**Automatics**

**Chapter 24 PD Control**

**Chapter 24.1 Introduction**The

**PD**controller belongs to the group of

**PID**controllers. Only the integration action has been disabled in it. Unfortunately, it has a basic defect of the

**P**controller. It does not completely eliminate the

**steady-state error**. The closed-loop gain

**Kz**and the steady-state

**error**gain

**Ke, Ke**are also the same as for

**P**-type control:

**Go back to the end of the previous**

Fig. 24-1

Fig. 24-1

**chapter**and you’ll know why. This means that, for example, for

**K=100**, the

**steady error**is approx.

**1%**. So what is

**PD**better than

**P**? Improves

**stability**and reduces

**control time**! For the P controller,

**stability**and a

**small steady-state**error are in contradiction with each other. A large

**Kp**provides a

**small steady-state**error, but

**oscillations**and

**control time**increase. The system may even become

**unstable**. Take a look at the typical

**P**-type control.

**Fig. 24-2**You received this response from the

**P**control system in the previous chapter (

**Fig. 23-9**). What can be done to make the oscillations smaller? The first brilliant idea – reduce the

**Kp**of the controller. E.g.

**Kp=5**. Great, the oscillations will be smaller. However, the

**steady error**will be

**twice**as large. Don’t go this way. So let’s not move

**Kp**, but let’s combine. The oscillations are because at the beginning of the

**x(t)**step the signal

**y(t)**increases too

**quickly**. In this way, the accelerated “pendulum”

**y(t)**goes to the “other side”. What if you stopped the pendulum during acceleration with

**braking**proportional to its

**speed**? Car hydraulic dampers work in a similar way. Maybe then he won’t go to the “other side”. Or “just a little”? Eventually

**y(t)**will settle. Then

**Kz**and

**Ke**in steady state will be consistent with

**Fig. 24-1**.

**Because the derivative unit in steady state is zero**. As if it was turned off! In this way, we come to some kind of control, which also depends on the

**speed**of the signal! So a differentiating

**unit**will appear. Let us recall the basic properties of this

**unit**, called

**D**->Differential.

**Chapter 24.2 The ideal differentiating D unit and the real differentiating D unit****Chapter 24.2.1 Introduction**We will examine the response to a

**step**and to a

**linearly increasing**signal. What’s the second for? Because the

**step**gives a

**Dirac**impulse in the

**differentiator**unit, which is difficult to analyze.

**Chapter 24.2.2 The ideal differentiating unit response to a x(t) step**

**Fig. 24-3**The

**x(t)**unit step was

**differentiated**. The result is a

**Dirac**hammer response

**y(t)**. Let me remind you that the differentiator “calculates” the speed of the signal

**x(t)**. And what is the speed of the signal

**x(t)**. For

**t<1sec x(t)=0**and

**for t>1sec x(t)=1**. For these

**t**the signal

**x(t)**is constant. So

**y(t)=velocity**is

**zero**. Therefore,

**y(t)=0**, as you can see in the attached picture.

And what is the velocity

**y(t)**for

**t=1**? Infinitely great! Summa sumarum the output

**y(t)**is a

**Dirac**pulse.

In

**control**, we try to avoid ideal

**differentiators**. They are very sensitive to disturbances, especially to rapid

**changes**, e.g. to electromagnetic disturbances. Rather, the so-called

**real differentiators**are used.

**Chap. 24.2.3 Real differentiating unit response to a x(t) step.**

**F****ig. 24-4**An

**inertial**unit with time constant

**T=0.1 sec**was added in series to the ideal

**differentiating**unit. It dampens the unit step before it enters the ideal

**differentiating**unit. We got something similar to a

**Dira**c impulse. Its height is no longer

**infinite**, but has the value

**y(t)=10**. You associate it somehow with the time constant

**T=0.1sec**? I guess so, it’s the inverse of

**0.1**. A unit step

**x(t)**gives a somewhat strange, hard-to-measure answer. Especially for the

**ideal differentiating**unit. Therefore, a

**linearly increasing**input signal

**x(t)**is better suited for their study. Another name is

**saw, ramp**….

**Chap. 24.2.4 Response of an ideal differentiating to a linearly increasing signal**

**Fig. 24-5**The

**differentiating**unit, as the name suggests,

**differentiates**the

**x(t)**signal to give

**y(t)**. For example, for students, the texts in the picture are trivial. For those who have problems with calculus, let it suffice that for the differentiator

**y(t)**is the velocity of black

**x(t)**. It is clear that in

**5**seconds the signal speed

**x(t)**

**doubled**. Therefore, the

**y(t)**as derivative of

**x(t)**also

**doubled**.

**Chap. 24.2.5 Response of a real differentiating to a linearly increasing signal x(t)**

**Fig. 24-6**That’s what we expected. Where the speed does not change, e.g. after

**2**and 6

**seconds**, it is calculated as a constant of

**0.5/sec**and

**1/sec**. However, when it changes (at

**1**and

**5**seconds), the

**real unit**needs some time to

**calculate**it. This can be a certain, though not decisive,

**disadvantage**. However, the

**advantage**will be resistance to “quick change” type

**disturbance**. They will simply be suppressed by the

**inertial unit**. Next, we will study the

**PD**units, which are a parallel combination of a

**proportional**and a

**differentiating**units (ideal or real). They are not yet

**controllers**, because the input is only a single signal

**x(t)**and not the error

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

**Chapter 24.3 The ideal and real PD unit****Introduction 24.3.1**

We will also examine the response to a** step** and a **linearly increasing** signal.**Chapter 24.3.2 Ideal PD unit response to a step **

Remember!

