**Chapter 1 Introduction
When You **wish to know “How how does automatics work?”, especially controllers king

**PID**–>this course is for You!

**You will test**different objects in real time! Mainly

**10…120 sec**..

**WE BUILD LABORATORY–>Chapters 2…5**

**You will**

– **Install** **free (!!!)** program **Scilab from internet**–>**Chapter 2 Program SCILAB installation***

– **Download** diagram blocks folder from internet–>**Chapter 3 Block diagrams files downloading***

**You:**

– read further chapter from the table of contents

– call block diagram and make an experiment (for example **oscillation block**).

**We start** with simple dynamic blocks –>**chapters. 6…14** and finish with different complicated automatics control structures –>**Cascade Regulation, Close-opened structures,… etc** –>**chapter. 33 Automatics control structures**.

Your job is only to analyze different parameters influence for process time.

***Note**

You can bypass these points and don’t build a laboratory. Then it’s normal book with pictures . But it’s pity. Can you imagine the engineering or technical college without laboratories?

**Chapter 4 Scilab shortly**

**Scilab** is a supercalculator. For equations solving, also differential equations. It does all what’s the math. It’s similar to payable **MATLAB. **The **Scilab 5.5.2** version is used.

**Chapter 5 Xcos shortly**

We will use **Xcos** only–>**Scilab’s** application. **Xcos** is so clever, that it changes the block diagram and input signal **x(t)** for differential equation. It will solve this equation and we will have beautifull output signal **y(t)**.

**PRINCIPAL DYNAMICS UNITS–>Chapters 6…14**

**Every** dynamic unit has its own chapter. You give input signal **x(t)** and observe the output signal **x(t)** .

**Input **signal **x(t)** is:

– unit step generator – often

– ramp generator

– hand operated virtual potentiometer

– dirac type (“short hammer hit”)

**Output** signal **y(t)** is observed by:

– oscilloscope (your computer monitor)

– virtual digital meter

– virtual analog meter-black bar

The main goal is the transfer function **G(s)** parameters connotation with the input **x(t)** and output **y(t)** signals.

You will easy forecast reactions two similar **G(s)** when you will finish chapters **6…14**.

**Fig. 1-1**

**A BIT OF MATHS–>chapters. 15…18
**

I wish the course will be also for those who don’t know differential calculus.

**Chapter. 15 Differentiation**

**Fig. 1-2**

Differential unit is good for differentiation understanding.

**Chapter. 16 Integration**

**Fig. 1-3**

Integration unit is good for integration understanding.

**Chapter. 17 Differential equations**

When you were 10 years old, you have deal with **differential equations**! Really? When you solved the mathematical task type “The train is going from town A to town B . The velocity is v….”

**Fig. 1-4**

We wil become acquainted with **differential equations** meeting the problems. The examples will be less trivial than “train is going…”:

**A**–**differential equation** filling the tank without hole

**B**-simplified **differential equation** filling the tank with hole

**C**-accurate **differential equation** filling the tank with hole

**Chapter. 18 Laplace transform**

**Fig. 1-5**

Every time function **f(t)** is connected with its Laplace transform **F(s)** and vice versa.–>**Fig. 1-5a.**

Parameter **s** is a **complex** number **s**. Don’t you know this subject? Don’t bother yourself. Consider **s** temporarily as a **real** number from the grammar school.

**Fig. 1-5b** is a **Fig. 1-5a** version for the particular time function **f(t)=sin(t)**.

How do we create Laplace transform **F(s)** on the base time function **f(t)**? It isn’t important. We assume that we have a very clever book with all possible pairs **f(t)<–>F(s)**.

Laplace transform has a fun asset. It would be worthless without this asset. It’s easy Laplace transform function derivative creation. It’s easy multiplication.

**Fig. 1-6**

The above mentioned statement enables easy **linear differential equations*** solving. They are changed for **algebraic equations** with the **polynomians**. These **polynomians** are a source of important informations of the **control theory** as stability, gain etc…

***Linear differential equations** are an **approximation** of the control system. The examples are **A**, **B**–>**Fig.1-4**. Equation **C** isn’t linear.

**GENERALLY ABOUT REGULATION–>Chapters. 19…25**

**Chapter. 19 More about regulation and blocks structures**

You will know that:

**1** Transfer function **G(s)** is an equivalent of the **linear differential equations** describing **particular** dynamic unit.

