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.

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.

Consider the following bilinear descriptor system:

$$\begin{pmatrix} 1 & -1\\ 0 & 0 \end{pmatrix}x_{k+1}=\begin{pmatrix} -0.5 & 1\\ -1 & 0 \end{pmatrix}x_{k}+\begin{pmatrix} 0.5 & 0.25\\ -1 & 0.5 \end{pmatrix}x_{k}u_{k}.$$

In the operation of a power plant superheater, exacting demands are made on the steam temperature maintenance at the outlet. For temperature control at the outlet of a superheater, the relevant system state is the temperature pattern along the superheater tube. This is described by a distributed-parameter system, which involves an infinite number of state variables. To derive a simplified model for control purposes, the superheater is divided into segments, and a lumped model is derived, which represents a finite number of intermediate temperatures.

Assuming that the pressure inside the tube is constant, the enthalpy of the steam satisfies the relation $dH = C_pdT(kcal/kg)$ , where $C_p(kcal/kg^{\circ}C)$ is the constant-pressure specific heat. Hence, we conclude that the heat supplied to the following fluid(steam) only increases its enthalpy, $dH = dQ$ , where $Q$ denotes the heat. In the above equations, it is assumed that convection is the exclusive heat transfer mode for the superheater. Hence the heat transfer from to metal $Q_{ms}(kcal/s)$ and from gas to metal $Q_{gm}(kcal/s)$ are expressed in terms of the heat transfer rates from gas to metal $\alpha_{gm}(kcal/m^2s^{\circ}C)$ and from metal to steam $\alpha_{ms}(kcal/m^2s^{\circ}C)$ and heating surface $S(m^2)$ :

$$\begin{align*} \alpha_{ms}S_1(T(l,t)-T(l,t)) &=Q_{ms} \\ \alpha_{gm}S_2(T_m(l,t)-T(l,t)) &=Q_{gm}. \end{align*}$$

It is also assumed that the heat transfer rates $\alpha_{gm}$ and $\alpha_{ms}$ are constants.

Now, to simulate the profile of superheated steam precisely, it is necessary to divide the superheater into $n$ segments as shown in the attached image.

In the first segment, the desuperheater is included and system is modified as follows:

$$\begin{align*} V_s\rho C_p\frac{{\mathrm d} x_1}{{\mathrm d}t} &={C_{p}}{T_{i}}{w_{i}}-{C_{p}}({w_{i}}+{w_{d}}){x_{1}} +{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}})+{C_{pd}}{T_{d}}{w_{d}}\\ M_mC_m \frac{{\mathrm d}z_1}{{\mathrm d}t} &={\alpha_{gm}}{S_{2}}(T{g_{1}}-{z_{1}})-{\alpha_{ms}}{S_{1}}({z_{1}}-{x_{1}}), \end{align*}$$

where $x=[x_1,x_2,\ldots,x_n]^{\mathrm T}=[T_1,T_2,\ldots,T_n]^{\mathrm T}$, $z=[z_1,z_2,\ldots,z_n]^{\mathrm T}=[T_{m1},T_{m2},\ldots,t_{mn}]^{\mathrm T}$, and $T_{mi}(^{\circ}C)$ are metal temperature, $T_i(^{\circ}C)$ are steam temperature, $i=1,\ldots,n$.