Passivity-based control (PBC) is a well-established technique that has shown to be very powerful to design robust controllers for physical systems described by Euler-Lagrange (EL) equations of motion. For regulation problems of mechanical systems, which can be stabilized "shaping" only the potential energy, PBC preserves the EL structure and furthermore assigns a closed-loop energy function equal to the di erence between the energy of the system and the energy supplied by the controller. Thus, we say that stabilization is achieved via energy balancing. Unfortunately, these nice properties of EL-PBC are lost when used in other applications which require shaping of the total energy, for instance, in electrical or electromechanical systems, or even some underactuated mechanical devices. Our main objective in this paper is to develop a new PBC theory which extends to a broader class of systems the aforementioned energy-balancing stabilization mechanism and the structure invariance. Towards this end, we depart from the EL description of the systems and consider instead port-controlled Hamiltonian models, which result from the network modelling of energy-conserving lumped-parameter physical systems with independent storage elements, and strictly contain the class of EL models. ?

No abstract

No abstract

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.