Robust stability and control of spacecraft formations
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The work of this thesis is focused on the robustness of control laws for space-craft formation. Robustness in this case refers to the ability of the system to withstand persistent perturbations, and to keep some of the (stability) characteristics of the unperturbed system. Chapter 2 makes up the theoretical framework for this thesis. Analogous to the definition of practical asymptotic stability in Chaillet and Loría (2006b), we define practical exponential stability. This definition is more restrictive than its asymptotic counterpart, but has the advantage of an exponentially decaying upper bound of the solution on the considered part of the state space. Lyapunov sufficient conditions are stated, both for general systems and systems which are interconnected on a cascaded structure. Systems can naturally show a cascaded structure, as e.g. a leader follower spacecraft formation, or they can be rewritten into a cascaded structure, which is a common approach for systems with an observer and certainty equivalence controller. Furthermore, we provide a theoretical framework that fits realistic challenges related to spacecraft formation with disturbances. We show that the input-to-state property of such systems guarantees some robustness with respect to a class of signals with bounded average-energy, which encompasses the typical disturbances acting on spacecraft formations. Robustness is considered in the sense that solutions are bounded by a converging function of time, up to an offset which is somewhat proportional to the considered average energy of disturbances. The proposed approach allows for a tighter evaluation of the disturbances.in.uence, which in turns allows for the use of more parsimonious control gains. In Chapter 3 the leader-follower spacecraft formation is modeled. This type of formation is chosen because of its simplicity. It is therefore, in the authors opinion, the type of formation most likely used for real applications in the .eld of spacecraft formation control in the nearest future. Both a model for relative translation and rotation is derived. The relative translation model is derived in a general setting, where we can choose the origin of the frame of reference as center of gravity of the leader spacecraft or some other convenient point. Chapter 4 is devoted to output tracking of relative translation. The follower spacecraft control law is derived under limited knowledge of the leader spacecraft. It is required that the leader spacecraft can either broadcast its position, or the follower spacecraft are equipped with devices that can take the necessary measurements. In addition, it is assumed that the control action and disturbances acting on the leader spacecraft is upper bounded. In deriving the control laws we make use of the theory for control of robotic manipulators and ocean vehicles, as they are systems with similar properties. Output attitude tracking is treated in Chapter 5. As opposed to the translational case in Chapter 4, we derive control laws for both the leader-and the follower spacecraft, and show stability properties under bounded disturbances. In Chapter 6 we analyse stability of the controllers of a spacecraft formation with respect to a class of bounded-energy signals, using the frame-work developed in Chapter 2. Our application shows that our framework is not only useful for systems perturbed by certain disturbances, but we also show that the reference trajectory of the leader spacecraft can be seen as a disturbance from the follower spacecraft point of view. As propulsion systems of spacecraft often do not provide continuous actuation, Chapter 7 is devoted to the analysis of such systems when the actuation is quantized or pulse width modulated.