The **PD** element is not a **controller** yet! It does not have a comparator that calculates the** error e(t)=x(t)-y(t)**. Also applies to real controller

**Fig. 24-7**At first, the

**time chart**is similar to

**Fig. 24-3**. Only when you look closely will it turn out that there is a

**proportional**component

**y(t)=1**. For the ideal

**derivative**from

**Fig. 24-3**, this component was

**y(t)=0**.

**Chapter 24.3.3 Real PD unit response to a step**

How is it different from ideal

**PD**? Compare with

**Fig.**

**24-7**.

**Fig. 24-8**You can see the

**P**component (unit step in 3 seconds) and the

**real differentiating**component (“stretched” peak with a height of 10=11-1).

**Chap. 24.3.4 Response of an ideal PD unit to a linearly increasing signal**

**Fig.24-9**The input

**x(t)**is a linearly increasing signal that doubles the rate of increase in

**5 second**. In the response

**y(t)**you can see the differential

**D**and the proportional

**P**component. Only in the case when

**Kp=1**,

**Td=1sec**, the

**P**component perfectly coincides with the

**x(t)**excitation. By the way, you see the definitions of the

**differentiation**time

**Td**. This is the time

**t=1sec**for the proportional component

**P**to equal the derivative component

**D**.

**Chap. 24.3.5 Response of a real PD unit to a linearly increasing signal**

**F****ig. 24-10**The response is similar to the ideal

**PD**unit. “Smoothing” after

**1**and

**5**seconds (then there is a change in the speed of the signal

**x(t)**) result from the fact that the real

**differentiator**needs about

**0.5**seconds to “recover” and calculate the correct

**speed**. In steady state, the

**proportional**component

**P**and

**D**is clearly visible, which is the same as for the ideal

**PD**unit. I emphasize – the same but only in a steady state.

**Chap. 24.4 PD controller with a two-inertial object****Chap. 24.4.1 Introduction**

We will study the control system with a **two-inertial** object, and then in **chapter 24.7** with **three-inertial**. And why don’t we start with a **one-inertial** one? Because to control this object it is enough to control **P** from **chapter 23**.

Here it can be proved, e.g. from the **Hurwitz criterion**, that for any large **gain**, even for **Kp=1,000,000**, the system will always be **stable**. Not only that, there are no **oscillations** and the response will be almost a **step**.

For other objects, adding **to** a proportional **P** unit the differentiating **D** unit dramatically improves the response. The **oscillations** may disappear and the system which was **unstable** with the **P**-type controller may become **stable**.**Note**

In fact, we will be examining a **real** and not an** ideal** **PD** controller. i.e. in the **PD** controller, instead of the ideal **D**, there will be a real differentiating **D**. It will give a response proportional to the speed of the input signal, which is calculated with some inertia. See **Fig. 27-18**. This structure will reduce disturbances from high-speed signals that are typical of an ideal **D** unit.

First, let’s examine the **two-inertial** object itself. You did this in **chapter 23** in **p.23.4.2**.**Chapter 24.4.2 Two-inertial unit as an open loop**

