** Chapter 27.1 Introduction
**The steady closed loop gain

**Kz**and closed loop

**Ke**of the

**PD**control are the same as for

**P**control type.

**Fig. 27-1**

Where is the

**PD controller**better if so? There are better dynamics parameter here. The

**stability**and

**small steady error**are in contradiction in

**P**and other type controllers. The

**bigger**is

**Kp**the system is

**less stable**. The

**PD**controllers, more strictly- its

**differential**component

**D**, enables the

**stability**increasing.

**Fig. 27-2**

You see the

**non-null**steady

**error**in

**P**control type. What to do to make smaller

**oscillations**? The answer is obviously-minimize the controller gain

**Kp**. It’s ok,oscillations will minimize but the

**non-null**steady

**error**will increase! Don’t go this way my friend.

So don’t touch the

**Kp**, but be up to this time. What is the oscillations reason? The output

**y(t)**signal is going

**to fast**its steady state determined by

**Kz**in

**Fig. 27-1**, like as a pendulum. Pendulum as a symbol of overshoot!

Let’s try to brake this pendulum. Not too intensive–>pendulum will accomplish the steady state with smaller oscillations but after a long time. And not too gently–>it will be similar to

**P**control with big oscillations.

**The most important**. The

**braking intensivity**must be

**proportional**to the output signal

**y(t) velocity**, other words its derivative

**y'(t)**. We hope that the dispelled pendulum will “pass to the other side” but less than without braking. The conclusion. The oscillations

**are smaller**so the system is more

**stable**! But the steady error

**e(t)**is the same because

**y'(t)=0**in steady state–>there is no breaking–>the differentiating unit

**D**is off. The system is more stable so we can increase gain

**Kp**without

**instability risk**. The effect is the lower steady state error

**e(t)**with less or equal oscillations.

Let’s remind the differentiating unit

**D**attributes.

**Chapter 27.2 Ideal and real differentiating units**

**Chapter 27.2.1 Introduction
**The input

**x(t)**will be:

– step type

– ramp type (linear function type)

**Chapter 27.2.2 Ideal differentiating unit and x(t) step type**

Call Desktop/PID/13_regulacja_typu_PD/01_rozniczkujacy_ideal_skok.zcos

**Fig. 27-3**

Click “start”

**Fig. 27-4
**The response

**y(t=1sec)**is Dirac type because velocity of the

**y(t=1sec)**is

**infinity**now and

**y(t>1sec)=0**because velocity is

**null**.

**Chapter 27.2.3 Real differentiating unit and x(t) step type**

Call Desktop/PID/13_regulacja_typu_PD/02_rozniczkujacy_real_skok.zcos

**Fig. 27-5**

Click “Start”

**Fig. 27-6
**It’s something similar to Dirac pulse, but

**y(t=1sec)=10**. It’s finite and the

**reciprocal**of the

**T=0.1sec.**The real

**differentiating**units are more popular than

**ideal**. Why? Ideal aren’t imprevious to noises.

**Chapter 27.2.4 Ideal differentiating unit and x(t) ramp type
**Step type

**x(t)**input signals aren’t comfortable to

**identify**the transfer function

**G(s)**of the differentiating type . The better is

**x(t)**

**ramp**type-other name

**linear function**type.

Call Desktop/PID/13_regulacja_typu_PD/03_rozniczkujacy_ideal_pila.zcos

**Fig. 27-7**

Click “start”

**Fig. 27-8**

The velocity of the ramp

**x(t)**is double increased in

**t=5sec**–>the

**y(t)=x'(t)**is doubled too!

I propose to return to

**Chapter 9 Differential Unit**for a moment. There is

**Td**parameter definition here.

**Chapter 27.2.5 Real differentiating unit and x(t) ramp type**

Call Dersktop/PID/13_regulacja_typu_PD/04_rozniczkujacy_real_pila.zcos

**Fig. 27-9**

Click “start”

