A Physics-Informed Neural Networks based method for Interconnection and Damping Assignment Passivity-Based Control

The Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is an advanced nonlinear control strategy founded on the passivity and stability principles of port-Hamiltonian (PH) systems. This approach provides a powerful tool for controlling underactuated nonlinear systems by assigning them a desired energetic structure, establishing a new equilibrium point and achieving asymptotic stability through energy dissipation. The main challenge in this context lies in solving complex, nonlinear Partial Differential Equations (PDEs) that arise from the desired energy shaping process. This challenge is often mitigated by identifying classes of systems for which these equations are solvable. To address these limitations, a Physics-Informed Neural Networks (PINNs) based approach for solving the nonlinear PDE related to the kinetic energy and the definition of a customized loss function is proposed. Furthermore, it is shown that, assuming the kinetic energy PDE is solved with minimal error, stabilization can be achieved through appropriate tuning of the damping matrix and by adopting a simple structure for the potential energy function, eliminating the need to solve the potential energy PDE. The proposed methodology is verified through numerical simulations.