**Fig. 24-11**Typical response of a multi-inertial object, here a

**two-inertial**one. We will further explore the

**PD**control with this object at

**Kp=10**and four different

**Td=0, 1, 5 and 20 sec**. Which

**Td**will give us the best response with as little

**oscillation**as possible and short

**control time**?

**Chap. 24.4.3 PD control Kp=10 Td=0 sec**

**Td=0**means that (real)

**D**differentiation is turned

**off**. So we have a typical

**P**control.

**Fig. 24-12****Kp=10 Td=0**So this is the typical

**P**control in

**Fig.23-9**from the previous

**chapter**. The steady-state gain

**Kz=y/x=0.91/1=0.91**is consistent with the

**time chart**and theory

**Chapter 24.4.4 PD control Kp=10 Td=1 sec**

The oscillations in

**Fig. 24-12**are a bit annoying. So let’s introduce a differentiation component

**D**into the fray. First carefully, a little differentiation. E.g.

**Td=1 sec**. I remind you that all the time we have a real

**differentiating unit**, not an

**ideal**one! Why? I explained earlier. And so it will continue.

What is the difference between the

**PD**controler above and the

**PD**element in e.g. Fig.

**24-8**? In addition to a different

**Kp**setting, of course. Only that the input is an error

**e(t)=x(t)-y(t)**and not a single

**x(t)**. The most important element of the controller appears here, the element that compares the

**setpoint x(t)**with the

**output**signal

**y(t)**. It is even more important than the rest of the computational part of the controller –

**P**and

**D**units. Because you can imagine a controller consisting only of a

**comparator**calculating the error

**e(t)=x(t)-y(t)**. It is simply a

**P**controller with

**Kp=1**setting. On the other hand, a controller without a comparison element is meaningless.

**Fig. 24-13****Kp=10 Td=1 sec**Didn’t I say that

**PD**improves dynamic properties? Compare only with

**Fig. 24-12**where there was no

**D**component. The

**oscillations**and

**control time**are radically less. And that was just a little differentiation of

**Td=1sec.**So what will happen when we increase

**Td**to, for example,

**Td=5 sec**? You will learn about it in a moment. As for the

**static**properties, the

**steady**state

**y(t)=0.91**is the same as for

**P**-control in

**Fig. 24-12**. This is obvious, because in

**steady**state the differentiation of

**D**gives a

**zero**output, i.e. it

**has no effect**on the control.

How to justify

**better dynamic**properties? In other words, a nicer approach of

**y(t)**to the setpoingt

**0.91**The easiest way is “Thus says the Queen of Sciences –

**Mathematics**. Exactly, it’s mathematical analysis.

And for the

**common sense**?

**1- Steady**state is the same as

**P**control in

**Fig. 24-12**.

**2-**The

**slew**rate

**y(t)**in the first second of the unit step

**x(t)**is

**faster**, because there is a

**differentiation**of the

**leading**edge

**x(t)**, which “boosts” the control signal

**s(t)**. In

**Fig. 24-14**, you will examine the same schema, just with different oscilloscope settings, giving you a full

**s(t)**view, not a level

**2**

**cut**view as shown in

**Fig. 24-14**.

It turns out that for

**t=3**(beginning of the step

**x(t)**) the control signal

**s(t)=110=100+10**!, where

**10**is the

**P**component and

**100**is the

**D**component. So

**PD**will kick up at the beginning of the step

**x(t)**

**11**times larger than the

**P**controller! This should bring it to steady state

**y(t)=0.91**faster. Not to mention that this kick is

**110**times greater than in an

**open system**, i.e. without a

**controller**.

**3-Okay**, the

**PD**regulator gave a powerful

**kick up**at the beginning of the

**x(t)**step. In this case, we should all the more expect large overshoots greater than

**ymax=1.2**in the

**P**control in

**Fig. 24-12**. And here it is just the opposite, a tiny overshoot of

**ymax=1**after

**5 seconds**.

How to explain it?

It turns out that the component

**D**not only “rushes” the object at the beginning of the

**x(t)**step, but then “brakes” it. Note that the

**D**component of

**x(t)**that dominates the control signal at the

**beginning decays**to

**zero**quite quickly after some time. This can be seen, for example, in

**Fig. 24-3**. Instead, there is a component

**D**of

**y(t)**acting as a “braking” component. At the beginning, this component is

**greatest**, because the velocity

**y(t)**is the greatest. Then when

**y(t)**settles, it will also disappear. But much later than the

**D**component of

**x(t)**! And the most important. The

**D**component of

**y(t)**is

**subtracted**from the

**D**component of

**x(t)**hence braking. Because that’s how the

**comparison**element works –

**e(t)=x(t)-y(t)**. And how does the

**blue**

**s(t)**brake! After all,

**s(t)**sometimes freezes with

**negative temperature**, although the set value is

**x(t)=+100°C**! But thanks to this, the system is very dynamic and quickly comes to a steady state. Differentiation

**y(t)**tries to prevent overshoots. Therefore, the

**PD**control time charts come to a

**steady state**faster and with

**less oscillation**.

**PD**control was not only created as a result of theoretical considerations of very smart people. The

**swordsman**must hit a specific location of the opponent. It does not have to do it with absolute

**accuracy**, e.g. to the

**millimeter**. It is enough to do it with an accuracy of

**5 cm**, but quickly! So it is guided not only by the

**location**of the target, but also by the

**speed of change**of the target’s

**location**. He acquires this skill subconsciously through many years of training, gradually optimizing the settings of his private

