On IDA-PBC with Maximum Energy Shapeability

TL;DR

This paper introduces the concept of maximum energy shapeability in IDA-PBC, enabling systematic control design by simplifying the PDE matching equations, and demonstrates its application on a magnetic levitation system.

Contribution

It defines maximum energy shapeability and shows how it simplifies IDA-PBC design, providing a new systematic approach and revealing that some existing methods implicitly use this concept.

Findings

Maximum energy shapeability allows systematic IDA-PBC design.

Sufficient conditions for maximum energy shapeability are provided.

Application demonstrated on a magnetic levitation system.

Abstract

Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a well-established stabilization technique for affine nonlinear systems. However, its application is generally hindered by the requirement of solving a set of partial differential equations (PDEs), i.e., the so-called matching equation. This paper introduces the notion of \emph{maximum energy shapeability} which describes the scenario that the homogeneous part of the matching equation admits $m$ independent solutions with $m$ the dimension of the control input. We demonstrate that the maximum energy shapeability enables a systematic procedure for the IDA-PBC design by transforming the matching equation into a set of easier-to-solve PDEs. Sufficient conditions for maximum energy shapeability are also provided. It is shown that some existing constructive IDA-PBC designs actually implicitly exploit the maximum energy shapeability. The proposed procedure for the IDA-PBC design is illustrated with the magnetic levitation system.