**Fig. 27-10
**The diagram is simalar to

**ideal differentiating unit**when signal

**y(t)**is steady. The

**real differentiating unit**needs some time to

**“calculate”**the

**y'(t)**in the moments

**t=1 sec**and

**t=5sec,**when

**x(t)**changes its velocity. You see the influence of the inertia

**T=0.1sec**.

**Chapter 27.3 Ideal PD and Real PD unit.**

**Introduction 27.3.1
**These units aren’t controllers yet! They haven’t the most important controller part-the

**subtractor node**. It realizes the function

**e(t)=x(t)-y(t)**and controller decides what to do then. We can imagine the controller with

**subtractor node**only. This is

**P controller**with the

**Kp=1**. But we can’t imagine the controller without

**subtractor node**!

The input signals will be used:

–

**x(t)**step type

–

**x(t)**ramp type

**Chapter 27.3.2 Ideal differentiating PD unit and x(t) step type
**Call Desktop/PID/13_regulacja_typu_PD/05_PD_ideal_skok.zcos

**Fig. 27-11
**There is unit:

–

**proportional**part (“naked” wire)

**Kp=1**here

–

**differentiating**part The adjustment parameter is

**Td=1sec**here.

Note that intensity of all the

**differentiating**depends of the both

**Td**and

**Kp**parameters!

Click “start”

**Fig. 27-12**

The diagram is the same as

**Fig. 27-4**at the first impression. But look more exactly! There is

**y(t>1sec)=1**and

**y(t>1sec)=0**in

**Fig. 27-4**.

**Chapter 27.3.3 Real differentiating PD unit and x(t) step type**

Call Desktop/PID/13_regulacja_typu_PD/06_PD_real_skok.zcos

**Fig. 27-13
**Compare with the ideal

**PD**unit

**Fig. 27-11**. Where is the difference?

Click “start”

**Fig. 27-14**

The figure notes clarify all.

**Chapter 27.3.4 Ideal differentiating PD unit and x(t) ramp type**

Call Desktop/PID/13_regulacja_typu_PD/07_PD_ideal_pila.zcos

**Fig. 27-15**

Click “start”

**Fig. 27-16
**There is

**x(t)**ramp type and you see better the

**P**and

**D**components of the

**y(t)**signal than

**Fig.**

**27-12**–

**x(t)**with the step type.

**Kp=1**and

**Td=1sec**–>

**P**component of the

**y(t)**is the same as

**x(t)**in this case.

We can read very easy the

**Td**parameter here. The

**P**and

**D**components are

**equal**after

**Td=1sec**when

**x(t)**ramp was started.

**Chapter 27.3.5 Real differentiating PD unit and x(t) ramp type**

Call Desktop/PID/13_regulacja_typu_PD/08_PD_real_pila.zcos

**Fig. 27-17**

No comments

Click”start”

**Fig. 27-18
**You see better the

**P**and

**D**components of the

**y(t)**signal than

**Fig. 27-14**.

**Kp=1**and

**Td=1sec**–>

**P**component of the

**y(t)**is the same as

**x(t)**in this case.

**Chapter. 27.6 PD controller with the two-inertial object.**

**Chapter 27.6.1 Introduction
**We will test

**PD controller**with the

**two-inertial**and

**three-inertial**

**Go(s)**objects. But what about

**one-inertial**? This object is so simply that the

**P controller**assures the best control.

There are

**Kp=10**and

**Kp=100**and some

**Td**parameters. What

**Kp**and

**Td**combinations does assure the most optimal* step response?

**Note**

We will test

**real PD**controller beacuse these are mainly used in the industry. The

**ideal PD controllers**aren’t noise resistant.

*optimal

**Kp,Td**– there are small oscillations and short setting time.

**Chapter 27.6.2 Two-inertial object in open loop**

Call PID/13_regulacja_typu_PD/09_obiekt_2T.zcos

**Fig. 27-19**

Click “start”

**Fig. 27-20**

Typical **multinertilal** object response with characheristic inflexion point.

