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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…”:
Adifferential 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 criterion
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 criterion
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 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 CHANGESr. So glance through this report occasionally
.