**PD**controller. Note that

**zero error**, i.e. absolute hit accuracy, is not required here.

I will show again the same time charts as in

**Fig. 24-14**with such settings of the oscilloscope so that the entire

**s(t)**control signal is visible.

**Fig. 24-14**It can be seen that

**s(t)=110**! But

**y(t)=0.91**in steady state, it’s almost on the time axis. The time chart shows us how large

**s(t)**can be. They are exactly the same as in

**Fig. 24-14**, just on a different scale.

Summary

**PD**at the beginning gives a large control signal

**s(t)**mainly from the differentiation of the

**x(t)**step so that y(t) comes to a

**steady**state faster. Then it

**brakes**with a

**differential**signal from

**y(t)**in order to reduce the pendulum effect, i.e. to reduce

**oscillations**. The result is a

**y(t)**time chart, that is clearly better than the

**P**control in

**Fig. 24-12**.

What if we could increase the intensity of

**differentiation**even more. E.g.

**Td=5 sec**.

**Chapter 24.4.5 PD control Kp=10 Td=5 sec**

**Fig. 24-15****Kp=10 Td=5 sec**

**Differentiation**is now

**5**times more powerful! It would seem that it cannot be better than in

**Fig. 24-14**. And here

**red**

**y(t)**is almost rectangular

**x(t)**! Compare with the

**red**

**y(t)**in

**Fig. 24-12**of the

**PD**control where

**Td=0**, i.e.

**P**-control. The signal

**s(t)**“cut” by the oscilloscope has the value

**smax=510**in

**3**seconds! So let’s go all the way and let

**Td=20 sec**. Maybe

**y(t)**even better, read-more “rectangular”?

**Chapter 24.4.6 PD control Kp=10 Td=20 sec**

**Fig. 24-16****Kp=10 Td=20 sec**We

**overdid**the differentiation. What too much is not healthy. Not only did

**oscillations**occur, but then the system reaches a steady

**state y=0.91**only after about

**50 seconds**. A

**s(t)**in

**3 seconds**has a value of

**2010**! This slow approach to the steady state can be explained by too strong “braking effect” of the

**D**signal coming from the

**y(t) differentiation**.

**Chapter 24.4.7 PD control Kp=100 Td=0**

The settings

**Kp=10**and

**D=5 sec**in

**Fig. 24-15**gave us a quick response, with the beginning of one oscillation barely visible. Only the steady signal

**y(t)=0.91**deviates too much from the setpoint

**x(t)=1**. Indeed, the error

**e(t)=1-0.91=0.09**does not bring

**immortal glory**. To reduce it, we increase the gain to

**Kp=100**. So we will study

**PD**control with a

**two-inertial**object with

**Kp=100**and four different

**Td = 0, 0.25, 2**and

**5 sec**. We will start as usual with

**Td=0**, i.e. with

**P**control (without differentiation).

**Fig. 24-17**

**Kp=100 Td=0 means**that

**D**differentiation is disabled. So we have a typical

**P**-control. We repeat the experiment in

**Fig. 23-11**from the previous

**chapter**.

**Note**

In previous time charts, the control signal

**s(t)**was a

**blue**line. However, this color is quite dominant and often muddied the drawing, a vivid example of which was, for example,

**Fig. 24-16**. Therefore, we will assign yellow to the signal

**s(t).**According to theory, the output signal

**y(t)**reached a fixed value of

**0.99**and the steady error

**e(t)**a value of

**0.01**. However,

**oscillations**and a long time to reach equilibrium are

**unacceptable**. We expect that the

**D**component will help again. As usual, we will start carefully with a small

**Td=0.25sec**.

**Chapter 24.4.8 PD control Kp=100 Td=0.25sec.**

**Fig. 24-18**

**Kp=100 Td=0.25 sec**

It is much faster and less “oscillatory” to reach the steady state

**y(t)=0.99**(and of course the steady state error

**e(t)=0.01**) than for the

**P**control (because

**Td=0**) in

**Fig. 24-17**. Or can it be better?

**Chapter 24.4.9 PD control Kp=100 Td=2sec**

**Fig. 24-19**

**Kp=100 Td=2 sec**

**y(t)**itself has a small oscillation. Instead, the control signal

**s(t)**is ho, ho! It is typical that the oscillations of the control signal

**s(t)**are much larger than

**y(t)**. Even when

**y(t)**seems to be asleep,

**s(t)**is still making some back and forth movements. The response is better than

**Td=0.25 sec**in

**Fig. 24-18**. So let’s increase

**Td**even more?

**Chapter 24.4.10 PD control Kp=100 Td=5sec**

**Fig. 24-20**

**Kp=100 Td=5 sec**

We overdid the

**differentiation**. The oscillations and control time are greater.

**Chap. 24.4.11 Conclusions from the PD control with a two-inertial object**

**1–The error**set for

**Kp=10**is

**0.09%**and for

**Kp=100**it is

**0.01%**and does not depend on

**Td**. So, as

**Kp**increases, the steady error

**e(t)**decreases. In steady state

**y(t)=0.99**and

**e(t)=0.01**.

**2–For**some optimal

**Td**, the

**D**component dramatically shortens the

**control time**and

**reduces oscillations**. For each gain

**Kp**there is an optimal

**Td**for which the response is

**optimal**. That is, with small oscillations (even without) and with a short control time. We found that for

**Kp=10,**

**Td=5 sec**is optimal and for

**Kp=100**

**Td=2 sec**. There are definitely better settings, because we only did a few experiments.

**3–It can**be proved, for example, using the

**Hurwitz**or

**Nyquist**criterion, that for any

**Kp**the

**control**with the

**two-inertial**unit is always

**stable**. Another thing is that

**oscillations**with a large

**amplitude**and long duration may then appear.

**Chapter 24.5 PD controller with a three-inertial object****Chapter 24.5.1 Introduction**We will do the same as with the

**two-inertial**. We expect more trouble. Indeed, the

**control time**and

**oscillations**will be greater. Moreover, with some gain, the system will become

**unstable**. We will study the control for gain

**Kp=10**and

**Kp=25**at different

**Td**. At

**Kp=25**the error will be smaller, of course, but at certain

**Td**the system will become unstable! This was not in the

**two-inertial.**

We’ll start with a bare object.

