\addcontentsline{toc}{chapter}{Introduction}

\chapter*{Introduction \label{ch_intro}}

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The idea of using kites for wind power generation that was first rigorously investigated by Loyd~\cite{Loyd1980} in the 1980's has never been closer to a large scale realization than now. The University of Torino has already tested a kite prototype~\cite{Canale2006, Ippolito2006} and plans to build a large scale \textit{Kite Wind Generator}, the Delft University of Technology is developing kites to generate energy with so-called \textit{Laddermills}~\cite{Ockels2001, Ockels2003}, and the SkySails company~\cite{SkySails2006, Wrage2005} is already using kites to pull large ships. All these concepts have in common that a kite is exerting a strong force on its cable that is either used to actuate a generator on the ground or to pull a large object. Recent work by several groups shows that kites can be used in an efficient way to generate wind power~\cite{Houska2007a, Houska2007, Ilzhoefer2006, Lansdorp2005, Lansdorp2005b, Williams2007, Williams2007a} or to propell a ship~\cite{Houska2006, Ilzhoefer2007, Naaijen2006, Ockels2007}. In comparison to conventional windmills, wind power generation with kites has many important advantages: 
\begin{itemize}
	\item Kites can use the stronger winds at higher altitudes.
	\item The costs of a kite system are expected to be much lower than for windmills.
	\item The fast-flying kite can intercept a much larger area.
\end{itemize}
However, one of the main challenges is to keep the kite automatically in the air under varying wind and weather conditions. Our aim in this thesis is to steer the kite securely on the orbit that is energy-optimal for the current wind conditions.\\
First, we are interested in optimized periodic orbits to achieve a maximal average power at the generator. To solve the corresponding optimal control problem numerically, we use the optimal control package MUSCOD-II~\cite{Leineweber1999, Leineweber2003a} that is based on the direct multiple shooting method, which was first presented by Bock and Plitt~\cite{Bock1984, Plitt1981}. Since the generated average power is increasing with the third power of the wind speed~\cite{Loyd1980}, it is evident that we have to find an optimized periodic orbit for every wind velocity in order to generate the maximum power. However, these energy-optimal orbits are not open-loop stable, what makes stabilizing feedback control necessary to steer the kite securely on its reference. The method of choice is nonlinear model predictive control (NMPC). We use the continuous-time linear quadratic regulator controller as precontroller for NMPC, and we even deduce a linear controller that is robust with respect to constraints by solving a special optimal control problem including information on the Dryden wind turbulence model~\cite{Beal1993}. We use this robust linear controller to robustify the NMPC algorithm with respect to constraints. The challenging issue of real-time applicability is tackled by using a real-time iteration scheme that was developed by Diehl~\cite{Diehl2002, Diehl2005}. NMPC has already been applied to the problem of controlling a kite on a reference orbit~\cite{Diehl2003e, Diehl2004e}. In comparison to this work, we use an advanced kite model~\cite{Houska2007a} that actually allows to generate energy with the kite as well as a realistic model for random wind disturbances~\cite{Beal1993}. Our method to robustify NMPC with respect to constraints is also a new aspect. Furthermore, we use moving horizon estimation (MHE)~\cite{Diehl2006e} to estimate the wind velocity and the direction of the wind vector from measurements of the position and velocity of the kite. We have to make sure that the estimation is as good as possible, because the optimal orbit should ideally be changed according to the wind conditions. In practice, arbitrary fast changes of the reference orbit are not advisable. Thus, we present a heuristic that allows to change the NMPC reference orbit according to the MHE wind estimation. MHE has been initially developed for chemical processes~\cite{Diehl2006a, Kraus2007, Kraus2006}, which are in general slowly changing over time, and therefore a deterministic dynamic system is assumed. We have to assume that the wind conditions are changing relatively fast, and the estimation of the wind can never match the real wind exactly, because there are always small random wind disturbances present. We consider this by adding process noise according to Dryden's wind turbulence model to the MHE. This thesis is organized as follows:
\begin{itemize}
	\item In Chapter~\ref{ch_offline} we present the offline optimization of kite orbits under constant wind conditions. Therefore, we specify our nonlinear kite model and the optimal control problem that is solved. We briefly explain the numerical algorithm based on direct multiple shooting which we use to solve this optimal control problem and we present the optimization results.
	\item In Chapter~\ref{ch_linear} we present two different ways to deduce a closed-loop linear feedback control law. Firstly, we linearize our kite model at the energy-optimal orbit and compute the continuous-time linear quadratic regulator controller. Secondly, we solve a periodic Lyapunov differential equation during the offline optimization to deduce a linear controller that is robust with respect to constraints under Dryden wind turbulences and stabilizes the kite on a simultaneously optimized orbit.
	\item In Chapter~\ref{ch_nmpc} we apply NMPC based on a real-time iteration scheme to our problem. The linear quadratic regulator controller is used as precontroller, and we robustify the NMPC algorithm with respect to constraints by using the robust linear controller. We show that NMPC is real-time applicable for our problem and enlarges the region of attraction significantly compared to the linear controllers.
	\item In Chapter~\ref{ch_mhe} we use -~in cascade with NMPC~- MHE to estimate the unknown wind velocity and direction. We show that this combination is still real-time applicable for our problem. The estimated wind can never match the real wind exactly, because there are always small random wind disturbances present. We consider this by adding process noise according to Dryden's wind turbulence model to the MHE, and we show that this actually improves the wind estimation under Dryden wind turbulences significantly.
	\item In Appendix~\ref{ch_kiteboat} we present a paper called \textit{Optimization of a Kite Boat} about numerical optimization studies for a boat that is propelled by a kite. It has been submitted recently by Andreas Ilzh\"ofer, Boris Houska, Moritz Diehl, and Wubbo Ockels to the \textit{IFAC Journal of Process Control}.
	\item In Appendix~\ref{ch_kiteview} we present the documentation of an OpenGL program called \textit{Kiteview}, which is a useful tool to visualize orbits of one kite or a kite system with several kites. This program was developed during a software practical for advanced students, supervised by Dr. Moritz Diehl, at the University of Heidelberg.
\end{itemize}

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