**Chapter 27.6.3 PD controller Kp=10 Td=0 sec
**

**Td=0**means that differentiation action is

**OFF,**or other words

**D**component of the

**PD**is off. So it’s

**P**controller see

**chapter 26.4.3**. Why do we do it? Because you will be confident that

**PD**control is much better than

**P**control.

Call PID/13_regulacja_typu_PD/10_2T_Kp10_D0.zcos

**Fig. 27-21**

**Kp=10 Td=0**

Click “start”

**Fig. 27-22**

There are big oscillations in

**P**type control. What are the

**PD**control effects?

**Note**

I remind that we have

**real PD**and not

**ideal PD**.

**Chapter 27.6.4 PD controller Kp=10 Td=1 sec**

Call PID/13_regulacja_typu_PD/11_2T_Kp10_D1.zcos

**Fig. 27-23**

**Kp=10 Td=1 sec**

Click “start”

**Fig. 27-24
**Compare with

**P**control

**Fig. 27-22.**Shocking improvement! Oscillations and the setting time are much better. The static attribute

**Kz**is the same as

**P**controller–>steady state

**y(t)=0.91**.

How to circumstantiate the

**PD**control this same static and better dynamics compared with

**P**control? The easiest answer is “There are such mathematics rules!”. Roma locuta causa finita.

But what does common sense say?

**1**–

**P**and

**PD**controllers haven’t distinctions in steady state, because

**D**component control signal

**s(t)=0**here.

**2**–

**PD**controller is “very clever” in non steady-state phase

**t=3…15sec**see

**Fig. 27-24**.

Let’s analize the

**direct**input signal to controller. Not set point

**x(t)**but error

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

**3**-The

**y(t=3sec)=0**and velocity of

**y(t=3sec)=y'(t=3sec)=0**–>

**s(t=3sec)=10+100=110**see

**Fig. 27-24**. Full steam ahead! You will see signal in the next experiment when

**y(t)**is not cutted by oscillosope .

The main conclusion. The

**D**component differentiates step

**x(t)**when it starts and doesn’t differentiate

**y(t)**in this time becacuse

**y'(t=3sec)=0**! More strictly-

**D**differentiates

**y(t)**but the effect is

**null**!

Different approach. There is open loop in the

**Fig. 27-23**when

**x(t)**starts in

**t=3sec,**because

**y(t)=0**then. It’s normal behaviour of all “normal” dynamic objects

**Go(s)**excluded

**proportional**and

**differentiating**units. Look at

**Fig. 27-14**now. The output

**y(t=3sec)**controller signal is

**11**when

**Kp=1**!

**If**

**Kp=10**–>

**y(t)=110.**

**4**-Let’s analize the rest of non steady-state phase

**t=3…15sec**. There is

**x(t)=1**steady signal

**t=3sec…infinity**. The driving force derived from

**x(t)**weakens (mainly its differentiating) but the

**braking force**derived from

**y(t)**increases initially and falls then to

**0**too. Why

**braking force**? Look at formula

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

**braking force**avoid oscillations and shortens the set time.

**5**-What about steady time

**t=15sec…infinity**? The

**y'(t)=0**now and it’s like

**P**controller. The steady signal

**y(t)=0.91**is obviously.

**Conclusions**when

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

The main influence for

**y(t)**has the differential component

**D**at the begining. This influence weakens then and remains only

**Kp**proportional component in steady state.

Let’s observe all the control signal

**s(t)**with the changed oscilloscope parameters.

Call Desktop/PID/13_regulacja_typu_PD/12_2T_Kp10_D1_pelny_widok.zcos

Fig is the same as **Fig. 27-23 **but other oscilloscope paramters.

Click “start”

**Fig. 27-25
**You see “full steam ahead”–>

**s(t)=110**at start. Compare with the

**y(t)=0.91**!

**Chapter 27.6.5 PD controller Kp=10 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/13_2T_Kp10_D5_opt.zcos

**Fig. 27-26**

**Kp=10 Td=5 sec**

We are braver now. The differentiation action is **5 **times more aggressive. Will be the response better?

