## Preliminary Automatics Course

**Chapter 32. Disturbances analysis
**

**Chapter 32.1 Introduction**

Control engineer has

**2**main goals:

**1. Set point x(t)**achievement by the output signal

**y(t)**.

Ideal is a steady state

**x(t)=y(t)**after a possibly short time with the small oscillations for example.

There are other

**optimal**benchmarks too.

**2.**Disturbances

**z(t)**suppresion.

**Note**

Optimal

**x(t)**response doesn’t assure optimal disturbances

**z(t)**suppresion and vice versa!

**Chapter 32.2 The set point x(t) and disturbance z(t) influence on the output signal **

– The **x(t)**–>**y(t)** influence is inclosed in the closed loop transfer function **Gz(s)=y(s)/x(s)**

– The **z(t)**–>**y(t)** influence is inclosed in the disturbance transfer function **Gzakl(s)=y(s)/z(s)**

**Fig. 32-1**

**Fig. 32-1a** The general closed loop system with the disturbance **z(t)**.

**Fig. 32-1b **The more detailed closed loop system example with the disturbance **z(t) **in the **Go(s) **object input.

I propose a bit of math now. The **R(s) **is a controller transfer function for example **PID**.

**Fig. 32-2**

Note that **Gz(s)** trannsfer function is the same as a “most important control theory” formula.

**Fig. 32-3
**The numerator of the

**Gzakl(s)**disturbance transfer function is other than numerator of the

**Gz(s)**closed loop transfer function.

It’s the reason that optimal

**PID**parameters based on set point

**x(t)**aren’t optimal based on disturbance

**zakl(s)**and vice versa.

But when disturbance

**zakl(s)**is in the “middle” as under?

**Fig. 32-4**

**Fig. 32-4**

**a**is more accurate than general

**Fig. 32-1**

**a**with the “middle” disturbance. Please assume that

**G1(s)**is a controller part and this “controller” transfer function is

**R(s)*G1(s)**and the new “object” is

**G2(s)**now. See at the

**Gz(s)**and

**Gzakl(s)**in the

**Fig. 32-2**again and the

**Fig. 32-4b**formulas are as clear as day.

**Conclusion**

The more is the

**z(t)**disturbance “right” the more distinct is

**y(t)**responce. You will be convinced in this

**chapter**.

**Chapter 32.3 Two-stage heat exchanger
Chapter 32.3.1 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 32.3.2 Two-stage heat exchanger as a opened loop**

**Fig. 32-5
**The exchanger is intended for control systems analysis, not for thermal engineering teaching. We do very big simplifications then.

**1**-The liquid doesn’t steam and freeze. All temperatures are allowed even

**+100000ºC**or

**-100000ºC**! The last temperature doesn’t exist in the world really but it exists in in this course world.

**2**-Positive voltage on the immersion resistant heater heats and negative cools! This is so-called

**Peltier element**known from the XIX century.

**3**-The mixers (not visible in the figure) are working all the time in the both tanks. It means the momentary liquid temperature

**Tc1**is the same in every

**tank1**place. The analogical is

**Tc2**in the

**tank1**. The

**Tc1**is mostly different than

**Tc2**. These temperature depend on the set pont

**x(t)**. The set point

**x(t)**voltage is a input signal to the

**tank 1**, because voltage gain of the power amplifier is

**ku=1**. The heat is transferred from

**tank1**to

**tank2**by the aid of coiled tubes and circulating pump. The momentary liquid temperatures in the

**2**coiled tubes are the same

**Tc1**. It means that

**Tc1**is an input signal to

**tank2**.

**Tc3**is a temperature measured by the thermometer.

**Tc3**is a little delayed by the

**inertia**of the metalic thermometer case. There is

**Tc1=Tc2=Tc3**in the steady state.

**4**– Step type set point

**x(t)=**

**+10V (**corresponding

**+100 ºC)**will cause after some time the

**Tc1=+100 ºC**and after some time

**Tc2=+100 ºC**and after some time

**Tc3=+100 ºC**as a

**+10V**on the thermometer.

**Note**that set point

**x(t)=**

**+10V**caused

**y(t)=**

**+10V**after some time. It’s like a amplifier

**K=1**with

**3 inertia.**All this complicated system is replaced by the simple electronic amplifier with inertia! The analysis is much easier now.

