### Automatics

**Chapter 29 Disturbances analysis**

**Chapter 29.1 Introduction**

Automatics has

**2**main tasks:

**1. Getting the output signal y(t)**to the setpoint

**x(t)**by a man –

**operator**of the technological process.

The ideal is a steady state where

**x(t)=y(t)**. This should also be a “nice” response to the unit step

**x(t)**. What “pretty” or “optimal” means is another matter.

For the

**first**, this will be the shortest regulation time, even with oscillations.

For the

**second**one, also the shortest time, but without oscillations.

The

**third**one already allows

**10%**oscillation.

The

**First, Second**and

**Third**is Mr. Technologist or Mr. Customer who gives automatics engineer to earn. Most of the experiments in this course are the study of the response of

**y(t)**to unit step

**x(t)**.

**2. Ensuring disturbances suppression z(t)**.

They are constantly trying to push the output

**y(t)**out of the virtue path of keeping

**y(t)**at the set point

**x(t)**. This is what the controller fights against. The ideal is a

**state**where, despite the disturbances,

**e(t)=0**is still present. This is only possible in theory. In practice, it is only after some time that the disturbance

**z(t)**will be completely

**suppressed**. You found out about it by examining the influence of

**z(t)**disturbances in

**I**,

**PI**and

**PID**control. Although the

**P**and

**PD**control suppresses the

**error**, it does not completely. It only reduces it to, for example,

**2%**, compared to the system without control.

**Note**

An

**optimal**step response

**x(t)**generally does not provide an

**optimal**response to a

**disturbance z(t)**and

**vice versa**.

**Chapter 29.2 Effect of setpoint x(t) and disturbance z(t) on output signal y(t)**

– The action of **x(t)** on **y(t)** is contained in the transmittance of the closed system **Gz(s)=y(s)/x(s)**

– The action **z(t)** on** y(t)** is included in the disturbance transmittance **Gzakl(s)=y(s)/z(s)Fig. 29-1**

**Fig. 29-1a**is used when we do not know exactly what the impact of disturbances

**z(t)**on the object is. We feel that the controller is trying to suppress the disturbance, but how it does it is up to it. In this form, it is impossible to check how

**y(t)**will respond to the disturbance step

**z(t)**. Therefore, a more accurate will be

**Fig. 32-1b**with a specific situation. Here, the disturbance

**z(t)**is added directly to the control signal from the controller

**s(t)**. For example, when the object is a furnace and

**s(t)**is the power supplied from the

**controller**to the heater. The positive interference

**z(t)**will then be the power supplied to a separate

**heater**located close to the control

**heater**.

Let’s introduce some more math. Here

**R(s**) is the transmittance, which can be a

**PID**controller.

Fig. 29-2

Fig. 29-2

Note that the transmittance of a closed system

**Gz(s)**is the same as “in the most important formula”

Fig. 29-3

Fig. 29-3

On the other hand, there is no

**R(s)**in the

**disturbance**transmittance

**nominator**

**Gzakl(s)**. So

**Gzakl(s)**has a little less inertia than

**Gz(s)**. Therefore, the response to

**z(t)**is usually a bit faster than

**x(t)**! Therefore, when we choose the

**PID**settings, we must make a decision.

What do we care more about?

On the optimal response to

**x(t)**or

**z(t)**?

If changes of the setpoint

**x(t)**occur frequently, we tune the controller due to the setpoint

**x(t)**, if rarely, due to

**z(t)**. More often, however, we are guided by tuning the controller due to the setpoint

**x(t)**, because it is more predictable than the disturbance

**z(t)**. It’s like the method of the guy who looks for the key under the lamppost just because he can see better. On the other hand, tuning differences due to

