Chapter 31.1 Introduction
You read the chapters 26..30 and you know the first  method of the PID controller tuning. This is “hand ” or “trial and error” method.
I will present a slightly oldfashioned  2 PID tuning methods invented by Ziegler and Nichols in 1940:
– step response method
– self oscillation method

Chapter 31.2 Step response method
Chapter 31.2.1 Introduction
It requires the substitute transfer function Gp(s) (see chapter 14) of the controlled object Go(s).
There are 4 stages of works:
1-Parameters K, T and To  Gp(sdesignating of the  of the object Go(s).  This is step response experiment.
2-Optimal  Kp, Ti and Td designating based on known K, T and To  parameters of the Gp(s). We use Ziegler and Nichols tables
3-Is a good step response now?
4-The attempts to find better Kp, Ti and Td by the hand method.

Chapter 31.2.2 Stage 1-Step response
Call Desktop/PID/17_dobór_nastaw_PID/01_obiekt_4T.zcos
31-1a
Fig. 31-1
Typical multi-inertial object–>four-inertial here. Imagine that you are on the industrial site and you don’t see K,T1,T2,T3 and T4
parameters. Many objects, especially in the chemical industry, are multi-inertial type.
Click “start”
31-2a
Fig. 31-2
How to obtain K, T and To?  See chapter 14 Fig. 14-9
31-3a
Fig. 31-3
K T and To parameters.

Chapter 31.2.3 Stage 2-Optimal Kp, Ti and Td designating
There are some different optimal Kp, Ti and Td, because there are different optimum benchmarks. One doesn’t accept the
oscillations, the another allows 20% overregulations and the next …etc. Our benchmark allows 20% overregulations.

31-4a
Fig. 31-4
Ziegler-Nichols table and the optimal parameters
The optimal Kp, Ti and Td parameters obtained from the a.m Ziegler-Nichols table. There is one restriction–> 0.15T<To<0.6T.
It’s fulfilled because  2.1 sec<4 sec<8.4 sec.

Chapter 31.2.4 Etap 3-Is a good step response now?
Call Desktop/PID/17_dobór_nastaw_PID/02_Kp4.2_I8_D1.6.zcos
31-5a
Fig. 31-5
PID control with the optimal parameters from Fig. 27-4.
Click “start”
31-6a
Fig. 31-6
My feelings are ambivalent. The overregulation and setting time are too big. I am not sure you will be convinced but I try to explain.  The chosen parameters are only approximately optimal. It’s the first step only. I will look now better parameters using “trial and
error” method. These PID parameters should be near Kp=4.2 Ti=8 sec and Td=1.6 sec.

Chapter 31.2.4 Stage 4-The attempts to find better Kp, Ti and Td by the hand method.
I made some experiments and there is proposed PID controller.
Call Desktop/PID/17_dobór_nastaw_PID/03_Kp3_I7_D3.zcos
31-7a
Fig. 31-7
Kp=3 Ti=7 sek Td=3 sek
Click “start”
31-8a
Fig. 31-8
The process is much better. But why the first shot  (Fig. 31-6) wasn’t accurate? The ambivalent feelings are remained.

Chapter 31.3 – Self oscillation method
Chapter 31.3.1 Introduction
We disconnect the I and D component first to obtain simple P controller. Then we will increase the Kp parameter up to oscillations with the steady amplitude appearance.

Chapter 31.3.2 Test no. 1
Call Desktop/PID/17_dobór_nastaw_PID/04_Kp3_wzbudzenie.zcos
31-9a
Fig. 31-9
The P controller because the I and D components are disconnected. I remind that the unstable systems may be immobile as
a vertical pencil on the table. There is required small  Dirac type pulse to unbalance the system.
Click “start31-10a
Fig. 31-10
The Dirac hammer is used but the system returned to its stable initial state. Let’s increase the Kp.

Chapter 31.3.3 Test no. 3
Call Desktop/PID/17_dobór_nastaw_PID/05_Kp5_wzbudzenie.zcos
The block diagram is the same but Kp=5
Click “start”
31-11a
Fig. 31-11
The oscillations are bigger but the system is stable yet.
Let’s increase gain for Kp=6.27

Chapter 31.3.4 Test no. 4
Call Desktop/PID/17_dobór_nastaw_PID/06_Kp6.27_wzbudzenie_graniczne.zcos
The block diagram is the same but Kp=6.27
Click “start”
31-12a
Fig. 31-12
There are steady amplitude oscillations with the period Tosc=16.3 sec.
Conclusion-the system is in the stability border state and Kp=Kkr=6.27 (Kkr-critical gain)
The parameters Kkr=6.27 and Tosc=16.3 sec are necessary to calculate the optimal Kp Ti and Td parameters in the chapter 31.3.6. We have  Kkr=6.27 and Tosc=16.3 sec  parameters and we don’t need to increase the Kp now.  But we do it because we are just
curious! Let’s increment a gain a little for Kp=6.5.

Chapter 31.3.5 Test no. 5
Call Desktop/PID/17_dobór_nastaw_PID/07_Kp6.5_wzbudzenie.zcos
The block diagram is the same but Kp=6.5
Click “start”
31-13a
Fig. 31-13
We expected this system behaviour. The amplitude is slowly growing up to +/-infinity.
It was an ideal linear system without saturations. The real systems have the finite amplitudes.

Chapter 31.3.6 Optimal Kp, Ti and Td designating
31-14a
Fig. 31-14
Fig. 31-14a-Formulas assure short setting time and no more than 30% overregulations.
Fig. 31-14b –The formulas result when Kkr=6.27 and  Tosc=16.3 sek from  chapter 31.3.4. I propose to check it by calculator.

Chapter 31.3.7 Is a good step response now?
Call Desktop/PID/17_dobór_nastaw_PID/8_Kp3.76_I8.15_2.04.zcos
31-15a
Fig. 31-15
Kp Ti and Td optimal parameters obtained by the self oscillation method
Click “start”
31-16a
Fig. 31-16
The process doesn’t bowl over but it’s much better than in the step response method in the Fig. 31-6.
My feelings aren’t so  ambivalent now. 

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