Immersion and invariance orbital stabilization of underactuated mechanical systems with collocated pre-feedback

In this note we study the generation of attractive oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, i.e., a partial linearizing state feedback, and an immersion and invariance orbital stabilization controller. The first step is adopted to simplify analysis and design, however, bringing an additional difficulty that the model loses its Euler-Lagrange structure after the collocated pre-feedback. To address this, we propose a constructive solution to the orbital stabilization problem via a smooth controller in an analytic form, and the model class identified in the paper is characterized via some easily apriori verifiable assumptions on the inertia matrix and the potential energy function

Speed Observation and Position Feedback Stabilization of Partially Linearizable Mechanical Systems

The problems of speed observation and position feedback stabilization of mechanical systems are addressed in this paper. Our interest is centered on systems that can be rendered linear in the velocities via a (partial) change of coordinates. It is shown that the class is fully characterized by the solvability of a set of partial differential equations (PDEs) and strictly contains the class studied in the existing literature on linearization for speed observation or control. A reduced order globally exponentially stable observer, constructed using the immersion and invariance methodology, is proposed. The design requires the solution of another set of PDEs, which are shown to be solvable in several practical examples. It is also proven that the full order observer with dynamic scaling recently proposed by Karagiannis and Astolfi obviates the need to solve the latter PDEs. Finally, it is shown that the observer can be used in conjunction with an asymptotically stabilizing full state-feedback interconnection and damping assignment passivity-based controller preserving asymptotic stability.</p

Control of Underactuated Mechanical Systems:Observer Design and Position Feedback Stabilization

We identify a class of mechanical systems for which a globally exponentially stable reduced order observer can be designed. The class is characterized by (the solvability of) a set of partial differential equations and contains all systems that can be rendered linear in (the unmeasurable) momenta via a (partial) change of coordinates. It is shown that this class is larger than the one reported in the literature of observer design and linearization. We also prove that, under very weak assumptions, the observer can be used in conjunction with an asymptotically stabilizing full state-feedback Interconnection and Damping Assignment Passivity-Based Controller, preserving stability.Caveat Emptor: This paper is a shortened version of the technical note [1] which can be obtained upon request from the authors.</p

A simplified IDA-PBC design for underactuated mechanical systems with applications

We develop a method to simplify the partial differential equations (PDEs) associated to the potential energy for interconnection and damping assignment passivity based control (IDA-PBC) of a class of underactuated mechanical systems (UMSs). Solving the PDEs, also called the matching equations, is the main difficulty in the construction and application of the IDA-PBC. We propose a simplification to the potential energy PDEs through a particular parametrization of the closed-loop inertia matrix that appears as a coupling term with the inverse of the original inertia matrix. The parametrization accounts for kinetic energy shaping, which is then used to simplify the potential energy PDEs and their solution that is used for the potential energy shaping. This energy shaping procedure results in a closed-loop UMS with a modified energy function. This approach avoids the cancellation of nonlinearities, and extends the application of this method to a larger class of systems, including separable and non-separable port-controlled Hamiltonian (PCH) systems. Applications to the inertia wheel pendulum and the rotary inverted pendulum are presented, and some realistic simulations are presented which validate the proposed control design method and prove that global stabilization of these systems can be achieved. Experimental validation of the proposed method is demonstrated using a laboratory set-up of the rotary pendulum. The robustness of the closed-loop system with respect to external disturbances is also experimentally verifie

Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one

Published versio

Adaptive IDA-PBC for underactuated mechanical systems with constant disturbances

This work investigates the control of nonlinear underactuated mechanical systems with matched and unmatched constant disturbances. To this end, a new control strategy is proposed, which builds upon the interconnection‐and‐damping‐assignment passivity‐based control, augmenting it with an additional term for the purpose of disturbance compensation. In particular, the disturbances are estimated adaptively and then accounted for in the control law employing a new matching condition of algebraic nature. Stability conditions are discussed, and for comparison purposes, an alternative controller based on partial feedback linearization is presented. The effectiveness of the proposed approach is demonstrated with numerical simulations for three motivating examples: the inertia wheel pendulum, the disk‐on‐disk system, and the pendulum‐on‐cart system

Energy-Based Control for the Cart-Pole System in Implicit Port-Hamiltonian Representation

This master thesis is devoted to the design, analysis, and experimental validation of an energy-based control strategy for the well-known benchmark cart-pole system in implicit Port-Hamiltonian (PH) representation. The control scheme performs two tasks: swingup and (local) stabilization. The swing-up controller is carried out on the basis of a generalized energy function and consists of bringing the pendulum trajectories from the lower (stable) position to a limit cycle (homoclinic orbit), which passes by the upright (unstable) position, as well as the cart trajectories to the desired point. The (local) stabilizing controller is designed under a novel algebraic Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) technique and ensures the upright (asymptotic) stabilization of the pendulum as well as the cart at a desired position. To illustrate the effectiveness of the proposed control scheme, this work presents simulations and real-time experiments considering physical damping, i.e., viscous friction. The results are additionally contrasted with another energy-based control strategy for the cart-pole system in explicit Euler-Lagrange (EL) representation.Diese Masterarbeit widmet sich dem Entwurf, der Analyse und der experimentellen Validierung einer energiebasierten Regelstrategie für das bekannte Benchmarksystem des inversen Pendels auf einem Wagen in impliziter Port-Hamiltonscher (PH) Darstellung. Das Regelungssystem erfüllt zwei Aufgaben: das Aufschwingen und (lokale) Stabilisierung. Das Aufschwingen erfolgt auf Grundlage der generalisierten Energiefunktion und besteht darin, sowohl die Trajektorien des Pendels von der unteren (stabilen) Position in einen Grenzzyklus (homokliner Orbit) zu bringen, wobei die (instabile) aufrechte Lage passiert wird, als auch den Wagen in einer gewünschten Position einzustellen. Die (lokale) Regelung zur Stabilisierung ist nach einer neuartigen algebraischen Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) Methode konzipiert und gewährleistet die aufrechte (asymptotische) Stabilisierung des Pendels sowie die Positionierung des Wagens an einem gewünschten Referenzpunkt. Um die Funktionalität des entworfenen Regelungssystems zu veranschaulichen, werden in dieser Masterarbeit Simulationen und Echtzeit-Experimente unter Berücksichtigung der physikalischen Dämpfung, d.h. der viskosen Reibung, vorgestellt. Die Ergebnisse werden zusätzlich mit einem weiteren energiebasierten Regelungsansatz für das System des inversen Pendels auf einem Wagen in expliziter Euler-Lagrange (EL) Darstellung verglichen.Tesi