Pid controller tuning methods

Pid controller theory, tuning rules including ziegler-nichols, e practical guidelines.

pidcontrollertuning

Overview

A PID controller continuously calculates an error value and applies a correction based on proportional (P), integral (I), and derivative (D) terms:

$u(t) = K_p \cdot e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d \frac{de(t)}{dt}$

where:

$e(t)$ = error = setpoint − process variable
$K_p$ = proportional gain
$K_i$ = integral gain
$K_d$ = derivative gain

Ziegler-Nichols Tuning (Open Loop)

Based on the process reaction curve with parameters: gain $K$ , dead time $L$ , time constant $T$ :

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Ziegler-Nichols open-loop tuning rules
Controller	Kp	Ti	Td
P	T / (K·L)	∞	0
PI	0.9·T / (K·L)	3.33·L	0
PID	1.2·T / (K·L)	2·L	0.5·L

Fonte: engineeringtoolbox.com

Ziegler-Nichols (Closed Loop / Ultimate Gain)

Using the ultimate gain $K_u$ and ultimate period $T_u$ :

Controller	$K_p$	$T_i$	$T_d$
P	$0.5 K_u$	—	—
PI	$0.45 K_u$	$T_u / 1.2$	—
PID	$0.6 K_u$	$T_u / 2$	$T_u / 8$

Effect of Each Term

Term	Effect of Increasing	Side Effect
P (proportional)	Faster response, reduces steady-state error	May oscillate
I (integral)	Eliminates steady-state error	May cause overshoot, slow settling
D (derivative)	Reduces overshoot, dampens oscillations	Sensitive to noise

Practical Tips

Start with PI control (set $K_d = 0$ ) for most processes
Add D term only for slow processes with significant dead time
Typical anti-windup: limit integrator output range
Derivative filter: use $K_d \cdot s / (1 + T_f \cdot s)$ with $T_f \approx T_d / 10$

Fonte: engineeringtoolbox.com