## Preliminary Automatics Course

**Chapter 31. PID controller tuning
**

**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(s**

**)**designating 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

**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”

**Fig. 31-2**

How to obtain

**K**,

**T**and

**To?**See

**chapter 14 Fig. 14-9**

**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**.

**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

**Fig. 31-5**

**PID control**with the optimal parameters from

**Fig. 27-4**.

Click “start”

**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

**Fig. 31-7**

**Kp=3 Ti=7 sek Td=3 sek**

Click “start”

**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

**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 “start

**The**

Fig. 31-10

Fig. 31-10

**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”

**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”

**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

**and**

**Kp Ti****parameters in the**

**Td****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”

**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

**finit**

**e**amplitudes.

**Chapter 31.3.6 Optimal Kp, Ti and Td designating**

**Fig. 31-14
**

**Fig. 31-14**

**a-**Formulas assure short setting time and no more than

**30%**overregulations.

**Fig. 31-14**

**b –**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

**Fig. 31-15
**

**Kp Ti**and

**Td**optimal parameters obtained by the

**Click “start”**

**self oscillation**method

**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.