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I have a linear control system defined by the following equations:

\begin{array}{l}\dot{x}_{1}(t)=-\frac{1}{2}\left(x_{1}(t)-x_{2}(t)\right) \\ \dot{x}_{2}(t)=\frac{1}{10}\left(2 x_{2}(t)-10 x_{1}(t)\right)+u(t) \\ y(t)=x_{2}(t)\end{array}

I would like to find the transfer function from a constant input $u(t)=\bar{u}$ to the output $y$.

I know that the transfer function is equal to $G(S)=\frac{\mathcal{L}(y(s))}{\mathcal{L}(\bar{u})}=\mathcal{L}(y(s))\cdot \frac{s}{\bar{u}}$

In order to find $\mathcal{L}(y(s))$ we solve the following system of equations:

\begin{array}{l}s\mathcal{L}(x_1(s))-\mathcal{L}(x_1(0))=-\frac{1}{2}\mathcal{L}(x_1(s))+\frac{1}{2}\mathcal{L}(x_2(s))\\s\mathcal{L}(x_2(s))-\mathcal{L}(x_2(0))=\frac{1}{10}(2\mathcal{L}(x_2(s))-10\mathcal{L}(x_1(s))+\frac{\bar{u}}{s}\end{array}

However, this can be tricky to solve, so we calculate the transfer function using the following formula: $G(s)=C(s I-A)^{-1} B$.

My question is: is the reasoning correct? Is the formula true for all types of control systems?

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    $\begingroup$ Since you have the system in the state-space form, I think you should go with the formula. This formula is applied to any linear system given in the state space form in order to convert it into the $s$ domain. There is also a more general version of the formula in case you have a $D$ matrix as well. $\endgroup$ – Teo Protoulis May 13 '20 at 11:35
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Yes, your reasoning is right and is applicable to all control systems with a valid state space representation. The formula to go from state-space to transfer function can be easily derived like so: $$\dot{x} = Ax +Bu$$ $$y = Cx + Du$$

Taking laplace transform on both equations one by one $$sX= AX + BU$$ i.e. $$(sI-A)X = BU$$ i.e. $$X = (sI-A)^{-1}BU\cdots(i)$$ $$Y = CX + DU...(ii)$$ put (i) in (ii) you get $Y = C(sI-A)^{-1}BU + DU$. We know transfer function is $$G(s) = \frac{Y(s)}{U(s)}$$ $$G(s) = C(sI-A)^{-1}B + D$$

Now your equations are: $$\begin{bmatrix}\dot{x_1} \\ \dot{x_2} \\ \end{bmatrix} = \begin{bmatrix} -0.5 & 0.5 \\ -1 & 0.2 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} + \begin{bmatrix}0\\1\end{bmatrix}u$$
$$y = \begin{bmatrix}0 &1\end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix}0 \end{bmatrix}u$$ using these equations in $G(s)$ we get $$G(s) = \frac{s + 0.5}{s^2 + 0.3s + 0.4}$$

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  • $\begingroup$ I can't see the reason why to write an answer which just derives a fundamental mathematical formula and of which the OP is already aware. Kindly speaking :) $\endgroup$ – Teo Protoulis May 13 '20 at 16:28
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    $\begingroup$ @Teo Protoulis By showing that the formula can be derived from only the state space and the Laplace transformation, without further assumptions, it was shown that this formula can be used for any control system in state space form indeed. In my opinion this answers OP's second question (though a conclusive sentence for this might help). $\endgroup$ – OpticalResonator May 13 '20 at 17:52
  • $\begingroup$ @OpticalResonator hadn't thought it this way. You are tight :) $\endgroup$ – Teo Protoulis May 14 '20 at 0:53

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