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Transfer Function Derivation

Converting the derived state-space representation into the frequency domain using Laplace transforms. This allows for classical control analysis, yielding G(s)=Vout(s)Vin(s)G(s) = \frac{V_{out}(s)}{V_{in}(s)}.
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The statement of the theorem

Given the state-space representation x⃗˙(t)=A⃗x⃗(t)+B⃗u⃗(t)\dot{\vec{x}}(t) = \vec{A}\vec{x}(t) + \vec{B}\vec{u}(t) with initial condition x⃗(0)=x⃗0\vec{x}(0) = \vec{x}_0, the Laplace transform yields the solution X⃗(s)\vec{X}(s): \vec{X}(s) = (s\vec{I} - \vec{A})^{-1} \left( \vec{x}_0 + \vec{B}\ring{\vec{U}}(s) \right) The transfer function G(s)G(s) relating the output Y⃗(s)\vec{Y}(s) to the input U⃗(s)\vec{U}(s) is then defined as: G(s)=Y⃗(s)/U⃗(s)=C⃗(sI⃗−A⃗)−1B⃗G(s) = \vec{Y}(s) / \vec{U}(s) = \vec{C}(s\vec{I} - \vec{A})^{-1} \vec{B}