**2** It’s easy to calculate stable state **gain** from the **G(s)**

**3**. The units connected:

– serial

– parallel

– with the feedback

may be replaced by individual transfer function **Gz(s)**

**Chapter. 20 How does feedback work?**

**Fig. 1-7**

**Fig. 1-7a** forms positive feedback. It’s something wrong. Why do we consider this problem? Because it’s easy for understanding. There are some interesting conclusions here. The positive feedback system may be stable, when gain **K** is low for example.

**Fig. 1-7b** forms negative feedback. It’s fundamental control theory block diagram. The output **y(t)** strives to follow input **x(t)** all the time. There is no ending battle beetwen **y(t)** and **x(t)**. When **y(t)** is too big –>”something” orders him to drop, when too small, it orders him to increase. The effect of the battle beetwen beetwen **y(t)** and **x(t)** is not always happy. The oscillations may be occur–> system is **unstable.**

**Fig. 1-7d** The result of this battle is a stable state **y(t)=Kz*x(t)**. When **K** aspires to big values, then **Kz** aspires to **Kz=1**. It performs every control engineer dream–>**y(t)=x(t)** (in stable state of course!)

**Fig. 1-7c** forms close loop transfer function **Gz(s)**

**Chapter. 21 Instability**

The delay signal **y(t)** towards **x(t)** is a reason of instability. You will see it in the experiment, when the unit **G(s)** is a “clear” delay

**Fig. 1-8**

When gain **K** performs:

**K<1** system is **stable**

**K=1** system is border on **stability**

**K>1** is **unstable**

It’s similar to Nyquist stability benchmark–>see next chapter

The real objects delay is slightly “fuzzy”. The above principle in this cases, isn’t so easy, but similar.

For example. When we replace the **delay** object (Fig. 1-8) for **3-inertial** object–>the benchmark will be:

When gain **K** performs:

**K** is **small**–>system is **stable**

**K** is **middle**–>system is border on **stability**

**K** is **big**–>system is **unstable**

**Conclusion**:

Big **gain** and **inertia** are a source of the **instability**.

**Chapter. 22. Nyquist stability benchmark**

Generally-**open loop system** is stable. It can be **unstable** when we make it **close loop system**–> more precisely **negative feedback system**. **Benchmark** tests **amplitude phase characteristic** of the **open loop** object **G(s)** and predicts the stability of the **closed loop **of the object **G(s)**. You become acquainted with the concept of the **amplitude phase characteristic** by the way.

**Chapter. 23. Hurwitz stability benchmark**

We test the denominator **M(s)** coefficients. The **M(s)** is a polynomian of the denominator of the object **G(s)=L(s)/M(s)**.

**Chapter. 24 ON-OFF control**

The principal ON-OFF control algorithm is very easy.

If output signal **y(t)** is:

– greater than input set point **x(t)** then decrease the control signal **s(t)**

– lower than input set point **x(t)** then increase the control signal **s(t)**

You will be very tired when You will be personally **Mister Controler**. But stupid **ON-OFF relay controler** will be much better.

**Chapter. 25 Continous control**

This control algorithm enables steady control **s(t)** and output **y(t)** signals in stable state. You will be also tired when control a process, but you will be much worse than a simple differential amplifier–>the **P** type controler. **Continuos controler** strives for state **y(t)=Kz*x(t)** where **x(t)** is a step signal and **Kz** see **Fig.1-7d.**

**CRÉME DE LA CRÉME I.E. PID CONTINUOS CONTROL**

The intuition is most important. I write for example “The controller observes the situation and it decides to do some actions…” I try to explain very exactly **PID** controler working and what especially **Kp, Ti, Td** settings are doing. You will be looking for the optimal settings **Kp, Ti, Td.**

Don’t be afraid of the **chapters. 25…29** big sizes, but all experiment are repetitive with different **Kp, Ti , Td** settings and you wiil be not tired.

We will be looking for optimal* settings for objects:

**1** one-inertial

**2** two-inertial

**3** three-inertial

We will test disturbances influence for control quality.

***** There are many optimal responses. Our response should be as fast as possible with small oscillations.

**Chapter. 26 P control**

You will be convinced that this control doesn’t assure absolute accuracy in stable state. Control **e(t)** error always exists. The bigger is amplification **Kp** setting–> the lower is **e(t)** error. The bigger **Kp** can cause oscillations and even instabilities

**Fig. 1-9**

**y(t)** and **e(t)** stable state signals when **x(t)** is step type:

**K=Ko*Kp** is a amlification of all the open loop. It includes object-**Ko** and controler **Kp** amplifications.

**Chapter. 27 PD control**

Stable state **e(t)** error is the same as in **P** control but differential part **D** exceptionally improves dynamic qualities and prevents instabilities.