**x(t)**or

**z(t)**are rarely shockingly large.

And when the disturbance is in the “center” of the

**Go(s)**object, as in

**Fig. 29-4a**?

Fig. 29-4

Fig. 29-4

**Fig. 29-4a**is a more accurate approximation

**Fig. 29-1a**, in which

**z(t)**appears in the “middle” of the object

**Go(s)=G1(s)*G2(s)**, i.e. between the transmittances

**G1(s)**and

**G2(s)**.

The formula in

**Fig. 29-4b**results directly from

**Fig. 29-2**. It is enough to assume that

**G1(s)**is part of the transmittance of the controller and is

**R(s)*G1(s)**and the object will be

**G2(s)**. The formula for the disturbance transmittance

**Gzakl(s)**shows that the closer the disturbance

**z(t)**is to the exit of the object, the more pronounced will be its impact on

**y(t)**. What does “closer to the exit” mean? The fact that the

**G2(s)**inertia is lower compared to the

**G1(s)**inertia. Otherwise, the transmittance

**G2(s)**becomes closer to the proportional term

**G2(s)=1**. So the disturbance

**z(t)**will be less “flattened” by

**G2(s)**. You will find out later in this chapter.

**Chapter 29.3 Two-stage heat exchangerChapter 29.3.1 Why do we visit the thermal engineering?**

**Two-stage**

**heat exchanger**is a good analysis example for:

**– Disturbance analysis**

**– Different control systems structure**–>useful in the next

**chapter**

**Chapter 29.3.2 Two-stage heat exchanger as a opened loop**

We will examine the object itself without control, i.e. in an

**open system**.

Fig. 29-5

Fig. 29-5

The scheme is to be used to learn the principles of automatics, not the thermal processes occurring in the exchanger. That is why we will make very large simplifications.

**1–**The liquid does not evaporate or freeze. So it can reach a temperature of e.g.

**+1000ºC**or even –

**1000ºC**, which is even more

**absurd**. A heating

**plus**on the heater is normal, but a cooling

**minus**? However, since the

**19th**century, the so-called the

**Peltier element**. Such a

**heater-cooler**makes it easier to understand the negative

**sPID(t)**control signal, which in some very short moments “wants” to cool the liquid to a temperature of e.g.

**-1200 ºC**!. What does “wants” mean? That it won’t cool down to that temperature right now. But if such a

**cooling**power (i.e.

**negative**) was fed to the

**heater-cooler**all the time, then after some time the liquid would reach

**-1200 ºC**.

**2-**In the tanks, the mixers are constantly working (they are not shown in the diagram), i.e. in every place of a

**given**tank there is the same temporary temperature

**Tc1**or

**Tc2**. Instantaneous

**Tc1**may differ from instantaneous

**Tc2**. It can increase or decrease depending on the position of the potentiometer

**slider**.

**3–**Heat is transferred from tank

**No. 1**to tank

**No. 2**through the

**output coil**of the

**tank. 1**and

**inpput coil**of the

**tank. 2**. Heat transport is facilitated by a

**circulation pump**, similarly to a pump in a building that forces water to flow through

**radiators**. The liquid in the

**input coil**of tank

**No. 2**has a temperature of

**Tc1**, i.e. the temperature of the liquid in tank

**No. 1**. Therefore,

**Tc1**is the input signal to tank

**No. 2**. Similarly, the voltage from the amplifier to the

**heater**is the input signal for tank

**No. 1**. In steady state,

**Tc1 = Tc2 =Tc3**.

**4-**The power on the heater is proportional to the voltage from the amplifier, not the square of the voltage. This, of course, is contrary to electrical engineering, but thanks to this, the temperature of the liquid will be proportional to the voltage from the adjuster and it will be easier to analyze the waveforms.

**5-**If there is e.g.

**+10V**on the setting unit, corresponding to the temperature of

**+100ºC**, then after some time in the tank

**No. 1**will show

**Tc1=+100 ºC**and then in tank.

**No. 2**will also be

**Tc2=+100 ºC**. This temperature “covering” the housing of the platinum sensor (platinum wire) will heat the wire and the transmitter will show the temperature

**Tc3=+100ºC**in the form of

**+10V**voltage.

Note that in steady state the input is

**+10V**and the output is

**+10V**. So the whole exchanger with the power amplifier, coils, etc. can be treated as an

**amplifier**with a gain of

**K=1**and a certain dynamics -> here a

**three-inertial term**. It makes the analysis so much easier!

**Similarly**:

**+5V**–>Tc1=Tc2=Tc3=+50ºC–>

**+5V**

**0V–>**Tc1=Tc2=Tc3=0ºC

**–>0V**

**-5V–>**Tc1=Tc2=Tc3=-50ºC

**–>-5V**

**-10V–>**Tc1=Tc2=Tc3=-100ºC

**–>-10V**

**6–**Zero initial conditions–>

**Tc1=Tc2=Tc3=0ºC**

**7 –**A power amplifier with a voltage gain of

**ku=1**ensures that there is a voltage

**x(t)**across the heater regardless of the load (or resistance) of the heater.

We will examine the influence of the location of the disturbance on the output signal

