The model describes the oscillations in enzyme kinetics. The state variable $x_1$ represents an enzyme concentration whose rate of synthesis is regulated by feedback control via a metabolite $x_3$, and $x_2$ regulates the synthesis of $x_3$. It is characterised by a rational kinetics consisting of a Hill-like term, and it is given by:

$$\begin{align*} \dot{x}_1 &= -bx_1 + \dfrac{a}{A+X_3^\sigma},\\ \dot{x}_2 &= \alpha x_1 - \beta x_2,\\ \dot{x}_3 &= \gamma x_2 - \delta x_3 \end{align*}$$

with $x_1(0) = 0.3617$, $x_2(0) = 0.9137$, $x_3(0)=1.3934.$

The following two-dimensional single-input single-output system represents a chemical reactor model

$$\begin{align*} \dot{x}_1 &= u(Ce -x_1) - rx_1 \\ \dot{x}_2 &= rx_1 - ux_2 \\ y &= x_1 - x_2, \end{align*}$$

where coefficients are in $\mathbb{R}$, $x_1$ and $x_2$ denote the reactant and product concentrations, respectively. The input $u$ corresponds to the input flow of reactant, $r$ and $C_e$ denote kinetic and reactor parameters.

Consider the model from Dynamic Model of Tumor Growth (1). The complete model formulation describes the phenomenology of tumor growth slowdown, as the tumor consumes its available support; stimulatory and inhibitory influences from the tumor cells; inhibition due to administered inhibitors; and the clearance of the administered inhibitor. In the simplified model, the latter effect is not described, only the serum level of the inhibitor to be maintained is represented, so a second-order system is to be analyzed:

$$\begin{align*} \dot{x}_{1} &=-\lambda x_{1}\ln\left(\dfrac{x_{1}}{x_{2}}\right) \\ \dot{x}_2 &= b_x1 - dx_1^{{2}\over{3}}x_2 - ex_2u \\ y&=x_1, \end{align*}$$

where $x_1$ is the tumor volume (mm$^3$), $x_2$ is the vasculature volume (mm$^3$), and $u$ is the serum level of the inhibitor (mg/kg). The last equation represents that tumor volume is the measured output of the system. The characteristics of the parameters for the Lewis lung carcinoma and the mice used in the experiment are: $\lambda = 0.192($day$^{-1})$, $b = 5.85 ($day$^{−1}),$ $d =0.00873 ($day$^{−1}$mm$^{−2}),$ while the parameter characteristic for the inhibitor (endostatin) is: $e = 0.66 ($day$^{−1} ($mg/kg$)^{−1}).$ Attached figure shows the nonlinear behavior of the simplified model.

In 1999, a research was carried out at the Harvard Medical University by Philip Hahnfeldt et al. to investigate experimentally and theoretically the effects of angiogenic inhibitors on tumor growth dynamics. They posed a quantitative theory for tumor growth under angiogenic stimulator/inhibitor control. In their experiments, mice were injected with Lewis lung carcinoma cells. The following equations comprise the entire model formulation:

$$\begin{align*} \dot{x}_1 &=-\lambda_1x_1\ln\left(\frac{x_1}{x_2}\right) \\ \dot{x}_2 &=bx_1-dx_1^{\frac{2}{3}}x_2-ex_2x_3 \\ \dot{x}_3 &=\int_0^tu(t^{\prime})\exp(-\lambda_{3}(t-t^{\prime})){\mathrm d}t^{\prime} \\ y &=x_{1}, \end{align*}$$

where $x_1$is the tumor volume (mm$^3$), $x_2$is the supporting vasculature volume (mm$^3$), $x_3$ is the inhibitor serum level (mg/kg), and $u$ is the inhibitor administration rate (mg/kg/day).

In 2-DOF helicopter, a coupled 2input-2output system can be achieved due to coupling between the pitch and yaw motor torques. The linear 2-DOF helicopter state-space matrices are

$$\begin{align*} A&=\left[\matrix{ 0 &0 &1 &0\cr 0 &0 &0 &1\cr 0 &0 &-{B_{p}\over J_{eq{\_}p}+m_{heli}{l_{cm}}^{2}} &0\cr 0 &0 &0 &-{B_{y}\over J_{eq{\_}y}+m_{heli}{l_{cm}}^{2}}}\right],\\ B &=\left[ \matrix{ 0 &0\cr 0 &0\cr \dfrac{K_{pp}u_{p}}{J_{eq{\_}p}+m_{heli}{l_{cm}}^{2}} & \dfrac{K_{py}u_{y}}{J_{eq{\_}p}+m_{heli}{l_{cm}}^{2}} \cr \dfrac{K_{yp}u_{p}}{J_{eq{\_}y}+m_{heli}{l_{cm}}^{2}} & \dfrac{K_{yy}u_{y}}{J_{eq{\_}y}+m_{heli}{l_{cm}}^{2}} \cr } \right], \\ C &=\left[\matrix{1 &0 &0 &0\cr 0 &1 &0 &0}\right], D=\left[\matrix{0 &0\cr 0 &0}\right], \end{align*}$$

where $\theta(t)$ is the pitch angle and $\psi(t)$ is the yaw angle. $u_p$ and $u_y$ are the control signals applied to pitch and yaw motors, respectively. The amounts of parameters used in this formula are written in the table below.