Discrete PID realization and roles of P, I, D

A digital PID controller is obtained by replacing the integral and derivative of the analog PID law with sums and differences of the sampled error $e(kT)$.

Analog PID law

$$m(t) = K\left[e(t) + \frac{1}{T_i}\int_0^t e(t)\,dt + T_d\frac{de(t)}{dt}\right]$$

Discretization

• Integral by the trapezoidal rule, derivative by the backward difference:

$$m(kT) = K\left\{e(kT) + \frac{T}{T_i}\sum_{h=1}^{k}\frac{e((h-1)T)+e(hT)}{2} + \frac{T_d}{T}\big[e(kT)-e((k-1)T)\big]\right\}$$

• Taking the z-transform:

$$\begin{aligned} \frac{M(z)}{E(z)} = G_D(z) &= K_P + \frac{K_I}{1-z^{-1}} + K_D(1-z^{-1})\\ K_P = K - \frac{KT}{2T_i},\quad K_I &= \frac{KT}{T_i},\quad K_D = \frac{KT_d}{T} \end{aligned}$$

Realization (velocity / recursive form)

Multiplying by $(1-z^{-1})$ gives an algorithm that needs only the last two errors and the last output:

$$m(k) = m(k-1) + (K_P+K_I+K_D)e(k) - (K_P+2K_D)e(k-1) + K_De(k-2)$$

          +--> [ KP ] -----------------+
          |                            v
 e(k) ----+--> [ KI/(1-z^-1) ] ---->( + )---> m(k)
          |                            ^
          +--> [ KD(1-z^-1) ] ---------+

The computer runs this equation once every sampling period, and the D/A converter with a zero-order hold applies $m(k)$ to the plant.

Role of each part

• P (proportional): gives an output proportional to the present error. A larger $K_P$ makes the response faster and reduces steady-state error, but too much gain causes overshoot and instability.
• I (integral): adds a pole at $z=1$ and accumulates past error. It removes steady-state error to a step (raises the system type by one) but adds phase lag, so it can make the response more oscillatory. Anti-windup is needed when the actuator saturates.
• D (derivative): responds to the rate of change of error, so it anticipates future error. It adds damping, reduces overshoot and improves stability, but it amplifies high-frequency measurement noise. It is often applied to the measured output instead of the error, with a filter.