**Chapter 25.5.2 Three-inertial unit as an open loop**

**Fig. 24-21**No comment.

**Chapter 24.5.3 PD control Kp=10 Td=0 sec**

**Td=0**means that we have

**P**-type control. So we repeat the experiment from the previous

**chapter Fig. 23-13**.

**Fig. 24-22**

**Kp=10 Td=0**, i.e.

**P**control

Time chart with multiple oscillations and long control time. So the

**P**control does not work for this object. The steady-state gain

**Kz=0.91**, or

**y(t)=0.91**in the steady state, is consistent with the time chart and theory.

**Chap. 24.5.4 PD control Kp=10 Td=0.5 sec**

We start carefully with a small differentiation

**Td=0.5 sec**.

**Fig. 24-23****Kp=10 Td=0.5 sec**Much better than without the derivative in

**Fig. 24-22**. So let’s increase

**Td**by

**1.5 sec**. Maybe it will be even nicer?

**Chap. 24.5.5 PD control Kp=10 Td=1.5 sec**

**Fig. 24-24****Kp=10 Td=1.5 sec**The time chart is clearly better than

**Fig. 24-23**, and even more so than

**Fig. 24-22**.

Or can it still be improved? E.g. for

**Td=5sec**.

**Chapter 24.5.6 PD control Kp=10 Td=5 sec**

**Rys. 24-25****Kp=10 Td=5 sec**Worse than

**Td=1.5 sec**. Larger amplitude

**y(t)**and more

**oscillations**. It can be assumed that further increase of

**Td**will only make the situation worse.

**Conclusions**for

**Kp=10**.

It is clearly visible that the object is more difficult to control than the

**two-inertial**one. The optimal differentiation parameter is

**Td=1.5 sec**. The big steady error of

**0.09**is a bit annoying. Therefore, let’s try to increase the gain to

**Kp=25**. Probably not for more. I’m afraid of instability.

**I turned off differentiation giving**

**Chapter 24.5.7 PD control Kp=25 Td=0**

**Td=0**. So, as usual, we start with the

**P**control.

**Fig. 24-26**

**Kp=25 Td=0**

The system has become

**unstable**. Why did I give the formula for the

**steady**gain

**Kz**? Does it make sense when the system is

**unstable**? He has some. Here

**Kz=0.96**is the constant component of this

**sine**wave

**y(t)**. Let’s introduce a small differentiation carefully.–>

**Td=0.5 sec**. Maybe that will stabilize the system?

**Chapter 24.5.8 PD control Kp=25 Td=0.5sec**

**Fig.24-27**

**Kp=25 Td=0.5 sec**

Here you can see the wonderful influence of the

**differential**component

**D**. The system has become stable! Maybe a little too much oscillation. What if you give

**Td=1sec**?

**Chapter 24.5.9 PD control Kp=25 Td=1sec**

**Rys. 24-28****Kp=25 Td=1 sek.**Probably better. Let’s keep looking for luck, maybe

**Td=5 sec**?

**Chapter 24.5.10 PD control Kp=25 Td=5sec**

**Fig. 24-29****Kp=25 Td=5 sek.**We overdid it. Unacceptable.

**Chapter 24.6 PD control with disturbances****24.6.1 Introduction**

The same** 2** objects as before will be controlled: **two-inertial** and **three-inertial**. Their inputs will be supplied additionally (apart from the setpoint signal **x(t)** from the **PD** controller) with the **disturbance** signal **z(t)=+0.5** or **z(t)=-0.5**. These are powerful disturbances! It is hard to imagine that the **network** jumps from **230 V** to **345 V** or to **135 V**. I have deliberately provided such a caricature of the network in order to show the main purpose of the **control**. – **disturbances suppression**. This is often more important than getting to the setpoint **x(t)** nicely. If you have a fridge you know why. The temperature **setpoint x(t)** in your fridge rarely changes (often you have it set at the factory all your life!) and disturbances happen all the time. In the case of **disturbances**, we chose the settings of **Kp** and **Td** determined by us earlier, for which the response to **x(t)** was optimal. In other words, “the prettiest”.**Note**

Note that the response to a setpoint **x(t)** will be much **faster** than the response to a disturbance **z(t)**. So the

“**closed**” transmittance **G(s)** (“main?”) and the **disturbances Gz(s)*** are different. This leads to an important conclusion. When designing a system in which you know that the setpoint changes rarely, you should be guided by the optimal disturbances transmittance **Gz(s)**.

