Non-linear LQG design in Quadratic KF

1. The dynamic estimator equation:

   $\dot{\hat{X}} = AX + Bu + L(CX + \sum_{i=1}^{m} e_i X^T M^i X + v - C(A\hat{X} + Bu) - \sum_{i=1}^{m} e_i (A\hat{X} + Bu)^T M^i (A\hat{X} + Bu))$

2. The system matrix equation:

   $\begin{bmatrix} \dot{X} \\ \text{Vec}(\dot{X}\dot{X}^T) \\ \dot{\hat{X}} \\ \text{Vec}(\dot{\hat{X}}\dot{\hat{X}}^T) \\ e_E \\ y \end{bmatrix} = \begin{bmatrix} A & 0 & 0 & 0 & I & 0 & B & 0 \\ 0 & (I \otimes A + A \otimes I) & 0 & 0 & I \otimes I & 0 & (I \otimes B + B \otimes I) & 0 \\ 0 & 0 & I(A - LC) & 0 & 0 & L & (B - LCB) & 0 \\ 0 & 0 & 0 & (I \otimes (A - LC) + (A - LC) \otimes I) & 0 & (I \otimes L + L \otimes I) & (I \otimes (B - LCB) + (B - LCB) \otimes I) & 0 \\ N_1 & 0 & N_2 & 0 & 0 & 0 & 0 & 0 \\ C & \sum_{i=1}^{m} e_i (M^i \otimes I) & 0 & 0 & 0 & I & 0 & 0 \end{bmatrix} \begin{bmatrix} X \\ \text{Vec}(XX^T) \\ \hat{X} \\ \text{Vec}(\hat{X}\hat{X}^T) \\ w \\ v \\ \text{Vec}(vv^T) \\ u \\ \text{Vec}(uu^T) \end{bmatrix}$

$\hat{Z}^S(k)= \mathbb{E}[Z(k)|y^S(1:k)]$