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Total energy shaping control of mechanical systems: simplifying the matching equations via coordinate changes. (English) Zbl 1136.93020

Bullo, Francesco (ed.) et al., Lagrangian and Hamiltonian methods for nonlinear control 2006. Proceedings from the 3rd IFAC workshop, Nagoya, Japan, July 19–21, 2006. Berlin: Springer (ISBN 978-3-540-73889-3/pbk). Lecture Notes in Control and Information Sciences 366, 147-156 (2007).
Summary: Total energy shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function – that qualifies as Lyapunov function for the desired equilibrium. The success of the method relies on the possibility of solving two PDEs which identify the kinetic and potential energy functions that can be assigned to the closed-loop. Particularly troublesome is the PDE associated to the kinetic energy which is nonlinear and non-homogeneous and the solution, that defines the desired inertia matrix, must be positive definite. In this paper we prove that we can eliminate or simplify the forcing term in this PDE modifying the target dynamics and introducing a change of coordinates in the original system. Furthermore, it is shown that, in the particular case of transformation to the Lagrangian coordinates, the possibility of simplifying the PDEs is determined by the interaction between the Coriolis and centrifugal forces and the actuation structure. The example of a pendulum on a cart is used to illustrate the results.
For the entire collection see [Zbl 1119.93006].

MSC:

93B51 Design techniques (robust design, computer-aided design, etc.)
93D20 Asymptotic stability in control theory
93D30 Lyapunov and storage functions
70Q05 Control of mechanical systems
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