Problem $2(60$ points$)$

Figure $($below$)$ shows a room heated with an electric heater. The inside of the room is at temperature $T_r$ and the walls are assumed to be at temperature $T_w.$

The capacitance between the room and wall is defined as $C_1$ while the capacitance between the wall and outside is defined as $C_2.$

a$.(20$ pts$)$ If the outside temperature is $T_o,$ develop a model of the system to show the relationship between the supplied heat $q$ and the room temperature $T_r.$

b$.(20$ pts$)$ Write the state$-$space model of the system in a matrix form. In other words, the state equation, $x^{}=Ax+Bu,$ is expressed in a matrix form as

$\left[\begin{array}{c}{T}_r^{}\\ T_w^{}\end{array}\right]=[A]\left[\begin{array}{c}{T}_r\\ T_w\end{array}\right]+[B]\left[\begin{array}{c}q\\ T_o\end{array}\right]$

c$.(20$ pts$)$ Find the matrices $A,B,x,$ and $u$ using the following values

$($Assume that $C_1=0\mathrm{.}5,C_2=1\mathrm{.}5,R_r=0\mathrm{.}5,R_w=1\mathrm{.}8,T_r=8,T_w=4,T_o=3,$ and $q=5)$

$$ Noting that the thermal resistance is

$R=\frac{d(T)}{dq}=\frac{T}{q}$

and the heating equation is given by

$q_2-q_1=C\frac{dT}{dt}$

Hint: find the heat flow from inside to the wall $(q_{rw})$ and then the heat flow from the wall to the outside $(q_{wo}).$