Consider the half-car active suspension system with disturbances shown in the attached image. The dynamic equations are given as follows:

$$\begin{align*} \dot{x}_1 &=x_2 \\ \dot{x}_2 &= \dfrac{1}{m_S}[-(B_f + B_r)x_2 + (aB_f - bB_r)x_4 \cos{x_3} -k_fx_5 \\ & + B_fx_6 - k_rx_7 + B_rx_8 + (f_f + f_r)] \\ \dot{x}_3 & = x_4 \\ \dot{x}_4 &= \dfrac{1}{J_y}[(aB_f - bB_r)x_2\cos{x_3} \\ & - (a^2B_f + b^2B_r)x_4\cos^2{x_3} + ak_fx_5\cos{x_3}\\ & -aB_fx_6\cos{x_3}-bk_rx_7\cos{x_3} \\ & +bB_rx_8\cos{x_3} + (-af_r+bf_r)\cos{x_3}] \\ \dot{x}_5 &= x_2 - ax_4\cos{x_3} - x_6 \\ \dot{x}_6 &= \dfrac{1}{m_{uf}}[-K_{tf}x_1 + B_fx_2 + aK_{tf}\sin{x_3} \\ & - aB_fx_4 \cos{x_3} +(k_f + K_{tf})x_5 - B_f x_6 + K_{tf}z_{rf} - f_t] \\ \dot{x}_7 &= x_2 + bx_4\cos{x_3} - x_8 \\ \dot{x}_8 &= \dfrac{1}{m_{ur}}[-K_{tf}x_1 + B_rx_2 - bK_{tf}\sin{x_3} \\ & + bB_rx_4 \cos{x_3} +(k_r + K_{tr})x_7 - B_r x_8 + K_{tr}z_{rr} - f_r] \\ y_1 &= x_1 + x_2 :=h_1 \\ y_2 &= x_3 + x_4 := h_2, \end{align*}$$

where $x_1 = z$ is the displacement of the center of gravity, $x_2 = \dot{z}$ is the payload velocity, $x_3 = \theta$ is the pitch angle, $x_4=\dot{\theta}$ is the pitch velocity, $x_5 = z_{sf} - z_{uf}$ the front wheel suspension travel, $x_6 = \dot{z}_{uf}$ is the front unsprung mass velocity, $x_7 = z_{sr} - z_{ur}$ is rear wheel suspension travel and $x_8=\dot{z}_{ur}$ is the rear unsprung mass velocity.

The physical parameters are defined as:

The helicopter which is the subject of this paper was developed by Humusoft as the 2 degrees of freedom educational model. The model is a multidimensional, unstable nonlinear system with two manipulated inputs and two measured outputs. It has also significant cross couplings. The system consists of the body, carrying two propellers driven by DC motors, and a massive support (See attached image). The body has two degrees of freedom. Both body position angles (horizontal and vertical) are influenced by rotation of propellers. The axes of a body rotation are perpendicular. Power amplifiers, with a pulse width modulation, drive the DC motors. Both angles are measured. Helicopter model is described by the non-linear state-space equations. The model has nine states, two inputs, which are the control signals for main and side propeller motors. The two outputs are the elevation and azimuth angles.

The dynamics of the helicopter are represented by the following non-linear continuous time state space model:

$$\begin{align*} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \dfrac{1}{I_\psi} (-\sin{x_1} \cdot \tau_g -x_2 b_{\psi} + a_1(x_3)^2 + b_1x_3 - k_{gyro} \cdot \cos{x_1} \cdot x_6 \cdot u_1) \\ \dot{x}_3 &= -\dfrac{1}{T_1}x_3 + \dfrac{1}{T_1}x_4 \\ \dot{x}_4 &= -\dfrac{1}{T_1}x_4 + \dfrac{1}{T_1}u_1 \\ \dot{x}_5 &= x_6 \\ \dot{x}_6 &= \dfrac{1}{I_{\phi}} \left(-x_6 \cdot b_{\phi} + a_2 (x_7)^2 + b_2 x_7 - x_9 - \dfrac{k_r t_{0r}}{t_{pr}}u_1\right) \\ \dot{x}_7 &= -\dfrac{1}{T_2}x_7 + \dfrac{1}{T_2}x_7 \\ \dot{x}_8 &= -\dfrac{1}{T_2}x_x + \dfrac{1}{T_2}u_2 \\ \dot{x}_9 &= -\dfrac{1}{t_{pr}}x_9 + \left(\dfrac{k_r}{t_{pr}} + \dfrac{k_r t_{0r}}{t_{pr}}\right)u_1, \end{align*}$$

where $I_{\psi}$, $b_{\psi}$, $\tau_g$, $k_{gyro}$, $I_{\phi}$, $b_{\phi}$, $k_{r}$, $t_{0r}$, $t_{pr}$, $a_1$, $b_1$, $a_2$, $b_2$, $k_{\psi}$, $k_{\phi}$ are the constant parameters.

