**Chapter 29.1 Introduction
**The previous chapter conclusion (see

**p. 28.8**) is:

**–**The

**P**and especially

**PD**control enables

**fast**reaction for

**x(t)**step type but it doesn’t enable

**null**steady control error

**e(t)=0**

–The

–

**I**control behaviour is inverse. The reaction is

**slow**but it enables

**null**steady control error

**e(t)=0**

The

**PI controller**conception is obvious. Let’s combine

**P**and

**I**controllers!

**Chapter 29.2 PI unit**

**Chapter 29.2.1 PI unit Kp=1 Ti= 1 sec**

Call Desktop/PID/15_regulacja_typu_PI/01_czlon_PI_Kp1_I1.zcos

**Fig. 29-1
**This is not

**PI controller**yet. It isn’t equipped with the most important controller device-the

**subtractor node**!

The input signal

**x(t)**is amplified first (here gain

**Kp=1)**and divided for

**2**components

**–**proportional

**P**(“naked wire”)

**–**integrating

**I**(here

**Ti=1 sec)**

Click “start”

**Fig. 29-2**

**Kp=1 Ti=1 sec**

The input signal is

**x(t)**step type. You see the

**yp(t)**proportional component and

**yi(t)**integral component of the

**y(t)**output signal. The

**yp(t)**

**=**

**x(t)**

**=1**because

**Kp=1**here. The

**yi(t)**is a ramp type and the both components

**yp(t)=yi(t)**are equal after

**Ti=1sec**.

For all

**PI**controllers both components

**yp(t)=yi(t)**are equal after so-called reset time

**Ti**. The other name of

**Ti**is a integral time.

**Note**

The

**y(t)**speed depends on the on the

**Kp**and

**Ti**parameters. But

**Ti**parameter doesn’t depend on

**Kp**. Please analyse

**Fig. 29-2**when

**Kp=0.5**for example.

**Chapter 29.2.2 PI unit Kp=1 Ti= 10 sec**

Call Desktop/PID/15_regulacja_typu_PI/02_czlon_PI_Kp1_I10.zcos

**Fig. 29-3**

**Kp=1 Ti=10 sec**

Click “start”

**Fig. 29-4
**The

**yi(t)**speed is

**10**times slower now, but

**y(t)**output signal was doubled after

**Ti=10 sec**too.

**Chapter 29.2.3 PI unit Kp=3 Ti=10 sec**

Call Desktop/PID/15_regulacja_typu_PI/03_czlon_PI_Kp3_I10.zcos

**Fig. 29-5**

**Kp=3 Ti=10 sec
**Click “start”

**Fig. 29-6**

The integration is

**3**times faster now, but

**y(t)**was doubled after

**Ti=10 sec**too. Conlusion-

**Kp**has no influence for

**Ti**.

**Chapter 29.3 Control System with the PI didactic controller
Chapter 29.3.1 PI didactic controller**

I realised the

**PI**

**didactic controller**. It demonstrates that the component :

– proportional

**P**assures shorter setting time

– integral

**I**assures

**null**steady state control error

**e(t)=0**

–

**PI**

**didactic controller**starts as

**P**controller and ends as a

**I**controller when

**x(t)**is a step type. The

**P–>I**transition is in a one specific moment. The real

**PI**controller

**P–>I**transition from

**P**is continuos. You will be convinced later.

Call Desktop/PID/15_regulacja_typu_PI/04_didactic_PI_Kp3_I35_od_0.zcos

**Fig. 29-7
**The controllers type depends on the contact position:

**Fig. 29-7a**– contact “high” –

**P**controller

**Fig. 29-7b**– contact “low” –

**I**controller

**Fig. 29-7c**–

**didactic PI**as a

**Xcos**model. The

**Xcos**enables the complicated multiblock replacing for the one block.

Click “Start”

