# The State Dependent Model of the Helicopter

## Model description:

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.

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## Publication details:

 Title Non-linear predictive control of 2 DOF helicopter model Publication Type Conference Paper Year of Publication 2003 Authors Dutka, A.S, Ordys A.W, and Grimble M.J. Conference Name Proceedings on Decision and Control, 2003. Date Published 12/2003 Publisher IEEE ISBN Number 0-7803-7924-1 Accession Number 7929673 Keywords aircraft control, helicopters, nonlinear control systems, predictive control, state-space methods, time-varying systems Abstract This paper presents the application of non-linear predictive control algorithm to a helicopter model. First, the model of the helicopter is discussed. Next, the nonlinear algorithm is introduced which is based on state-space GPC controller. The non-linearity is handled by converting the state-dependent state-space representation into the linear time-varying representation. The predictions of the future controls are used to calculate predictions of the future states and of the future time varying system parameters. Applied to the helicopter model, the algorithm performs well. It is capable of the stabilizing the system for maneuvers for which it's linear counterpart fails. DOI 10.1109/CDC.2003.1271768