Simplified Schmid pendulum:

$$\begin{align*} \ddot{\psi} + a_{21}\omega - a_{11}\sin{\psi} &= -b_1u, \\ \dot{\omega} + a_{22}\omega + a_{12}\sin{\psi} &= b_2u, \end{align*}$$

where $\psi$ is the pendulum angle; $\omega$ is the wheel angular rate; $u$ is the controlling voltage, applied to the motor; $a_{11},a_{21},a_{12},b_1,b_2$ are positive constants, depending on the design parameter of the pendulum. It is assumed that the upper (unstable) equilibrium point corresponds to $\psi=0$.

Self-excited nonlinear oscillator:

$$\begin{align*} \dot{x}_1 &=x_2,\\ \dot{x}_2 &=-\omega_1^2\sin{y_1}-\varrho x_2 + k_p \arctan(k_c y_2),\\ \dot{x}_3 &=\omega_2 \cdot (x_1 - x_3), \\ y_1 &=x_1, \\ y_2 &=x_1-x_3, \end{align*}$$

where $y(t)\in \mathbb{R}^2$ is the sensor output vector (to be transmitted over the communication channel), $\omega_1, \omega_2, \varrho, k_p, k_c$ are system parameters, $x=[x_1,x_2,x_3]^{\mathrm T}\in\mathbb{R}^3$ of the system state vector $x(t)$ based on the signals $y_1(t),y_2(t)$, transmitted over the communication channel.

The system has the form

$\dot{x}(t)=Ax(t)+\varphi(y(t)),$ $y(t)=Cx(t),$

where

$A=\begin{bmatrix}0 & 1 & 0 \\ 0 & -\varrho & 0 \\ \omega_2 & 0 & -\omega_2\end{bmatrix},$

$C = \begin{bmatrix}1, & 0, & 0\\ 1, & 0, & -1\end{bmatrix},$

$ \varphi(y) = \begin{bmatrix} 0\\ \omega_1^2 \sin{y_1} + k_p \arctan{(k_c y_2)} \\ 0 \end{bmatrix}.$

Attached image depicts the model of this helicopter. The dynamics of this model simulate the attitude dynamics of a helicopter. Using Lagrange’s equations, one may readily show that the dynamical model of this system is given by:

$$\begin{align*} T_p &= I_x\ddot{\phi}-(I_y-I_z)\dot{\psi}^2S_{\phi}C_{\phi}+mgAC_{\phi} \\ T_y &= (I_yS_{\phi}^2 + I_zC_{\phi}^2)\ddot{\psi} + 1(I_y - I_z)\dot{\psi}\dot{\phi}S_{\phi}C_{\phi}, \end{align*}$$

where $\phi$ is the pitch angle in radians, $\psi$ is the yaw angle in radians, $I_x, I_y, I_z$ are inertia constants about the point of rotation, m is the total mass of the system, and $T_p, T_y$ are the pitch and yaw control torques, respectively.

The above can be written in general form:

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

where

$M(q,\delta) = \begin{bmatrix} I_x & 0\\ 0 & I_yS_{\phi}^2+I_zC_{\phi}^2\end{bmatrix}$

$G(q,\delta)=\begin{bmatrix}mgAC_{\phi}\\0\end{bmatrix}$

$C(q,\dot{q},\delta)=\begin{bmatrix}0&-(I_y-I_z)\dot{\psi}S_{\phi}C_{\phi} \\ (I_y-I_z)\dot{\psi}S_{\phi}C_{\phi} &(I_y-I_z)\dot{\phi}S_{\phi}C_{\phi}\end{bmatrix} $

$q=\begin{bmatrix}\phi \\ \psi\end{bmatrix}$

$u=\begin{bmatrix}T_p \\ T_y\end{bmatrix}$

The nominal values of the model parameters are $m=0.5719 kg$, $A = 0.0801 m^2$, $I_x = 0.0762 kg\cdot m^2$, $I_y = 3.86\times10^{-4}kg\cdot m^2$, $I_z = 0.0766 kg\cdot m^2,$ and $g = 9.81 m/sec^2$. It is straight forward to show that $[\dot{M}(q)-2C(q,\dot{q})]$ is skew symmetric.