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 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):

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

The model describes the first multi-gene loop identified in the Arabidopsis circadian clock that comprises a negative feedback loop, in which two partially redundant genes, Late Elongated Hypocotyl (LHY) and Circadian Clock Associated 1 (CCA1), repress the expression of their activator, Timing of CAB Expression 1 (TOC1). A minimal mathematical representation of the system requires 7 coupled differential equations and 29 parameters. The differential equations involve Michaelis-Menten kinetics that describe enzyme-mediated protein degradation, and Hill functions that describe some transcriptional activation terms. The model is given by:

$$\begin{align*} \dot{x}_1 &= n_1 \dfrac{x_6}{g_1+x_6} - m_1 \dfrac{x_1}{k_1 + x_1} + q_1x_7u(t),\\ \dot{x}_2 &= p_1x_1 - r_1x_2 + r_2x_3 - m_2 \dfrac{x_2}{k_2 + x_2},\\ \dot{x}_3 &= r_1x_2 - r_2x_3 - m_3 \dfrac{x_3}{k_3 + x_3},\\ \dot{x}_4 &= n_2\dfrac{g_2^2}{x_2^2+x_3^2} - m_4 \dfrac{x_4}{k_4 + x_4},\\ \dot{x}_5 &= p_2x_4 - r_3x_5 + r_4x_6 - m_5 \dfrac{x_5}{k_5 + x_5},\\ \dot{x}_6 &= r_3x_5 - r_4x_6 - m_6 \dfrac{x_6}{k_6 + x_6},\\ \dot{x}_7 &= p_3 - m_7 \dfrac{x_7}{k_7 + x_7} - (p_3 + q_2x_7)u(t) \end{align*}$$

with $x_i(0)=0, i=1,...,7$.

The system, that could describe a biochemical reaction network, is represented by twenty differential equations, twenty two parameters, and all the states are assumed to be measured:

$$\begin{align*} \dot{x}_1&=-\dfrac{v_{max}x_1}{(k_m+x_1)}-p_1x_1 + u(t),\\ \dot{x}_{2}&=p_{1}x_{1}-p_{2}x_{2},\\ \dot{x}_{3}&=p_{2}x_{2}-p_{3}x_{3},\\ \dot{x}_{4}&=p_{3}x_{3}-p_{4}x_{4},\\ \dot{x}_{5}&=p_{4}x_{4}-p_{5}x_{5},\\ \dot{x}_{6}&=p_{5}x_{5}-p_{6}x_{6},\\ \dot{x}_{7}&=p_{6}x_{6}-p_{7}x_{7},\\ \dot{x}_{8}&=p_{7}x_{7}-p_{8}x_{8},\\ \dot{x}_{9}&=p_{8}x_{8}-p_{9}x_{9},\\ \dot{x}_{10}&=p_{9}x_{9}-p_{10}x_{10},\\ \dot{x}_{11}&=p_{10}x_{10}-p_{11}x_{11},\\ \dot{x}_{12}&=p_{11}x_{11}-p_{12}x_{12},\\ \dot{x}_{13}&=p_{12}x_{12}-p_{13}x_{13},\\ \dot{x}_{14}&=p_{13}x_{13}-p_{14}x_{14},\\ \dot{x}_{15}&=p_{14}x_{14}-p_{15}x_{15},\\ \dot{x}_{16}&=p_{15}x_{15}-p_{16}x_{16},\\ \dot{x}_{17}&=p_{16}x_{16}-p_{17}x_{17},\\ \dot{x}_{18}&=p_{17}x_{17}-p_{18}x_{18},\\ \dot{x}_{19}&=p_{18}x_{18}-p_{19}x_{19},\\ \dot{x}_{20}&=p_{19}x_{19}-p_{20}x_{20},\\ \end{align*}$$

where $x_i, i = 1, …, 20$ are the states. The unknown parameters are $p_i, i = 1, …, 20$, plus the two Michaelis-Menten parameters $ν_{max}$ and $k_m$. All the states are assumed to be measured.