# Orbital stabilization of nonlinear systems via Mexican sombrero energy shaping and pumping-and-damping injection

## TL;DR

This paper extends passivity-based control methods to achieve orbital stabilization of nonlinear systems by shaping energy functions and using adaptive damping, demonstrated on induction motors.

## Contribution

It introduces a novel modification to energy shaping and damping injection techniques for orbital stabilization of nonlinear systems.

## Key findings

- Successfully stabilizes nonlinear systems on desired orbits.
- Applies the method to induction motor control, matching industry standards.
- Demonstrates effectiveness through simulation examples.

## Abstract

In this paper we show that a slight modification to the widely popular interconnection and damping assignment passivity-based control method---originally proposed for stabilization of equilibria of nonlinear systems---allows us to provide a solution to the more challenging orbital stabilization problem. Two different, though related, ways how this procedure can be applied are proposed. First, the assignment of an energy function that has a minimum in a closed curve, i.e., with the shape of a Mexican sombrero. Second, the use of a damping matrix that changes "sign" according to the position of the state trajectory relative to the desired orbit, that is, pumping or dissipating energy. The proposed methodologies are illustrated with the example of the induction motor and prove that it yields the industry standard field oriented control.