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. ?

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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.

In this paper, we give a tutorial account of several of the most recent adaptive control results for rigid robot manipulators.Our intent is to lend some perspective to the growing list of adaptive control results for manipulators by providing a unified framework for comparison of those adaptive control algorithms which have been shown to be globally convergent. In most cases we are able to simplify the derivations and proofs of these results as well.

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Abstract-A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective-this is in contrast with most existing adaptive designs that (relying on certain matching conditions) treat these terms as disturbances to be rejected. It is interesting to note that our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.

Abstract-The dynamics of many physical processes can be suitably described by Port-Hamiltonian (PH) models, where the importance of the energy function, the interconnection pattern and the dissipation of the system is underscored. To regulate the behavior of PH systems it is natural to adopt a Passivity-Based Control (PBC) perspective, where the control objectives are achieved shaping the energy function and adding dissipation. In this paper we consider the PBC techniques of Control by Interconnection () and Standard PBC. In the controller is another PH system connected to the plant (through a power-preserving interconnection) to add up their energy functions, while in Standard PBC energy shaping is achieved via static state feedback. In spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems, the current version of imposes a severe restriction on the plant dissipation structure that stymies its practical application. On the other hand, Standard PBC, which is usually derived from a uninspiring and non-intuitive "passive output generation" viewpoint, is one of the most successful controller design techniques. The main objectives of this paper are: (1) To extend the method to make it more widely applicable-in particular, to overcome the aforementioned dissipation obstacle. (2) To show that various popular variants of Standard PBC can be derived proceeding from a unified perspective. (3) To establish the connections between and Standard PBC proving that the latter is obtained restricting the former to a suitable subset-providing a nice geometric interpretation to Standard PBC-and comparing the size of the set of PH plants for which they are applicable.