## Preliminary Automatics Course

**Chapter 34. Nonlinearity influence for regulation
**

**Chapter 34.1 Introduction**

The most important influence for the control process is a

**saturation**of the power amplifier. The more general name of the power amplifier is a

**final control element**or

**actuator**. This element transforms control signal for power signal. Please note that all the remaining signals e.i. set point

**x(t)**, measured output signał

**y(t)**or control signal

**y(t)**doesn’t require much energy. And the nonlinearity of the power amplifier is a

**saturation**. Thus the chapter title should be more unambiguous “The final control element saturation influence for the control process”

I propose the reminder about transfer functions now. When you use the transfer it means that you are living in the linear objects world. The

**y(t)**output is proportional to

**x(t)**input in steady states.

**Fig. 34-1
**The linear object characteristic in steady state.

But non steady state? There are ruling

**linear**differential equations now. All the coefficients are constant, for example

**Fig. 34-2**

Linear differential equation

**Fig. 34-3**

Non-linear differential equation example. The

**red circles**are the bugs. The reason that differential equation isn’t linear.

**Fig. 34-4**

The linear differential equation

**Fig. 34-2**and a/m

**G(s)**transfer function are equivalent. I propose you to associate differential equation coefficients

**Fig. 34-2**with transfer function parameters. Have you any problems? Go to

**Fig. 19-8**chapter

**19**.

We used mainly the linear objects in this course. And only the transfer function

**Go(s)**makes a sense for these objects. The linear differential equation theory is deeply and good established now. It enables problems solving as stability,

**PID**controllers adjusting etc. But there is one condition only. The object must be linear and we forgot it often. The one nonlinear element can break down–> all the system is nonlinear.

**Chapter 34.2 Nonlinear elements**

**Chapter 34.2.1 Introduction**

**Fig. 34-5
**The most every real object nolinearity is

**saturation**. It’s

**+/-15V**for amplifiers or maksimal

**valve**flow. Sorry for unscientific names of the typical elements.

**Fig. 34-5a**– “progressively growth with the saturation”

**Rys. 34-5b**– “linear with the saturation”

**Rys. 34-5c**– “more and more slowly with the saturation”

**Rys. 34-5d**– histeresis

**Rys. 34-5e**– “linear with the dead band and saturation”

We will test only

**2**elements. The input is a

**black sine wave x(t)**and the output is a

**red**

**y(t)**.

**Chapter 34.2.2 The linear element with the saturation**

Call Desktop/PID/20_regulacja_i_nasycenia/01_nasycenie.zcos

**Fig. 34-6**

**x(t) **is a sine wave

Click “start”

**Fig. 34-7**

The saturation when **x(t)=+/-0.5**.

It acts as a amplifier **k=1 **before saturation and **y(t)=x(t)
After saturation**

for

**x(t)>0.5**–>

**y(t)=+0.5**

dla

**x(t)<0.5**–>

**y(t)=-0.5**

Chapter 34.2.3 The linear element with the dead band and saturation

Chapter 34.2.3 The linear element with the dead band and saturation

Wywołaj PID/20_regulacja_i_nasycenia/2_strefa_martwa

**Fig**

**. 34-8**

**x(t)**is a sine wave.

Click “start”

**Rys. 34-9**

The dead band is for

**x(t)**

**-0.5….+0.5**.

**Chapter 34.3 How does the ideal control system work?
**E.i the linear control system. It’s most popular experiment in this course but it does no harm repetition.

**Fig. 34-10**

Typical

**PID**control system with

**2-inertial**object.

We expect steady state

**y(t)=x(t)=+1**.

Click “start”

**Fig. 34-11**

It may be temperature control system and

**+1**means

**+100°C**.

The integral

**I**component made

**null**control error, as usual. The process is very fast because of the big

**sPID(t)**control signal amplitudes. The driver uses strongly gas pedal and brake pedal. The effect is a dynamic drive. The controller job is the same. The gas pedal is a big positive

**sPID(t)**and the brake pedal is a big negative

**sPID(t)**.

Let’s change the oscilloscope parameters and observe the real

**sPID(t)**(not “cutted”) signals.

Call PID/20_regulacja_i_nasycenia/04_PID_ideal_pelen_widok.zcos

The block diagram is the same but there are other oscilloscope parameters. You will see all the

**sPID(t)**signal.

Click “start”

