### Automatics

**Chapter 1 Introduction**

**Introduction ****Chapter 1 **

**If you** want to know how a regulator works, including the **king of regulators – PID**, **then this course is for you!** Especially if you don’t feel confident in the world of integrals and derivatives. **You will independently** explore various objects in real time! It is important that the waveforms will be of “normal” duration – usually in the range of **10 … 120 sec**. You can see everything moving and crawling! Not so when the waveform lasts only **30 µs**, or vice versa **30 years**. It is also completely different than time charts in books, e.g. regarding the **inertial unit**. That is, to put it simply, observing the increase in the temperature of the furnace when I give the heater abruptly (“suddenly”) **230V** voltage. **You’ll notice** that the rate of temperature **increase** (not the temperature **itself**!) is greatest at the beginning, then gradually decreases. Eventually, the temperature will stop at, for example, **300°C**. When I repeat the experiment with half the previous power, the course will be similar, but instead of **300°C** it will be **150°C**.

The **inertial element** as a furnace model is very simplified. The time chart of a real furnace will be slightly different. For example, the initial rate of temperature increase is **zero**. Also at half the power will be, for example, +**170°C**. and not +**150°C** But that’s a completely different story…**There is very little in the course for “word of honor”. We try to check everything.**For example. We check according to the

**Hurwitz criterion**whether a given object

**G(s)**is stable, and then by touching the input with an impulse

**x(t)**, we try to throw it off balance. Learning consists in developing individual schemes of

**Control Systems**and examining their properties.

The further into the forest, the more trees there are. We start by examining the simplest dynamic units –>

**Chap. 2…10**and we end up with more interesting topics such as

**Cascade Control**,

**Closed-Open System**,

**Ratio Regulation**…

Your job is just to study the influence of various parameters on the time charts and draw appropriate conclusions. You will feel like a dispatcher in the Refinery Control Room , watching the technological process on monitors.

Nay. You can do forbidden things! For example, drill a hole in the column through which the distillate escapes. This example is one of the basic concepts of so-called automation

**disturbance**. The

**control system**will compensate for the disturbance (escaping distillate) with an additional inflow. You can also change the settings of the

**PID**controller so that the oscillations do not go out. In this way, you safely gain experience, like a pilot practicing dangerous situations on a

**Jumbojet simulator**.

**I emphasize**. These will not be ordinary “static” drawings, e.g. response to a unit step of an inertial unit, but a real cinema with

**mp4**files! I performed all the experiments in the

**SCILAB**program, more precisely in its main application

**Xcos**. Then I recorded their waveforms using the video program

**Active Presenter**(“cinema”). So you don’t have to go into

**SCILAB**, just press the

**start video**button. I guarantee you that you will feel the dynamics of the process better than on ordinary charts.

**Note!**

If you’ve had little to do with automatics so far, the rest of this chapter, which summarizes the entire course, may be a bit intimidating. Then don’t worry. Read it to the end and move on to

**Chapter. 2 Proportional term**.

**Basic Dynamic Units → Chapters. 2…10**

**Each** chapter is an examination of a separate dynamic unit, from the simplest to the more complex. It consists in providing the appropriate input signal **x(t)** and observing the output signal **y(t)**.**The source** of the input signal **x(t)** can be:**– Unit step function** generator – most often**– Ramp signal** generator**– Virtual potentiometer** slider**– Dirac impulse**, i.e. a “short tap of the input with a hammer”**The output** signal **y(t)** can be observed by:**-an oscilloscope**, i.e. your monitor – most often**-digital virtual** meter**-bar graph**, i.e. an analog virtual meter in the form of a black vertical line

You will associate the transmittance parameters **G(s)** with the response **y(t)** to the input** x(t)**. The most common is a **single step**.**Animation example**

**Fig. 1-1Oscillatory Unit** when the input is a

**unit step function**

To start the animation, click on the video triangle.

