Design and implementation of nonlinear and robust control for Hamiltonian systems: the passivity-based control approach

Abstract

Recently, control techniques that adopt the geometrical structure and physical properties of dynamical systems have gained a lot of interest. In this thesis, we address nonlinear and robust control problems for systems represented by port-controlled Hamiltonian (PCH) models using the interconnection and damping assignment passivity-based control(IDA-PBC) methodology, which is the most notable technique facilitating the PCH framework.

In this thesis, a novel constructive framework to simplify and solve the partial differential equations (PDEs) associated with IDA-PBC for a class of underactuated mechanical systems is presented. Our approach focuses on simplifying the potential energy PDEs to shape the potential energy function which is the most important procedure in the stabilization of mechanical systems. The simplification is achieved by parametrizing thedesired inertia matrix that shapes the kinetic energy function, thus achieving total energy shaping. The simplification removes some constraints (conditions and assumptions) that have been imposed in recently developed methods in literature, thus expanding the class of systems for which the methods can be applied including the separable PCH systems(systems with constant inertia matrix) and non-separable PCH systems (systems with non-constant inertia matrix). The results are illustrated through software simulations and hardware experiments on real engineering applications.

We also propose an integral control and adaptive control schemes to improve the robustness of the IDA-PBC method in presence of uncertainty. We first provide some results for the case of fully-actuated mechanical systems, and then extend those results to underactuated systems which are more complex. Integral action control on both the passive and non-passive outputs in the IDA-PBC construction, a strategy to ensure the robustness of the systems by preserving its stability in face of external disturbances, is introduced, establishing the input-to-state stability (ISS) property. The results are applied to both the separable and non-separable PCH systems and illustrated via several simulations. The extension to the non-separable case exhibits more complicated design as we need to take into account the derivative of the inertia matrix.

Finally, the IDA-PBC method is employed to solve an important nonlinear phenomenon called ‘pull-in’ instability associated with the electrostatically actuated microelectromechanical systems (MEMSs). The control construction is an output-feedback controller that ensures global asymptotic stability and avoids velocity measurement which may not be practically available. Furthermore, the integral, adaptive and ISS control schemes proposed in this thesis for mechanical systems are extended to facilitate the stabilization of electromechanical systems which exhibit strong coupling between different energy domains.

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