We consider the attached image where the VARIO helicopter mounted on an experimental platform is represented. It is important to say that in this particular case the helicopter is in an OGE condition. The effects of the compressed air in take-off and landing are then neglected. The model has the form

$$M(q)\ddot{q}+C(q,\dot{q})\dot{q}+G(q)=Q(u),$$

where $M(q)\in\mathbb{R}^{3\times3}$ is the inertia matrix, $C(q,\dot{q})\in\mathbb{R}^{3\times3}$ is the Coriolis matrix, $G(q)\in\mathbb{R}^3$ is the vector of conservative forces, $Q(u)=\begin{bmatrix}f_z & \tau_z & \tau_\gamma \end{bmatrix}^{\mathrm T}$ is the vector of generalized forces, $q = \begin{bmatrix} z & \phi & \gamma \end{bmatrix}^{\mathrm T}$ is the vector of generalized coordinates and $u=\begin{bmatrix}h_M & h_T \end{bmatrix}^{\mathrm T}$ is the vector of control inputs. Here $f_Z, \tau_Z$ and $\tau_{\gamma}$ are the vertical forces, the yaw torque and the main rotor torque, respectively. The height $z < 0$ upwards, $\phi$ is the yaw angle and $\gamma$ is the main rotor azimuth angle.

$M(q)=\begin{bmatrix} c_0 & 0 & 0 \\ 0 & c_1 + c_2 \cos^2{(c_3\gamma)} & c_4\\ 0 & c_4 & c_5 \end{bmatrix},$

$C(q,\dot{q})=\begin{bmatrix} 0 & 0 & 0\\ 0 & c_6\sin{(2c_3\gamma)}\dot{\gamma} & c_6\sin{(2c_3\gamma)}\dot{\phi} \\ 0 & -c_6\sin{(2c_3\gamma)}\dot{\phi} & 0\end{bmatrix},$

$G(q)=\begin{bmatrix}c_7 \\ 0 \\ 0 \end{bmatrix},$

where $c_i$'s $i = 0, ..., 7$ are the physical constants given in the table below.

The generalized forces vector is given by

$Q(u)=\begin{bmatrix} c_8\dot{\gamma}^2u_1 + c_9\dot{\gamma} + c_{10} \\ c_{11}\dot{\gamma}^2u_2\\ (c_{12}\dot{\gamma}^2 + c_{13})u_1 + c_{14}\dot{\gamma}^2 + c_{15} \end{bmatrix}$

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.

Consider the model described in Two van der Pol oscillators coupled via a bath (1).

The current model is using a slightly different notation:

$$\begin{align*} \dot{\xi}_{1}^{1} &= \dot{x}_1 \\ \dot{\xi}_{2}^{1} &= \dot{x}_2 \\ \dot{\xi}_{1}^{2} &= \dot{x}_3 \\ \dot{\xi}_{2}^{2} &= \dot{x}_4 \\ \dot{\eta}_{1} &= \dot{x}_5 \end{align*}$$

Note that this system is decouplable by static state feedback because the decoupling matrix of this system is

$D_{1}(\xi,\eta)=\left[\matrix{1 &0 \cr 0 &1}\right]$

The authors have proposed the following Yuz and Goodwin type approximate model which is more accurate than the Euler model.

$$\begin{align*} x_{1,k+1}&=x_{1,k}+T_{x_{2,k}}+\frac{T^2}{2}\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{2,k+1}&=x_{2,k}+T\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{3,k+1}&=x_{3,k}+T_{x_{4,k}}+\frac{T^2}{2}\{u_{2,k}-x_{1,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{1,k})\} \\ x_{4,k+1}&=x_{4,k}+T_{x_{4,k}}+T\{u_{2,k}-x_{2,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{3,k})\} \\ x_{5,k+1}&=x_{5,k}+T\{k(x_{1,k}-x_{5,k})+k(x_{3,k}-x_{5,k})\} \\ y_{1,k}&=x_{1,k} \\ y_{2,k}&=x_{3,k} \end{align*}$$