The process dynamic model consists of six nonlinear ordinary differential equations:

$$\begin{align*} \dot x_{11} &= b_{11} x_{12} \\ \dot x_{12} &= b_{12} u_1 \\ \dot x_{21} &= b_{21} x_{22} + \phi _{21} \left({x_{11},x_{21} } \right) + \Phi x_{31} \\ \dot x_{22} &= b_{22} u_2 + \phi _{22} \left({x_{21},x_{22} } \right) \\ \dot x_{31} &= b_{31} x_{32} + \phi _{31} \left({x_{11},x_{12},x_{21},x_{31} } \right) + \Psi w \\ \dot x_{32} &= b_{32} u_3 + \phi _{32} \left({x_{31},x_{32} } \right) \\ y &= \left[{y_1,y_2,y_3 } \right] = \left[{x_{11},x_{21},x_{31} } \right], \end{align*}$$

where

$\eqalignno{b_{11} &= 1,b_{12} = 1,b_{21} = {{UA} \over {\rho c_p V}},b_{22} = {{F_{j2} } \over {V_j}},b_{31} = {{UA} \over {\rho c_p V}}\cr b_{32} &= {{F_{j1} } \over {V_j}},\Psi = {{F_0 } \over V},\Phi = {{F + F_R } \over V} \cr \phi _{21} &= {{F + F_R } \over V}T_1^d - {{F + F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}\left({x_{11} + C_{A2}^d } \right)e^{- \left({{E \over {R\left({x_{21} + T_2^d } \right)}}} \right)}\cr &\quad - {{UA} \over {\rho c_p V}}\left({x_{21} + T_2^d - T_{j2}^d } \right) \cr \phi _{22} &= {{F_{j2} } \over {V_j}}\left({T_{j20}^d - x_{22} - T_{j2}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{21} + T_2^d - x_{22} - T_{j2}^d } \right) \cr \phi _{31} &= {{F_0 } \over V}T_0^d - {{F + F_R } \over V}\left({x_{31} + T_1^d } \right) + {{F_R } \over V}\left({x_{21} + T_2^d } \right)\cr &\quad - {{\alpha \lambda } \over {\rho c_p}}C_A e^{- \left({{E \over {R\left({x_{31} + T_1^d } \right)}}} \right)} - {{UA} \over {\rho c_p V}}\left({x_{31} + T_1^d - T_{j1}^d } \right) \cr \phi _{32} &= {{F_{j1} } \over {V_j}}\left({T_{j10}^d - x_{32} - T_{j1}^d } \right)\cr &\quad + {{UA} \over {\rho _j c_j V_j}}\left({x_{31} + T_1^d - x_{32} - T_{j1}^d } \right) \cr C_A &= {V \over {F + F_R}}\Bigg(x_{12} + {{F + F_R } \over V}({x_{11} + C_{A2}^d })\cr &\quad + \alpha ({x_{11} + C_{A2}^d })e^{- \Big({{E \over {R({x_{21} + T_2^d })}}} \Big)} \Bigg). }$

The values of the process parameters are

$\eqalignno{& \alpha = {\rm 7}{\rm .08} \times {\rm 10}^{{\rm 10}} {\rm h}^{- 1},\quad \rho = 800.9189\,{\rm kg/m}^{\rm 3}\cr & \rho _j = 997.9450\,{\rm kg/m}^3,\quad \lambda = - 3.1644 \times {\rm 10}^{\rm 7} {\rm J/mol}\cr & R = 1679.2\,{\rm J/(mol} {\cdot} {}^{\circ} {\rm C)},\quad E = 3.1644 \times 10^7 {\rm J/mol}\cr & c_\rho = 1395.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)},\quad c_j = 1860.3\,{\rm J/(kg} {\cdot} {}^{\circ} {\rm C)}\cr & U = 1.3625 \times 10^6{\kern1pt} {\rm J/(h} {\cdot} {\rm m}^{\rm 2} {\cdot} {}^{\circ} {\rm C)},\quad F_0 = F_2 = F = 2.8317\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_R = 1.4158\,{\rm m}^{\rm 3}\!{\rm /h},\quad F_{j1} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h}\cr & F_{j2} = 1.4130\,{\rm m}^{\rm 3}\!{\rm /h},\quad T_0^d = 703.7\,{}^{\circ} {\rm C},\quad T_1^d = 750\,{}^{\circ} {\rm C}\cr & T_2^d = 737.5\,{}^{\circ} {\rm C},\quad T_{j1}^d = 740.8\,{}^{\circ} {\rm C},\quad T_{j2}^d = 727.6\,{}^{\circ} {\rm C}\cr & T_{j10}^d \! = \! 629.2\,{}^{\circ} {\rm C},\quad T_{j20}^d \!=\! 608.2\,{}^{\circ} {\rm C},\quad C_{A0}^d \!=\! 18.3728\,{\rm mol/m}^{\rm 3}\cr & C_{A1}^d = 12.3061\,{\rm mol/m}^{\rm 3},\quad C_{A2}^d = 10.4178\,{\rm mol/m}^{\rm 3}\cr & V_1 = V_2 = V = 1.3592\,{\rm m}^{\rm 3},\quad V_{j1} = V_{j2} = V_j = 0.1090\,{\rm m}^{\rm 3}\cr & A = 23.2\,{\rm m}^{\rm 3} . }$