The parameter

**K=2**and

**T=2sec**can be read from the time chart

After reading chapters

**2…10**, you can easily predict, for example, the behavior of

**2**similar-looking dynamic members in

**Fig. 1-2**.

**No Higher Math!**

Fig 1-2

Fig 1-2

**A bit of maths → Chapters. 11…14**

If you are familiar with the topic, skip to

**Chapter 15.**

**I tried to make the course useful also for people who are not familiar with higher math.**

Therefore, it contains several chapters on derivatives, integrals, simple differential equations and operational calculus. Because how can you not understand integration and differentiation when you have:

**– differentiator**which differentiates the input signal,

**– integrator**that integrates the input signal

**Chapter 11 Differentiation**If you have a problem with the derivative, you will be enlightened by examining the

**differentiator**.

**Fig.1-3**The input signal is the parabola

**x(t)=t²**.

It is differentiated twice by the differentiator. The experiment shows that the first derivative of x'(t)=

**2t**and the second derivative of

**x”(t)=2**. It agrees with the theory.

**Chapter. 12 Integration**Why exactly a unit step

**x(t)=1(t)**? Because it is difficult to find a simpler function and it is easy to calculate the

**integral**denoted as the

**field S**from it. Let’s treat the

**definite integral**as the output

**y(t)**of the

**integrating unit**whose input is

**x(t)=1(t)**. It turns out that

**y(t)=t(t)**.

**Note:**

**t(t)**this is a function

**y(t)=0**for

**t<0**

**y(t)=t**for

**t>0**

**Fig.1-4**Also, the integral as the area under the function will become obvious after examining the integrating unit. The area under the function

**x(t)=1**is

**y(t)=t(t)**. And that’s nothing but the

**integral of x(t)=1(t)**.

**Chapter 13 Differential Equations**You didn’t know it then. But you have already dealt with

**differential equations**in primary school by solving a problem like “a train goes from city

**a**to city

**b**with speed

**v**“… After all, speed is a

**derivative of the distance**, so…

**Fig.1-5**

We will learn differential equations on the basis of less trivial examples:

**A**– differential equation of filling the tank without a hole.

**B**– simplified differential equation of filling a tank with a hole.

**C**– exact differential equation of filling a tank with a hole.

I will show animations as solutions to the above differential equations.

**Chapter. 14 Operational Calculus**

**Fig.1-6**Each time function

**f(t)**can be assigned a transform

**F(s)**and vice versa–>

**Fig. 1-6a**. The parameter

**s**is the so-called a complex number. If you don’t know

**complex numbers**, treat them temporarily as

**real numbers**. In other words – don’t worry.

**Fig. 1-6b**is a version of

**Fig. 1-6a**for the specific function

**f(t)=sin(t)**.

How is the transform

**F(s)**derived from

**f(t)**?

Never mind! Suppose there is such a smart book with all possible pairs

**f(t)<–>F(s)**.

The

**Operational Calculus**has one nice feature. He would be worthless without her. It is easy to calculate the transform from the derivative of the

**f(t)**function, i.e. the transform from

**f'(t)**. Simply multiplying

**F(s)**by

**s**like this:

How is the transform

**F(s)**derived from

**f'(t)**?

**This makes it easier to solve**

Fig.1-7

Fig.1-7

**linear differential equations***. They are converted into ordinary algebraic equations in which

**polynomials**of the

**nth**degree of the variable

**s**occur. And from them it is already possible to draw conclusions about automatic control systems.

What are the waveforms, what is the steady state output, what is the stability? e.t.c..

***Linear differential equations**are an approximation to most control systems. Examples of these are differential equations

**A**and

**B**in

**Fig.1-5**. Equation

**C**, however, is not

**CONTROL IN GENERAL → Chap. 15…22**

**Chap. 15 More about transmittance and connecting blocks**You will learn that:

The transmittance

**G(s**) is equivalent to the

**differential equation**describing a given dynamic object from the parameters

**G(s)**we can easily determine the steady-state gain

**K**

Connected block transmittances:

– in series

– in parallel

– with negative feedback,

can be replaced by a single equivalent transmittance

**Gz(s)**.

**Chap. 16 How does feedback work?**You read the manual of a washing machine, cell phone or something else and you have problems. Although the guy writes wisely and in beautiful language, but something is missing. Exactly. The person writing the instructions is deep in the subject, because that’s all he does. For him, the operation of the device is as obvious as the operation of a flail. He simply does not want to offend the User with an