Click “start”

**Fig. 27-27
**I am prestidigitator now. Where is the applause? Compare with the

**P**controller response

**Fig. 27-22**(Strictly-There is

**PD**controller with

**D**action

**off**). Are you convinced that

**PD**is ok? Let’s go baldheaded and set

**Td=20sec**! Will be the

**y(t)**more ideal as a rectangle?

**Note**

The “cutted” by oscilloscope control signal is

**s(t)=510**!

**Chapter 27.6.6 PD controller Kp=10 Td=20 sec**

Call Desktop/PID/13_regulacja_typu_PD/14_2T_Kp10_D20_przesada.zcos

**Fig. 27-28**

**Kp=10 Td=20 sec**

Click “start”

**Fig. 27-29
**The differentiation is too aggressively! All the exaggerations are bad. The

**y(t)**is coming very slowly to its steady state

**y(t)=0.91**after

**50 sec**. This is a braking effect of the

**D**component.

**Chapter 27.6.7 PD controller Kp=100 Td=0 sec****
**The

**Kp**increment will decrease the steady error

**e(t)**. What about dynamics?

We start

**PD**with disconnected

**D**component. This is

**P**controller now.

Call Desktop/PID/13_regulacja_typu_PD/15_2T_Kp100_D0.zcos

**Fig. 27-30**

**Kp=100 Td=0**

Click “start”

**Fig. 27-31**

**Note**

The control

**s(t)**colour is changed from blue to yellow. Yellow is less dominated than blue and all signals are more visible.

This is

**P**controller because

**D**is off. You wil see the the

**PD**advantage then

**Chapter 27.6.8 PD controller Kp=100 Td=0.25 sec
**We start very carefully with differentiation.

**Td=0.25**

**sec**is small.

Call Desktop/PID/13_regulacja_typu_PD/16_2T_Kp100_D_0.25.zcos

**Fig. 27-32**

**Kp=100 Td=0.25 sec**

Click “start”

**Fig. 27-33
**We were very carefull with the differentiation. But it’s much better than

**P**control

**Fig. 27-31.**May be better yet?

**Chapter 27.6.9 PD controller Kp=100 Td=2 sec**

Call Desktop/PID/13_regulacja_typu_PD/17_2T_Kp100_D2_opt.zcos

**Fig. 27-34**

**Kp=100 Td=2 sec**

Click “start”

**Fig. 27-35
**It’s better.

**Y(t)**has small oscillations, but yellow

**s(t)**-wow! It’s typical that control signals

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

**y(t)**. The

**y(t)**“sleeps” after

**4 sec**but control

**s(t)**decreasing oscillations are yet.

**Chapter 27.6.10 PD controller Kp=100 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/18_2T_Kp100_D5_przesada.zcos.

**Fig. 27-36**

**Kp=100 Td=5 sec**

Click “start”

**Fig. 27-37
**Is better? More oscillations but shorter setting time. Not mind. De gustibus non est disputandum.

**Chapter. 27.6.11 PD controller with the two-inertial object-Conclusions
**

**1**– The static characteristic as

**P**controller–>

**PD**steady error the same as

**P**controller.

**2**– Better dynamics

**3**– Optimal

**PD**controller adjustments

**Kp=10**and

**Td**=

**5 sec**when

**e=0.09**

**Kp=100**and

**Td**=

**2 sec**when

**e=0.01**

These are “hand made” adjustments with the small experiments amount only. It’s possibly to have better adjustments of course.

**Chapter. 27.7 PD controller with the three-inertial object**

**27.7.1 Introduction
**The object is more complicated and we are expecting the troubles if so. Instability for example.

**Chapter 27.6.3 Three-inertial object in open loop**

Call Desktop/PID/13_regulacja_typu_PD/19_obiekt_3T.zcos

**Fig. 27-38**

Click “start”

**Fig. 27-39**

The **y(t) **initial speed is smaller than for **two-inertial **in **Fig. 27-20**

**Chapter 27.7.3 PD controller Kp=10 Td=0 sec**

Wywołaj PID/13_regulacja_typu_PD/20_3T_Kp10_D0.zcos.