**Fig. 29-8**

It’s

**P**controller up to

**25 sec**because contact is “high”and

**y(t)=Kp*e(t)=3*1=3**. The contact is “low” then and the controler changes to

**I**type now. The component

**I**starts integration process from the initial state

**s(t)=0**as every moral integration unit! The

**I**and

**P**components are equal after

**Ti=35 sec**as in the classical

**PI**controller but this discrete

**s(t)**drop is strange a little. The controller developed the

**s(t)**signal based on the earlier

**e(t)**process after all. The conclusion is only one. The integration must start from the last

**P**controler state that is from the

**s(t)=3**. Let’s improwe the

**PI**didactic controller!

Call/Desktop/ PID/15_regulacja_typu_PI/05_didactic_PI_Kp3_I35_od_3.zcos

**Fig.29-9
**The controller structure is the same as

**Fig. 29-7**. The difference is in the algorthm. The

**s(t)**starts from the last

**P**state when is the change

**P–>I**. It means

**s(t)**starts from

**s(t)=3**and not from

**s(t)=0**as

**Fig. 26-7**)

Click “Start” and all will be clear

**Fig. 29-10
**The controller transition

**P–>I**appears in

**25**sec. If input

**e(t)**jumped immediately in

**40 sec**up to

**1.5 (**for example), the

**s(t)**speed will increase as

**red hash**line. There isn’t

**s(t)**jump because the

**I**component is active only. The

**P**is off.

**Chapter 29.3.2 Control System with the didactic PI controller**

The experiment demonstrates that:

– **P **component is accountable for fast **y(t) **value achievement. We know that this value is near the set point **x(t)**.

– **I **(integral) component is accountable for steady state **x(t)=y(t) **achievement. In short. It assures **null **control error. The every control engineer goal.

Call Desktop/ PID/15_regulacja_typu_PI/06_didactic_PI_Kp10_I50_z_inercyjnym

**Fig. 29-11**

**PI** didactic controller **Kp=10** and **Ti=50 sec**.

**Fig. 29-11a**

**PI **before transition, that is before **13 sec**. The **P **component is active only, contact is “high”. Controller changes for **I **then, contact is “low”.

**Fig. 29-11b
**as

**Fig. 29-11a**but

**Xcos**model

Click “Start”

**Fig. 29-12**

**Fig. 22-12a**

It’s difficult to see that

**SP(t)**and

**SI(t)**components are time separated.

**Fig. 22-12b**

There are separate

**SP(t)**and

**SI(t)**processes and one common

**SPI(t)**controll signal.

**It’s clearly visible that:**

–

**P**component is accountable for fast

**y(t)**growing before

**13 sec**., but

**P**control can’t make state

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

**.**

–

**I**component is accountable for steady state

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

**13 sec**.

**Chapter 29.3.3 PI Didactic and classical** **P controller comparison
**See

**Rys. 29-12a**.

**P**controller is a

**PI Didactic**

**special case without changing**

**P–>I**. This same

**y(t)=0.91**signal will be up to Doomsday. –>no

**null**controll error

**e(t)**.

**PI Didactic**controller enables

**y(t)=x(t)**steady state–>

**null**controll error

**e(t)=0**.

**Chapter 29.3.4 PI Didactic and classical ****I**** controller comparison**

Call Desktop/PID/15_regulacja_typu_PI/07_porown_didactic_PI_1T_KP10_I145_I16.zcos

**Fig. 29-13
**These same inertial objects are controlled by:

–

**PI didactic**controller

– Calssical

**I**controller

Who is the winner? It seems like a rhetorical question.

Click “start”

**Fig. 29-14**

Score

**5:0**for

**PI**

**Didactical controller**. The

**y1(t)**setting time is lower than

**y2(t)**and the oscillations are lower too. The

**PI**

**Didactical**controller starts as

**P**and and quickly reaches the steady otput value

**y1(t)=0.91**. It can’t do more–>to do

**y1(t)=x(t)**. There is a job for

**I**component now. The controller changes for

**I**type in

**13 sec**and finish the job–>reduce the control error up to

**0**–>

**y1(t)=x(t)**. This job is easy because the start point is

**y1(t)=0.91**and not

**0**! The start point for

**I**controller is

**0**–>The

**y2(t)**setting time is much longer. Score

**5:0**.

The real

**PI**controller is better than

**PI**

**Didactical controller**yet!

**Chapter. 29.4 PI controller with the one-inertial object**

** Chapter 29.4.1 Introduction
**Let’s go to the real

**PI**controller. These are

**p.29.4**,

**p.29.5**and

**p.29.6**subchapters

**Chapter 29.4.2 One-inertial object in open loop**

Call Desktop/PID/15_regulacja_typu_PI/08_1T_otwarty.zcos

**Fig. 29-15**

**T=10 sec** **K=1**.

Click “start”