**Fig. 34-12**

The main

**x(t), y(t)**signals are small and hardly visible now but you see all the

**sPID(t)**control signal majesty! Compare please

**x(t), y(t)**circa

**0…+1**range and

**sPID(t)**range

**-5…+160**. This is a reason of a very good

**y(t)**dynamic–>”brake and gas” control algorithm.

The real control system has power limits. Any design engineer will not give

**160 kW**power when steady state process requires

**1 kW**only! The negative power (cooling for example) is also rarely used.

There is a difference between theory and practise*. The real system have the control signals limits. There are saturations

**0…+1.5**instead of

**-5…+160**. The range is even more narrow

**0.2…+1.5**! The power

**off**causes

**y(t)**temperature decrease up to ambient temperature

**+20°C**and not to

**0°C**.

The linear systems are smart, relatively easy and there are answers for many general questions. Stability, Hurwitz etc… The real systems are different. The problem is how big is this difference.

*I have seen the poster in the worthy control systems company

**P**

**ractitioner-**All is operating but he he don’t know why

**Theoretician-**Nothing is operating but he knows why

Our company combines

**practice**and

**theory**. Nothing is operating and we don’t know why. I took to them quickly.

**Chapter 34.4 How does the real control system work?**

**Chapter 34.4.1 Introduction
**

**Fig. 34-13**

I try to argue that the main nonlinearity in the real systems is a

**power amplifier-**other names

**final element**or

**actuator**.

We will analyse

**power amplifier**,

**PID**controller and the

**object**.

**Chapter 34.4.2 Power amplifier with the saturation
**The

**power amplifier**is just a difference between

**Fig. 34-10**. This is a linear unit with the saturation

**Fig. 34-5b**. It makes that all the object is nonlinear. We can’t calculate the closed loop transfer function for example!

**Chapter 34.4.3 PID controller**

This almost a linear unit and its **transfer function** is:

**Fig. 34-14**

The appropriate **Fig.34-13** controllers parameters are:

**Kp=10**

**Ti =7 sec**

**Td=2 sec**

**Note** This is a** real** differentiation, not **ideal**!

**Comment** to word “almost a linear unit”

The ** PID controllers **are mostly the microprocessor type devices. They can calculate every **sPID(t) **control signal **digital **value–>there are no limits here. Some people will complain that this value is quantum timed. But these quantum times are very small compared to object inertia and **PID **is for us a pure tranfer function **Gpid(s)**.

**Chapter 34.4.4 G(s) Object
**Object is a furnace, rectifying column or rocket. There are weaker linearity arguments than

**PID**. The most popular static object

characteristic is

**Fig. 34-5c**type–> “more and more slowly with the saturation”.

What’s the conclusion?

The

**G(s)**nominator

**e.i.**unit static gain

**k=1**only for

**null**working point, then is lower. The dynamic parameters

**T**time constants also may change due to working point. It means that the real parameters are “fuzzy”. They change in the some range.

**Fig. 34-15**

**Fig. 34-15a**– the ideal

**G(s)**

**2-inertial**object with parameters

**k=1**,

**T1=10 sec**and

**T2=5 sec**.

**Fig. 34-15b**-The real parammeters are variable in the ranges. These parameters depend on working point and our identification methods. The real parameter values are inside the ranges

**k=0.95…1.05**,

**T1=9.5…10.5 sec and T2 4.5…5.5 sec**.

Many objects, especially

**multi-inertial**may be approximate as

**Substitute Transfer Functions**with

**K**,

**T**and

**To**parameters.

The first question. What for? The better is

**exactly**than

**approximately**model. It’s better to be rich, young and healthy than poor, old and ill.

**Firstly**– It’s easier to experimentally determine

**K**,

**T**and

**To**parameters than the real

**multi-inertial**object parameters.

**Secondly**– There are ready optimal

**PID**parameters dependend of Substitute Transfer Functions

**K**,

**T**and

**To**parameters–>chapter

**31**

The

**Substitute Transfer Function**was discussed in chapter

**31.**I remind the step response.

**Fig. 34-16**

Substitute Transfer Function step response.

