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Given a discrete-time linear time-varying system (LTV)

$$x(k+1) = A(k) x(k) + B(k) u(k)$$

where $A(k)$ and $B(k)$ are generated by a stationary random process. Is there an equivalent linear time-invariant (LTI) system which will calculate the expected trajectory of $x(k)$?

$$\mathbb E[x(k+1)] = z(k+1) = A_{\text{eq}} z(k) + B_{\text{eq}} u(k)$$

If so, how is it calculated?

If not, are there conditions where this can be calculated?

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  • $\begingroup$ $$\mathbb E (\mathrm x_{k+1}) = \mathbb E (\mathrm A_{k}) \, \mathbb E (\mathrm x_{k}) + \mathbb E (\mathrm B_{k}) \, \mathrm u_{k}$$ $\endgroup$
    – Rodrigo de Azevedo
    Commented Jan 9, 2017 at 10:08
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    $\begingroup$ Not really. $A(k)$ and $x(k)$ are not independent. $\endgroup$
    – ofer zeitouni
    Commented Jan 9, 2017 at 15:24

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