**exact**translation of

**how it works**.

The same is true with

**negative feedback**. After all, all this follows from the formula shown below

**Fig. 1-8c**! It concerns the transmittance

**Gz(s)**of a system with negative feedback. In addition, this formula is very easy to derive. But how does

**negative feedback**really work? The same

**y(t)**signal at the output and input? A snake eating its own tail? Why the output signal

**y(t)**“tries” to imitate the input signal, so called

**set value x(t)**?

A good understanding of this problem is understanding the essence of automation. Without it, various Nyquists, Hurwitzes, state spaces… are worth nothing.

So let’s move on to the principle of operation of the flail.

**The chapter begins with the positive feedback in**

Fig.1-8

Fig.1-8

**Fig. 1-8a**, which is easy to understand. You will learn that not always a system with

**positive feedback**is

**unstable**. It becomes it only with higher

**K*** amplification.

You will understand why in the

**negative**feedback circuit in

**Fig. 1-8b**, the

**y(t)**output “tries” to follow the

**x(t)**input. The most important thing is to understand the principle that the system tends to a steady state in which:

**y(t)=K*e(t)**

Like a pendulum that tends to its lowest point. You will see that the values of

**K*e(t)**and

**y(t)**are attracted to each other. At the beginning of the

**x(t)**jump, the difference between

**K*e(t)**and

**y(t)**is large, then it decreases until the equilibrium state where

**y(t)=K*e(t)**!

If the “clinging” waveforms

**K*e(t)**and

**y(t)**are obvious to you, then you are feeling negative feedback! From the equilibrium state

**y(t)=K*e(t)**follows the formula for the gain

**Kz**of a closed system in steady state

**Fig.1-8d**. It is, moreover, a special case of the transmittance

**Gz(s)**of the closed system in

**Fig. 1-8c**.

The

**y(t)**does not always end with a state of equilibrium, but with undying vibrations. The system is unstable. Then the formula for

**Kz**makes little sense, but not entirely. Namely, the oscillations have a constant component

**c=Kz*x(t)**where

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

*

**K**is the gain of the object

**G(s)**in steady state.

**Chap. 17. Instability, or how oscillations are created?**

We will conclude that the cause of the instability is the delay between the input and output signal. The easiest way to see this is to control pure delay.

**Fig.1-9**It will be found that when the

**K**amplification is:

**– K<1**then the system is

**stable**

**– K=1**the system is

**on the verge of stability**

**– K>1**the system is

**unstable**

**Fi**** g. 1-10**An example of an unstable system when

**K>1**

A short impulse x(t) throws the system out of balance

**Isn’t this similar to the Nyquist Criterion, which will be discussed in the next chapter?**

Most often we deal with

**multi-inertial**objects in which the delay is

**“fuzzy”**.

The principle is no longer so simple, but similar.

If in

**Fig. 1-9**instead of “pure” delay there will be, for example, a

**three-inertial unit**, then we will conclude that when:

**– K**is

**small**, then the system is

**stable**

**– K**is average, the system is

**on the verge of stability**

**– K**is large, the system is

**unstable**

Conclusion:

Conclusion:

**High gain**and

**inertia**of the object favors

**instability**.

**Chapter 18. Amplitude-Phase Characteristics**You will learn the concept of

**amplitude-phase characteristics**in the most natural way. You will determine it experimentally for a specific

**inertial unit**by feeding

**sinusoids**of different frequencies to its input.

**Chapter 19. Nyquist Stability Criterion**An

**open**system is generally stable. It can become unstable only after being closed with a feedback loop. The

**Nyquist Criterion**can predict, based on the

**amplitude-phase characteristics**of an open system (which is “easier” than a closed one), the

**stability**of a

**closed**system.

**Nyquist**belongs to the

**frequency criteria**.

**Chapter 20. Hurwitz Stability Criterion**Unlike

**Nyquist**, the

**Hurwitz**Criterion is

**algebraic**. We study the stability of the transmittance

**G(s)**based on the coefficients of its

**denominator**-polynomial

**M(s)**. The

**Hurwitz Criterion**is not difficult, but quite tiring. So you can skip this topic and move on. The most important thing about this course is everything else–>

