Abstract EN:

This work considers the constructive control and observer design of underactuated mechanical systems exploiting the notions of passivity, immersion and invariance. The modelling framework that we adopt for our derivations is that of analytical mechanics with the dynamic equations of Euler-Lagrange and Hamilton. In the first part of this work the controlled Euler-Lagrange equations and the controlled Hamiltonian equations are reviewed and some new conditions, necessary or sufficient, for the partial linearization of the dynamics either by a feedback or a coordinate transformation are presented. A complete characterization of the different classes of mechanical systems that satisfy these conditions follows. In the second part, we consider the stabilization problem using the novel approach of Immersion and Invariance (I&I) that relies on finding a manifold in state-space that can be rendered invariant and attractive, with internal (target) dynamics a copy of the desired closed-loop dynamics, and on designing a control law that steers the state of the system close to this manifold and we provide a constructive approach to obviate the solution of the partial differential equations deriving from the invariance condition. We also solve the stabilization problem of a new crane mechanism called 2D-SpiderCrane by the Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) technique. In the third part, the problem of observer design for a class of mechanical systems is solved constructively using I&I perspective, where now the manifold is described in a parameterized manner and the observer dynamics are suitably chosen to render the manifold invariant.