**Chapter 34.4.5 Conclusions
**–

**PID**controller is a linear unit

**Gpid(s)**–>

**Fig. 34-14**.

**–**

**Object**is a linear object too but with some toleration dose.

–

**Power amplifier**is a nonlinear unit–>linear with the saturation. This unit (other names-

**final control element**,

**actuator**). This element “perverts” all the closed system unit, makes it nonlinear.

The

**actuator**exists always (almost) in closed loop systems. It means that all (almost!) closed loop systems are nonlinear and all the theory (

**G(s)**units configuration,Nyquist, Hurwitz…) isn’t necessary. Fortunately the situation isn’t so bad.

**Chapter 34.5 Ideal and Real control systems comparison.**

**Chapter 34.5.1 Introduction
**We will test:

– step responses

**x(t)=+1**

**– disturbances**

**z(t)=+0.4**and

**z(t)=-0.4**responses

The saturations are

**0…+1.5**and

**0…+5**

The common sense says that:

– the lower are input signals

**x(t)**and

**z(t)**the more is a system similar to ideal

– the wider is saturation range the more is a system similar to ideal.

**0…+5**is better than

**0…+1.5**.

**We have**

Chapter 34.5.2 Saturation 0…+1.5, x(t)=+1, z(t)=+0.4 (heating)

Chapter 34.5.2 Saturation 0…+1.5, x(t)=+1, z(t)=+0.4 (heating)

**2**identical tanks with identical liquid. Controllers parameters are identical too. The different are the heater power limits-saturations.

–

**Ideal System**– unlimited heater power. The heating power

**+100 000 kW**and cooling power

**-1000 000kW**are possibly when required.

–

**Real System**– limited heater power

**0…+1.5 kW**

The additional heater

**+0.4 kW**is a disturbance.

It’s certainly the

**ideal**is better than a

**real**system. But how better?

**100:0**or

**5:0**only.

Call Desktop/PID/20_regulacja_i_nasycenia/5_PID_nas0…1.5_skok+1_zak+0.4.zcos

**Fig. 34-17**

The

**ideal**system hasn’t saturations. It means that all control signals are possible, the negative (cooling) values too.

The

**real**control signal

**sPID(t)**enables temperatures

**0…**when ambient temperature is

**+150°C****.**

**0°C**Click “start”

**Fig. 34-18**

The set point

**x(t)=1**step is in

**3 sec**and the disturbance

**z(t)=+0.4**is in

**60 sec**.

**IDEAL SYSTEM**without saturations.

The

**PID**has control signal with

**-infinity…+infinity**amplitudes at disposal. The step

**yi(t)**response is fast and assures

**null**control error. It’s possible beacuse big

**sPIDi(t)**control signal has big amplitudes and “brake and gas” control algorithm is used here–>see

**Fig. 34-12**–>

**blue sPIDi(t)**–>“brake=

**-5**and “gas=

**+160**“.

**REAL SYSTEM**with saturations

**0…+1.5**

The

**PID**has control signal with

**0…+1.5**amplitudes at disposal. It means that “brake and gas” effect is weaker than for

**ideal**system–>

**real**–>

**green sPIDr(t)**–>“brake=

**0.9**and “gas=

**+1.5**“. The set point

**x(t)=1**response isn’t fast as

**ideal**. There is even

**30%**overregulation here! On the other hand. The “brake and gas” effect is circa

**100 times**weaker than for

**ideal system**. But the setting time isn’t

**100 times**worse! And the most inportant. There is

**null steady error**.

The main conclusion. The nonlinearity (saturation here) impairs the quality of the control systems but “not to bad”.

**Note**that

**z(t)**disturbance

**supressions**are the same for

**ideal**and

**real**systems! The real

**green sPIDr(t)**control signal doesn’t exceed the saturations now and that’s a reason.

**Chapter 34.5.3 Saturation 0…+1.5, x(t)=+1, z(t)=-0.4 (cooling)**

Call Desktop/PID/20_regulacja_i_nasycenia/06_PID_nas0…1.5_skok+1_zak-0.4

The block diagram is similar to **Fig. 34-17 **but **z(t)=-0.4 **(cooling)

Click “Start”

**Fig. 34-20**

The set poin **x(t)=1 **response is the same and it’s obviously. The **z(t)=-0.4 **(cooling) was compensating by the **green sPIDr(t)** control signal increase (heating)

**Chapter 34.5.4 Saturation 0…+5, x(t)=+1, z(t)=+0.4 (heating)
**This and the next block diagram differs in saturation only. There is

**0…+5**instead

**0…+1.5**. We expect that

**x(t)=+1**step response will be better because system is “more ideal” now. Our power amplifier-actuator has more power now–> it’s expensiver.