**chapters 21…31**.

**Chapter 21. On-Off Control**The control algorithm is very simple.

**When**the output signal

**y(t)**is:

–

**greater**than the set value

**x(t)**then

**reduce**the control signal

**s(t)**

–

**smaller**of the set value

**x(t)**then

**increase**the control signal

**s(t)**

**Fi****g. 1-11**

**On-Off**Control with

**z(t)=+30°**disturbance (additional heating).

For up to

**35**seconds, the system tries to maintain the temperature around the set value

**x(t) = +50°C**. At

**35**seconds, a disturbance occurred, i.e. additional heating

**z(t)=+30°C**. As you can see, the disturbance has been suppressed. I.e. the average temperature y

**(t)=+50°C**is still maintained. The

**heating**periods when

**s(t)=100°C**are shorter than the

**cooling**periods

**s(t)=0°C**.

**Note:**

The

**Fig 1-11**time chart represents the above algorithm but corrected by so-called hysteresis.

**Chapter 22. Continuous Control**This type of control already provides a constant

**steady-state**control signal from the controller and a constant

**steady-state**output signal. You will also get tired as

**Mr. Continuous Controller**and you will also get complexes in relation to the stupid differential amplifier that acts as a

**controller**.

The

**Continuous Controller**will strive for the

**state of equilibrium**:

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

where

**x(t)**is the unit step and

**Kz**is the gain in

**Fig.1-8d**.

**CRÉME DE LA CRÉME THAT IS CONTINUOUS PID CONTROL ****Chapters 23…27**

Generally, I rely on intuition when writing, for example, “the regulator thinks that something is there and reacts like this and not like that”. I am trying to explain the operation of the regulator very thoroughly. and especially what the **Kp, Ti** and **Td** settings are responsible for. Manual tuning of the regulators will certainly help here. You will be looking for the **optimal*** response to a unit step in the input signal **eithe**r to a disturbance. The conjunction **either** did not appear by chance.

Don’t be intimidated by the size of **chapters 23…27**, but all experiences are reproducible with different parameters.

We will look for optimal settings for the objects:**– one inertial** unit**– two inertial** unit**– three inertial** unit

We will also study the effect of **z(t)** disturbances which will be really powerful, rather unheard of in real systems. It will be **z(t)=+0.4** (heating) and **z(t)=-0.4**(cooling)*****There are different optimality criteria. We want the system to reach a steady state relatively quickly, even at the expense of small oscillations.

**Chapter 23 P Control**You will find that this control is not accurate, there is always a control error

**e(t)**. This can be reduced by increasing the only setting of this regulator –

**Kp**gain. Oscillations and even instability may then occur. At equilibrium, the output signal

**y(t)**will be reached with the following value:

**Fig.1-12**Here

**K=Ko*Kp**is the static gain of the entire open loop, i.e. taking into account the steady-state gain of the object

**Ko**and the gain

**Kp**of the controller. The formulas show the steady state for the

**x(t)**step. For large

**K**, they approach very desirable in automatics formulas:

**– y(t)=x(t)**

– e(t)=0

Then the output signal

– e(t)=0

**y(t)**mimics the input signal

**x(t)**in steady state, or what comes to the same thing, we have

**zero**error

**e(t)**.

**Fig.1-13**Example of

**P**control with negative disturbance

**z(t)=-0.2**

See how the disturbance

**z(t)=-0.2**is suppressed. That is compenasated by additional heating in

**s(t)**control signal. But you can always choose

**Kp**and

**Td**so that it is the other way around. I.e. It responds quickly to

**z(t)**but slower to

**x(t)**.

**Note**that the control error is

**not zero**as is typical for

**proportional control**.

**Chapter 24 PD Control**It gives the same control error as the

**P-control**, but the differentiating component

**D**greatly improves the dynamic properties. It reduces oscillations and successfully fights instability. You’ll find out in a moment.

