# Matching in the method of controlled Lagrangians and IDA-passivity based control

G. Blankenstein, Romeo Ortega, Arjan van der Schaft

Research output: Chapter in Book/Report/Conference proceedingChapterAcademicpeer-review

## Abstract

This paper reviews the method of controlled Lagrangians and the interconnection and damping assignment passivity based control (IDA-PBC)method. Both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an Euler-Lagrange system, respectively Hamiltonian system, by searching for a stabilizing structure preserving feedback law. The conditions under which two Euler-Lagrange or Hamiltonian systems are equivalent under feedback are called the matching conditions (consisting of a set of nonlinear PDEs). Both methods are applied to the general class of underactuated mechanical systems and it is shown that the IDA-PBC method contains the controlled Lagrangians method as a special case by choosing an appropriate closed-loop interconnection structure. Moreover, explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms. The $\lambda$-method as introduced in recent papers for the controlled Lagrangians method transforms the matching conditions into a set of linear PDEs. In this paper the method is extended, transforming the matching conditions obtained in the IDA-PBC method into a set of quasi-linear and linear PDEs.
Original language Undefined Nonlinear and Adaptive Control: Tools and Algorithms for the User A. Astolfi Singapore World Scientific 79-114 36 1-86094-617-8 Published - 2006

### Publication series

Name World Scientific 500-266

## Keywords

• METIS-237838
• EWI-8860
• IR-66801