Machine learning based state observer for discrete time systems evolving on Lie groups
This work addresses a specific challenge in control theory for systems on manifolds, offering a more efficient observer design, though it is incremental as it builds on existing machine learning approaches for such systems.
The paper tackles the problem of designing state observers for systems evolving on Lie groups without requiring multiple charts or switching between models, by proposing a deep learning method that predicts an error term on the Lie algebra and maps it to the group, validated through Monte Carlo simulations on a rigid body rotation and translation system.
In this paper, a machine learning based observer for systems evolving on manifolds is designed such that the state of the observer is restricted to the Lie group on which the system evolves. Conventional techniques involving machine learning based observers on systems evolving on Lie groups involve designing charts for the Lie group, training a machine learning based observer for each chart, and switching between the trained models based on the state of the system. We propose a novel deep learning based technique whose predictions are restricted to a measure 0 subset of Euclidean space without using charts. Using this network, we design an observer ensuring that the state of the observer is restricted to the Lie group, and predicting the state using only one trained algorithm. The deep learning network predicts an ``error term'' on the Lie algebra of the Lie group, uses the map from the Lie algebra to the group, and uses the group action and the present state to estimate the state at the next epoch. This model being purely data driven does not require the model of the system. The proposed algorithm provides a novel framework for constraining the output of machine learning networks to a measure 0 subset of a Euclidean space without chart specific training and without requiring switching. We show the validity of this method using Monte Carlo simulations performed of the rigid body rotation and translation system.