Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning
This work addresses a specific problem in control theory for nonlinear systems, offering an incremental improvement by combining physics-informed machine learning with continuation techniques to handle challenging gradients.
The paper tackles the problem of feedback linearization for nonlinear discrete-time dynamical systems by proposing a physics-informed machine learning (PIML) scheme that finds the nonlinear transformation law in one step, ensuring stability via pole placement, and shows it outperforms traditional numerical methods in terms of numerical approximation accuracy on a benchmark example with steep gradients.
We present a physics-informed machine learning (PIML) scheme for the feedback linearization of nonlinear discrete-time dynamical systems. The PIML finds the nonlinear transformation law, thus ensuring stability via pole placement, in one step. In order to facilitate convergence in the presence of steep gradients in the nonlinear transformation law, we address a greedy-wise training procedure. We assess the performance of the proposed PIML approach via a benchmark nonlinear discrete map for which the feedback linearization transformation law can be derived analytically; the example is characterized by steep gradients, due to the presence of singularities, in the domain of interest. We show that the proposed PIML outperforms, in terms of numerical approximation accuracy, the traditional numerical implementation, which involves the construction--and the solution in terms of the coefficients of a power-series expansion--of a system of homological equations as well as the implementation of the PIML in the entire domain, thus highlighting the importance of continuation techniques in the training procedure of PIML.