Talk:PlanetPhysics/2D With Drag Airship Optimal Control

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: airship optimal control, 2D with drag %%% Primary Category Code: 02.30.Yy %%% Filename: 2DWithDragAirshipOptimalControl.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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\begin{document}

\section{ 2D with Drag and No Wind }

{\bf Disclaimer}: This is a \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} in progress... \\

The \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} due to drag in 2D is given by

$$F_d = \frac{1}{2} C_d \rho A v^2$$

Unlike the 1D case, which uses the absolute value, the direction of the \htmladdnormallink{drag force}{http://planetphysics.us/encyclopedia/CoriolisEffect.html} in 2D is taken into account by the \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} of the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} times the component velocities such that the x and y components of the drag force are

$$F^x_d = -\frac{1}{2} C_d \rho A \dot{x} \sqrt{\dot{x}^2 + \dot{y}^2}$$ $$F^y_d = -\frac{1}{2} C_d \rho A \dot{y} \sqrt{\dot{x}^2 + \dot{y}^2}$$

The equations of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} for this problem are

$$\dot{x} = \dot{x}$$ $$\dot{y} = \dot{y}$$ $$\ddot{x} = F^x_d + \frac{T}{m} \cos \alpha$$ $$\ddot{y} = F^y_d + \frac{T}{m} \sin \alpha$$

where T is the applied Torque in Newtons, $\alpha$ is the angle of applied torque counter clockwise from the negative x-axis and $m$ is the \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of the airship.

The performance measure for the minimum \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} problem assuming there is no effort required to change the angle of attack

\begin{equation} J = \int_0^{t_f} \frac{T^2}{m^2} dt \end{equation}

The \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} for this optimal control problem

\begin{equation} \mathcal{H} = \frac{T^2}{m^2} + p_1 \dot{x} + p_2 \dot{y} + p_3 \left ( F^x_d + \frac{T}{m} \cos \alpha \right) + p_4 \left ( F^y_d + \frac{T}{m} \sin \alpha \right ) \end{equation} The neccessary conditions for the unconstrained control inputs

$$ \frac{\partial \mathcal{H}}{\partial \alpha} = 0 $$ $$ \frac{\partial \mathcal{H}}{\partial T} = 0 $$ \\ $$ \frac{\partial \mathcal{H}}{\partial \alpha} = -p_3 \frac{T}{m} \sin \alpha + p_4 \frac{T}{m} \cos \alpha = 0$$ $$ \frac{p_4 T}{m}\cos \alpha = \frac{p_3 T}{m} \sin \alpha $$ $$ \frac{p_4}{p_3} = \frac{\sin \alpha}{\cos \alpha} $$ \begin{equation} \tan \alpha = \frac{p_4}{p_3} \end{equation}

$$ \frac{\partial \mathcal{H}}{\partial T} = \frac{2T}{m^2} + \frac{p_3}{m} \cos{\alpha} + \frac{p_4}{m}{sin \alpha} = 0 $$ $$ \frac{2T}{m} + p_3 \cos{\alpha} + p_4{sin \alpha} = 0 $$ $$ \frac{2T}{m} = -\left (p_3 \cos{\alpha} + p_4{sin \alpha} \right ) $$ \begin{equation} T = -\frac{m}{2}\left (p_3 \cos{\alpha} + p_4{sin \alpha} \right ) \end{equation}

The neccessary conditions for the costates are

\begin{equation} \dot{p_1} = -\frac{\partial \mathcal{H}}{\partial x} = 0 \end{equation} \begin{equation} \dot{p_2} = -\frac{\partial \mathcal{H}}{\partial y} = 0 \end{equation} \begin{equation} \dot{p_3} = -\frac{\partial \mathcal{H}}{\partial \dot{x}} = -p_1 + \frac{1}{2} p_3 C_d \rho A \sqrt{\dot{x}^2 + \dot{y}^2} + \frac{p_3 C_d \rho A \dot{x}^2}{\sqrt{\dot{x}^2 + \dot{y}^2}} + \frac{p_4 C_d \rho A \dot{x} \dot{y}}{\sqrt{\dot{x}^2 + \dot{y}^2}} \end{equation} \begin{equation} \dot{p_4} = -\frac{\partial \mathcal{H}}{\partial \dot{y}} = -p_2 + \frac{1}{2} p_4 C_d \rho A \sqrt{\dot{x}^2 + \dot{y}^2} + \frac{p_4 C_d \rho A \dot{y}^2}{\sqrt{\dot{x}^2 + \dot{y}^2}} + \frac{p_3 C_d \rho A \dot{x} \dot{y}}{\sqrt{\dot{x}^2 + \dot{y}^2}} \end{equation}

\end{document}