilate as in

**ON-OFF**control type!

**Chapter 28.3.2 “Slow” servomechanism**

Call/Desktop/PID/14_regulacja_typu_I/04_model_serwo_wolny.zcos

**Fig. 28-10
Ti=4 sec
**Click “start”

**Fig. 28-11**

**Conlusion**

Closed loop

**integral unit**is

**inertial**type.

**Note**

**x(t)=y(t)**in steady state!

**Chapter 28.3.3 “Fast” servomechanism**

Call/Desktop/PID/14_regulacja_typu_I/05_model_serwo_szybki.zcos

**Fig. 28-12**

**Ti=1 sec**

Click “start”

**Fig. 28-13**

Make your own conclusions.

**Chapter. 28.4 I controller with the inertial object.**

**Chapter 28.4.1 Introduction
**We will test

**I controller**with the I

**nertial**object.

**Chapter 28.4.2 Inertial object in open loop**

**Fig. 28-14**

Inertial unit **K=1** **T=10 sec**

Click “start”

**Fig. 28-15**

No comment

**Chapter 28.4.3 I controller Ti=36 sec**

Call Desktop/PID/17_regulacja_typu_I/07_1T_I36

**Fig. 28-16
I** controller has only a a single parameter

**Ti**. We start with the “slow”

**Ti=36sec**. , because we are afraid of the instability.

Click “start”

**Fig. 28-17**

Our concerns were bloated. There aren’t oscillations and the setting time is very long. The steady state

**x(t)=y(t)**isn’t seen in the figure, because steady state is after

**120 sec**.

**Chapter 28.4.4 I controller Ti=16 sec**

The previous system was slow. The faster integration **Ti=16** should make better.

Call Desktop/PID/14_regulacja_typu_I/08_1T_I16_opt.zcos

**Fig. 28-18**

**Ti=16 sec**

Call “start”

**Fig. 28-19
**The system is faster but there are oscillations now.

**Chapter 28.4.5 I controller Ti=8 sec**

Will be better?

Call Desktop/PID/14_regulacja_typu_I/09_1T_I8.zcos

**Fig. 28-20**

**Ti=8 sek**

Click “start”

**Fig. 28-21**

To much oscillations. The best parameter is **Ti=16 sec** if so.

**Chapter. 28.5 I controller with the Two-inertial object.**

**Chapter 28.5.1 Introduction
**We will test

**I controller**with the

**Two-i**

**nertial**object. This object is more complicated than

**Inertial**. We are almost sure that it’s more difficult to control–>more oscillations and long setting time.

**Chapter 28.5.2 Two-inertial object in open loop**

Call Desktop/PID/14_regulacja_typu_I/10_2T_otwarty.zcos

**Fig. 28-22**

**K=1**, **T1=3 sec** and **T2=5 sec**

Click “start”

**Fig. 28-23
**No comment

**Chapter 28.5.3 I controller Ti=25 sec
**Call Desktop/PID/14_regulacja_typu_I/11_2T_I25.zcos

**Fig. 28-24**

**Ti=25 sek**

Click “start”

**Fig. 28-25**

Surprice. The response is much better than for a “easier”

**one-inertial**unit as

**Fig. 28-17**!

**Chapter 28.5.4 I controller Ti=15 sec**

Call Desktop/PID/14_regulacja_typu_I/12_2T_I15_opt.zcos

**Fig. 28-26**

**Ti=15 sek**

Click “start”

**Fig. 28-27
**We are braver and the integration

**Ti=15 sec**faster than previopus

**Ti=15 sec**. The system is faster but there are oscillations now. Which is better? I don’t know.

**Chapter 28.5.5 I controller Ti=6 sec**

Let’s go baldheaded and give **Ti=6 sec**

Call Desktop/PID/14_regulacja_typu_I/13_2T_I6.zcos

**Fig. 28-28**

**Ti=6 sek**

Click “start”

**Fig. 28-29
**The oscillations are not to accept. The optimal integration

**Ti**for

**two-inertial**at

**Fig. 28-22**is

**Ti=15sec**.

**Chapter. 28.6 I controller with the Three-inertial object.**

**Chapter 28.6.1 Introduction
**We will test

**I controller**with the

**thrre-inertial**object.

**Chapter 28.6.2 Three-inertial object in open loop
**Call Desktop/PID/14_regulacja_typu_I/14_3T_otwarty.zcos

**Fig. 28-30**

**K=1**

**T1=0.5 sek**,

**T2=3 sek**and

**T3=5 sek**

Click “start”

**Fig. 28-31**

No comment

**Chapter 28.6.3 I controller Ti=30 sec**

Call PID/14_regulacja_typu_I/15_3T_I30.zcos

**Fig. 28-32**

**Ti=30 sek**

Click “start”

**Fig. 28-33**

The careful control effect. Long setting time and without oscillations. Control signal **s(t)** is a little bigger than **y(t) **only.

As a little demanding teacher-**s(t)** from the student-**y(t)**.

**Chapter 28.6.4 I controller Ti=10 sec**

Call Desktop/PID/14_regulacja_typu_I/16_3T_I10_opt.zcos

**Fig. 28-34**

**Ti=10 sek**

Click “start”

**Fig. 28-35**

Better? The answer is difficult.

**Chapter 28.6.5 I controller Ti=5 sec**

Call Desktop/PID/14_regulacja_typu_I/17_3T_I5.zcos

**Fig. 28-36**

**Ti=5 sek**

Click “start”<

**Fig. 28-37**

Terrible oscillations. The **Ti=10 sec** is optimal for three-inertial if so.