*It’s “**closed**” too but with additional disturbance z(t).**Chapter 24.6.2 Positive disturbance with a two-inertial object z(t)=+0.5 Kp=10 Td=5sec**The disturbance

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

**30 seconds**.

**Fig. 24-30**Up to

**30**seconds, i.e. until the appearance of a disturbance, the

**time chart**is the same as in

**Fig. 24-13**for obvious reasons. The disturbance

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

**y(t)**to increase

**to +0.95**. So the

**disturbance**was not completely

**suppressed**. Unfortunately, that’s the charachteristic of

**P**or

**PD**control. But without control, i.e. in an open system, the signal

**y(t)**would jump from

**1.0**to

**1.5**!

Note that the

**setpoint x(t)**response is

**fast**and the

**disturbance**response is

**slow**.

**Because the disturbance transmittance Gz(s) is different than that of the closed system G(s)!**

**Note**

**When**the setpoint

**x(t)**changes

**rarely**, select the controller settings due to the disturbance

**z(t).**

**When**the setpoint

**x(t)**changes

**frequently**, select the controller settings due to the disturbance

**z(t).**

**Chapter 24.6.3 Negative disturbance with a two-inertial object z(t)=-0.5 Kp=10 Td=5sec**

**Fig. 24-31**The disturbance

**z(t)=-0.5**caused a

**decrease**of

**y(t)**to

**0.86**. Note that the control signal

**s(t)**tries to

**compensate**for the disturbance

**z(t)**by acting in the

**opposite**direction to the

**disturbance**.

**This is how any well-designed control system works!**

**Chapter 24.6.4 Positive disturbance with a two-inertial object z(t)=+0.5 Kp=100 Td=2sec**

The disturbance

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

**30 seconds**.

**Fig. 24-32**.

The disturbance **z(t)=+0.5** resulted in an increase of** y(t)** to **0.995**. There is almost no disturbance effect! It was almost compensated by the decrease in the **yellow** control signal **s(t)**.**Note**

Signals established** before** and **after** the disturbance **z(t)****e(t)=0.01** and **e(t)=0.005** are close to **e(t)=0**

and**y(t)=0.99** and **y(t)=0.995** are close to **x(t)=1**

This effect is more visible in **Fig. 24-30** when the gain **Kp** of the controller is lower, i.e. **Kp=10**.**Chapter 24.6.5 Negative disturbance with a two-inertial object z(t)=-0.5 Kp=100 Td=2sec**The disturbance

**z(t)=-0.5**will appear in

**30 seconds**.

**Fig. 24-33**

The disturbance **z(t)=-0.5** resulted in a decrease of **y(t) to 0.985**. The impact of the disturbance is barely visible. Compare with **Fig.24-31** when **Kp=10**.**Chapter 24.6.6 Positive disturbance with a three-inertial object z(t)=+0.5 Kp=10 Td=1.5sec**The disturbance

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

**30 seconds**..

**Fig. 24-34**

Compare with **Fig. 24-30**. I wanted to write that the system reacts **slower** to **x(t)** and **z(t)**, and it turns out that it does. It reacts **slower** to a jump in the setpoint** x(t)**, but faster to a disturbance **z(t)**! And yet a **three-inertial** object is more difficult to control than a **two-inertial** one! Agreed, but as we already know, the optimal **Kp,** **Td** setting for the **x(t)** be suboptimal for **z(t)** and vice versa. For a **two-inertial** object and for **Td** less than **5 seconds**, the response to** z(t)** will be faster (read – **better**), while the response to **x(t)** will be more oscillatory (read – **worse**).**Chapter 24.6.7 Negative disturbance with a three-inertial object z(t)=-0.5 Kp=10 Td=1.5sec**The disturbance

**z(t)=-0.5**will appear in

**30 seconds**..

**Fig. 24-35**For a

**negative**disturbance

**z(t)**(e.g. cooling), the controller correctly reacted with an increase in the control signal

**s(t)**(heating).

**Chapter 24.6.8 Positive disturbance with a three-inertial object z(t)=+0.5 Kp=25 Td=1sec**

The disturbance

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

**30 seconds**.

**Fig. 24-36**

The** y(t)** signal swings a bit. Especially as a response to** x(t)**, because the response to a disturbance **z(t)** is pretty good. To a **positive** disturbance (heating), the controller reacted with a **decrease** in the control signal s(t) (cooling). Thanks to the greater gain **Kp=25**, the error **e(t)** in the steady state is** smaller** than for **Kp=10** in **Fig. 24-34**.**Chapter 24.6.9 Negative disturbance with a three-inertial object z(t)=-0.5 Kp=25 Td=1sec**The disturbance

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

**30 seconds**.

**Fig. 24-37**

No comment.

**Chap. 24.7 Comparing the PD controller with the P controller****24.7.1 Introduction**

It’s just like **FC Barcelona** and and **small country football club**. We will **simultaneously** give the same unit **step x(t)=1** with disturbance** z(t)=+0.5** in **30** seconds. Can you guess which one is **Barcelona**? We will examine **2** cases with the setting** Kp=10** and** Kp=100** for a **two-inertial** object.**Chap. 24.7.2 PD controller Kp=10, D=5 sec and P controller Kp=10****Fig. 24-38Kp=10 Td=5 sec**Scheme

