# Muscle-knee state space model

## Model description:

The state model of the knee-quadriceps can be expressed as

\begin{cases} \begin{align*} \dot{x}_1 &= \left[ s_0 \alpha K_m + s_v q\dfrac{s_0\alpha F_mx_1 - s_ux_2x_1}{1 + px_1 - s_vqx_2}\right] u_{ch} - s_ux_1u_{ch} - \dfrac{s_v ax_1 r_p x_4}{L_0 (1+px_1-s_vqx_2)}\\ \dot{x}_2 &= \left[ \dfrac{s_0\alpha F_m - s_ux_2}{1 + px_1 - s_vqx_2} \right]u_{ch} + \dfrac{bx_1r_px_4 - s_vax_2r_px_4}{L_0(1+px_1-s_vqx_2)}\\ \dot{x}_3 &= x_4\\ \dot{x}_4 &= \dfrac{1}{I}[x_2r_p - \lambda x_3 - \mu x_4 - mgl_c \cos{x_3}], \end{align*} \end{cases}

where $\textbf{x}=[x_1, \ldots, x_4]^{\mathrm T} = [K_c, F_c, \theta, \dot{\theta}]^{\mathrm T}$ is the state vector and $\textbf{u}=[u_{ch},\alpha ]^{\mathrm T}$ the control vector. The variable $\theta$ represents the knee joint angle and the variables $K_c, F_c, u_{ch}, \alpha$ represent the state variables of the quadriceps muscle model.

## Publication details:

 Title Toward lower limbs movement restoration with input-output feedback linearization and model predictive control through functional electrical stimulation Publication Type Journal Article Authors Mohammed, S., Poignet P., Fraisse P., and Guiraud D.

# 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

# Bilinear system

## Model description:

The time-invariant bilinear system is given by

$$Y(t) = 1.5X(t) + 1.2X(t-1) - 0.2X(t-2) + 0.7X(t-1)Y(t-1) - 0.1X(t-2)Y(t-2) + \epsilon(t),$$

where $A=0, \alpha=0, B=\begin{bmatrix}1.5 &1.2 &-0.2\end{bmatrix}, C = \begin{bmatrix}0.7 &0 &-0.1\end{bmatrix}$. Note that $\Theta = \begin{bmatrix}B & C\end{bmatrix}^{\mathrm T}.$

2

## Publication details:

 Title Identification of bilinear systems using Bayesian inference Publication Type Conference Paper Authors Meddeb, S., Tourneret J.Y., and Castanie F.