**Fig. 29-16**

No comment

**Chapter 29.4.3 PI controller Kp=3 Integration OFF
I.e **we have typical

**P**controller

Call Desktop/PID/15_regulacja_typu_PI/09_1T_Kp3_bez_calkowania.zcos

**Fig. 29-17**

We test

**P**control to compare it with the

**PI**control. You will know advantage of the

**PI**then

Click “start”

**Fig. 29-18**

Typical

**P**controll process. The non

**null**steady error

**e(t)=0.25**. This is a weakness of course. I remind you the main rule of the control system by the way. The steady state is achieved when

**s(t)=y(t)**. The input object signal=output object signal. Control signal

**s(t)**is a object input signal here.

**Chapter 29.4.4 PI controller PI Kp=3 Ti=12 sec**

Call Desktop/PID/15_regulacja_typu_PI/10_1T_Kp3_I12.zcos

**Fig. 29-19**

**Kp=3 Ti=12 sec**

Click “start”

**Fig. 29-20**

Yes, yes, yes!!! Steady state **y(t)=x(t)** after **40 sec**. Null control error. The** Kp** and **Ti **parameters are “shy” slightly. Do the more “aggressive” **I **integration. Will be the process better?

**Chapter 29.4.5 More detailed analysis of the PI controller**

Call Desktop/PID/15_regulacja_typu_PI/11_1T_Kp3_I12_P_I.zcos

**Fig. 29-21**

**Kp=3 Ti=12 sek**

The previous control system but we analyse proportional **sP(t) **and integral **sI(t) **components additionally.

Click “start”

**Fig. 29-22
**We are interested in

**sP(t)**and

**sI(t)**components only, therefore the main controller signal

**sPI(t)**has a less pronounced yellow colour.

**sP(t)=3**and

**sI(t)=0**in

**t=3sec**when

**x(t)**started–>

**sPI(t)=sP(t)**

**sP(t)=0**and

**sPI(t)=0**after

**t=55 sec**(steady state)–>

**sI(t)=sPI(t)**

Conclusion

PIcontroller starts as

Conclusion

PI

**P**controller and ends as a

**I**controller. It’s similar to

**PI**

**Didactical**controller in the present case. But betweentimes? The

**sP(t)**proportional components drops continuosly from

**3**to

**0**and

**sI(t)**arises from

**0**to

**1**. It means that

**PI**controller changes continuosly its character from

**P**controller to

**I**controller. It’ s less similar to the

**PI Didactical**controler. This one changes character in one moment in

**13 sec**–>

**Fig. 29-12**.

**Chapter 29.4.6 PI controller PI Kp=3 Ti=4 sec**

Call Desktop/PID/15_regulacja_typu_PI/12_1T_Kp3_I4_opt.zcos

**Fig. 29-23**

**Kp=3 Ti=4 sec**

Integration **I **is more “aggresive” now. Will be better?

Click “start”

**Fig. 29-24
**There is a small small initial 10% over-regulation but it’s incontestably better. So try to be more aggresive and give

**Ti=2 sec**.

**Chapter 29.4.7 PI controller PI Kp=3 Ti=2 sec**

Call Desktop/PID/15_regulacja_typu_PI/13_1T_Kp3_I2.zcos

**Fig. 29-25**

**Kp=3 Ti=2 sec**

Click “start”

**Fig. 29-26
**More oscillations and a longer setting time. Previous

**Ti=4sec**is better.

Let’s give further

**Kp=10.**

**Chapter 29.4.8 PI controller Kp=10 Integration OFF**

Call Desktop/PID/15_regulacja_typu_PI/14_1T_Kp10_bez_calkowania.zcos

**Fig. 29-27**

**Kp=10** integration** off–>P **controller

Click “start”

**Fig. 29-28**

This is **P **controller now. This is under theory that **Kp** increament causes system faster and steady error isn’t **null**. The bigger is **Kp **the lower is steady error **e(t)**–>compare **Kp=3 **in **Fig. 29-18**. The system system is faster too.

**Chapter 29.4.9 PI controller PI Kp=10 Ti=15 sec**

Calll Desktop/PID/15_regulacja_typu_PI/15_1T_Kp10_I15.zcos

**Fig****. 29-29**

**Kp=10 Ti=15 sec**

Click “start”

**Fig. 29-30**

It’s better than **Kp=3** in **Fig. 29-20**. Let’s increase the integration speed.

