Fazal-ur-Rehman and L. Khan

Stabilizing feedback control, non-holonomic systems, Lie bracket extension, controllability Lie algebra, logarithmic coordinates

This paper presents two diﬀerent kinds of attitude regulating control strategies for a rigid spacecraft model in actuator failure mode, which is an example of drift less non-holonomic control systems. The ﬁrst strategy presents piece-wise constant, states dependent feedback control laws. The method is based on the construction of a cost function V (not a Lyapunov function), which is sum of two semi-positive deﬁnite functions V1 and V2 . The semi-positive deﬁnite function V1 is dependent on ﬁrst m state variables which can be steered along the given vector ﬁelds and V2 is dependent on the remaining n − m state variables which can be steered along the missing Lie brackets. The values of the functions V1 and V2 allow in determining a desired direction of system motion and permit to construct a sequence of controls such that the sum of these functions decreases in an average sense. The second strategy presents a time-varying feedback law based on the model reference approach, where the trajectory of the extended system is chosen as the model reference trajectory. The controllers are designed in such a way that after each time period T , the trajectory of the rigid body model intersects the trajectory of the model reference, which can be made asymptotically stable. The proposed feedback law is as a composition of a standard stabilizing feedback control for a Lie bracket extension of the original system and a periodic continuation of a speciﬁc solution to an open loop control problem stated for an abstract equation on a Lie group. m z= ˙ i=1 gi (z) ui , z ∈ n (1)

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