This paper proposes local exponential observers for systems on linear Lie groups. We study two different classes of systems. In the first class, the full state of the system evolves on a linear Lie group and is available for measurement. In the second class, only part of the system's state evolves on a linear Lie group and this portion of the state is available for measurement. In each case, we propose two different observer designs. We show that, depending on the observer chosen, local exponential stability of one of the two observation error dynamics, left or right invariant error dynamics, is obtained. For the first class of systems these results are developed by showing that the estimation error dynamics are differentially equivalent to a stable linear differential equation on a vector space. For the second class of system, the estimation error dynamics are almost linear. We illustrate these observer designs on an attitude estimation problem.
This paper proposes two local exponential observers for left-invariant systems on linear Lie groups, where the full state of the system is available for measurement. We show that, depending on the observer chosen, local exponential stability of one of the two estimation error dynamics, left or right invariant error dynamics, is obtained. Our proposed observers are noteworthy because their estimation error dynamics are differentially equivalent to a linear and stable differential equation on the Lie algebra. We illustrate our observer designs for an attitude estimation problem on the special orthogonal group SO (3).
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