**Chapter 29.4.10 PI controller PI Kp=10 Ti=5 sec**

Call Desktop/PID/15_regulacja_typu_PI/16_1T_Kp10_I5_opt.zcos

**Fig. 29-31**

**Kp=10 Ti=5 sek**

Click “start”

**Fig. 29-32
**The small initial over-regulation but the system is much faster. Let’s go this way and make integration more intensive.

**Chapter 29.4.11 PI controller PI Kp=10 Ti=1.75 sec**

Call Desktop/PID/15_regulacja_typu_PI/17_1T_Kp10_I.75.zcos

**Fig. 29-33**

**Kp=10 Ti=1.75 sec**

Click “start”

**Fig. 29-34
**The response is very fast but all depends of our Client decision. If this intial over-regulation doesn’t hurt Client technology then the parameters

**Kp=10**and

**Ti=1.75**sec are optimal for the

**Fig. 29-15**object.

**Chapter. 29.5 PI controller with the two-inertial object**

**Chapter 29.5.1 Two-inertial object in open loop**

Call Desktop/PID/15_regulacja_typu_PI/18_obiekt_2T.zcos

**Fig. 29-35**

**T1=3 sek**, **T1=5 sek**, **K=1**.

Click “start”

**Fig. 29-36**

No comments

**Chapter 29.5.2 PI controller Kp=3 Integration OFF**

Call Desktop/PID/15_regulacja_typu_PI/19_2T_Kp3_bez_calkowania.zcos

**Fig. 29-37**

**Kp=3** Integration **OFF**

Click “start”

**Fig. 29-38**

Typical **P **control with non steady error **e(t)=0.25**. The response is with oscillations because the object is more complicated than in the **Fig. 29-18**

**Chapter 29.5.3 PI controller PI Kp=3 Ti=32 sec**

Call Desktop/PID/15_regulacja_typu_PI/20_2T_Kp3_I32.zcos

**Fig. 29-39**

**Kp=3 Ti= 32 sec
**We start with a very subtle integration as usual.

Click”start”

**Fig. 29-40**

**Fig. 29-40a**

The integration is to shy and there isn’t steady error

**e(t)=0**before

**60 sec**yet. But this state will be achieved later. See

**Fig. 29-40b**

Fig. 29-40b

The experiment time is

Fig. 29-40b

**180 sec**now. The steady error

**e(t)=0**was achieved finally. We were very shy with the integration so be brave and do

**Ti=8sec**.

**Chapter 29.5.4 PI controller PI Kp=3 Ti=8 sec**

Call Desktop/PID/15_regulacja_typu_PI/21_2T_Kp3_I8_opt.

**Fig. 29-41**

**Kp=3 Ti= 8 sec**

Wciśnij “start”

**Fig. 29-42
**The courage pays off. The response is much better. Let’s go on this way.

**Chapter 29.5.5 PI controller PI Kp=3 Ti=5 sec**

Call PID/15_regulacja_typu_PI/22_2T_Kp3_I5.zcos

**Fig. 29-43**

**Kp=3 Ti= 5 sec**

Click “start”

**Fig. 29-44**

It’s worse than before.

**Chapter 29.5.6 PI controller Kp=10 Integration OFF**

Call Desktop/PID/15_regulacja_typu_PI/23_2T_Kp10_bez_calkowania.zcos

**Fig. 29-45**

**Kp=10 Integration OFF**

Click “start”

**Fig. 29-46
**There are oscillations but the steady

**non null**control error is consistent with the theory. We expect troubles with

**I**component.

**Chapter 29.5.7 PI controller Kp=10 Ti=20 sec**

Call Desktop/PID/15_regulacja_typu_PI/24_2T_Kp10_I20.zcos

**Fig. 29-47**

**Kp=10 Ti=20 sec
**Click “start”

**Fig. 29-48**

We started carefully with

**Ti=20 sec**(low integration speed) but there are oscillations and the setting time is long, more than

**60 sec**. The steady error is

**e(t)=0**. This a

**I**component job!

By the way. The

**s(t)**oscillations are bigger than

**y(t)**oscillations and it’s typical for control systems. The driver steering wheel movements are bigger than a car course oscillations. The car is going by the “almost” straight line but driver steering wheel movements are quite visible.