**Fig.1-14**Example of

**PD**control with positive disturbance

**z(t)=+0.5**

I told you so? Compare with the

**P**Control in

**Fig. 1-13**. Non-zero static error as for

**P**Control. On the other hand, it greatly improves the course of regulation. Here, especially per unit step

**x(t)**.

**Chapter 25 I Control**Its

**operating algorithm**is easier than the

**P**-regulation and

**similar**to the

**on-off**control:

when

**x(t)>y(t)**–>

**increase**the control signal

**gradually**

when

**x(t)<y(t)**–>

**decrease**the control signal

**gradually**

The

**decrease/increase velocity**is proportional to the error

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

**e(t)=0**, the velocity is

**zero**, i.e. it is in a

**steady state**. Unlike

**on-off**control, the steady-state control signal is

**constant**. The most important feature is

**zero**control error in

**steady state**. Often novice automation specialists have a problem with this. The input is

**zero**and the output is

**“not zero”**?

**I Control**is very slow and hardly used in practice. It is mainly used in teaching

**Fig.1-15I** control.

It does indeed reduce the error to

**zero**. But how bovine! Compare with

**PD**in

**Fig. 1-14.**

**Chapter 26 PI Control**

The

**P**control is fast but does not bring the error to

**zero**, while the

**I**control brings the error to

**zero**but is slow. Then let’s create a combination of

**P**and

**I**called

**PI**that will be

**fast**(but not as fast as

**P**) but brings the error

**e(t)**to

**0**. We’ll start with a didactic

**PI**that I tinkered with. It starts as

**P**and changes to

**I**at some point.

**Fig.1-16**

**PI**control with positive disturbance

**z(t)=+0.5**

The steady error is

**zero**and comes to a steady state relatively quickly.

**Chapter 27 PID Control**After adding the derivative component

**D**to the

**P**I controller, we obtain a

**PID**controller which:

-also reduces the error to

**0**:

-however, it does it much faster than

**PI**, much less

**P**and even “all the more”

**I**

The

**D**component prevents instabilities and overshoots.

At unit step each

**PID**controller starts as

**PD**, then the

**P**and

**D**components gradually decay to

**zero**and the

**I**component increases from

**0**to steady state

**y(t)=x(t)**, in steady state it always ends up as an

**I**controller.

**Fig.1-17**Example of

**PID**control.

**Zero**error and the

**fastest**of all. The good example of the

**z(t)**disturbance damping.

**Chapter 28 Controller Settings**

You learned the **first** method, i.e. **manual**, in **chapters 23…27**. It was mainly used to make you feel what this setting is, so that you know what the **Kp, Ti, Td** settings do. There are people who assemble cabinets, regulators, valves and connect it all with cables in accordance with the automation design.**There are also those who run these systems, remove assembly and software errors. They make the Control System come alive.**I do not want to claim that this is the elite of automation. This may not be correct, but some people think so. And it is these people who tune the regulators so that the system reacts as quickly and accurately as possible. Then the system works for years. It can be accurate and fast, or inaccurate and slow. And people are surprised that the product of one company is good and expensive and the other is trashy and cheap. That is why the work of

**piano tuners**is so responsible, pardon the

**PID controllers tuners**.

I will present some old-fashioned methods invented during World War II by a certain Ziegler and Nichols:

**-Step Response method.**

**-Limit Cycle method.**

Each is an example of a different Philosophy. The first requires a mathematical description, i.e. the

**Go(s)**transmittance of the object. The second is only to examine the object, or more precisely to measure the period of vibration at a certain

**critical**amplification. So she is less fussy about the exact knowledge of the object.

To finish the topic, many modern controllers (most?) have a built-in self-tuning mechanism. Observing the input and output, the controller itself changes the parameters

**Kp, Ti, Td**. They are so smart that you better not approach without a stick. Let’s just trust that numerous doctoral and postdoctoral theses written in the

**PID controller**software are ok.

### OTHER THINGS YOU NEED TO KNOW –>29…31

There are still a few important topics that are needed to understand the essence of **automation**.

