**Chapter 30.1 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

–

**PI**control enables

**null**steady control error

**e(t)=0**but it is relatively slow.

The

**PID controller**conception is obvious. Let’s add

**D**component to obtain

**PID**controller!

**Chapter 30.2 PID unit**

**Chapter 30.2.1 PID unit Kp=1 Ti=10 sec Td=1 sec**

Call Desktop/PID/16_regulacja_typu_PID/01_czlon_PID_Kp1_I10_D1_skok.zcos

**Fig. 30-1**

**PID **unit has **3** components:

– proportional **sP(t)**

– integrating **sI(t)****
**– differential

**sD(t)**

Oscilloscop shows

**sD(t)**,

**sI(t)**and output controller signal

**sPID(t)**

But where is

**sP(t)**?

**sP(t)=x(t)**because

**Kp=1**here.

Click”start”

**Fig. 30-2**

**Kp=1 Ti=10 sec Td=1 sec**

Input

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

The output controller signal

**sPID(t)**is doubled after

**Ti=10 sec**. You dont see this moment at oscilloscope.

Other

**Ti**definition–>

**Ti=10 sec**–>time when

**sI(t)=sP(t)**.

There is a real

**differentiation**and not ideal. The

**Td**interpretation is complicated a little. It’s easier when

**x(t)**is a ramp type.

**Chapter 30.2.2 PID unit when x(t) is a ramp type**

Call Desktop/PID/16_regulacja_typu_PID/02_czlon_PID_Kp1_I10_D1_pila.zcos

**Fig. 30-3**

This same **PID **but **x(t)** is a ramp type.

Click “start”

**Fig. 30-4**

**Kp=1 Ti=10 sec Td=1 sec**

**x(t)** ramp type

– Proportional** sP(t) **is a copy of the **x(t) **because **KP=1**.

– Integral **sI(t) **is a quadratic function as a integral of the linear function

– Real differential **sD(t) **is a **x(t) **speed “calculated” with inertia **0.1sec**. There is state **sP(t)=sD(t)** after **Td=1sec**.

**Chapter 30.3 PID controller with the one-inertial object**

**Chapter 30.3.1 Introduction
**The

**one-inertial object**is controlled by the

**PID**

**Chapter 30.3.2 PID controller Kp=3 Ti=4 sec Td=0 Differentiation OFF**

Call Desktop/PID/16_regulacja_typu_PID/03_1T_Kp3_I4_D0.zcos

**Fig. 30-5**

**Kp=3 Ti=4 sec Td=0 sec**

**E.i. **This is **PI controller**

Click “start”

**Fig. 30-6**

The process is the same as **chapter 29 Fig. 29-24** because there are the same control systems. Let’s make the real **PID control**.

We start with the careful differentiation **Td=0.5 sec**.

**Chapter 30.3.3 PID controller Kp=3 Ti=4 sec Td=0.5 sec
**Call Desktop/PID/16_regulacja_typu_PID/04_1T_Kp3_I4_D0.5.zcos

**Fig. 30-7**

**Kp=3 Ti=4 sec Td=0.5 sec**

Click “start”

**Fig. 30-8**

Where is the positive effect of the

**D**component? The setting time is even longer!

**Chapter 30.3.4 PID controller Kp=3 Ti=4 sec Td=1 sec**

Call Desktop/PID/19_regulacja_typu_PID/05_1T_Kp3_I4_D1

**Fig. 30-9**

**Kp=3 Ti=4 sec Td=1 sec
**Click “start”

**Fig. 30-10**

It’s worse!