**Chapter 29.5.7 PI controller Kp=10 Ti=10 sec**

Call Desktop/PID/15_regulacja_typu_PI/25_2T_Kp10_I10opt.zcos

**Fig. 29-49**

**Kp=10 Ti=10 sec**

Click “start”

**Fig. 29-50
**The process is decently, but the parameters

**Kp=3**and

**Ti=8 sec**are better–>

**Fig. 29-42**! The life is full of surprises. The higher

**Kp**was better by now. Let’s decrease

**Ti**. Will be better?

**Chapter 29.5.9 PI controller Kp=10 Ti=5 sec**

Call Desktop/PID/15_regulacja_typu_PI/26_2T_Kp10_I5.zcos

**Fig. 29-51**

**Kp=10 Ti=5 sec**

Click “start”

**Fig. 29-52**

It’s worse.

**Chapter. 29.6 PI controller with the three-inertial object**

**Chapter 29.6.1 Three-inertial object in open loop**

Call Desktop/PID/15_regulacja_typu_PI/27_obiekt_3T.zcos

**Fig. 29-53**

**K=1**, **T1=0.5 sec** **T2=3 sec** and** ** **T3=5 sec**

Click “start”

**Fig. 29-54
**Compare with the

**two-inertial**–>

**Fig. 29-36**. The different isn’ very distinct. But the more “

**multi”**is a

**multi-inertial**unit, the more distinct is

**To**delay parameter.

**Chapter 29.6.2 PI controller Kp=3 Integration OFF**

Call Desktop/PID/15_regulacja_typu_PI/28_3T_Kp3_bez_calkowania.zcos

**Fig. 29-55**

**Kp=3 Integration OFF**

Click “start”

**Fig. 29-56**

Compare with the **Fig. 29-38**. The same **P **controller but the object is simpler-**two-inertial**. The steady error **e(t)=0.25**. But the over-regulation (first oscillation) is bigger here.

**Chapter 29.6.3 PI controller Kp=3 Ti=16 sec**

Call Desktop/PID/15_regulacja_typu_PI/29_3T_Kp3_I16.zcos

**Fig. 29-57**

**Kp=3 Ti=16 sec**

Click “start”

**Fig. 29-58
**Component

**I**made steady error

**e(t)=0**after

**60 sec**. But it’s very slowly. Let’s make

**I**more aggresive

**Ti=10 sec.**

**Chapter 29.6.4 PI controller Kp=3 Ti=10 sec**

Call Desktop/PID/15_regulacja_typu_PI/30_3T_Kp3_I10opt.zcos

**Fig. 29-59**

**Kp=3 Ti=10 sec
**Click “start”

**Fig. 29-60**

Good job! May be better? Let’s minimize on

**Ti**.

**Chapter 29.6.5 PI controller Kp=3 Ti=4 sec**

Call Desktop/PID/15_regulacja_typu_PI/31_3T_Kp3_I4.zcos

**Fig. 29-61**

**Kp=3 Ti=4 sec**

Click “start”

**Fig. 29-62
**It’s worse. And what about

**Kp=10?**

**Chapter 29.6.6 PI controller Kp=10 Integration OFF**

Call PID/15_regulacja_typu_PI/32_3T_Kp10_bez_calkowania.zcos

**Fig. 29-63**

**Kp=10 **** Integration OFF**

Click “start”

**Fig. 29-64
**The steady error is

**e(t)=0.09**and is lower than for

**Kp=3**. But long setting time and oscillations are terrible.

**Chapter 29.6.7 PI controller Kp=10 Ti=16 sec**

Call Desktop/PID/15_regulacja_typu_PI/33_3T_Kp10_I16opt.zcos

**Fig. 29-65**

**Kp=10 Ti=16 sec**

Click “start”

**Fig. 29-66**

It’s worse!