Call Desktop/PID/20_regulacja_i_nasycenia/07_PID_nas0…+5_skok+1_zak+0.4

Click “start”

**Fig. 34-21**

The set point

**x(t)**response is faster (better) than by

**0…+1.5 saturation**. The

**z(t)=+0.4**disturbance suppresion is the same as ideal because

**sPIDr(t)**doesn’t exceed

**0**saturation value.

**Chapter 34.5.5 Saturation 0…+5, x(t)=+1, z(t)=-0.4 (cooling)**

Call Desktop/PID/20_regulacja_i_nasycenia/08_PID_nas0…+5_skok+1_zak-0.4

Click “start”

**Fig. 34-22**

The diffrence is in the **z(t)=-0.4 **disturbance suppresion.

**Chapter 34.5.6 Conclusions
**It’s obviously that the ideal system is better than real system with final element power limits. It’s obviously too that the “more ideal” is the system-the more powerfull is the actuator–>the more expensiver it’s. But the real system isn’t to bad as come from saturations. Let’s go to football analogy. The difference between

**ideal**and real

**system**is more similar to

**Germany-Austria**than to

**Germany-Liechtenstein**.

**Chapter 34.6 The set point x(t) exceeds the saturations levels.**

**Chapter 34.6.1 The first approach – signals x(t), yi(t) and yr(t) only
**Let’s assume that

**PID**“wishes” to heat the liquid up to

**+105 °C**. But the

**actuator**(or other name

**power amplifier**)

**has a maksimal power**

**+0.95 kW**and it enables

**+95 °C**only. The

**PID**can’t do this job. It’s impossible as a leader who demands from his employee more than employee professional competence.

Call Desktop/PID/20_regulacja_i_nasycenia/09_PID_nas0…+0.95_skoki_-1…2_simple

**Fig. 34-23**

We compare

**ideal**and

**real**systems behaviours. The set point

**x(t)**is multistep type signal and

**x(t)**exceeds the high

**+0.95**real

**PID**saturation level sometimes. Can you predict the

**yi(t)**and

**yr(t)**processes?

Click “start”

**Fig. 34-24**

**Period 1 x(t)=+0.3**

The

**x(t)=+0.3**is lower than

**+0.95**saturation. The feedback will be active and the behaviours are discussed in the

**Chapter 34.5**. The real

**yr(t)**response is slower than ideal

**yi(t)**and with the overregulation but

**null**control error condition is fulfilled.

**Period 2 x(t)=+1.05**

The ideal

**yi(t)**is obvious because there are no

**PID**control signal limits. The real

**yr(t)**can’t attain

**x(t)=1.05**because it exceeds the saturation

**0.95**level of the real

**PID**. The feedback doesn’t work and the response is as for open loop system.

**Period 3 x(t)=+0.5**

The setpoint

**x(t)**falls rapidly up to

**+0.5**. The ideal system reacted immediately and the steady state

**yi(t)=x(t)**is quickly attained. The real steady state

**yr(t)=x(t)**is slowly attained. The feedback works again because the

**sPIDr(t)**control signal is in the inside of the saturation range

**0…+0.95**again. But what’s about the “dead” time? It wasn’t in the period

**1**. I propose to test

**sPIDi(t)**and

**sPIDr(t)**control signals.

**Chapter 34.6.2 The second more accurate approach
**We will observe the

**sPIDi(t)**and

**sPIDr(t)**control signals of the

**PID**. Do we discover the “dead” time reason?

Call desktop/PID/20_regulacja_i_nasycenia/10_PID_nas0…+0.95_skoki_-1…2.zcos

**Fig. 34-25**

Click “start”

**Fig. 34-26**

The

**ideal**system is obvious.

The

**sPIDr(t)**of the real system in the period

**2**is limited by the saturation and

**sPIDr(t) =x(t)=0.95**. The feedback doesn’t act now the

**yr(t)**is an opened loop type now. But the “dead” time in the period

**3**remains mysterious. I propose to observe

**sPIDir(t)**signal before

**power amplifier**.

**Chapter 34.6.3 The third even more so accurate approach**

Call Desktop/PID/20_regulacja_i_nasycenia/11_PID_nas0…+0.95_skoki_-1…2_more.zcos

**Fig. 34-27**

Click “start”

