Ziegler–Nichols method

The Ziegler–Nichols tuning method is a heuristic method of tuning a PID controller. It was developed by John G. Ziegler and Nathaniel B. Nichols. It is performed by setting the I (integral) and D (derivative) gains to zero. The "P" (proportional) gain, $K_{p}$ is then increased (from zero) until it reaches the ultimate gain $K_{u}$ , at which the output of the control loop has stable and consistent oscillations. $K_{u}$ and the oscillation period $T_{u}$ are used to set the P, I, and D gains depending on the type of controller used:

Ziegler–Nichols method^[1]
Control Type	$K_{p}$	$T_{i}$	$T_{d}$
P	$0.5K_{u}$	-	-
PI	$0.45K_{u}$	$T_u/1.2$	-
PD	$0.8K_{u}$	-	$T_u/8$
classic PID^[2]	$0.6K_{u}$	$T_u/2$	$T_u/8$
Pessen Integral Rule^[2]	$0.7K_{u}$	$T_u/2.5$	$3 T_u/20$
some overshoot^[2]	$0.33K_{u}$	$T_u/2$	$T_u/3$
no overshoot^[2]	$0.2K_{u}$	$T_u/2$	$T_u/3$

The ultimate gain (Ku) is defined as 1/M, where M = the amplitude ratio

These 3 parameters are used to establish the correction $u(t)$ from the error $e(t)$ via the equation:

u(t)=K_{p}\left(e(t)+{\frac {1}{T_{i}}}\int _{0}^{t}e(\tau )d\tau +T_{d}{\frac {de(t)}{dt}}\right)

which has the following transfer function relationship between error and controller output:

u(s)=K_{p}\left(1+{\frac {1}{T_{i}s}}+T_{d}s\right)e(s)=K_{p}\left({\frac {T_{d}T_{i}s^{2}+T_{i}s+1}{T_{i}s}}\right)e(s)

Evaluation

The Ziegler-Nichols tuning creates a "quarter wave decay". This is an acceptable result for some purposes, but not optimal for all applications.

This tuning rule is meant to give PID loops best disturbance rejection.^[2]

It yields an aggressive gain and overshoot^[2] – some applications wish to instead minimize or eliminate overshoot, and for these this method is inappropriate.

References

↑ Ziegler, J.G & Nichols, N. B. (1942). "Optimum settings for automatic controllers" (PDF). Transactions of the ASME. 64: 759–768.
1 2 3 4 5 6 Ziegler-Nichols Tuning Rules for PID, Microstar Laboratories

Bequette, B. Wayne. Process Control: Modeling, Design, and Simulation. Prentice Hall PTR, 2010.

Co, Tomas; Michigan Technological University (February 13, 2004). "Ziegler-Nichols Closed Loop Tuning". Retrieved 2007-06-24.

External links

https://web.archive.org/web/20080616062648/http://controls.engin.umich.edu:80/wiki/index.php/PIDTuningClassical#Ziegler-Nichols_Method

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[1] Ziegler, J.G & Nichols, N. B. (1942). "Optimum settings for automatic controllers" (PDF). Transactions of the ASME. 64: 759–768.

[microstar-2] 1 2 3 4 5 6 Ziegler-Nichols Tuning Rules for PID, Microstar Laboratories