**Fig. 27-40**

**Kp=100 Td=0** e.g. **P **controller

Click “start”

**Fig. 27-41
**

**P**control doesn’t assure good dynamics. Very long setting time and oscillations.

**Chapter 27.7.4 PD controller Kp=10 Td=0.5 sec**

Call Desktop/PID/13_regulacja_typu_PD/21_3T_Kp10_D0.5.zcos

**Fig. 27-42**

**Kp=10 Td=0.5 sec**

Click “start”

**Fig. 27-43**

**PD **is better than **P **control **Fig. 27-41**. Is this **PD **controller** **last word?

**Chapter 27.7.5 PD controller Kp=10 Td=1.5 sec**

Call Desktop/PID/13_regulacja_typu_PD/22_3T_Kp10_D1.5_opt.zcos.

**Fig. 27-44**

**Kp=10 Td=1.5 sec**

Click “start”

**Fig. 27-45
**Better!

**Chapter 27.7.6 PD controller Kp=10 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/23_3T_Kp10_D5.zcos

**Fig. 27-46**

**Kp=10 Td=5 sec**

Click “start”

**Fig. 27-47
**Previous experiment was better. The differentiation action

**D**was too big!

**Chapter 27.7.7 PD controller Kp=25 Td=0**

Call PID/13_regulacja_typu_PD/24_3T_Kp25_D0.zcos

**Fig. 27-48**

**Kp=25 Td=0 **–> **D **component is disconnected–>it’s **P **controller. The oscilloscope amplification is **10 times **decremented. You will know why at the moment.

Click “start”

**Fig. 27-49**

The **Kp=25 **is a reason of the instabililty.

**Chapter 27.7.8 PD controller Kp=25 Td=0.5 sec**

Call Desktop/PID/13_regulacja_typu_PD/25_3T_Kp25_D0.5.zcos.

**Fig. 27-50**

**Kp=25 Td=0.5 sec**

Click “start”.

**Fig. 27-51
**The wonderful influence of the

**D**component. System is stable. Let’s try to improve the time diagram by the

**Td**increasing.

**Chapter 27.7.9 PD controller Kp=25 Td=1 sec**

Call Desktop/PID/13_regulacja_typu_PD/26_3T_Kp25_D1_opt.zcos

**Fig. 27-52**

**Kp=25 Td=1 sec**

Click “start”

**Fig. 27-53
**It’s better.

**Chapter 27.7.10 PD controller Kp=25 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/27_3T_Kp25_D5.zcos.

**Fig. 27-54**

**Kp=25 Td=5 sec**

Click “start”.

**Fig. 27-55
**Stable but not acceptable.

**Chapter 27.8 PD controller with the disturbances**

**Chapter** **27.8.1 Introduction
**The noise suppression is the main job of the controller. There will be

**2**disturbances positive

**z(t)=+0.5**and negative

**z(t)=-0.5**. They are really powerfull! It’s difficult to imagine that the voltage in the socket jumps from

**230 V**to

**345 V**or to

**135 V**. The earlier hand best adjusted

**Kp**and

**Td**parameters are used.

Note that

**x(t)**setting point response is better than noise

**z(t)**here because transfer functions

**G(s)=y(s)/x(s)**and

**Gzakl(s)=z(s)/x(s)**are different! There is an important conclusion for control system design. When the setting point

**x(t)**changes are often you adjust

**Kp**and

**Td**considering the

**x(t)**and disturbance

**z(t)**when

**z(t)**is often.

Chapter

Chapter

**27.8.2 Two-inertial z(t)=+0.5, Kp=10 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/28_2T_Kp10_Td5_zakl+.zcos

**Fig. 27-56**

The disturbance

**z(t)=+0.5**will occur in the

**30 sec**.

Click “start”

**Fig. 27-57**

The the positive disturbance

**z(t)=+0.5**(for example additional heating) is suppressed by the control signal

**s(t)**diminishment. Please note that setting

**x(t)**response is faster than

**z(t)**response.

**Chapter** **27.8.3 Two-inertial z(t)=-0.5, Kp=10 Td=5 sec**

Call Desktop/PID/13_regulacja_typu_PD/29_2T_Kp10_Td5_zakl-.zcos

**Fig. 27-58
**The disturbance

**z(t)=-0.5**will occur in the

**30 sec**.

Click “start”

