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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?

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Model your deterministic input as a $2 \times 1$ matrix :

$$ \begin{bmatrix} u \\ T_a \\ \end{bmatrix} $$

Then your state space equations will be:

$$ \begin{bmatrix} \dot x_1 \\ \dot x_2 \\ \dot x_3 \end{bmatrix} = \begin{bmatrix} -\frac 13p & \frac 13p & 0 \\ p & -(p+ \frac{c}{10}) & \frac{c}{10} \\ 0 & \frac{c}{30} & -(\frac{c}{30}+\frac a3) \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ v & 0 \\ 0 & a/3 \end{bmatrix} \begin{bmatrix} u \\ T_a\end{bmatrix} $$

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  • $\begingroup$ Thanks! Can I somehow create a system output y= cx+du out of this? $\endgroup$ – Roxell Nov 25 '19 at 19:19
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    $\begingroup$ Didn't you define $x_1 = y$ as already your output ? Yet you can also define it with C =[1,0,0] and D = [0,0] with X = [y,To,Tc]' and U = [u,Ta]' $\endgroup$ – Fat32 Nov 25 '19 at 19:27

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