**y(t)**. It turns out that the closer the disturbance

**z(t)**is to the exit of the object, the more pronounced its influence. The disturbance

**z(t)**will enter the

**three-inertial**object

**Go(s)**, an example of which is a

**heat exchanger**.

The slider of the setpoint

**x(t)**in

**Fig. 29-5**will now be rapidly moved from the middle position to the upper position. In this way, you will step the temperature from

**x(t)=0ºC**to

**x(t)=+100ºC**. This corresponds to the model and animation.

**Fig.29-6**A step of

**0ºC**to

**+100ºC**which corresponds to a step of

**0**to

**1**

Model of the exchanger in the previous drawing. There is no power amplifier in it because

**ku=1**.

Given the small time constants, our tanks are the size of glasses. Therefore, a decent

**amplifier-emitter follower should**be enough to control the

**heater**. Here part of the power is lost in the transistor. For larger exchangers, pulse amplifiers are used. Let’s not delve too much into whether there are such small

**Pt100**thermometers in the housing. Suppose there are such

**midgets**.

The

**three-inertial**nature of the signal

**Tc3=y(t)**is clearly visible and how the temperature

**Tc1**leads

**Tc2**, and

**Tc3**. Due to the small time constant of the thermometer,

**Tc3**is almost

**Tc2**.

Our assumptions were confirmed that:

–

**instantaneous**temperatures

**Tc1, Tc2 and Tc3**may be different

-in

**steady state**all temperatures are equal->

**Tc1=Tc2=Tc3=100ºC**

**Chapter 29.3.3 Exchanger in a closed system**

Classic automatic control system without disturbances.

Fig. 29-7

Fig. 29-7

The

**flowchart**at the top and the corresponding

**block diagram**at the bottom.

The

**input**to the

**tank**

**1**is the signal from the

**PID**controller, and more precisely from its

**power amplifier**, and the output is the temperature

**Tc1**of the

**output**

**coil**, the same as the temperature of the liquid in the

**tank 1**. It is also an input signal to

**tank 2**, because in its coil there is also temperature

**Tc1**.

**Fig. 29-8**

If you look closely, you have already studied this diagram–>**Chapter 27 Fig.27-29**. Then only you did not associate it with **2 heat exchangers** and with the voltage output **Tc3 0 … + 10V** on the **Pt100** thermometer. Actually, on its **converter**, which converts changes in the resistance of a** platinum wire** (hence the name **PT100**) into **voltage** changes.**General note**

There are a lot of time charts here, from which you can get nystagmus. To make the task easier, I suggest analyzing it “piece by piece”.**1.** First, the most important, i.e. **input** and **output**, i.e. **x(t)** and **y(t)**.**2.** Error **e(t)=x(t)-y(t)**–>Check if it is true for each** t**.**3. SPID(t)** control signal. Or would you do it manually?**4.** The next “softening” signals** Tc1, Tc2** and **Tc3**. Note that you have already analyzed **Tc3** as **y(t)=Tc3**.

Apply this rule later as well.

**Fig. 29-9**

The previous time chart just a different oscilloscope scale, so you also see what was “crop” in **Fig. 29-8**.

The setpoint **x(t)** forces a step in** 3** sec. **+10V**. So it “wants” that after some time the **thermometer** will also be **+10V**. So** y=Tc3=+100ºC**. We remember that the initial temperature of the liquid in the tanks **Tc1=Tc2=0ºC**. In **3** seconds (at the moment of jump) error **e(t)=+100ºC**. The **P** component of the** PID** controller then forces the control signal corresponding to the power which in **steady** state (if it did not change) gives **+1000ºC**! (here **Tc1=10**) Shock? It could be even bigger. After all, the **D** component differentiates the jump and gives =**+200** . In total, at the moment of the jump, the controller gives the signal **sPID(t)=210**, which corresponds to the temperature of **+21,000ºC**. More than in the sun! And so we are lucky that in the PID controller the **differentiating** component **D** is real, i.e. “flattened” by the **inertial unit**. If it were a perfect **differentiator**, then the power on the heater would correspond to an **infinite temperature** for a while! The **hydrogen bomb** is a piece of cake. And all this happens in **tank 1** smaller than a glass.**Why am I scaring my readers?** To be aware that so far we have modeled** ideal** objects. There were no restrictions. And in life, and even more so in electronics, they occur. Our power amplifier – the emitter follower enters the saturation resulting from its power supply, e.g. **-15V…+15V**. In **ideal** systems, **negative** voltage **cools** with temperature. In **real** systems, a negative **SPID(t)** signal only **turns off** power to the heater. So the temperature **Tc3** will drop only to the ambient temperature. And besides, even in ideal systems there are no “**hydrogen bombs**” – powerful energies. So what if there are very **high** powers for very **short** moments. But** energy** can be normal now because it is **power*time**.