**Fig. 27-59**

The the negative disturbance

**z(t)=-0.5**(for example additional cooling) is suppressed by the control signal

**s(t)**increment.

**Chapter** **27.8.4 Two-inertial z(t)=+0.5, Kp=100 Td=2 sec**

Call Desktop/PID/13_regulacja_typu_PD/30_2T_Kp100_Td2_zakl+.zcos

**Fig. 27-60
**The disturbance

**z(t)=+0.5**will occur in the

**30 sec**.

Click “start”

**Fig. 27-61**

The disturbance

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

**s(t)**decreasing. The disturbance influence is almost invisible.

**Chapter** **27.8.5 Two-inertial z(t)=-0.5, Kp=100 Td=2 sec**

Wywołaj PID/13_regulacja_typu_PD/31_2T_Kp100_Td2_zakl-.zcos

**Fig. 27-62**

The disturbance **z(t)=-0.5** will occur in the **30 sec**.

Click “start”

**Fig. 27-63**

No comment.

**Chapter** **27.8.6 Three-inertial z(t)=+0.5, Kp=10 Td=1.5 sec**

Call Desktop/PID/13_regulacja_typu_PD/32_3T_Kp10_Td1.5_zakl+.zcos

**Fig. 27-64**

Click “start”

**Fig. 27-65**

No comment

**Chapter** **27.8.7 Three-inertial z(t)=-0.5, Kp=10 Td=1.5 sec**

Call Desktop/PID/13_regulacja_typu_PD/33_3T_Kp10_Td1.5_zakl-.zcos

**Fig. 27-66**

Click “start”

**Fig. 27-67**

No comment.

**Chapter** **27.8.8 Three-inertial z(t)=+0.5, Kp=25 Td=1 sec**

Call Desktop/PID/13_regulacja_typu_PD/34_3T_Kp25_Td1_zakl+.zcos

**Fig. 27-68**

Click “start”

**Fig. 27-69**

No comment

**Rozdz. 27.8.9 Zakłócenie ujemne z(t)=-0.5, Kp=25 Td=1 sek **

Wywołaj PID/13_regulacja_typu_PD/35_3T_Kp25_Td1_zakl-.zcos

**Fig. 27-70**

Click “start”

**Fig. 27-71**

No comment.

**Chapter 27.9 PD and P controllers comparison**

**27.9.1 Introduction
**Score in a match

**PD**controller contra

**P**

10:0

10:0

**Chapter 27.9.2 PD controller Kp=10, D=5 sec and P controller Kp=10**

Call Desktop/PID/13_regulacja_typu_PD/36_porownanie_PD_P_2T_KP10_D5.zcos

**Fig. 27-72**

**Kp=10**and

**Kp=10, Td=5 sek**

These same signals

**x(t)=1**and

**z(t)=+0.5**are acting for

**P**and

**PD**control systems.

Click “start”

**Fig. 27-73**

Note that the

**PD**and

**P**controllers

**stable states**are the same.

**PD**dynamic is much better. Some people say that

**PD**disturbance suppresion isn’t much better than

**P**controller.

**Kp**and

**Td**parameters were adjusted considering setting point

**x(t)**. There will be less worse

**y(t)**response for

**x(t)**(but much better than

**P**control) and much better reaction for

**z(t)**when considering disturbance

**z(t)**.

**Chapter 27.9.3 PD controller Kp=100, D=2 sec and P controller Kp=100**

Call Desktop/PID/13_regulacja_typu_PD/37_porownanie_PD_P_2T_KP100_D2.zcos.

