Recurrent Trainable Neural Network

The RTNN model is described bythe following equations:

$$\begin{align*} X(k+1) &= JX(k) + BU(k)\\ Z(k) &= S[X(k)]\\ Y(k) &= S[CZ(k)]\\ J &\doteq \mathrm{blockdiag}(J_i); |J_i| <1, \end{align*}$$

here $X(\cdot)$ is a $n$-state vector of the RTTN; $U(\cdot)$ is a $m$-input vector; $Y(\cdot)$ is a $l$-output vector; $Z(\cdot)$ is an auxiliary vector variable with $l$ dimension; $S(\cdot)$ is a vector-valued smooth activation function (sigmoid, $tanh$, saturation) with appropriate dimensions; $J$ is a weigh-state block-diagonal matrix with $(1 \times 1)$ and $(2 \times 2)$ blocks; $J_i$ is an $i-th$ block of $J$ and $|J_i|<1$ is a stability condition.