Identification of Differential Flat Systems with Artifical Neural Networks

The property of differential flatness in dynamic systems leads to advantages in the field of analysis and control. These properties are widely elaborated [1]. Flatness means, that the whole system dynamics can be described by a flat output and a finite number of its derivatives. The flatness property defines a diffeomorphism from the system manifold to a trivial one. We use the bijective property of this map to design artifical neuronal networks accordingly. The weights of the designed networks directly correspond to the parameterization of the diffeomorphism. Training these networks result in a parameter estimation of the observed system. One main objective is to get more insights in how to choose network topologies for a given problem formulation. Parameter estimation of mechanical models with neural networks is performed in [2], where the authors estimate parameters of a dynamic aircraft system model directly from the weights of a feed forward neural network. Using a flatness based method leads in practice often to a static diffeomorphism which can exactly be reproduced by feed forward neural networks. The considered hidden layers correspond to the parameterization of measurements by the flat output. Therefore several learning algorithms (e.g. Levenberg-Marquardt, gradient descent with and without momentum, ADAM) and initialisation values of the weights are evaluated. It can be shown that it is not suitable to use a constant and overall learning rate if parameters are not in similar domains. To overcome the necessity of using all flat outputs and their derivatives as inputs for the neural networks we integrate an algebraic derivative estimation into the net. Several models have been tested to show the potential of this approach, for example, a first order linear system up to a single track model of a vehicle. Finally the method has been compared to a model based machine learning linear regression approach.