**Chapter 28.6.6 The integration speed Ti exaggeration ****
**For example

**Ti = 1.5 sec**

Call Desktop/PID/14_regulacja_typu_I/18_3T_I1.5_niestabilny.zcos

**Fig. 28-38**

**Ti=1.5 sec**

The oscilloscope parameters are changed to see big signals. Do you conjecture why?

Click “start”

**Fig. 28-39**

Beautilful instability! The amplitude is growing up to

**+/-infinity**.

**Chapter 28.7 How does I controller suppress the disturbances?**

**Chapter 26.7.1 Introduction
**The

**one**,

**two**and

**threeinertial**objects are used as before. The additional disturbance

**z(t)=+0.5**or

**z(t)=-0.5**occures at their inputs in

**130 sec**. The big patience is required because there is long experiment time-

**4 min**. It’s typical that

**I control**is very slowly and it’s very rarely (never?) used in practice. It’s didactic important for me only.

**Chapter 28.7.2 One-inertial object Ti = 16 sec and positive disturbance z(t)==+0.5**

Call Desktop/PID/14_regulacja_typu_I/19_1T_I16_opt_zakl+.zcos

**Fig. 28-40**

Disturbance **z(t)=+0.5** appeares in **130 **sec.

Click “start”

**Fig. 28-41
**The time chart is the same as in

**Fig. 28-19**up to

**130 sec**. But please note that there are different oscilloscope scales. The disturbance

**z(t)=+0.5**caused

**y(t)**growth but the controller

**I**started decreasing the

**s(t)**for value

**Δs(t)=-0.5**in steady state.

Resultat-The disturbance

**z(t)=+0.5**was compensated by

**Δs(t)=-0.5**and the

**y(t)**is the same as before the disturbance appearance! This is a main task of the controller.

**Chapter 28.7.3 One-inertial object Ti = 16 sec and negative disturbance z(t)==-0.5**

Call Desktop/PID/14_regulacja_typu_I/20_1T_I16_opt_zakl-.zcos

**Fig. 28-42**

Disturbance **z(t)=-0.5** appeares in **130 **sec.

Click “start”

**Fig. 28-43**

The disturbance **z(t)=-0.5 **was compensated by **Δs(t)=+0.5**

**Chapter 28.7.4 Two-inertial object Ti = 15 sec and positive disturbance z(t)==+0.5**

Call Desktop/PID/14_regulacja_typu_I/21_2T_I15_opt_zakl+.zcos

**Fig. 28-44**

Disturbance **z(t)=+0.5** appeares in **130 **sec.

Click “start”

**Fig. 28-45**

The disturbance **z(t)=+0.5 **was compensated by **Δs(t)=-0.5**

**Chapter 28.7.5 Two-inertial object Ti = 15 sec and negative disturbance z(t)==-0.5**

Call Desktop/PID/14_regulacja_typu_I/22_2T_I15_opt_zakl-.zcos

**Fig. 28-46
**Disturbance

**z(t)=-0.5**appeares in

**130**sec.

Click “start”

**Fig. 28-47**

The disturbance

**z(t)=-0.5**was compensated by

**Δs(t)=+0.5**

**Chapter 28.7.6 Three-inertial object Ti = 10 sec and positive disturbance z(t)==+0.5**

Call Desktop/PID/14_regulacja_typu_I/23_3T_I10_opt_zakl+.zcos

**Fig. 28-48**

Disturbance **z(t)=+0.5** appeares in **130 **sec.

Click “start”

**Fig. 28-49**

Disturbance **z(t)=+0.5** appeares in **130 **sec.

**Chapter 28.7.7 Three-inertial object Ti = 10 sec and negative disturbance z(t)==-0.5**

Call Desktop/PID/14_regulacja_typu_I//24_3T_I10_opt_zakl-.zcos

**Fig. 28-50**

Disturbance **z(t)=-0.5** appeares in **130 **sec.

**Fig. 28-51
**The disturbance

**z(t)=-0.5**was compensated by

**Δs(t)=+0.5**increase.

Click “start”

**Chapter 28.8 I, P and PD controllers comparison
28.8.1 Introduction
**These same

**x(t)=1**and

**z(t)=+0.5**signals

**are acting for**

**PD, P**and

**I**control systems.

Compare the outputs

**y1(t), y2(t)**and

**y(3)**signals. Especially the steady states and setting times.

**Chapter 28.8.2 Positive disturbance z(t)=+0.5**.

Call Desktop/PID/14_regulacja_typu_I/25_2T_Porown_P_PD_I_zakl+.zcos

**Fig. 28-52
**

Click “start”

**Fig. 28-53**

**controller ensures steady error**

**P-blue****9%**.

**PD**controller ensures steady error

**-green****9%**too but dynamic (oscillations and setting time) is much better here!

**I**controller ensures steady error

**-red****0%**. This is its main advantage–>compare

**P**and

**PD**steady errors. But dynamic characheristics? Waste your breath! The

**I**control is so lazy, because doesn’t exist the initial ” peak” type signal as in the

**P**and especially in

**P**

**D**controller type.

**Conclusion**

The pure

**I**controller is possible of course but is rarely used in practics. The integration

**I**component is used in the

**PI**and

**PID**controllers. See

**chapters 29**and

**30**.

**Chapter 28.8.3 Negative disturbance z(t)=-0.5**.

Call Desktop/PID/14_regulacja_typu_I/26_2T_Porown_P_PD_I_zakl-.zcos

**Fig. 28-54**

**z(t)=-0.5**

Click “start”

**Fig. 28-55**

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