By the way. Control systems with a** Peltier** element are used only in special cases, when very **fast** temperature changes are required.**What is the conclusion?** Ideal **time charts** give very strong control signals for very **short** periods. At the beginning of the jump** x(t)** the control signal **sPID(t)=210** which corresponds to the temperature of **+21000ºC**! The temperature of the liquid, of course, does not have time to reach this value, but it increases very quickly. Feedback reduces this signal even faster, even for a moment it wants to **cool** the liquid to **-1200ºC**! This can be seen in **Fig. 29-9**. Real runs have **limitations** and will therefore be **slower**. In addition, the setpoint** x(t)** is changed rather rarely. Most often, the regulator works in the disturbances suppression mode **z(t)** and not when the setpoint **x(t)** changes. Disturbances **z(t)** are usually much smaller than spikes **x(t)**. Then the control signals **sPID(t)** are less saturated and the waveforms will be closer to **ideal**. You will learn more about this in **chapter 31**.

**Chapter 29.4 Exchanger temperature control with disturbances in different places****Chap. 29.4.1 Introduction****Fig. 29-4b** shows that the closer the disturbance** z(t**) is to the output** y(t)**, the stronger its influence will be. We will check it in our exchanger. We will introduce a disturbance** z(t)** before the** first, second** or **third** inertia on the control system in **Fig. 29-7**. The disturbance will be the insertion of an **additional** heater. We’ll see how the **PID** controller handles it. Finally, we will introduce the most malicious disturbance – directly to the output of the **thermometer**.**Chapter 29.4.2 Disturbance before first inertia – auxiliary heater at tank heater 1**

Fig. 29-10

Fig. 29-10

The disturbance will be a

**+4V**voltage step (contact short) on the

**additional**heater right next to the

**main**heater in

**tank 1**. This will cause

**additional**heating. The

**resistances**of the

**heater**controlled by the

**amplifier**and the

**additional heater**are the same. That is, the same voltage across the

**two heaters**causes the same

**increase**in the temperature of the surrounding

**liquid**.

**Fig. 29-11**In steady state, the disturbance

**z(t)**caused by

**40% heating**was compensated by the

**PID**with a voltage drop (cooling) of

**40%**. As a result, the temperature

**y(t)**barely moved. Let me remind you that

**1**on the chart corresponds to the temperature

**+100ºC.**

**Chapter 29.4.3 Disturbance before secondinertia – additional heater at Tank 2 coil**

Fig.29-12

Fig.29-12

**Fig. 29-13**The response of

**y(t)**to the setpoint

**x(t)**is the same as before. It is obvious

The reaction

**y(t)**to the disturbance

**z(t)**is more rapid and shorter than before. The reaction of the

**PID**, i.e. the

**blue**

**sPID(t)**, is also stronger.

**Chap. 29.4.4 Disturbance before the third inertia – the heater acts directly on the thermometer part of the tank No. 2**

You will see how malicious a

**disturbance**acting directly on the

**output**or very close to the output can be. The disturbance z(t) is the voltage step on the

**heater**, which is a resistance wire wound on the housing of the

**Pt100**thermometer.

Fig. 29-14

Fig. 29-14

Several turns of the resistance wire of the heater “circle” a piece of the thermometer housing.

Therefore, the thermometer is affected by:

– liquid temperature

**Tc2**in

**tank 2**

– a

**heater**“circling” a piece of

**thermometer**

We also assume that the parameters “heating the thermometer” from the heater are the same as the parameters “heating the thermometer” from the liquid flowing around the part of the thermometer not covered by the heater.

The disturbance will be a voltage jump

**z(t)=+4V**on the heater.

**Fig. 29-15**

In steady state, the disturbance** z(t)** caused by the **40%** additional **heating** of the thermometer was compensated by the **PID** with a **40%** power **drop**. The **disurbance** effect is even stronger than before because the **disurbance** is closer to the **PID** input. Therefore, the controller reacts more vigorously to **z(t)**. This is visible in the **blue** **sPID(t)**. We’re glad it’s okay. **Tc3** has returned to its previous value. But what is **Tc3**? This is the voltage on the **thermometer**. And for us, the temperature of the liquid **Tc2** in the second tank is important, not some voltage. Let’s examine the time chart of **Tc2**.