Consider the following relative degree two MIMO nonlinear systems:

$$\begin{align*} \dot x_1 &= x_2 + \vartheta _1 x_1 \sin \left(t \right) + \Delta _1 \left({x_1 } \right) \\ \dot x_2 &= u + \vartheta _2 \left[{\matrix{{\left({x_1 + x_{2,1} } \right)\sin ^3 \left(t \right)} \cr {x_{2,1} + 2x_{2,2} } \cr}} \right] + \left[{\matrix{1 \cr {x_1 + x_{2,2} } \cr}} \right]\Delta _2 \left({x_{2,1} } \right) \\ y &= \left[{x_1,x_{2,2} } \right]^{\mathrm T}, \end{align*}$$

where $\Delta_1(x_1)=d_1\sin{(r_1x_1)}$ and $\Delta_2(x_{2,1})=d_2\tan{(r_2,x_{2,1})}.$ $\vartheta_1, \vartheta_2,\Delta_1,\Delta_2$ satisfy

$\displaylines{2 \le \vartheta _1 \le 4, - 4 \le \vartheta _2 \le - 1, \left\vert {\Delta _1 \left({x_1 } \right)} \right\vert \le \delta _1 = 40\cr \left\vert {\Delta _2 \left({x_{2,1} } \right)} \right\vert \le \delta _2 = 20. }$

The initial conditions are assumed to be $x_1(0)=0.5,x_{2,1}(0)=0$ and $x_{2,2}(0)=0.2$.

The actual plant parameters are

$\theta _1 = 3$, $\theta _2 = - 3$, $d_1 = - 30$, $d_2 = - 15$, $r_1 = 2$, $r_2 = 0.05.$

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 $y_1=x_7,$ $y_2=x_{10} + x_{13},$ $y_3=x_9,$ $y_4=x_1+x_2+x_3,$ $y_5=x_2,$ $y_6=x_{12}$.

This is a pure polynomial form of the model given in the Arabidopsis Thaliana Model (1):

$$\begin{align*} \dot{x}_1 &= n_1x_6x_8 - m_1x_1x_9 + q_1x_7u(t), \\ \dot{x}_2 &= p_1x_1 - r_1x_2 + r_2x_3 - m_2x_2x_{10},\\ \dot{x}_3 &= r_1x_2 - r_2x_3 - m_3x_3x_{11},\\ \dot{x}_4 &= n_2g_2^2x_{12} - m_4x_4x_{13},\\ \dot{x}_5 &= p_2x_4 - r_3x_5 + r_4x_6 - m_5x_5x{14},\\ \dot{x}_6 &= r_3x_5 - r_4x_6-m_6x_6x_{15},\\ \dot{x}_7 &= p_3-m_7x_7x_{16} - (p_3 + q_2x_7)u(t),\\ \dot{x}_8 &= -\dot{x}_6x_8^2,\\ \dot{x}_9 &= -\dot{x}_1x_9^2,\\ \dot{x}_{10} &= -\dot{x}_2x_{10}^2,\\ \dot{x}_{11} &= -\dot{x}_3x_{11}^2,\\ \dot{x}_{12} &= -2x_3\dot{x}_3x_{12}^2,\\ \dot{x}_{13} &= -\dot{x}_4x_{13}^2,\\ \dot{x}_{14} &= -\dot{x}_5x_{14}^2,\\ \dot{x}_{15} &= -\dot{x}_6x_{15}^2,\\ \dot{x}_{16} &= -\dot{x}_7x_{16}^2, \\ x_0(p)&=\left(0,0,0,0,0,0,0,\dfrac{1}{g_1},\dfrac{1}{k_1},\dfrac{1}{k_2},\dfrac{1}{k_3},\dfrac{1}{g_1^2},\dfrac{1}{k_4},\dfrac{1}{k_5},\dfrac{1}{k_6},\dfrac{1}{k_7}\right). \end{align*}$$