Problem $1$:

Consider an oven with a heating element and a thermocouple for sensing$.$

$K{T}^{}=-(T-T_{\text{a}\text{m}\text{b}})+U_{\text{h}\text{e}\text{a}\text{t}}+U_{\text{d}\text{i}\text{s}\text{t}}$

where K is the thermal conductivity constant, T is the temperature of the oven, $T_{\text{a}\text{m}\text{b}}$ is the ambient air temperature $($which is constant$),U_{\text{h}\text{e}\text{a}\text{t}}$ is the thermal heat energy, and $U_{\text{d}\text{i}\text{s}\text{t}}$ is the disturbance from the impatient person who keeps opening the oven.

Equation $(2)$ is the error in the temperature and equation $(3)$ is the error in the PD controller being used. Assume $U_{\text{n}\text{o}\text{m}}$ is equal to zero.

tilde$(T)=T-T_{\text{a}\text{m}\text{b}}$

tilde$(U{)}_{\text{h}\text{e}\text{a}\text{t}}=-k_p**tilde(T)-k_d**tilde(T{)}^{}$

Find the equation of motion and draw a block diagram for the closed$-$loop control system.