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I have seen the following formulation of state space models:

$$ z_{t + 1} = A z_t + B u_t + \epsilon_t \\ y_t = C z_t + D u_t + \delta_t $$

But sometimes it is also written as follows: $$ z_{t} = A z_{t-1} + B u_t + \epsilon_t \\ y_t = C z_t + D u_t + \delta_t $$ Why are there different formulations for computing the next state?

When to use which formulation?

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