LandonZeKepitelOfGreytBritn Asks:

I do know what the difference in meaning is between a transfer function (output over the input) and a state transition matrix $Phi$ (describes the unforced response of the system). Yet when looking closer at the mathematics to me it seems like they are both the same. Could someone clarify?

*What is the mathematical difference between a transfer function and a state transition matrix?*I do know what the difference in meaning is between a transfer function (output over the input) and a state transition matrix $Phi$ (describes the unforced response of the system). Yet when looking closer at the mathematics to me it seems like they are both the same. Could someone clarify?

$$dot{q} = Aq(t)+Bu(t)$$ where A is the state matrix, q the state vector, B the input matrix and u the input vector.

$$sQ(s) = AQ(s) + BU(s)$$ $$ sQ(s)- AQ(s) = BU(s)$$ $$ Q(s)(sI-A) = BU(s) $$ $$ Q(s) = (sI-A)^{-1}BU(s) $$

where $(sI-A)^{-1} = Phi$ ie the state transition matrix.

$$ Q(s) = Phi BU(s)$$ $$ Phi = frac{Q(s)}{BU(s)}$$

$frac{Q(s)}{BU(s)}$ looks to me like the representation of a transfer function and based on the math it looks like the state transition matrix in fact equals a transfer function, yet that doesn’t correspond to the interpretation I have of both of those things. Could somebody please elaborate a bit on that?

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