**Chapter. 29 Disturbances Analysis**

The main signal acting on the **Control System** is the set **value x(t)**. But there are other so-called disturbances **z(t)**, which are constantly scheming, as if to harm here. More precisely, so that, for example, the output temperature **y(t)** instead of the one set by **x(t)=+100°C** was, for example, **y(t)=+96°C**. Steady state of course!

We are only interested in the disturbance acting on the **Go(s)** object. Because if the disturbance **z(t)** acts on the regulator and adds to the set value **x(t)**, it is scary to even think! It was as if a spy had sneaked into the enemy army’s headquarters and started issuing orders. Fortunately, it is easier to secure a **headquarters** than an **entire army**, just as it is easier to protect against disturbances a tiny **PID** controller than a large **Go(s) **object.

Rys.1-18

**Chapter. 30 Control Systems Structures**The quality of control can be improved not only by tuning the

**PID**, but also by changing the structure of the control system.

Typical Structures are:

**1– Open loop**

**2– Open loop with the disturbance compensation**

**3– Closed loop**

**4– Closed loop with the disturbance compensation**

**5– Cascade control**

**6– Ratio control**

The entire course so far has been about

**Open**and

**Closed**systems. Therefore, we will briefly discuss the others.

**2. Open loop with the disturbance compensation**You control an electric furnace in which the main disturbance – fluctuations of the power network is easy to measure. Therefore, this disturbance should be measured, e.g.

**+37V**of the amplitude and compensated with

**-37V**of the amplitude. The furnace itself “thinks” that nothing has happened and you will not even see the impact of the disturbance. Note that there is no feedback here, hence the other name

**“feed forward”**as opposed to the good old

**“feedback”.**

**4. Closed loop with the disturbance compensation**

Compensated Open System does not suppress residual disturbances (which are not measured). So the whole thing should be closed with a simple feedback loop. And so the main disturbance-network fluctuactions be suppressed very quickly by

**compensation**, and the remaining ones will be slowly suppressed by ordinary feedback.

Another name for this control system is

**Closed-Open**.

**5. Cascade control**This control is when it is possible to isolate a certain part of the object affected by the disturbance. And in this it is similar to the previous one –

**Closed loop with the disturbance compensation**. However, it is not necessary to measure the disturbance. So this part can be closed with an internal feedback loop with a separate

**slave controller**. An ordinary

**P**controller is often sufficient here. We close the whole thing with an external loop with a

**master controller**– most often of the

**PID type**. Note that the

**P**controller will quickly suppress the disturbance in the inner loop before the information about it reaches the

**PID**controller through the large inertia of the remaining part of the object. The

**PID**controller suppresses all other disturbances, including any imperfections in the damping of the

**P**controller.

The principle is therefore similar to the operation of a company in which the

**President-master regulator**suppresses all disturbances and the

**Manager-slave controller**suppresses only “private” disturbances of his department. A good manager will suppress them so quickly that it often doesn’t reach the

**CEO**.

And rightly so, because

**Mr. President**, he is responsible for great matters.

**6. Ratio control**Example. The color of a given paint depends on the ratio of its

**2**basic components. So, fill the mixing tank with

**2 flows**of different paints, the flow ratio of which is constant regardless of disturbances.

**Chapter 31 Effect of Nonlinearity on Control**So far we have studied linear systems. In them, the

**PID**controller can give as much signal as it needs for fast operation. In the course, you will repeatedly encounter a situation where the furnace needs, for example,

**1 kW**to reach the temperature

**y(t)=+100°C**in a steady state. But in transient states

**PID**gives short spikes even after

**1000 kW**, in order to quickly reach these

**+100 ° C**.

**Real systems**have limitations. The furnace can only give a maximum of

**3 kW**, more would be economically unjustified. It is known that the answer

**y(t)**in this particular case of

**3 kW**will not be so good. But will it be much worse?

The answer to this question is precisely the purpose of this chapter. It is also to show that the main non-linearity negatively affecting the quality of regulation are the limitations of the

**Power Amplifier**. This amplifier also has another name –

**Final Control Element**.