**2**of

**P**and

**PD**controls with the same

**two-inertial object**.

**Fig. 24-39**

**PD**is unbeatable for the set value

**x(t)**. However, with the disturbance

**z(t)**, the result is debatable, but rather with an indication of

**PD**. But you can always set the

**PD**so that it suppresses disturbances

**better**, at the expense of control for the setpoint

**x(t)**. The states established for the

**PD**and

**P**controllers are, of course, the same. In the

**PD**controller, the time to reach a steady state is much

**shorter**(read-better) than for the

**P**regulator.

**Chap. 24.7.3 PD controller Kp=100, D=2sec and P controller Kp=100**

Fig. 24-40

Fig. 24-40

**Kp=100**

**Td=2 sec**

Scheme

**2**of

**P**and

**PD**control with the same

**two-inertial**object

**Fig. 24-41**A crushing advantage of

**red ypd(t)**over

**green yp(t)**. That is,

**PD**over

**P**in response to a unit step

**x(t)**. Especially since

**P**gives terrible

**oscillations**. As for the

**suppression**of the powerful disturbance

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

**30**seconds, for

**PD**and

**P**it is so good at

**Kp=100**that the influence of the disturbance is almost invisible. So let’s look again at these time charts with a different oscilloscope range

**of 0.98…1.02**. So we’re going to use a l

**oupe**.

**Fig. 24-42**Here it is clear that the response to a

**x(t)**and

**z(t)**is clearly

**better**for the

**PD**controller.

**Chapter 24.8 PD controller with separate differentiation of the output signal y(t)****Chap. 24.8.1 Introduction**

The name of the **PD** controller in the title is quite long. Let it be abbreviated as **PDy**. Let’s look at the figure below.

**Fig. 24-43**So far, we have used a classic

**PD**controller such as in

**Fig. 24-43a**. The error

**e(t)=x(t)-y(t)**is amplified and differentiated. This is the case in many textbooks and automatics courses. It turns out that when there are

**slow**processes, e.g. in the

**chemical**industry, we more often encounter

**PDy**regulators from

**Fig. 24-43b**. Here

**only**the output signal

**y(t)**is differentiated, not the error

**e(t)**.

By the way – I use the term “

**differentiated**” for short, instead of “

**differentiated with an inertia of 0.1 sec**“. Let me remind you that this

**inertia**is used to

**suppress**fast-changing

**disturbances**.

Let’s get back to the topic.

**Let’s compare**the

**2**above-mentioned

**structures**in 2 different

**states**.

**1- Steady**state when

**x(t),y(t)**and

**z(t)**are

**constant**.

Then the

**differentiating**component

**D**is

**zero**. As if the

**differentiating**branch of the

**PD**controller disappeared. In this state,

**both**controllers behave as

**P**controllers and give an output signal

**y=Kz*x**and an error

**e=Ke*x**. Here, the parameters

**Ke**and

**Kz**are the

**gains**from

**Fig. 24-1**. The

**steady**state

**times**for each structure will of course be

**different**.

**2- When a**disturbance

**z(t)**occurred in the steady state.

**Example**– The temperature in the furnace reached, e.g., after

**15**minutes, the setpoint

**x(t)**value, and after

**25**minutes, someone turned on the additional

**heater**in the furnace

**manually**. I emphasize not the controller, but someone-a man turned on the

**additional**heater. Typical

**positive**disturbance

**z(t)**, to which the

**controller**should react by reducing the power supplied to the furnace. In Fig. 20-81, it will turn out that both regulator structures (in

**Fig. 24-43a**and

**24-43b**) will react identically to a

**disturbance**.

I will try to explain it “

**with common sense**” and “

**with the operational calculus**“.

**With common sense**

The beginning of the jump

**x(t)**occurred a long time ago. Then we can assume that the

**x(t)**signal is

**constant**. So it does not affect the differentiating branch of the controller

**D**. Then the

**D**by the

**y(t)**signal, or rather –

**y(t)**(with a minus sign!), and even more precisely

**-Kp*y(t)**.

This corresponds exactly to

**Fig.24-43b**.

**With the operational calculus****The calculated**

Fig. 24-44

Fig. 24-44

**Laplace**transform

**S(S)**of the control signal

**S(t)**shows that for constant

**x(t)**the schemes

**Fig. 24-44a**and

**Fig. 24-44b**are equivalent.

**Chapter 24.8.2 The PDy control of a two-inertial object with disturbance z(t)=+0.5**

**F****ig. 24-45**Implementation of the scheme

**Fig. 24-43b**for

**Kp=10 and Td=5sec**.

This is the response of the

**PDy**control, i.e. the

**PD**control with separate

**differentiation**. The signal

**y(t)**very slowly reaches the steady state

**y=0.91**and barely made it before the disturbance. The controller reacted correctly to the disturbance

**z(t)=+0.5**by lowering the control

**s(t)**. Compare the

**time chart**in

**Fig. 24-15**with the classic structure of the

**PD**controller. That is, with the version from

**Fig. 24-43a**. Until a disturbance

**z(t)**appears, i.e. up to

**30 sec**. the system reacts

**very slowly**! And that’s because it was deprived of the differentiating kick from the setpoint

**x(t)**in

**3 seconds**. Only proportional kick

**s(t)=10**works, not

**s(t)=110**as in

**Fig. 24-15**. Then the weakening signal from the proportional component

**P**works (because

**e(t)**decreases). In addition, the differentiating component

**D**brakes powerfully! It stopped braking only after

**30**seconds, when y(t) became

**stable**.

**Conclusions**

In

**Fig. 24-15**we had a classic version of the

**PD**controller. For it, we selected such settings

**Kp=10**and

**Td=5sec**so that the respone

**y(t)**was

**optimal**.

We’re getting into a different topic here. What does optimal mean? There are many definitions on this. Intuition should be enough for us. The answer

