PID Tuning Code using Zeigler-Nicholas Method in MATLAB

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, Kp is then increased (from zero) until it reaches the ultimate gain Ku, at which the output of the control loop has stable and consistent oscillations. Ku and the oscillation period Tu are used to set the P, I, and D gains depending on the type of controller used:

Ziegler–Nichols method^[1]
Control Type	$K_{p}$	$T_{i}$	$T_{d}$	$K_{i}$	$K_{d}$
P	$0.5K_{u}$	–	–	–	–
PI	$0.45K_{u}$	$T_u/1.2$	–	$0.54K_{u}/T_{u}$	–
PD	$0.8K_{u}$	–	$T_u/8$	–	$K_{u}T_{u}/10$
classic PID^[2]	$0.6K_{u}$	$T_u/2$	$T_u/8$	$1.2K_{u}/T_{u}$	$3K_{u}T_{u}/40$
Pessen Integral Rule^[2]	$7K_{u}/10$	$2T_{u}/5$	$3 T_u/20$	$1.75K_{u}/T_{u}$	$21K_{u}T_{u}/200$
some overshoot^[2]	$K_{u}/3$	$T_u/2$	$T_u/3$	$0.666K_{u}/T_{u}$	$K_{u}T_{u}/9$
no overshoot^[2]	$K_{u}/5$	$T_u/2$	$T_u/3$	$(2/5)K_{u}/T_{u}$	$K_{u}T_{u}/15$

The ultimate gain (Ku) is defined as 1/M, where M = the amplitude ratio, Ki = Kp / Ti and Kd = Kp . Td.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 (represented by the ‘Classic PID’ equations in the table above) 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. In this case, the equations from the row labeled ‘no overshoot’ can be used to compute appropriate controller gains.

References

Ziegler, J.G & Nichols, N. B. (1942). “Optimum settings for automatic controllers” (PDF). Transactions of the ASME. 64: 759–768.
Jump up to:^a^b^c^d^e^f Ziegler–Nichols Tuning Rules for PID, Microstar Laboratories.

Bequette, B. Wayne. Process Control: Modeling, Design, and Simulation. Prentice Hall PTR, 2010. [1]
Co, Tomas; Michigan Technological University (February 13, 2004). “Ziegler–Nichols Closed Loop Tuning”. Retrieved 2007-06-24.