**Analogically**

**+5V**–>Tc1=Tc2=Tc3=+50ºC–>

**+5V**

**0V**–>Tc1=Tc2=Tc3=0ºC–>

**0V**

**5**– Null initial state

**Tc1=Tc2=Tc3=0ºC**

6– Power amplifier

6

**ku=1**(emitter follower for example) ensures constant voltage on the resistant immense heater, regardless of its resistance.

The set point slider is set immediately from low to up position. It’s typical step type

**x(t)=+100ºC**signal. What are the

**Tc1, Tc2**and

**Tc3**signals?

Call Desktop/PID/18_analiza zakłóceń/01_3T.zcos

**Fig. 32-6**

Two-stage heat exchanger model. You don’t see power amplifier because

**ku=1**.

Click “start”

**Fig. 32-7**

**Tc3**follows

**Tc2**and

**Tc2**follows

**Tc1.**

The

**Tc3**delay is small because thermometer inertia is small.

There is

**Tc1=Tc2=Tc3=100ºC**in steady state.

**Chapter 32.3.3 Heat exhanger as a closed loop
**Classical

**PID**control system without disturbances.

**Fig. 32-8**

The higher system and the lower model are equivalent.

Call Desktop/PID/18_analiza zakłóceń/02_3T_Kp10_I7_D2.zcos

**Fig. 32-9**

You tested it earlier—>

**chapter 30 Fig.30-50**. But you didn’t know that it was a heat exchanger. The signal

**y(t)=**

**Tc3**is after temperature/voltage transformer. It transforms

**Tc3**temperature

**0…+100 ºC**for voltage

**0…+10**

**V**.

Click “start”

**Fig. 32-10**

You see “cutted”

**sPID(t)**and not

**“cutted”**

**Tc1(t)**,

**Tc2(t)**(unhappy “fuzzy yellow Tc2(t)) and

**y(t)=Tc3(t)**temperatures on the

**Fig. 32-10**

**a**and all the not cutted

**Tc1(t)**,

**Tc2(t)**and

**Tc3(t)**temperatures on the

**Fig. 32-10**

**b.**

Set point

**x(t)=+10V**–>

**+100ºC**

**step type in**

**t=3sec**t “wishes” a

**y(t)=Tc3(t)=**–>

**+100ºC****+10V**after some time.

The initial

**Tc1(t)=Tc2(t)=Tc3(t)=0**. The initial controller

**sPID(t)=210**is very big. Remind please the big initial

**proportional**and especially very big

**differential**components

**PID**!

Initial

**sPID(t)=210**“wishes”

**+21 000ºC**in tank no.

**1**in steady state , more than sun temperature! But after circa

**1 sec**. it “wishes” to cool the liquid up to

**-1200**! These big

**ºC****sPID(t)**changes enables good dinamic. Remember that our system as ideal. There are power limits. The real systems aren’t so ideal and this problem will be analised in

**chapter 34.**

**Chapter 32.4 The temperature control of the heat exchanger with the disturbance in different places.**

**Chapter 32.4.1 Introduction
**Return to the formulas in the

**Fig.**

**32-4b**,

**especially to the conclusion – “The more is the**

**z(t)**disturbance “right” the more distinct is

**y(t)**responce”. The next subchapters confirm it.

The positive disturbance-additonal heater will be given before first, second and third inertia in

**Fig. 32-8**. Finally will be given the most mean disturbance-directly for output

**y(t)**e.i. for thermometer output.

**Chapter 32.4.2 Disturbance before first inertia-additional heater by the heater of the tank no. 1.**

**Fig. 32-11**

The positive **z(t)=+0.4 **disturbance is a contact short circuit e.i **+4 V **on the additional resistance heater next to main heater in the tank no. **1**. It causes additional heating. The main and additional heater resistances are equal. E.i. these same voltage steps cause this same temperature liquid increase.