**Chapter. 28 I control**

Algorithm is similar to **ON-OFF** control:

**x(t)>y(t)**–>increase **s(t)** control signal

**x(t)<y(t)**–>decrease **s(t)** control signal

The increase/decrease velocity is proportional to **e(t)=x(t)-y(t)**. When **e(t)=0**–>velocity=0–>control **s(t)** is steady! (on the contrary to **ON-OFF** control). The main **I** control advantage is an absolutely accuracy **e(t)=0** in stable state! Control **I** is slowly and practically not used.

**Chapter. 29 PI control**

**Control P** is reasonably fast but it doesn’t assure null control error **e(t)=0** in stable state. **Control I** assures null control error **e(t)=0** but it is very slowly. Let’ s make a combination of the both controls **P** and **I** and create a **control PI**. This control will be fast (but not fast as **Control P**) and will assure null control error **e(t)=0**. I tinked a didactic **controler PI**. It starts as a **controler P** and changes in **controler I** in one moment. You will see then, that the real **PI** works similary, but the changing process **P–>I** is continuous.

**Chapter. 30 PID control**

When we add to **PI** differential part **D**, we create a **PID controler** which:

– assures control error **e(t)=0**

– is faster than **PI controler**

**D** part prevents instabilities.

**Chapter. 31 PID controller tuning**

You have known the first e.g. **manual tuning** method in the **chapters 25…30**. It was completely “non-scientific” method, but you can feel good what are doing **P,I,D** parts .

I will introduce a little old fashioned (Second War II) Ziegler and Nichols methods

– **Step response method**

You test the step response of the open loop object **G(s)** and build a substitute simplified object **Gs(s) **with parameters **K,T,To** (To-delay)

– **Ultimate Cycling method**

First-you disconect **I** and **D** actions and we have clear **P** controller then. We close the loop and increase the **Kp** setting of the **PID** controler. You stop **Kp** increasing, when the stable oscillatons appeare. You measure the oscillations period **Tp** then. Your job is only to read the **Kp, Ti, Td** settings of the **PID** from a special table.

Modern **PID** microprocessor type controlers are very clever. They “feel” the object **G(s)** behavior and continously tune the **Kp,Ti,Td** settings. Many doctoral thesis and other wisdoms are included in the **PID **controller microprocessor memories.

**Chapter. 32 Noise analysis **

The set point **x(t)** is the main control system input **signal**. But there are the others–>so-called input **noise signals**. They are up to the output signal **y(t)** impair. Strictly speaking, to make for example output temperature **y(t)=+96°C** instead of **100°C** as set point **x(t)=100°C.**

The operating for object **G(s)** noise **z(t)** is important for us only. Why? The operating for controller **z(t)** noise is terrible! It is like a spy giving orders in enemy army headquarter! But it’s easier to assure small **controller** noise immunity than for a big object **G(s)**. Analogously – it’s easier to assure headquarter security than all the army soldiers security .

Let’ come back to our object **Go(s)**. I will show that the moving to right **z(t)** noise causes more distinct effects.

**Fig. 1-10**

**Chapter. 33 Control Systems Structures**

Control quality may be improved not only by PID controller tuning but also by control structure modifing:

Typical control stuctures:

**1**– Opened loop

**2**– Opened loop with the noise compensation

**3**– Closed loop

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

**5**– Cascade control

**6**– Ratio control

**Chapter. 34 Nonlinearity influence for regulation**

We tested **linear** control systems only until now*. When **PID** controller needs to provide very big output control signal it wiil provide big signal. There were many situations when controler needs **1 kW** output power signal in stable state , but the signal is **1000 kW** in intermediate state short state!

There are limits in the real systems. The boiler needs **1 kW** output power signal only to reach **y(t)=+100°C**. The controller assures **3 kW** only. The **1000 kW** in short transcient state isn’t economically accepted. It means that output **y(t)** will be not so beautiful as for linear!. The real system will be worse of course. But how much “worse”? The answer is in this chapter.

We show that the main nonlinearity is the output controller **power amplifier**. The other name of this amplifier is a **final control element.**

The second conclusion is that linear and real systems behaviours are similar.

*except chapter 24

**LIST OF CHANGES
**I don’t suppose there are serious control theory heresies. But small heresies? Also, I can come to a conclusion that some subjects may be better described. I.e. all the cours will be all the time improved. I count on readers notes at the end of each chapter. The changes and improvments will be signed in in the

**LIST OF CHANGES**r. So glance through this report occasionally

**.**