**Fig. 29-16**

Contrary to the previous diagram, we can also observe the temperature of the liquid **Tc2** in **tank 2**. The real temperature of the liquid is **Tc2**. **Note**– **Tc2** is **yellow** and this color is barely visible on a white background! It turns out that in steady state, the liquid in **tank 2** has **Tc2=60ºC**! Simply, **PID**, like any decent controller, tries to equate the output signal **y(t)=Tc3** with the set value **x(t)** corresponding to the temperature of** 100ºC**. And he succeeds** 100%** of it, as you can see in the **red** **y(t)**. It’s just that the thermometer readings are falsified by **additional heating** of the thermometer housing. The controller “thinks” that the temperature of the liquid is **Tc3=100ºC**. Meanwhile, in steady state **Tc3=Tc2+40ºC=100ºC**. So **Tc2=60º**C as you can see in the attached picture.

There were **specific** mining accidents. A **methane sensor** was specially blown with fresh air. The **methane** concentration at the sensor itself was fine and machines with sparking electric motors were not shut down. There was a mining bonus and **100** times the **Russian** roulette was successful, but the 101st time … The measurement result was falsified, just like in our example with heating the thermometer housing.**Chapter 29.4.5 Most Malicious Disturbance – Direct to Output! (behind the thermometer)**

Fig. 29-17

Fig. 29-17

This is a similar disturbance to the previous one, but more venomous. Here we do not heat the thermometer, but we add voltage to the thermometer output! That is total fraud. The controller receives a “false” high temperature

**y(t)=Tc3 + z(t)**. In a steady state, the regulator will do its job, i.e. it will equal

**x(t)=y(t)**, but the real temperature of the liquid will be equal to

**Tc3(t)=y(t)-z(t)=100ºC-40ºC=60ºC.**

**Fig. 29-18**

When** z(t)=+0.4** appears in** 25** seconds, the controller thinks that there is now **y(t)=+140ºC** in the tank (although **Tc3=+100ºC**). Therefore, trying to bring the error **e(t)** to** zero**, it will reduce the heating, i.e. **sPID(t)**, so that** y(t) is 100ºC** again. And he will succeed only that then the steady state will be **Tc3=+60ºC**.

Compare the response of** y(t)** to a disturbance with **Fig. 29-16**. Here the jump **y(t)** to **1.4** (i.e. to **+140ºC**). Why. Because here the voltage is added to the thermometer, and in **Fig. 29-16** the thermometer was **heated**.

**Chapter 29.5 Key Disturbances Conclusions**

Fig. 29-19**1. The closer** to the **output, **the more pronounced the impact of the disturbance.According to

**Fig. 29-19c**, the disturbance

**z2(t)**, which is “closer” to the output

**y(t)**, will cause a stronger control signal

**s(t)**and more pronounced changes in

**y(t)**than the disturbance

**z1(t)**.

The justification is the formula in

**Fig. 29-19b**showing the relationship between the disturbance

**z(t)**and the input

**x(t)**and the output

**y(t)**. The more

**z(t)**moves to the right to output

**y(t)**, the more the inertia of

**G2(s)**decreases. And what does that mean? The “inertia” of the

**disturbance**transmittance

**Gzakl(s)**decreases because its

**nominator**tends to

**1**. So the response

**z(t)**is clearer and shorter. This is confirmed by the time charts

**y(t)**in

**Fig. 29-11, 29-13, 29-15, 29-16**and

**29-18**, where

**z(t)**“moves” from

**left**to

**right**.

**2.**

**Disturbance**directly to the output – the most perfidious

An example is the sensor in

**chapter 29.4.4 and 29.4.5.**It gives the controller false information, to which the controller reacts as if it were true. This is not only in

**automatics**. The most malicious are

**disturbances**, i.e. falsified signals that directly affect the

**decision-making**process. Have you read

**“The Needle”**by Ken Follett? Here British intelligence tipped Hitler’s best agent (that is, Hitler’s sensor) the information that the

**1944**invasion would be

**here**, and not

**there**. And Adolf stopped his best divisions for a few days.

A direct disturbance to the output is also a

**disturbance**to the input (with a changed sign). Another example is

**z(t)**in

**Fig. 29-18**. The controller “thinks” that the setpoint is

**x(t)+z(t)**and tries to adjust the value of

**y(t)**to it. It’s as if

**spy**disguised himself as

**Hitler**and gave orders. That is why engineers try to protect the

**tiny**controller against distuurbances as much as possible. And this is easier than securing the entire control system, especially the

**big**industrial facility.