1. Physical System Modeled as Differential Equation

let

• $T(t)$: temperature of the heated object
• $T_a$: ambient temperature
• $u(t)$: heater input (duty ratio, 0-1)
• $P(t)$: heater rated electrical power
• $C$: lumped thermal capacitance
• $R$: lumped thermal resistance to ambient
• $w(t)$: un-model disterbance or process noise

The energy balance gives:

$$

C \frac{dT}{dt} = P(t) + \frac{T_a - T}{R} + w(t)

$$

As we can control our heating power from control input u(t) = [0,1]

$$ P(t) = P_{max}u(t) $$

Assumed

$$ w(t) = 0 $$

This can be rewritten as:

$$

C \frac{dT}{dt} = P_{max}u(t) + \frac{T_a - T}{R} + w(t)

$$

$$

\frac{dT}{dt} = \frac{P_{max}u(t)}{C} + \frac{T_a - T}{RC} + \frac{w(t)}{C}

$$

$$

\frac{dT}{dt} = k\,u(t)-\frac{1}{\tau}(T - T_a) + \frac{w(t)}{C}

$$

With PI control