**Chapter 29.6.8 PI controller Kp=10 Ti=8 sec**

Call Desktop/PID/15_regulacja_typu_PI/34_3T_Kp10_I8.zcos

**Fig. 29-67**

**Kp=10 Ti=8 sec**

Click “start”

**Fig. 29-68
**It’s much worse. The parameters

**Kp=3**i

**Ti=10 sec**for

**three-inertial**object are better! But we are determinded and we will increase integration speed for

**Ti=2.5 sec**.

**Chapter 29.6.9 PI controller Kp=10 Ti=2.5 sec**

Call Desktop/PID/15_regulacja_typu_PI/35_3T_Kp10_I2.5_niestabilny.zcos

**Fig. 29-69**

**Kp=10 Ti=2.5 sec**

Why is there Dirac (pseudo Dirac-strictly) instead of the softie **x(t) **step type pule as usually?

Click “start”

**Fig. 29-70**

The short hammer-Dirac caused oscillations up to **+/- infinity**. By the way. The instability doesn’t mean immobility always. This system is instable but it still up to **3 sec**. As a vertical positioned pencil on the table.

**Chapter 29.7 How does PI controller suppress the disturbances?
Chapter 29.7.1 Introduction
**The

**one**,

**two**and

**three-inertial**objects are used as before. The additional disturbance

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

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

**The optimal**

**70 sec**.**Kp=10**and

**Ti=5 sec**parameters are choosed. They are optimal on the grounds of the

**x(t)**input, not

**z(t)**input. The

**z(t)**response will be better if they are choosed on the grounds of the

**z(t)**disturbance!

**Chapter 29.7.2 One-inertial object, Kp=10 Ti=5 sec and positive disturbance z(t)=+0.5**

Call Desktop/PID/15_regulacja_typu_PI/36_1T_Kp10_I5opt_zakl+.zcos

**Fig. 29-71**

Disturbance **z(t)=+0.5** occures in **70** sec.

Click “start”

**Fig. 29-72
**The process is the same as in

**Fig. 29-32**up to

**70 sec**. Remember that there are different oscilloscope time scales. The

**z(t)=+0.5**(additinal heating for example) disturbance is compensated by the controller output

**Δs=-0.5**drop. (oven power drop for example). There is a steady controll error

**e(t)**. The main goal of the control system. This is a good job of the

**I**controller component!

**Chapter 29.7.3 One-inertial object, Kp=10 Ti=5 sec and negative disturbance z(t)=-0.5**

Call Desktop/PID/15_regulacja_typu_PI/37_1T_Kp10_I5opt_zakl-.zcos

**Fig. 29-73**

Disturbance **z(t)=-0.5** occures in **70** sec.

Click “start”

**Fig. 29-74**

The **z(t)=-0.5** (additinal cooling for example) disturbance is compensated by the controller output **Δs=+0.5 **growth. (oven power growth for example).

**Chapter 29.7.4 Two-inertial object, Kp=3 Ti=8 sec and positive disturbance z(t)=+0.5**

Call Deskto/PID/15_regulacja_typu_PI/38_2T_Kp3_I8opt_zakl+.zcos

**Fig. 29-75**

Disturbance **z(t)=+0.5** occures in **70** sec.

Click “start”

**Fig. 29-76**

Object is more complicated than before and you see the effects.

**Chapter 29.7.5 Two-inertial object, Kp=3 Ti=8 sec and negative disturbance z(t)=-0.5**

Call Desktop/PID/15_regulacja_typu_PI/39_2T_Kp3_I8opt_zakl-.zcos

**Fig. 29-77**

Disturbance **z(t)=-0.5** occures in **70** sec.

Click “start”

**Fig. 29-78**

No comments

**Chapter 29.7.6 Three-inertial object, Kp=3 Ti=8 sec and positive disturbance z(t)=+0.5**

Call Desktop/PID/15_regulacja_typu_PI/40_3T_Kp3_I10opt_zakl+.zcos

**Fig. 29-79**

Disturbance **z(t)=+0.5** occures in **70** sec.

Click “start”

**Fig. 29-80**

More complicated object yet and the dynamic response parameters are worse–>bigger oscillations and setting time.

**Chapter 29.7.7 Three-inertial object, Kp=3 Ti=8 sec and negative disturbance z(t)=-0.5**

Call Desktop/PID/18_regulacja_typu_PI/41_3T_Kp3_I10opt_zakl-.zcos

**Fig. 29-81**

Disturbance **z(t)=-0.5** occures in **70** sec.

Click “start”

**Fig. 29-82**

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Students understand the effects of proportional, integral, and derivative control actions, together with their combinations on system response.