In this paper, we consider the application of a new formulation of passivity-based control (PBC), known as interconnection and damping (IDA) assignment to the problem of stabilization of underactuated mechanical systems, which requires the modification of both the potential and the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yields a smooth asymptotically stabilizing controller with a guaranteed domain of attraction. The class is given in terms of solvability of certain partial differential equations. One important feature of IDA-PBC, stemming from its Hamiltonian (as opposed to the more classical Lagrangian) formulation, is that it provides new degrees of freedom for the solution of these equations. Using this additional freedom, we are able to show that the method of "controlled Lagrangians"-in its original formulation-may be viewed as a special case of our approach. As illustrations we design asymptotically stabilizing IDA-PBCs for the classical ball and beam system and a novel inertia wheel pendulum. For the former, we prove that for all initial conditions (except a set of zero measure) we drive the beam to the right orientation. Also, we define a domain of attraction for the zero equilibrium that ensures that the ball remains within the bar. For the inertia wheel, we prove that it is possible to swing up and balance the pendulum without switching between separately derived swing up and balance controllers and without measurement of velocities.

This paper discusses the matching conditions resulting from the controlled Lagrangians method 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, respectively Hamiltonian, system. In the context of mechanical systems with symmetry, the original controlled Lagrangians method is reviewed, and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given. Secondly, both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled Lagrangians method is contained in the IDA-PBC method. The ¶-method as described in recent papers for the controlled Lagrangians method, transforming the matching conditions (a set of non-linear PDEs) into a set of linear PDEs, is discussed. The method is used to transform the matching conditions obtained in the IDA-PBC method into a set of quadratic and linear PDEs. Finally, the extra freedom obtained in the IDA-PBC method (with respect to the controlled Lagrangians method) is used to discuss the integrability of the closed-loop system. Explicit conditions are derived under which the closed-loop Hamiltonian system is integrable, leading to the introduction of gyroscopic terms.

In this paper we study the notion of symmetry for implicit generalized Hamiltonian systems, which are Hamiltonian systems with respect to a generalized Dirac structure. We investigate the reduction of these systems admitting a symmetry Lie group with corresponding conserved quantities. Main features in this approach concern the projection and restriction of Dirac structures, generalizing the corresponding theory for symplectic forms and Poisson brackets. The results are applied to the theory of symmetries and reduction in nonholonomically constrained mechanical systems. The main result extends the reduction theory for explicit Hamiltonian systems and constrained mechanical systems to a general unified reduction theory for implicit generalized Hamiltonian systems.

This paper develops the reduction theory of implicit Hamiltonian systems admitting a symmetry group at a singular value of the momentum map. The results naturally extend those known for (explicit) Hamiltonian systems described by Poisson brackets.

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