Ziegler-Nichols metoden
Man sætter $I$ og $D$ til 0, og så stiger $K_p$ fra 0 indtil den får den ultimative $K_u$ Så bruger man $K_u$ og $T_u$ til at finde alle de andre $P$, $I$ og $D$ den ultimative $K_u$ kan defines som 1/M og M er amplitude ratioen. $K_i = K_p/T_i$ og $K_d = K_pT_d$
| Control Type | $K_p$ | $T_i$ | $T_d$ | $K_i$ | $K_d$ |
|---|---|---|---|---|---|
| P | $0.5K_u$ | – | – | – | – |
| PI | $0.45K_u$ | $0.8\overline3T_u$ | – | $0.54K_u/T_u$ | – |
| PD | $0.8K_u$ | – | $0.125T_u$ | – | $0.10K_uT_u$ |
| classic PID | $0.6K_u$ | $0.5T_u$ | $0.125T_u$ | $1.2\frac{K_u}{T_u}$ | $0.075K_uT_u$ |
| Pessen Integral Rule | $0.7K_u$ | $0.4T_u$ | $0.15T_u$ | $1.75\frac{K_u}{T_u}$ | $0.105K_uT_u$ |
| some overshoot | $0.3\overline{3}K_u$ | $0.50T_u$ | $0.3\overline3T_u$ | $0.6\overline6\frac{K_u}{T_u}$ | $0.1\overline1K_uT_u$ |
| no overshoot | $0.20K_u$ | $0.50T_u$ | $0.3\overline3T_u$ | $0.4\frac{K_u}{T_u}$ | $0.06\overline6K_uT_u$ |
These 3 parameters are used to establish the correction ${\displaystyle u(t)}$ from the error ${\displaystyle e(t)}$ via the equation:
De her 3 parameter er brugt til at finde den rigtige
$u(t) = K_p \left( e(t) + \frac{1}{T_i} \int_0^t e(\tau) , d\tau + T_d \frac{de(t)}{dt} \right)$
which has the following transfer function relationship between error and controller output:
${\displaystyle u(s)=K_{p}\left(1+{\frac {1}{T_{i}s}}+T_{d}s\right)e(s)=K_{p}\left({\frac {T_{d}T_{i}s^{2}+T_{i}s+1}{T_{i}s}}\right)e(s)}$