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Generally, in nonlinear state estimation the state space model is defined by the following pair of difference equations in discrete-time:

\begin{equation} \begin{aligned} x_k & = f(x_{k-1},u_{k-1},\theta)+q_{k-1}\\ y_k & = h(x_k,u_{k},\theta)+r_k \end{aligned} \end{equation}

Can some one explain the difference between a nonlinear system with direct feedthrough and a nonlinear system without direct feedthrough, and to what genere belong the first system (generally used), if possible some references?

system with direct feedthrough: \begin{equation} \begin{aligned} x_{k+1} & = f(x_{k},u_{k},\theta)+q_{k-1}\\ y_k & = h(x_k,u_{k},\theta)+r_k \end{aligned} \end{equation}

system without direct feedthrough: \begin{equation} \begin{aligned} x_{k+1} & = f(x_{k},u_{k},\theta)+q_{k-1}\\ y_k & = h(x_k,\theta)+r_k \end{aligned} \end{equation}

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