# Time invariant stochastic bilinear system

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# A nonlinear system

## Model description:

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*}

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung

# T-S fuzzy model

## Model description:

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.

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung

# Tension leg platform system

## Model description:

The present study of tension leg platform is the first commercial application of a revolutionary design of offshore production platform developed by well-known oil company. Intended for oil and gas production in water depths beyond the reach of traditional fixed structures, the tension leg platform was designed as a rectangular shaped floating platform which was connected to the ocean floor by 16 vertical steel tethers or legs, four per corner. The legs were kept in tension so that vertical movement was suppressed, while limited horizontal movement may occur.

The estimation model is:

\begin{align*} y(k) &= 0.590y(k − 3)+1.0598y(k − 1) − 1.0931y(k − 2) + 121.13u(k − 1)u(k− 1)u(k − 9) \\ &− 116.54u(k − 6)u(k − 6)u(k− 6) − 19.797u(k − 4)u(k − 8)u(k − 8) \\ &+ 214.04u(k − 5)u(k − 9) − 34.877u(k− 1)u(k − 1)u(k − 1) − 3.7983u(k − 1)u(k− 2)u(k − 7) \\ &− 25.04u(k − 4)u(k − 8)u(k− 11) + 165.93u(k − 2)u(k − 3)u(k − 4) \\ &− 173.85u(k − 6)u(k − 7) − 69.693u(k− 4)u(k − 12) + 203.12u(k − 5)u(k − 6)u(k − 6) \\ &+ 727.86u(k − 2)u(k − 3)u(k − 5) − 11.107u(k− 3)u(k − 10)u(k − 11) \\ &+ 11.506u(k − 6)u(k− 6)u(k − 12) − 68.607u(k − 2)u(k − 4)u(k− 6) \\ &− 366.75u(k − 3)u(k − 5)u(k − 6)− 25.696u(k − 4)u(k − 8)u(k − 12) \\ &+ 137.86u(k − 1)u(k − 2)u(k − 5)− 142.24u(k − 2)u(k − 2)u(k − 9) \\ &+ 101.44u(k− 1)u(k − 6)u(k − 9) − 9.0283u(k − 3)u(k− 3)u(k − 12) \\ &− 168.30u(k − 2)u(k − 5)u(k − 6)+ 30.295u(k − 5)u(k − 6)u(k − 8) \\ &− 0.158u(k− 1)u(k − 2)u(k − 2) − 433.21u(k − 2)u(k− 2)u(k − 4) \\ &+ 39.88u(k − 3)u(k − 8)u(k − 11)− 162.26u(k − 1)u(k − 4)u(k − 11) \\ &− 212.08u(k− 1)u(k − 1)u(k − 5) − 438.7u(k − 3)u(k− 3)u(k − 5) \\ &+ 162.15u(k − 2)u(k − 4)u(k − 11)− 3.607u(k − 4)u(k − 4)u(k − 11) \\ &+ 13.262u(k− 6)u(k − 9)u(k − 9)+448.4u(k − 3)u(k − 4)u(k− 6) \\ &− 46.475u(k − 4)u(k − 4)u(k − 9) + 119.95u(k − 1)u(k − 1)u(k − 2) + noise \:terms. \end{align*}

Here, the input is the wave, and the output is the pitch. Sample rate is 2.2473 Hz

## Publication details:

 Title Non-linear pitch motion identification and interpretation of a tension leg platform Publication Type Journal Article Authors Liu, Jui-Jung, Huang Yun-Fu, and Lin Hung-Wei

# Time varying stochastic bilinear system with nonlinear feedback

## Model description:

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

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

## Publication details:

 Title Random parameter discrete bilinear system stability Publication Type Conference Paper Authors Yang, Xueshan, Mohler R.R., and Chen Lung-Kee

# Time invariant stochastic bilinear system

## Model description:

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.

2

## Publication details:

 Title Random parameter discrete bilinear system stability Publication Type Conference Paper Authors Yang, Xueshan, Mohler R.R., and Chen Lung-Kee