A Lie algebraic approach to dynamical system prediction

In this paper we study the problem of the prediction of autonomous continuous-time dynamical systems from discrete-time measurements of the state variables. In this case, the predictor of the system needs to be an invertible map from the state-space into the predictor itself. The problem then becomes one of how to approximate those invertible maps. We show that standard approximation schemes do not guarantee the property of invertibility. We therefore propose a new approximation scheme which is based on the composition of invertible basis functions and which preserves invertibility. This approach can be cast in a Lie algebraic framework where the approximation is based on the use of the Baker-Campbell-Hausdorff formula. The method can be implemented in a neural-like form and we illustrate it by an example. We then present a general implementation which we call “MLP in dynamics space”. And we also extend the method to nonlinear affine control systems