**Fig. 34-28**

**Ideal System
**The

**yellow**

**sPIDi(t)**control signal (fuzzy colour!) is obvious and doesn’t require the comment.

**Real System**

We observe the

**green**

**sPIDir(t)**before the power amplifier. The

**i**letter in the name

**sPIDir(t)**suggests that the signal is after the ideal part of the real

**PID**cpontroller.

**Period 1 x(t)=+0.3**

The

**green**

**sPIDir(t)**is bigger than saturation

**+0.95**, therefore the

**blue sPIDr(t)**control signal will be cutted. There isn’t feedback now and

**yr(t)**arises up to

**+0.95**as an opened loop system. The feedback shows after

**6.5 sec**and the

**yr(t)**attains steady state

**yr(t)=x(t)=0.3**after circa

**25 sec**.

**Period 2 x(t)=+0.95**

The

**sPIDir(t)**

**=+0.95**all the

**period 2**and

**yr(t)**is going to

**+0.95**as opened loop system.

**Period 3 x(t)=+0.5**

Something is clear with the “dead time”. The control signal

**blue PIDr(t)=+0.95=saturation**during the “dead time”! It’s steady and it means that feedback doesn’t work here! It works after the “dead time” and the steady state

**yr(t)=x(t)=0.5**is attained after circa

**110 sec**. Is the “dead time” secret hidden in the

**green sPIDir(t)**signal? Let’s look at this not cutted by the oscilloscope signal.

**Chapter 34.6.4 This same experimen but the bigger oscilloscope range.**

Call Desktop/PID/20_regulacja_i_nasycenia/12_PID_nas0…+0.95_skoki_-10…140_more.zcos

This same block diagram but other oscilloscope range **-100…+140**, before was **-1…+2**. All the **green sPIDir(t)** (before power amplifier) will be seen.

Click “start”

**Fig. 34-29**

The signals

**x(t), sPIDr(t), yi(t), yr(t)**are very small, almost unvisibile. But you see all the

**sPIDi(t) i sPIDir(t)**control signals.

Whoa! I see the

**green**real control signal before the power amplifier. But where is the

**sPIDir(t)****ideal control signal? It’s covered by the**

**sPIDi(t)****green**.

**sPIDir(t)****Period 1 x(t)=+0.3**

See

**Fig. 34-28**. All the real

**blue**, output

**sPIDr(t)****yr(t)**and a piece of the

**are seen here. You haven’t seen the falling**

**sPIDir(t)****sPIDir(t)**up to

**6.5 sec**. The feedback works after

**6.5**

**sec**and output

**yr(t)**attained steady state

**yr(t)=x(t)=0.3**. Nothing new and interesting in this period.

**Period 2 x(t)=+0.95**

The feedback doesn’t work (see

**Fig. 34-28**) and the ideal

**PID**part “thins” that it’s opened loop. It causes the typical arising of the

**green**signal. The typical

**sPIDir(t)****PID**step response. But the

**blue**is steady because of the saturation=

**sPIDr(t)=0.95****0.95**. We can say that this

**green**signal is unnecessary beacuse it doesn’t cause output

**sPIDir(t)****yr(t)**reaction.

**Period 3 x(t)=+0.5**

See

**Fig. 34-28**again. You see a piece of the falling

**green**signal. The ideal part of the real

**sPIDir(t)****PID**controller “thinks” that is opened loop (saturation=

**0.05**) and it’s reaction for the

**x(t) 0.95/0.5**signal drop (negative step) is typical. It falls up to

**+0.95**value and this is a reason of the “dead time”. (

**Fig. 34-28**) The feedback is again after the “dead time” and the steady state

**yr(t)=x(t)=+0.5**is attained atfter the

**115 sec**.

**Chapter 34.6.5 Conclusions**

**1. **The **PID** control is possible when the set point **x(t)** signal is inside the of the **power amplifier **(other name is **final control element** or **actuator**) **range. **It’s obviously. It’s impossible to heat a bath of water up to **+100°C **when you dispose a small electrical immersion “glass” heater.

**2. **When the et point **x(t)** signal is inside the **range** of the **power amplifier** –> the integral **I **of **PID **is able to assure **null **control error.

**3. **The wider is **power amplifier** output signal range–>the more similar to **ideal **is a **real **system.

**4. **The “dead time” of the **PID **control reason is a **exiting **from the **PID **saturation state. This is an harmful effect because there are **time delays**.* How to counteract? The **PID “**feels” that it’s in the saturation state (The signal from valve end contact or output **y(t) signal** is steady when **PID **control signal arises) and changes its structure to **PD **controller. The integration doesn’t act.

**5. **The smaller is the set point **x(t) **signal the more are the** real** and** ideal** systems similar.

**6. **The **real** and **ideal **systems disturbances suppresions are similar often. The similarities are closer when the **z(t) **disturbances are smaller.

*** **The time delay **To **is always harmfull in the control systems. An example. There is **To=1 sec **delay between steering wheel and the wheels. How is it made? No important. Is this a secure car?