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
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.
How to obtain K, T and To? See chapter 14 Fig. 14-9
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.
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?
PID control with the optimal parameters from Fig. 27-4.
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.
Kp=3 Ti=7 sek Td=3 sek
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
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.
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
The block diagram is the same but Kp=5
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
The block diagram is the same but Kp=6.27
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
The block diagram is the same but Kp=6.5
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
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?
Kp Ti and Td optimal parameters obtained by the self oscillation method
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.