I got stack in the process of deriving a state-space representation of the following system:
There is an electrical oven described as follows:
control of the power supply $u$,
heating efficiency constant $v$,
oven-product convection constant $p$,
oven-cover convection constant $c$,
cover-air convection constant $a$,
heater heating power $H$,
heating entering the product $P$,
heating entering the oven’s cover $C$,
heat loss to the surroundings $S$,
temperature of the product $y$,
temperature of the oven $T_o$,
temperature of the oven’s cover $T_c$,
temperature of the surrounding air $T_a$,
$$P = p(T_o−y),$$
$$C = \frac{c(T_o−T_c)}{10}, $$
$$S = a(T_c −T_a),$$
$$H = vu,$$
$$3\dot y = P,$$
$$\dot T_o = H−P−C,$$
$$3\dot T_c = C−S$$
State variables are $y, T_o, T_c$,
Input is $u$.
On the left side I placed derivatives of state variables:
$$\dot y = \frac 13 pT_o - \frac 13 py$$
$$\dot T_o = uv - pT_o + py - \frac{c}{10}T_o + \frac{c}{10}T_c$$
$$\dot T_c = \frac{c}{30}T_o - \frac{c}{30}T_c - \frac a3T_c + \frac a3T_a$$
Let $x_1 = y, x_2=T_o, x_3=T_c$
$$\dot x_1 = - \frac 13 px_1 + \frac 13 px_2 $$
$$\dot x_2 = px_1 - (p+\frac{c}{10})x_2 + \frac{c}{10}x_3 + uv$$
$$\dot x_3 = \frac{c}{30}x_2 - (\frac{c}{30}+\frac a3)x_3 + \frac a3T_a$$
$$ \begin{bmatrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \end{bmatrix} = \begin{bmatrix} -\frac 13p & \frac 13p & 0 \\ p & -(p+ \frac{c}{10}) & \frac{c}{10} \\ 0 & \frac{c}{30} & -(\frac{c}{30}+\frac a3) \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 0 \\ v \\ ??? \end{bmatrix} u $$
The question is: what should be placed in ??? ?
Should it be $\frac a3T_a$? It is not connected with $u$.
And what to do with $\frac a3T_a$? Is it another input or should I erase it?
