First Method

Figure 5.2: Reaction curve [1]

Ziegler-Nichols rule determines the values of gain $K$, integral time $\tau _i$ and derivative time $\tau _d$ based on the step response characteristics of a given plant. In this method, one can experimentally obtain the response of a plant to a step input, as shown in figure 5.2. This method is applicable only when the response to the step input exhibits S-shaped curve [3].
As shown in figure 5.2, by drawing the tangent line at the inflection point and determining the intersection of the tangent line with the time axis and the line $c(t)= K$ ,we get two constants, namely, delay time $L$ and time constant $T$.

Ziegler and Nichols suggested to set the values of $K, \tau_i , \tau_d$ according to the formula shown in table 5.1.

Table 5.1: Ziegler-Nichols tuning rule based on step response of plant

Type of controller	$K$	$\tau _i$	$\tau _d$
$P$	$\frac{1}{RL}$	$\infty$	0
$PI$	$\frac{0.9}{RL}$	$3L$	0
$PID$	$\frac{1.2}{RL}$	$2L$	$0.5L$

Notice that the PID controller tuned by the Ziegler-Nichols rule gives,

$$G_c(s) =K_p\left(1+\frac 1{T_is}+T_ds\right)$$ (5.10)

$$=1.2\frac {T}{L}\left( 1+\frac 1{2Ls}+0.5Ls\right)$$ (5.11)

$$=0.6T\frac{\left(s+\frac 1{L}\right)^2}{s}$$ (5.12)

Thus, the PID controller has a pole at the origin and double zeros at $s= -1/L$.