**y(t)**as

**quickly**as possible with a

**relatively small**one

**oscillation**. The

**Kp**and

**Td**settings, adopted by

**trial**and

**error**, roughly provided us with optimality defined in this way. Another definition may be adopted. E.g. the shortest time chart

**y(t)**without oscillation. It does not swing, but the control time will be longer.

**Let’s go back**to the conclusions. The settings

**Kp = 10**and

**Td = 5 sec**in

**Fig. 24-45**ensured the shortest control time with the setpoint

**x(t)**. Even then, it might have come as a surprise that the answer to

**z(t)**was suboptimal. A little long. What it comes from? It’s just that the signal path between

**z(t)**and

**y(t)**is different than between

**x(t)**and

**y(t)**. So the

**input**transmittance (between

**y(t)**and

**x(t)**) is different than the

**disturbance**transmittance (between

**y(t)**and

**z(t)**). And this means that settings that are optimal for

**input**transmittance are not

**optimal**for interference and vice versa.

And here we come to the most important conclusion.

**1 – When the setpoint**value

**x(t)**changes frequently, we use the

**PD**controller version, i.e.

**Fig. 24-43a**

**2 – When the setpoint**value

**x(t)**rarely changes, we use the

**PDy**controller version, i.e.

**Fig. 24-43b**

**Example 1**is a toy-radio-controlled car. The

**input**is the steering wheel, which you have on the control panel with the radio, and the output is the turning angle of the car wheels proportional to the movement of the steering wheel.

**Example 2**is a refrigerator. It is often the case that people do not change the set temperature value

**x(t)**for years. This is an extreme example, but in the chemical industry, for example, the setpoint value x(t) changes rarely, while disturbances occur all the time. It is not important to quickly reach the setpoint

**x(t)**, but to quickly respond to the

**z(t)**disturbance.

I forgot about the other important plus of the

**PDy**regulator. The time charts

**y(t)**are smooth. The control signal

**s(t)**is also gentler. It may not be so important in the picture, but when, for example, a valve on the pipeline during overregulation constantly “beats the lid” from

**opening**to

**closing**, it will not live long! In the next experiment, we will set the

**Td**setting in the

**PDy**controller due to the optimality of the time charts from the disturbance

**z(t)**.

**Chap. 24.8.3 PDy control of a two-inertial object with disturbance z(t)=+0.5 with optimal setting TD=1 sec**

**Fig. 24-46**This is the implementation of the scheme

**Fig. 24-43b**for

**Kp=10**and

**Td=1sec**. We gave a smaller differentiation

**Td=1 sec**due to the optimal attenuation of the disturbance

**z(t)**(and not the

**x(t)**jump as before).

Probably a faster response to the

**z(t)**disturbance than before. In addition, the response to the jump

**x(t)**improved, although it is not as good as in

**Fig. 24-43a**for the classic

**PD**controller. But for the rarely changing set value

**x(t)**it is not the most important.

**Chap. 24.8.4 Comparison of the classic PD control with the PDy control with separate differentiation**

Let’s compare the classical

**PD**control optimal for

**x(t)**(which is non-optimal for

**z(t)**) with the

**PDy**control optimal for

**z(t)**(which is suboptimal for

**x(t)**). A little cloudy, but we know what the author was trying to say.

**The unit step**

Fig. 24-47

Fig. 24-47

**x(t)**simultaneously affects the classical

**PD**control system and the control system with separate

**PDy**differentiation. These are optimal systems due to the setpoint

**x(t)**(upper diagram) and disturbance

**z(t)**(lower diagram).

**Fig. 24-48**We only observe the

**x(t), y1(t) and y2(t).**The disturbance z(t)=+0.5 (which is not visible!) will appear in

**30**seconds.

**Conclusions**

**1-In the control**with separate differentiation of

**PDy**, we are less concerned with the response to

**x(t)**in which changes occur relatively rarely. Therefore, we can focus on optimization due to disturbances

**z(t)**that occur often. So the basic role of the

**PDy**controller is optimal

**z(t)**disturbance suppression, and not optimal (read –

**nice**) reaching the value determined by

**x(t)**.

**2- An additiona**l advantage of the

**PDy**control is the smoother control time charts

**s(t)**.

**Chap. 27.9 Conclusions from the PD control****1- Steady** state error the same as for the** P** control. Non zero!**2- Much better** dynamic properties than the **P** control**3- There **are **PD** controllers with error differentiation **e(t)=x(t)-y(t)** and** PDy** controllers with **only y(t)** differentiation.