Since the delay and lag are not noticeable in the step response, we use a first-order model for the process.

$$ G(s) = \frac{K}{1+\tau s} $$

Combine with PI controller

$$ C(s) = K_P + \frac{K_I}{s} $$

The open-loop transfer function becomes:

$$

L(s) = C(s) \cdot G(s) = \left(K_P + \frac{K_I}{s}\right) \cdot \frac{K}{1 + \tau s} = \frac{K (K_P s + K_I)}{s(1 + \tau s)}

$$

The closed-loop transfer function is:

$$

T(s) = \frac{L(s)}{1 + L(s)} = \frac{K (K_P s + K_I)}{\tau s^2 + (1 + K K_P) s + K K_I}

$$

This is a second-order system, where:

• $\tau$ : influences inertia
• $K_P, K_I$ : determine closed-loop pole placement

Controller Tuning (Critical Damping)

Critical damping is the desired response for a heater system because it can only provide positive control input (active heating but not active cooling).

To achieve a critically damped response with desired speed $\omega_n$, we match the closed-loop denominator to:

$$ s^2 + 2\omega_n s + \omega_n^2 = (s+\omega_n)^2 $$

which gave 2 real poles at $-\omega_n$

Matching terms gives:

$$ T(s) = \frac{K (K_P s + K_I)/ \tau}{ s^2 + (1 + K K_P) s/\tau + K K_I/\tau} $$