**Chapter 30.3.5 PID controller Kp=10 Ti=5 sec Td=0 Differentiation OFF**

Call PID/16_regulacja_typu_PID/06_1T_Kp10_I5_D0.zcos

**Fig. 30-11**

**Kp=10 Ti=5 sec Td=0 sec**

Click “start”

**Fig. 30-12
**The best process so far!

**Chapter 30.3.6 PID controller Kp=10 Ti=5 sec Td=0.5 sec**

Call Desktop/PID/16_regulacja_typu_PID/07_1T_Kp10_I5_D0.5.zcos

**Fig. 30-13**

**Kp=10 Ti=5 sec Td=0.5 sec**

Click “start”

**Fig. 30-14
**The differentiation doesn’t work out the situation.

**Chapter 30.3.7 PID controller Kp=10 Ti=5 sec Td=1 sec**

Call Desktop/PID/30_regulacja_typu_PID/08_1T_Kp10_I5_D1.zcos

**Fig. 30-15**

**Kp=10 Ti=5 sec Td=1 sec**

Click “start”

**Fig. 30-16
**It’s worse. Conclusion. The optimal control for

**one inertial object**is a

**PI**control. I expect that the better is a

**P**control even, but it

requires

**Kp=infinity**. It’s a difficult problem.

**Chapter 30.4 PID controller with the two-inertial object**

**Chapter 30.4.1 Introduction
**The

**two-inertial object**is controlled by the

**PID**. We tested this object in the

**chapter 29 Fig. 29-38**.

**Chapter 30.4.2 PID controller Kp=3 Ti=8 sec Td=0 – Differentiation OFF
**Call Desktop/PID/16_regulacja_typu_PID/09_2T_Kp3_I8_D0.zcos

**Fig. 30-17**

**Kp=3 Ti=8 sec Td=0 sek**–

**Differentiation OFF**

This is

**PI controller**the same as in

**Chapter 29 Fig 29-23**.

Click “start”

**Fig. 30-18**

**PID**as

**PI controller**

**Chapter 30.4.3 PID controller Kp=3 Ti=8 sec Td=0.5 sec**

Call Desktop/PID/16_regulacja_typu_PID/10_2T_Kp3_I8_D0.5.zcos

**Fig. 30-19**

**Kp=3 Ti=8 sec Td=0.5 sec**

Click “start”

**Fig. 30-20
**You observe a good job of the

**D**component. Furthermore it was a very careful differentiaton. I remind you the principle of the

**D**job. There are two opposing actions here.

**D**differentiates the setpoint

**x(t)**first–>

**y(t)**arises very quickly at the begining. But this quickly arising

**y(t)**causes braking effect because

**e(t)=x(t)-y(t)**–> minus by

**y(t)**! It prevents from overregulations and generally process is more stable.

Let’s look at this process by the other oscilloscope parameters. You will see all the

**sPID(t)**control signal.

**Fig. 30-21**

This “needle” type

**sPID(t)**is an effect of the setpoint front edge differentiation. It causes quickly

**y(t)**arising. You see the braking effect between circa

**8…14**sec. This very careful differentiation gives good effects. Let’s be a bit more aggresive and set the

**Td=1 sec**.

**Chapter 30.4.4 PID controller Kp=3 Ti=8 sec Td=1 sec**

Call Desktop/PID/16_regulacja_typu_PID/11_2T_Kp3_I8_D1_opt.zcos

**Fig. 30-22**

**Kp=3 Ti=8 sec Td=1 sec**

Click “start”

**Fig. 30-23
**Marvel! The process is quick and without oscillations almost. Go this way and increase

**Td**.

**Chapter 30.4.5 PID controller Kp=3 Ti=8 sec Td=5 sec**

Call Desktop/PID/16_regulacja_typu_PID/12_2T_Kp3_I8_D5.zcos

**Fig. 30-24**

**Kp=3 Ti=8 sec Td=5 sec**

Click “start”

**Fig. 30-25
**Every exaggeration is bad. The braking

**D**effect is too strong and the setting time is longer. But some people like this process. There are no overregulations! Let’s increase the

**Kp**parameter up to

**Kp=10**. Will be better?

**Chapter 30.4.6 PID controller Kp=10 Ti=10 sec Td=0 sec Differentiation OFF**

Call Desktop/PID/16_regulacja_typu_PID/13_2T_Kp10_I10_D0.zcos

**Fig. 30-26**

**Kp=10 Ti=10 sec Td=0 sec – Differentiation OFF**

Click “start”

