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 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. 27-12x(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
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?
We will test real PD controller beacuse these are mainly used in the industry. The ideal PD controllers aeren’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?
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 control? The easiest answer is “There are such mathematics rules!”. Roma locuta causa finita.
But what does common sense say?
1P and PD controllers haven’t distinctions in steady state, because D component control signal s(t)=0 here.
2PD 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?
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
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 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

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

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.

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=5 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=5 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

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

Classical PD controller. The control signal s(t) is calculated considering the error e(t)=x(t)-y(t)
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.

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. 27-79
This is  Fig. 27-77b realization  Kp=10 i Td=5sek.
Click “start”
Fig. 27-80
The setting point x(t) response is very “lazy” now for t=3…30sec.
There are 2 reasons
– there isn’t peak  component in s(t) which is derived from differentiation. There is only smaller Kp component .
– the differentiating “braking” effect from growing y(t) lasted
The y(t) reaction for disturbance z(t)  for t=30…60 sec is the same as fo 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. 27-81
The big Td=10 sec parameter brakes to hard the y(t) signal. Let’s Td=1 sec. The braking should be smaller now. It’s optimization considering disturbance z(t) and not setting point x(t) as before.
Click “start”
Fig. 27-82
We have better reaction for disturbance z(t) for x(t) too by the way.

Rozdz. 27.10.4 Classical PD and PDy control comparison
Call Desktop/PID/13_regulacja_typu_PD/40_porownanie_PD_PDy.zcos
Fig. 27-83
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. 27-84
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 additional 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.
4The PDy controllers with separate output y(t) differentiation are often used.

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