# PID controller tuning: Ziegler-Nichols methods

The classical Ziegler-Nichols methods, introduced in 1942, are some of the most known and applied tuning methods for PID controllers.

In case you don’t know what is PID, I recommend to read the post about the subject before continue.

### Ziegler-Nichols for step response

Ziegler-Nichols methods were develop empirically, with observation and practice in many different plants. The plant is considered a black box, don’t need to know it’s equations. But, must know the transient response curve in output when there is set point change.

Was shown in post about PID, outputs of proportional, integral and derivative in time domain. Converting to s domain, these output are as shown below. In s domain, equations of PID controller become:

$G_{c}(s)=Kp[1+\frac{1}{sT_{i}}]\cdot [1+sT_{d}])=[Kp+\frac{Ki}{s}]\cdot [Kp+sKd]$

$G_{c}(s)=Kp[1+\frac{1}{sT_{i}}+sT_{d}]=Kp+\frac{Ki}{s}+sKd$

Because integral $Ki$ and derivative $Kd$ gains are:

$Ki=\frac{Kp}{Ti}$

$Kd=Kp\cdot Td$

Ziegler-Nichols methods are a set of equations to calculate values of proportional gain $Kp$, integral time $T_{i}$ and derivative time $T_{d}$. Considerate response curve showed below. L is delay time of response and T is time constant.

This is the equation set to obtain $Kp$, $T_{i}$ and $T_{d}$. This method can be used in open loop processes, by that, without feedback. Parameters tuning with this method is only applicable if process response to step signal is a curve which reminds a “S”, as shown above.

### Critical gain for oscillation method

This is Ziegler-Nichols second method, must determine system’s critical gain to output have a stable oscillatory response. Can only be applied in closed loop systems, because feedback is a prerequisite to create oscillations. How to find out the critical gain $K_{CR}$? I and D blocks must be off and proportional gain must be increased until obtain oscillation in output. Obviously, it is made with simulators to avoid damages to physical processes. Other alternative is use computational tools as rlocus from MATLAB. After find $K_{CR}$, must apply equation set shown in table below to calculate parameters. $P_{CR}$ is critical period of output wave. With both methods, can obtain an overshoot in maximum 25% of set point value. Can bring goods result of stability and speed to majority of systems. 