**Fig. 30-27
**This is

**PI**control. Big oscillations! Does

**D**component help?

**Chapter 30.4.7 PID controller Kp=10 Ti=10 sec Td=0.5 sec
**We start with the small-careful differentiation

**Td=0.5 sec**as usually.

Call Desktop/PID/16_regulacja_typu_PID/14_2T_Kp10_I10_D0.5.zcos

**Fig. 30-28**

**Kp=10 Ti=10 sek Td=0.5 sec**

Click “start”

**Fig. 30-29**

Good job. The smaller oscillations and shorter setting time. Isn’t

**D**component beautiful? Go on this way and increase

**Td**.

**Chapter 30.4.8 PID controller Kp=10 Ti=10 sec Td=1 sec**

Call Desktop/PID/16_regulacja_typu_PID/15_2T_Kp10_I10_D1.zcos

**Fig. 30-30**

**Kp=10 Ti=10 sec Td=1 sec**

Click “start”

**Fig. 30-31
**It’s better.

**Chapter 30.4.9 PID controller Kp=10 Ti=10 sec Td=2 sec**

Call Desktop/PID/16_regulacja_typu_PID/16_2T_Kp10_I10_D2.zcos

**Fig. 30-32**

**Kp=10 Ti=10 sec Td=2 sec**

Click “start”

**Fig. 30-33
**There aren’t overregulations but steady error

**e(t)=0**is only after

**25 sec**. It’s too slowly. Is the braking

**D**effect dominant? The previous

**Td=1 sec**was better.

Really? The error

**e(t)=0.05**was accomplished quickly after circa

**5 sec**. It achieves very slowly null error

**e(t)=0**then after circa

**25 sec**. It means that interration

**I**action is too careful! let’s make it a bit more aggresive e.i

**Ti=7sec**. We will diminish the

**Td**up to

**Td=1.5 sec**. by the way. Why is this

**Td**? I don’t know. My nose is my adviser.

**Chapter 30.4.10 PID controller Kp=10 Ti=7 sec Td=1.5 sec**

Call Desktop/PID/15_regulacja_typu_PID/17_2T_Kp10_I7_D1.5optall.zcos

**Fig. 30-34**

**Kp=10 Ti=7 sec Td=1.5 sec**

Click “start”

**Fig. 30-35**

Author!! author! The parameters ** Kp=10 Ti=7 sek Td=1.5 sek** are optimal for our **two-inertial **object. Compare this process with **PI **control **Fig. 30-27**. Shock!

**Chapter 30.5 PID controller with the three-inertial object**

**Chapter 30.5.1 Introduction
**The

**three-inertial object**with

**K=1**

**T1=0.5 sec T2=3 sec i T3=5 sec**was tested in

**chapter 29 Fig. 27-53.**

**Chapter 30.5.2 PID controller Kp=3 Ti=10 sec Td=0 – Differentiation OFF**

Call Desktop/PID/16_regulacja_typu_PID/18_3T_Kp3_I10_D0.zcos

**Fig. 30-36**

**Kp=3 Ti=10 sec Td=0 sec – Differentiation OFF**

This same **PI **control as in **chapter 29 Fig. 29-52**.

Click “start”

**Fig. 30-37**

Typical **PI **“slow” control but the main goal **e(t)=0 **is achieved.

**Chapter 30.5.3 PID controller Kp=3 Ti=10 sec Td=0.5 sec**

Call Desktop/PID/16_regulacja_typu_PID/19_3T_Kp3_I10_D0.5.zcos

**Fig. 30-38**

**Kp=3 Ti=10 sec Td=0.5 sec
**The careful differentiation

**Td=0.5 sec**at start as usually

Click “start”

**Fig. 30-39**

Even this small differentiation component

**D**gives positive effect. Go on and set

**Td=1 sec**.

**Chapter 30.5.4 PID controller Kp=3 Ti=10 sec Td=1 sec**

Call Desktop/PID/16_regulacja_typu_PID/20_3T_Kp3_I10_D1.zcos

**Fig. 30-40**

**Kp=3 Ti=10 sec Td=1 sec**

Click “start”

