CPS Model Thermostat

Physical Component

Given a room that has a temperature whose dynamics are defined by the function $T \in \mathbb{R}$ and a device that can control this temperature, such as an HVAC System. We can model the physical component of this system by the following differential equation.

$$\dot{T} = (\alpha - \delta S) T + T_R$$

Where the parameter $\alpha$ is the rate of heat exchange given the thermal load in that specific space, the parameter $T_R$ is the rate of influence of the temperature outside that room and the $\delta$ is the capacity of heat exchange of the equipment in question, times the state of the equipment itself, where we are assuming two states, on $(1)$ and off $(0)$.

Cyber Component

Now let's define the finite state machine model for a thermostat, where the set of state $Q$, is defined by the thermostat operating mode defined by: $q \in Q = { On, Off}$, set from which the values of output $\zeta$ are also defined. The input is defined by the continuous temperature variable, discretized by the analog-to-digital converter, and limited by the maximum and minimum temperature restrictions: $v \in \Sigma = T \in [ T_{min},T_{ max}]$. Is a transition function $\delta$ that defines the rule for state transition for a specified combination of state and input.

$$\delta(v,q) = \begin{cases} ON & \text{$v \ge T^{max}$ and q=OFF}\\ OFF & \text{$v \le T^{Min}$ and q=ON} \end{cases}$$