Problem $3.(10$ points$)$

Consider the following model for a thermostat system.

The thermostat has been designed to maintain the temperature of a room at $TC.$ The model has two states:

cooling and heating. When the system is in the cooling state and the temperature of the room goes below $TC,$

the system generates a signal to switch on a heater and moves to the heating state. When the temperature of

the room goes over $TC,$ the system generates a signal to switch off the heater and moves to the cooling state.

$($a$)$ Represent the system as an actor that takes the current temperature as input and produces a signal to

control the heater. The actor uses the set point T as a parameter.

$($b$)$ Identify a design problem in the model.

$($c$)$ Provide two different remedies to address the problem $-$ one event triggered and one time triggered.

$($d$)$ Compare your proposed two solutions in terms of ease of implementation and guaranties on the system

behavior.

$($e$)$ Provide a timed automaton model of your time$-$triggered solution.