**Fig. 27-74**

**Kp=100**

**Td=2 sek**

Click “start”

**Fig. 27-75**

The

**PD**controller advantage

**green ypd(t)**over

**P**controller

**cred yp(t)**is shocking here! The next figure shows this same time graph, but with the different oscilloscope range

**0.98…1.02**. You can observe the

**z**

**(t)**response.

**Fig. 27-76**

**PD**is much better.

**Chapter 27.10 PD controller with the separate y(t) signal differentiating **

**Chapter 27.10.1 Introduction**

The name of this **PD **type controller is long. Let’s call it **PDy**.

**Fig. 27-77
Fig.27-77a**

Classical

**PD**controller. The control signal

**s**

**(t)**is calculated considering the error

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

**Fig.27-77b**

Modified

**PDy**controller.

The control signal

**s**

**(t)**consists of

**2**separate parts.

– Proportional part calculates. The input is

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

– Differential part The input is

**-y(t)**only

By the way. We use the name “differentiating” instead of “differentiating with inertia” or “real differentiating”. I remind that inertia 0.1 sec inreases fast noises resistanse.

The

**PD**and

**PDy**controllers disturbance

**z(t)**suppresion is the same. The

**s(t)**signals peaks are lower for

**PDy**than for

**PD**setting point

**x(t)**is a step type. The energy consumption is lower too. This ia an advantage of the

**PDy**. The

**y(t)**setting time is longer now and this is weakness of course but it isn’t important when

**x(t)**changes are rare.

**Chapter 27.10.2 PDy controller with the two-inertial object and positive disturbance z(t)=+0.5**

Call Desktop/PID/13_regulacja_typu_PD/38_PD_osobne_rozniczkowanie.zcos

**Fig. 27-78
**This is

**Fig. 27-77b**realization

**Kp=10**i

**Td=5sek**.

Click “start”

**Fig. 27-79
**–

**The setting point**

**x(t)**response is very “lazy” now for

**t=3…30sec**. It isn’t important when the set point

**x(t)**changes are rare. This is typical for many industrial sites.

– There isn’t peak component in

**s(t)**which is derived from differentiation. It’s good for devices life time. The valves lifetime for example.

– The

**y(t)**reaction for disturbance

**z(t)**for

**t=30…60 sec**is the same as for classical

**PD**controller.

**Chapter 27.10.3 PDy controller with the two-inertial object and positive disturbance z(t)=+0.5. The more optimal Td=1sec parameter**

Call Desktop/PID/13_regulacja_typu_PD/39_PD_osobne_rozniczkowanie_opt.zcos

**Fig. 27-80
**The separate

**D**action is used when the set point changes are rare. So we should optimize the

**y(t)**response acc. to disturbance

**z(t)**and not acc. to

**x(t)**. Let’s try to set

**Td=1 sec**.

Click “start”

**Fig. 27-81**

We have better reaction for disturbance **z(t) **and for **x(t) **too by the way. Compare with the **Fig. 27-79**.

**Rozdz. 27.10.4 Classical PD and PDy control comparison**

Call Desktop/PID/13_regulacja_typu_PD/40_porownanie_PD_PDy.zcos

**Fig. 27-82
**The upper

**PD**classical system is optimal considering the setting point

**x(t)**

The lower

**PDy**system is optimal considering the disturbance

**z(t)**

The disturbance

**z(t)=+0.5**will occure in

**30 sec**. You don’t see it on the oscilloscope.

Click “start”

**Fig. 27-83**

Conclusions for

**PDy**control

**1-**We don’t worry about setting point

**x(t)**reaction but reaction for disturbance

**z(t)**is important.

**2-**The

**PDy**reaction for

**x(t)**slower. It maybe a fault, but not important when

**x(t)**changes are rarely.

**3-**The advantage of the

**PDy**control are smaller conrol signals

**s(t)**amplitudes. It’s economically important and has a positive influence for so-called

**final**

**elements**. Valves life time for examply.

**Rozdz. 27.11 PD control main conclusions**

**1-** The same static attributes as **P **control–>the same steady error **e(t)=x(t)-y(t)**

**2-** Much better dynamic–>shorter setting time

**3- **More impervious to instabilities. It enables bigger **Kp** and smaller steady errors.

**4****– **The **PDy **controllers with separate output **y(t) **differentiation are often used.

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