**Fig. 30-41
**The oscillations are smaller but the setting time is longer. Let’s increase

**Td**.

**Chapter 30.5.5 PID controller Kp=3 Ti=10 sec Td=1.5 sec**

Call Desktop/PID/16_regulacja_typu_PID/21_3T_Kp3_I10_D1.5.zcos

**Fig. 30-42**

**Kp=3 Ti=10 sec Td=1.5 sec**

Click “start”

**Fig. 30-43
**The improvent is questionable. I suppose that the

**Ti**decreament will do better as for

**two-inertial**in

**Fig. 30-34**. I recommend to do experiment with lower

**Ti**for more ambitious readers.

**Chapter 30.5.6 PID controller Kp=10 Ti=10 sec Td=0 – Differentiation OFF**

Call Desktop/PID/16_regulacja_typu_PID/22_3T_Kp10_I10_D0.zcos

**Fig. 30-44**

**Kp=10 Ti=10 sec Td=0 sec– Differentiation OFF**. In fact, this is **PI **control.

Click “start”

**Fig. 30-45
**We expected the bigger oscillations because

**Kp**was increased. Let’s calm the process by the

**D**component.

**Chapter 30.5.7 PID controller Kp=10 Ti=10 sec Td=1 sec **

Call Desktop/PID/16_regulacja_typu_PID/23_3T_Kp10_I10_D1.zcos

**Fig. 30-46**

**Kp=10 Ti=10 sec Td=1 sec**

Click “start”

**Fig. 30-47
**This is an good example “how differentiation

**D**component is good!” Compare with the

**Fig. 30-45**.

**Chapter 30.5.8 PID controller Kp=10 Ti=10 sec Td=2 sec**

Call Desktop/PID/16_regulacja_typu_PID/24_3T_Kp10_I10_D2.zcos

**Fig. 30-48**

**Kp=10 Ti=10 sec Td=2 sec
**Click “start”

**Fig. 30-49**

It’s better. But can we improve the setting time by the integration

**I**intensification? For example

**Ti=7 sec**?

**Chapter 30.5.9 PID controller Kp=10 Ti=7 sec Td=2 sec**

Call Desktop/PID/15_regulacja_typu_PID/25_3T_Kp10_I7_D2_optall.zcos

**Fig. 30-50**

**Kp=10 Ti=7 sec Td=2 sec**

Click “start”

**Fig. 30-51
**It’s better. The

**Kp=10 Ti=7 sec Td=2 sec**parameters are optimal

**PID**controller parameters for this

**three-inertial**object.

These parameters were “hand” adjusted. I am sure that there are more optimal parameters but that’s quite different matter.

**Chapter 30.6 How does PID controller suppress the disturbances?
Chapter 30.6.1 Introduction
**The

**two**and

**three-inertial**objects are used as before. We don’t test simple

**one-inertial**because

**PI**control is better than

**PID**here.

**The additional disturbance**

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

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

**The optimal**

**70 sec**.**Kp,**

**Ti**and

**Td**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 30.6.2 Two-inertial object, Kp=10 Ti=7sec Td=1.5 sec and positive disturbance z(t)=+0.5**

