Pointing Control of Spacecraft using Two SGCMGs via LPV Control Theory

Paper number

IAC-09.C1.6.6

Author

Mr. Sangwon Kwon, Osaka Prefecture University, Japan

Coauthor

Prof. Hiroshi Okubo, Osaka Prefecture University, Japan

Coauthor

Prof. Takashi Shimomura, Osaka Prefecture University, Japan

Year

2009

Abstract

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A control moment gyro (CMG) is a kind of actuator for spacecraft attitude control. In the case of smaller-sized satellites with limited resources, it is not a suitable option to increase hardware resources. Therefore, the problem of attitude control using a reduced number of CMGs has received considerable attention for decades and many studies to this problem have been performed in the previous decades in the area of underactuated spacecraft control [1],[2]. By using the law of angular momentum conservation, the objectives of pointing control of spacecraft using two single-gimbal control moment gyros (SGCMGs) are described as follows.
\begin{align*}
\omega &\rightarrow 0 \\
\delta_{e} \triangleq \delta &- \delta_{f} \rightarrow 0 \\
\phi_{e} \triangleq \phi &- \phi_{f} \rightarrow 0
\end{align*}
where $\omega$ is the angular velocity of the spacecraft, $\delta$ is the vector of the gimbal angles of two SGCMGs, and $\phi$ is an Euler angle. The subscript $f$ denotes the final state of parameters. To this control problem, we developed a switching controller that consists of a nonlinear controller based on the Lyapunov stability theory and an LQR controller in [1]. However, this switching controller is too complex, because it consists of multiple-steps. Therefore, in this paper, we develop another controller being simpler and more suitable than the former switching controller. In the development of this new controller, we propose a new method of pointing control using two SGCMGs via linear parameter-varying (LPV) control theory. The LPV control has advantages such that it provides guaranteed stability and performance over a wide range of varying parameters [3]. The nonlinear model of spacecraft using two SGCMGs in six degrees of freedom can be represented as an LPV system as follows.
\begin{align*}
\dot{x} &= A(\delta,\psi)x + B(\delta,\psi)u \\
u &= -K(\delta,\psi)x
\end{align*}
where the gimbal angle vector $\delta$ and the Euler angle $\psi$ are both scheduling parameters. The state feedback controller is developed on the basis of a gain-scheduled control technique for the LPV system. The gain-scheduled control approach is to find a Lyapunov function which guarantees overall stability and performance of the close-loop system. The design condition of such a controller is described by a set of linear matrix inequalities (LMIs). The original non-convex problem is transformed into a convex one and the nonlinear parameter relationship is described as a bound of a convex region in a parameter space [4]. Approximating this region by a set of successive LMIs, the state feedback gain $K(\delta,\psi)$ is successfully obtained. A numerical simulation demonstrates that the proposed method is highly effective for fast and stable pointing control of spacecraft.
\vspace{5mm}
\begin{center}
\bf References
\end{center}
\vspace{-14.5mm}
\begin{thebibliography}{4}
\bibitem{1}Kwon, S. and Okubo, H.,“Angular Velocity Stabilization of Spacecraft Using Two Single-Gimbal Control Moment Gyros”, \textit{The 26th International Symposium on Space Technology and Science}, 2008-d-16, 2008.
\bibitem{2}Yoon, H. and Tsiotras, P.,“Spacecraft Line-of -Sight Control Using a Single Variable-Speed Control Moment Gyro”, \textit{Journal of Guidance, Control, and Dynamics}, Vol. 29, No. 6, 2006, pp. 1295-1308.
\bibitem{3}Wu, F. and Prajan, S.,“A New Solution Approach to Polynomial LPV System Analysis and Synthesis”, \textit{Proceedings of the American Control Conference}, 2004, pp. 1362-1367.
\bibitem{4}Shimomura, T., Fujita, Y., and Okubo, H.,“Simultaneous Structure/Controller Optimal Design of Flexible Space Structures: Sensor/Actuator Placement and Control Design - Tangential-Line Linearizing Constraints”, \textit{Journal of Japan Society for Aeronautical and Space Sciences}, Vol. 56, No. 652, 2008, pp. 239-243.
\end{thebibliography}

Abstract document

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