$$\Sigma _{S_{1}}: \cases{ \begin{align*} \dot{x}_{1,1} &=f_{1,1}(\bar {x}_{1,1},\bar {x}_{2,1})+g_{1,1}(\bar {x}_{1,1},\bar{x}_{2,1})x_{1,2} \\ \dot{x}_{1,2} &=f_{1,2}(X)+g_{1,2}(\bar{x}_{1,1},\bar{x}_{2,1})u_{1} \\ \dot{x}_{2,1} &=f_{2,1}(\bar {x}_{1,1},\bar {x}_{2,1})+g_{2,1}(\bar{x}_{1,1},\bar{x}_{2,1})x_{2,2} \\ \dot{x}_{2,2} &=f_{2,2}(X,u_{1})+g_{2,2}(\bar{x}_{1,1},\bar {x} _{2,1})u_{2} \\ y_{j} &=x_{j,1}, \quad j=1,2, \end{align*}}$$

where $X = [\bar{x}_{1,2}^{\mathrm T}, \bar{x}_{2,2}^{\mathrm T}]^{\mathrm T}$ with $\bar{x}_{j,2}=[x_{j,1},x_{j,2}]^{\mathrm T},j=1,2$.

Consider a nonlinear system

$$\begin{align*} x_{1}(t+1) &=x_{1}(t)-x_{1}(t)x_{2}(t)+(5+x_{1}(t))u(t) \\ x_{2}(t+1) &=-x_{1}(t)-0.5x_{2}(t)+2x_{1}(t)u(t) \end{align*}$$

Consider a T-S fuzzy model

Plant Rule $i$: If $x_1(t)$ is $F_1^1(x_1(t))$

Then $x(t+1) = A_ix(t)+B_iu(t),$

where

$\begin{align*} A_1 &=\left[\matrix{-a & 2\cr -0.1 & b}\right], A_2=\left[\matrix{-a & 2\cr-0.1 & b }\right], A_3=\left[\matrix{-0.9 & 0.5\cr -0.1 & -1.7}\right] \\ B_1 &=\left[\matrix{b\cr 4}\right], B_2=\left[\matrix{b\cr 4.8}\right], B_3=\left[\matrix{3\cr 0.1}\right]. \end{align*}$

The parameters $a$ and $b$ are adjusted to compare the relaxation of stabilization conditions.

Consider the following time varying stochastic bilinear system with nonlinear feedback.

$$\begin{align*} \begin{bmatrix} x_1(t+1) \\ x_2(t+1) \end{bmatrix} &= \left\{\begin{bmatrix}0.1 & 0.2 \\ 0.5 & -0.3\end{bmatrix} + \begin{bmatrix}0.36 & -0.3 \\ 0.2 & 0.42\end{bmatrix}\omega(t) \right\}\begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix} \\ &+\begin{bmatrix}0.1 & 0.9 \\ 1.5 & 1.2\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}u(t) + \begin{bmatrix}-0.3t^2\exp{(-t)} \\ 0.4t\exp{(-t)}\end{bmatrix}u(t), \\ \begin{bmatrix}y_1(t) \\ y_2(t)\end{bmatrix} &= \begin{bmatrix} 0.7\sin{t} & -0.9 \\ 0.8 & -0.6\cos{t}\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}, \end{align*}$$

where

$u(t)=0.2\sin{(y_1(t) + y_2(t))} + 0.3[y_1(t)+y_2(t)].$

Consider the following time invariant stochastic bilinear system:

$$\begin{align*} \begin{bmatrix}x_1(t+1)\\x_2(t+1)\end{bmatrix} &= \left\{ \begin{bmatrix}0.2 & 0.4 \\ 0.5 & -0.3\end{bmatrix} + \begin{bmatrix}0.3 & 0.2 \\ -0.3 & 0.4\end{bmatrix}\omega(t) \right\} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}+ \begin{bmatrix}2 & 5 \\ 3 & 9\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}u(t) + \begin{bmatrix}-0.3 \\ 0.4\end{bmatrix}u(t), \\ \begin{bmatrix}y_1(t) \\ y_2(t)\end{bmatrix}& = \begin{bmatrix} 0.7 & 0.8 \\ -0.9 & -0.6\end{bmatrix} \begin{bmatrix}x_1(t) \\ x_2(t)\end{bmatrix}, \end{align*}$$

where

$u(t)=0.24[y_1(t) + y_2(t)] + 0.32[y_1(t-1) + y_2(t-1)]$

and $\omega(t)$ is a white noise with zero mean and variance 0.2.