Call Desktop/PID/18_analiza zakłóceń/03_3T_pierwsza_inercja_+zakl.zcos

**Fig. 32-12**

Click “start”

**Fig. 32-13
**I remind that

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

**+100ºC**.

The positive

**40%**additional heating was compensating by the

**40%**power decrease. The disturbance is hardly visible.

**Chapter 32.4.3 Disturbance before second inertia-additional heater by the coil tube heater of the tank no. 2.**

**Fig. 32-14**

Call Desktop/PID/18_analiza zakłóceń/04_3T_druga_inercja_+zakl.zcos

**Fig. 32-15**

Click “start”

**Fig. 32-16
**The

**x(t)**response is the same as before. It’s obviously. But the

**y(t)**suppresion is more rapid now. It confirms the

**Fig. 32-4b**conclusion. The more is the

**z(t)**disturbance “right” the more distinct is

**y(t)**responce. The

**blue**

**sPID(t)**controller reaction is more rapid too.

**Chapter 32.4.4 Disturbance before second inertia-additional heater acts directly for a thermometer
**The disturbances acting directly for output or near output are most mean. The resistance wire is is winded on the metalic thermometer case here.

**Fig. 32-17**

There are

**2**influences on thermometer

– Liquid temperature

**Tc2**of the tank no.

**2**

– “Winded” resistance heater – the

**z(t)=+4V**disturbance

Call Desktop/PID/18_analiza zakłóceń/05_3T_trzecia_inercja_+zakl.zcos

**Fig. 32-18**

Click “start”

**Fig. 32-19
**The system behaviour is similar but more rapid than before. The positive

**40%**additional heating was compensating by the

**40%**power decrease and

**y(t)=Tc3(t)**returned to its previous value

**y(t)=1**–>

**+100ºC**. It’ absolutely true but

**Tc3(t)=**isn’t a liquid

**+100ºC****Tc2(t)**temperature in steady state! Let’s test

**Tc2(t)**time process.

Call Desktop/PID/18_analiza zakłóceń/06_3T_trzecia_inercja_Tc2_+zakl.zcos

**Fig. 32-20**

We observe

**Tc2(t)**temperature too.

Click”start”

**Fig. 32-21**

**Note**-The

**Tc2**temperature is a “fuzzy” yellow colour!

The

**PID**, as every moral controller, tries to compensate the disturbance and do

**x(t)=y(t)=Tc3**. And controller succeed in his job–>

**y(t)=x(t)**! But the themometer value is falsified by additional themometer value. Thermometer “thinks” that is a

**Tc3=Tc2+40ºC=100ºC**but the real liquid temperature is

**!**

**Tc2=60ºC****Chapter 32.4.5 The most mean disturbance-Directly on output, after themometer!
**

**Fig. 32-22**

It seems like a sabotage! Somebody connected to the transmission wire a special device-adder. Controller thinks that the temperature is now

**y(t)=Tc3+z(t)**. Remember that

**PID**tries to do steady state

**x(t)=y(t)**so the real temperature will be

**Tc3=x(t)-z(t)**.

Call Desktop/PID/18_analiza zakłóceń/07_3T_za_-trzecia_+zakl.zcos

**Fig. 32-23**

This is

**Fig. 32-22**system

**Xcos**model.

Click “start”

**Fig. 32-24**

**PID**tries to do steady state

**x(t)=y(t)**so the real temperature will be

**Tc3=x(t)-z(t)=**

**100ºC**–**. The temperature**

**40ºC=+60ºC****Tc3**-unhappy “fuzzy” yellow colour, confirms it.

**Chapter 32.5 The most important disturbance suppresion conclusions**

**Fig. 32-25
1. The more right is the disturbance the more is distinct
**The more “right” is the disturbance

**z(t)-**the “lower” is

**G2(s)**. Other words,

**G2(s)**is more similar to

**proportional**unit and that disturbance transfer function

**Gzakl(s)**has less inertia. The response

**y(t)**is more rapid then. The

**Fig. 32-13**,

**16**,

**19**,

**21**and

**24**confirm it.

**2. ****The most mean disturbance-directly on output or on controller input
**The

**Fig. 32-21**is an example when disturbance

**z(t)**is near the output and the

**Fig. 32-24**when

**disturbance**

**acts directly on output. This phenomenon is known not only in the control theory. The most mean are the activities directly for a decision-making processes. Do you know Ken Follet-author “Eye of the Needle” -World War Two times? The german spy “Needle” informed Hitler that D-day in 1944 will be here and not here. Hitler made fatal decision in connection with falsified information. “Needle” was a bad sensor-the analogy is quite visible here.**

**z(t)**The disturbance

**z3(t)**acting directly on controller input

**Fig. 32-25c**is mean too. The controller “thinks” that the set point

**x(t)**is changed now.