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

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

The attached image depicts the kinematic car in the horizontal plane. Let us suppose that the Ackermann steering assumptions hold true, hence all wheels turn around the same point (denoted by P) which lies on the line of the rear axle. It follows that the kinematics of the car can be fully described by the kinematics of a bicycle fitted in the middle of the car (see attached image. The coordinates of the rear axle midpoint are given by $x$ and $y$. The orientation of the car with respect to the axis of $x$ is denoted by 9. The angle of the front wheel of the bicycle with respect to the longitudinal symmetry axis of the car is denoted by $φ$ . One may consider $φ$ or its time derivative $u_2=\dot{φ}$ as input. The longitudinal velocity of the rear axle midpoint is denoted by $u_1$ if it is a control input (two input case) and by $v_{car}$ if not (one input case). All lengths involved in the kinematic calculations, and in particular $l$, equal to one.

$$\begin{align*} \dot{x} &= u_1 \cos{\theta},\\ \dot{y} &= u_1 \sin{\theta},\\ \dot{\theta} &= u_1 \tan{\varphi},\\ \dot{\varphi} &= u_2. \end{align*}$$

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).

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

The overall model consists of hydraulics (cylinders, valves, pump, etc.) and two degree of freedom linkage. The equations can be combined to form a MIMO state space system. The state vector, $x$, is defined in the table below. The input is the current in two valve solenoids as follows: $u = [i_1, i_2]^T$ . The output is given as, $y = [θ_1, θ_{21}]$. The states of the model are summarized in the table below. The dynamic equations for the linkage and electrohydraulic system in terms of state variables can be written as follows:

$$\begin{align*} \dot{x}_1 &= x_3\\ \dot{x}_2 &= x_4\\ \begin{bmatrix} \dot{x}_3 \\ \dot{x}_4 \end{bmatrix} &= M^{-1}(\tau(x_9, x_{10}, x_{11}, x_{12}, x_{13}) - h(x_1, x_2, x_3, x_4))\\ \dot{x}_5 &= \dfrac{\beta}{V_p}(\omega_px_6/2\pi - x_5K_{Lp}-(Q_{P A,1}(x_5, x_8, x_9) + Q_{P B, 1}(x_8, x_{10} + Q_{P A,2}(x_5, x_{11}, x_{12}) + Q_{P B,2}(x_5, x_{11}, x_{13})))\\ \dot{x}_6 &= [x_7 - x_5 + P_{margin}] G_p\\ \dot{x}_7 &= (\max(x_9,x_{10},x_{12},x_{13})-x_7)1/\tau_p\\ \dot{x}_8 &= (-x_8 + G_vu_1)1/\tau_v\\ \dot{x}_9 &= \dfrac{\beta}{V_{A,1}(x_1)}(Q_{PA,1(x_5,x_8,x_9}) + Q_{TA,1}(x_8,x_9-\dot{V}_{A,1}(x_3))\\ \dot{x}_{10} &= \dfrac{\beta}{V_{B,1}(x_1)}(Q_{PB,1(x_8,x_{10}}) + Q_{TB,1}(x_8,x_{10}-\dot{V}_{B,1}(x_3))\\ \dot{x}_{11} &= (-x_{11} + G_vu_2)1/\tau_v \dot{x}_{12} &= \dfrac{\beta}{V_{A,12}(x_2)}(Q_{PA,2(x_5,x_{11},x_{12}}) + Q_{TA,2}(x_{11},x_{12}-\dot{V}_{A,2}(x_4))\\ \dot{x}_{13} &= \dfrac{\beta}{V_{B,2}(x_2)}(Q_{PB,2(x_5,x_{11},x_{13}}) + Q_{TB,2}(x_{11},x_{13}-\dot{V}_{B,2}(x_4))\\ \end{align*}$$