The plant is

$$y(k)=1.2y(k-1)-0.8y(k-2)+0.2y(k-1)u(k-1)+u(k-1)+0.6u(k-2) + d(k),$$

where $d(k)$ is a disturbance.

Consider the following unknown discrete nonlinear dynamic system:

$$\begin{align*} y(k+1)&=p[{\bf q}(k), u(k)]=0.2\cos[0.8(y(k)+y(k-1))] \\ & +0.4\sin[0.8(y(k)+y(k-1))+2u(k)+u(k-1)] \\ &+0.1[9+y(k)+y(k-1)]+\left[{2(u(k)+u(k-1))\over 1+\cos(y(k))}\right] \end{align*}$$

for $k=0,1,2,\ldots$ with $y(k)=0,u(k)=0$, for $k=0,-1,-2,\ldots$, $\Delta t := t(k+1)-t(k)=0.02sec$, for $k=0,1,2,\ldots$.

The suggested tracker scheme is tested with a 2-input 2-output nonlinear system given by:

$$\begin{align*} y_{1} (k) & = 0.21y_{1} (k-1)-0.12y_{2} (k-2) \\ & + 0.3y_{1} (k-1)u_{2} (k-1)-1.6u_{2} (k-1) \\ & + 1.2u_{1} (k-1), \\ y_{2} (k) & = 0.25y_{2} (k-1)-0.1y_{1} (k-2) \\ &- 0.2 y_{2} (k-1)u_{1} (k-1)-2.6u_{1} (k-1) \\ &-1.2u_{2} (k-1). \end{align*}$$

Under the aforementioned assumptions, dynamics of the 3DOF motion of “Helicopter” in a general form may be described by the following equations:

$\dot{\overrightarrow{K}} + \overrightarrow{\omega} \times \overrightarrow{K} =\overrightarrow{M},$ $\overrightarrow{K}=J\overrightarrow{\omega}$

$\omega_x=\dot{\theta}$, $\omega_y=\dfrac{\dot{\lambda}}{\cos{\epsilon}} - \dot{\theta}\tan{\epsilon},$ $\omega_z=\dot{\epsilon}$,

where $\overrightarrow{K}$ is a kinetic moment, $J$ is the inertia tensor, $\overrightarrow{M}$ is the sum of the moment of the propeller torques, the gravitational forces (bar plus the two motors) and the viscous friction torque. The vector $\omega$ is expressed in mobile coordinate system of the Helicopter.

The model describes nonlinear system with cross-talk coupling. At the stage of control law design, it is reasonable to make further simplifying assumptions. Since the propeller torque about the pitch axis does not depend on the travel and elevation angles (see attached image), and the moments of inertia about the travel and elevation axes are similar, the pitch motion may be considered independently of the other ones. In this way the following simplified model is obtained:

$\begin{align*} \ddot \theta &= - a_{m_x}^{\omega _x}\dot \theta - a_{m_x}^\theta \sin (\theta (t) - \theta _0) + a_{{m_x}}^v({v_f} - {v_r}) \\ \ddot \omega &= - a_{{m_z}}^{{\omega _z}}\dot \varepsilon - a_{{m_z}}^\varepsilon \sin (\varepsilon (t) - {\varepsilon _0}) - a_{{m_z}}^{{\omega _y}}\sin (2\varepsilon){{\dot \lambda }^2} + a_{{m_z}}^v({v_f} + {v_r})\cos \theta \\ \ddot \lambda &= - a_{{m_y}}^{{\omega _y}}\dot \lambda + a_{{m_y}}^v({v_f} + {v_r})\sin \theta. \end{align*} $

Parameters $a_j^i$ are considered to be unknown and subjected to estimation by means of adaptive identification algorithm. The constants $\theta_0$ and $\epsilon_0$ stand for pitch and elevation balance angles. The value of $\epsilon_0$ depends on the weight $M_x$ position (see attached image) and may be varying in different experiments at the operators commands (the “ADO option”). The term with $\lambda^2$ in the second equation describes influence of the “Helicopter” rotation about the vertical axis on the elevation caused by the centrifugal force. This term may be neglected if the travel motion is “sluggish”.

The lumped-parameter model in the attached images can be described using eleven physical parameters.

1. Four masses $m_b,m_k,m_1$ and $m_2$ corresponding to the mass of the beam, the moving head and actuators $X_1$ and $X_2$.
2. Five friction coefficients $C_{g1}, C_{g2}, C_y, C_{b1}$ and $C_{b2}$, corresponding to the viscous friction coefficients of $X_1, X_2$, and $Y$ actuators, and to the damping coefficients of the flexible joints.
3. Two stiffness coefficients $k_{b1}$ and $k_{b2}$ of the flexible joints of the beam to actuator junctions.

The beam and the head have lengths $L_b$ and $L_h$. The head's position is measured from the center of mass of the beam and is denoted by $Y$ and $d$, In the attached image, $X$ denotes the linear position of the center of mass of the beam and $\Theta$ is the yaw angle of the beam. They are defined as

$\begin{bmatrix} X \\ \Theta \\ Y \end{bmatrix} = \begin{bmatrix} 1/2 &1/2 & 0 \\ 1/L_{b} & -1/L_{b} & 0 \\ 0 & 0 &1 \end{bmatrix}\begin{bmatrix} X_1 \\ X_2 \\ Y \end{bmatrix}$

The Lagrange-Euler formalism is applied to derive the motion equations of the system.

$$M\ddot{q}+H\dot{q}+C\dot{q}+Kq=F,$$

where $M, C$ and $K$ are the inertia, viscous damping and stiffness matrices, $H$is the coriolis and centripetal acceleration matrix, $F$ is the vector of forces and $q$ is the vector of coordinates composed of $X, \Theta$ and $Y$ (8).

$\begin{align*} M &=\begin{bmatrix} M_{11} & M_{12} & m_{h}\sin(\Theta) \\ M_{12} & J_{t}+m_{h}Y^{2} &-m_{h}d \\ m_{h}\sin(\Theta) & -m_{h}d & m_{h} \end{bmatrix}, \\ H &=\begin{bmatrix} 0 & If_{12}\dot{\Theta} & -2m_{h}\dot{\Theta} & \cos(\Theta) \\ 0 & 2m_{h}Y\dot{Y} & 0 \\ 0 & -m_{h}Y\Theta & 0 \end{bmatrix}, \\ C &= \begin{bmatrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{22} & 0 \\ 0 & 0 & c_{y} \end{bmatrix},\ K=\begin{bmatrix} 0 & 0 & 0 \\ 0 & K_{22} & 0 \\ 0 & 0 & 0 \end{bmatrix}, \\ F &=[F_{c}\ M_{c}\ F_{y}]^{\mathrm T},\ q=[X \ \Theta \ Y]^{\mathrm T }. \end{align*}$