Call Desktop/ PID/16_regulacja_typu_PID/26_2T_Kp10_I7_D1.5_+zakl.zcos

**Fig. 30-52
**Disturbance

**z(t)=+0.5**heating type occures in

**70 sec**Click

**“start”**

**Fig. 30-53**

You see the fast set point

**x(t)**response and the good postive disturbance

**z(t)=+0.5**suppresion. The additional heating

**z(t)=+0.5**was compensated by the control signal drop

**ΔsPID(t)=-0.5**in steady state.

**Note**

line

**Control signal sPID(t)****covers**

**disturbance z(t)=+0.5**line after

**80 sec**here.

**Chapter 30.6.3 Two-inertial object, Kp=10 Ti=7sec Td=1.5 sec and negative disturbance z(t)=-0.5**

Call Desktop/PID/16_regulacja_typu_PID/27_2T_Kp10_I7_D1.5_-zakl.zcos

**Fig. 30-54
**Disturbance

**z(t)=-0.5**cooling type occures in

**70 sec**Click “start”

**Fig. 30-55**

The disturbance additional cooling type

**z(t)=-0.5**was compensated by the control signal increase

**ΔsPID(t)=+0.5**in steady state. (power increase)

**Chapter 30.6.4 Three-inertial object, Kp=10 Ti=7sec Td=2 sec and positive disturbance z(t)=+0.5**

Call Desktop/PID/16_regulacja_typu_PID/28_3T_Kp10_I7_D2_+zakl.zcos.

**Fig. 30-56**

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

Click “start”

**Fig. 30-57**

Object is more complicated than before but the control system behaviour is surprisingly good.

**Chapter 30.6.5 Three-inertial object, Kp=10 Ti=7sec Td=2 sec and negative disturbance z(t)=-0.5**

Call Desktop/PID/16_regulacja_typu_PID/29_3T_Kp10_I7_D2_-zakl.zcos

**Fig. 30-58
**Disturbance

**z(t)=-0.5**cooling type occures in

**70 sec**Click “start”

**Fig. 30-59**

No comments

**Chapter 30.7 P, PD, PI i PID controllers comparison**

**Chapter 30.7.1 Introduction
**This same set point

**x(t)**step type will be given for

**P, PD, PI and**

**PID**control systems.

There will be controlled the objects

–

**two-inertial**

**–**

**three-inertial**

The opimal (or “pseudoptimal”)

**Kp, Ti**and

**Td**parameters were “handy” chosen in the earlier experiments.

**Chapter 30.7.2 Control system with the two-inertial objects**

Call Desktop/PID/16_regulacja_typu_PID/30_por_2T_P_PI_PID.zcos

**Fig. 30-60**

You tested these systems in the chapters:

**– 26**

**P control**

**– 27**

**PD control**

**– 29**

**PI control**

**– 30**

**PID control**

There are the only poor versions with one input

**x(t)**and objects outputs

**yP(t), yPD(t), yPI(t) i yPID(t)**.

Click “start”

**Fig. 30-61**

**Black**of the

**yP(t)****P control**is the worst. The oscillations and the setting time are the biggest.

There isn’t steady

**null**control error

**e(t)=0**.

**Green yPD(t)**of the

**PD control**has the best dynamics. The response is almost rectangle.

There isn’t steady

**null**control error

**e(t)=0**too.

**Blue yPI(t)**of the

**PI**guarantes steady

**null**control error

**e(t)=0**, but the dynamics isn’t impressive.

There are oscillations and a big setting time.

**Red yPID(t) of the PID control**is the best. It assure steady

**null**control error

**e(t)=0**, small oscillations and short setting time.

These dynamical qualities are a little worse than

**PD control**.

Finally compare

**PID**and his poor relation

**P**

**Chapter 30.7.3 Control system with the three-inertial objects**

Call Desktop/PID/16_regulacja_typu_PID/31_por_3T_P_PI_PID.zcos

**Fig. 30-62**

Click “start”

**Fig. 30-63**

The object is more complicated now and and there is worse dynamic than previous.

**Chapter 30.8 PID control summary
**

**PID**is all the controllers king. Even the word

**PID**or

**controller**is the same for some people. As vacuum cleaner and electroluks.

The

**PID**idea isn’t the effect of the finespun mathematical considerations but the effect of the life observation. The pilot of the ship tries to provide ship course despite the disturbances as squall or waves.

The man uses subconsciously

**PID**algorithm in every day life .

–

**P**component – immediate reaction typical for young people

–

**I**component – reaction considering the history. Don’t be hot-headed. Something is small but if this something is in a long time we are reacting and waiting for effects. This behaviour is typical for old experienced people.

–

**D**component- foresight. Something is small but its speed is big. We are feeling trend and we make some decisions.