# 3 DOF model of a helicopter

## Model description:

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”.

## Publication details:

 Title Adaptive Identification of Angular Motion Model Parameters for LAAS Helicopter Benchmark Publication Type Conference Paper Authors Peaucelle, Dimitri, Fradkov A.L., and Andrievsky B.

# Lumped-parameter model of dual–drive gantry stages

## Model description:

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*}

## Publication details:

 Title Decoupling basis control of dual-drive gantry stages for path-tracking applications Publication Type Conference Paper Authors Garciaherreros, I., Kestelyn X., Gomand J., and Barre P.-J.

## Model description:

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*}

 State Symbol Description Units 1 $x_1$ Tilt cylinder position cm 2 $x_2$ Lift cylinder position cm 3 $\dot{x}_1$ Tilt cylinder velocity cm/sec 4 $\dot{x}_2$ Lift cylinder velocity cm/sec 5 $P_p$ Pump pressure MPa 6 $D_p$ Pump displacement cm$^3$ 7 $P_{LS}'$ Load sense pressure MPa 8 $s_1$ Tilt function spool valve position mm 9 $P_{A,1}$ Tilt cylinder cap end pressure MPa 10 $P_{B,1}$ Tilt cylinder cap end pressure MPa 11 $s_2$ Lift function spool valve position mm 12 $P_A,2$ Lift cylinder cap end pressure MPa 13 $P_B,2$ Lift cylinder rod end pressure MPa

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## Publication details:

 Title Robust control design for a wheel loader using mixed sensitivity h-infinity and feedback linearization based methods Publication Type Conference Paper Authors Fales, R., and Kelkar A.

# Second order diagonal recurrent neural network

## Model description:

The model structure of the SDRNN have been shown in the attached image, second-order nonlinear system model is assumed as:

$$y(k+1)=\dfrac{y(k)y(k-1))[y(k)+4.5]}{1+y^2(k)+y^2(k-1)}+u(k).$$

The SDRNN(2, 7, 1) is used in simulation, that is, the input layer has 2 neurons $u(k)$ and $y(k)$, 7 neurons in hidden layer, 1 neuron $y(k +1)$ in output layer. The activation function is sigmoid function in hidden layer: this function is the commonly used bipolar function $\rho(x)=\dfrac{1-e^{-x}}{1+e^{-x}}$, initial weight is random value between -1 and 1, the learning rate $\eta=0.45$, momentum factorγ = 0.1.

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## Publication details:

 Title Application of Second Order Diagonal Recurrent Neural Network in Nonlinear System Identification Publication Type Conference Paper Authors Shen, Yan, Ju Xianlong, and Liu Chunxue

# A fourth-order heat exchanger process

## Model description:

The system under study consists of two sets of single shell heat exchangers filled with water, placed in parallel and cooled by a liquid saturated refrigerant flowing through a coil system, as it is illustrated in the attached image. The saturated vapour generated in the coil system is separated from the liquid phase in the stages $S1$ and $S2$, both of neglected volumes. This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling system, and only the saturated liquid portion is used for cooling purposes. Table below provides the fluid properties and equipment dimensions. The temperature of the refrigerant remains constant at $T_C$ as the liquid is saturated, and the energy exchanged with water is used to vapourise a small portion of the refrigerant fluid. The idividual heat exchanger energy balances can be expressed in terms of deviation variables to define the following LTI system:

\begin{align*} A &= \begin{bmatrix} -\dfrac{1+\nu_A}{\tau_1} & \dfrac{1}{\tau_1} & 0 & 0\\ 0 & -\dfrac{1+\nu_A}{\tau_2} & 0 & 0 \\ 0 & 0 & -\dfrac{1+\nu_B}{\tau_3} & \dfrac{1}{\tau_3} \\ 0 & 0 & 0 & -\dfrac{1+\nu_B}{\tau_4} \\ \end{bmatrix} \\ B &= \begin{bmatrix} 0 & 0 & 0 & 0\\ \dfrac{k_1}{\tau_1} & 0 & \dfrac{k_3}{\tau_2} & 0\\ 0 & 0 & 0 & 0\\ 0 & \dfrac{k_2}{\tau_t} & 0 & \dfrac{k_4}{\tau_4}\\ \end{bmatrix}, C = \begin{bmatrix} \mu & 0 & \mu & 0\\ 0 & \mu & 0 & \mu\\ \mu & 0 & \mu & 0\\ 0 & \mu & \mu & 0\\ \end{bmatrix}, D = \begin{bmatrix} 0 \end{bmatrix}, \end{align*}

where: $\nu_A \triangleq (hA/C_p(\dot{m}_1 + \dot{m}_3)), \\ \nu_B \triangleq (hA/C_p(\dot{m}_2 + \dot{m}_4)),\\ \tau_1 \triangleq (M_1/(\dot{m}_1 + \dot{m}_3)),\\ \tau_2 \triangleq (M_2/(\dot{m}_1 + \dot{m}_3)),\\ \tau_3 \triangleq (M_3/(\dot{m}_2 + \dot{m}_4)),\\ \tau_4 \triangleq (M_4/(\dot{m}_4 + \dot{m}_3)),\\ k_1 \triangleq(\dot{m}_1/(\dot{m}_1 + \dot{m}_3)),\\ k_2 \triangleq(\dot{m}_2/(\dot{m}_2 + \dot{m}_4)),\\ k_3 \triangleq(\dot{m}_3/(\dot{m}_1 + \dot{m}_3)),\\ k_4 \triangleq(\dot{m}_4/(\dot{m}_1 + \dot{m}_3)),\\ \mu \triangleq(hA/h_{lv}).\\$

This system was discretised with a sampling time of $T = 1 s$ and was discretised while rounding the input and output delays to the closest integer-multiples of $T$.

 $C_p$ 4.217 kJ/kg K water specific heat $h_{lv}$ 850 kJ/kg refrigerator heat vapourisation $T_S(0)$ 40$^{\circ}$C initial temperature in $E_s$ $T_{jin}(0)$ 40$^{\circ}$C initial water inlet temperature $j$ $T_c$ 40$^{\circ}$C refrigerant temperature $\dot{m}_j$ 1 kg/s water mass flow $j$ $M_s$ 50 kg mass of water in $E_s$ $hA$ 8 kJ/kg overall surface heat transfer $V_1$ 0.5 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 1 $V_2$ 2 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 2 $V_4$ 1.5 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 4 $\rho$ 1000 kg/m$^3$ water density

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## Publication details:

 Title Discretisation of continuous-time dynamic multi-input multi-output systems with non-uniform delays Publication Type Journal Article Authors Kassas, Z.M.