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.

The behavior of phytoplankton cells in a continuous reactor is usually described by the Droop model. Cell growth is limited by a nutrient with concentration $S$. The biomass has a concentration $N$ and $Q$ represents the cell quota of assimilated nutrient, expressed as the amount of intracellular nutrient per biomass unit. The dilution rate $D$ corresponds to the flow rate of renewal medium over the volume of the reactor, and $D$ is the input of the system.

We denote $D = D_0 + u$, and the system fits

$$\sum_D \begin{cases} \dot{x}_i = f(x) + ug(x)\\ y=h(x_1) \end{cases}$$

with

$f(x)=\begin{pmatrix} a_2\left(1-\dfrac{1}{x_2}\right)x_1 - D_0x_1\\ a_3\dfrac{x_3}{a_1+x_3} - a_2(x_2 - 1)\\ D_0(1-x_3)-\dfrac{x_1x_3}{a_1+x_3} \end{pmatrix}$

$g(x)=\begin{pmatrix} -x_1\\ 0\\ 1-x_3 \end{pmatrix}$, and $h(x_1)=x_1$, where

$ x_1 = (\rho_m N/S_i);\\ x_2 = (Q/K_Q);\\ x_3 = (S/S_i);\\ a_1 = (K_{\rho}/S_i);\\ a_2 = \mu_m;\\ a_3 = (\rho_m/K_Q). $

Consider the system

$$\begin{align*} \dot{x}_1 &= p_{13}x_3 + p_{12}x_2 - p_{21}x_1+u \\ \dot{x}_2 &= -p_{12}x_2 + p_{21}x_1 \\ \dot{x}_3 &= -p_{13}x_3 \\ y &= x_2, \end{align*}$$

where $x=[x_1, x_2, x_3]$ is the state vector, e.g. $x_1,x_2,x_3$ are drug masses in compartment 1, 2 and 3, respectively; initial conditions are $x_1(0)=0$, $x_2(0)=0$, $x_3(0)=0$; $u$ is the drug input; $y$ is the measured drug output; $p=[p_{12},p_{21},p_{13}]$ is the rate parameter vector (assumed constant).

The models used to describe the growth of phytoplanktonic cells (biomass $x_2$) on a substrate (of concentration $x_1$) assume usually that the growth is a function of a variable ($x_3$) called internal quota, representing the nutrient stored in the cells:

$$\begin{align*} \dot{x}_1 &= u(t)(1-x_1)-\rho(x_1)x_2\\ \dot{x}_2 &= (\mu(x_3)-u(t))x_2\\ \dot{x}_3 &= \rho(x_1)-\mu(x_3)x_3. \end{align*}$$

The input $u(t)$ is the dilution rate of the continuously stirred bioreactor (we suppose $u(t) \geq u \geq 0$). The functions $\rho$ and $\mu$ represent the absorption rate and the growth rate:

$\rho(x_1)=a_1\dfrac{x_1}{a_2+x_1};$ $\mu(x_3)=a_3\left(1-\dfrac{a_4}{x_3}\right)$.

The model of the NF$\kappa$B regulatory module, as proposed by Lipniacki et al, is characterised by two compartment kinetics of the activators $IKK$ and $NF-kB$, the inhibitors $A20$ and $IkB\alpha$, and their complexes. The model is described by the differential system:

$$\begin{align*} \dot{x}_1 &= k_{prod}-k_{deg}x_1 - k_1x_1u(t),\\ \dot{x}_2 &= -k_3x_2 - k_{deg}x_2 - a_2x_2x_{10}+t_1x_4 - a_3x_2x_{13} + t_2x_5 + (k_1x_1 - k_2x_2x_8)u(t),\\ \dot{x}_3 &= k_3x_2 - k_{deg}x_3+k_2x_2x_8u(t),\\ \dot{x}_4 &= a_2x_2x_{10}-t_1x_4,\\ \dot{x}_5 &= a_3x_2x_{13}-t_2x_5,\\ \dot{x}_6 &= c_{6a}x_{13}-a_1x_6x_{10}+t_2x_5-i_1x_6,\\ \dot{x}_7 &= i_1kvx_6-a_1x_{11}x_7,\\ \dot{x}_8 &= c_4x_9-c_5x_8,\\ \dot{x}_9 &= c_2+c_1x_7-c_3x_9,\\ \dot{x}_{10} &= -a_2x_2x_{10}-a_1x_{10}x_6 + c_{4a}x_{12} - c_{5a}x_{10}-i_{1a}x_{10}+e_{1a}x_{11},\\ \dot{x}_{11} &= -a_1x_{11}x_7+i_{1a}kvx_{10}-e_{1a}kvx_{11},\\ \dot{x}_{12} &= c_{2a}+c_{1a}x_7 - c_{3a}x_{12},\\ \dot{x}_{13} &= a_1x_{10}x_6 - c_{6a}x_{13}-a_3x_2x_{13}+e_{2a}x_{14},\\ \dot{x}_{14} &= a_1x_{11}x_7 - e_{2a}kvx_{14},\\ \dot{x}_{15} &= c_{2c}+c_{1c}x_7 - c_{3c}x_{15}.\\ \end{align*}$$

In their paper, Lipniacki et al. fixed some of the model parameters by using values from the literature. In order to assign values to the following unknown parameters:

$\mathbf{p}=[t_1,t_2,c_{3a},c_{4a},c_5,k_1,k_2,k_3,k_{prod},k_{deg},i_1,e_{2a},i_{1a}]^{\mathrm T}.$

They used experimental data from previous works by Lee et al. and Hoffmann et al which corresponds to the observation of $y_1=x_7,$ $y_2=x_{10} + x_{13},$ $y_3=x_9,$ $y_4=x_1+x_2+x_3,$ $y_5=x_2,$ $y_6=x_{12}$.