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

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

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

D-decomposition method for stabilization of inverted pendulum using fractional order PD controller

This paper deals with stability problem of inverted pendulum controlled by a fractional order PD controller. Ddecomposition method for determining stability region in controller parameters space is hereby presented. The D decomposition problem for linear systems is extended for linear fractional systems and for the case of linear parameters dependence. Knowledge of stability regions enables tuning of the fractional order PD controller

Design and Implementation of a Furuta Pendulum Device for Benchmarking Non-Linear Control Methods

Furuta pendulum is an academic benchmark example for evaluating non-linear control algorithms. The main aim of this dissertation is to study this physical system, showing its dynamic model and several strategies for its control. An assortment of swing-up and upright control approaches is reported with its design and simulations. Besides, this document describes the project development which is being done at the FabLab of the Obuda University, whose objective is to design and manufacture a demonstration device that is capable to test and display various control strategies. Requirements and specications of the design, used tools and future work are described. The dissertation is structured in eight different chapters: (1) History of the Furuta pendulum, describing the origin of this system; (2) State-of-the-art in non-linear control, giving a background for the different control strategies; (3) Dynamic model of the Furuta pendulum; (4) Swing-up by energy control, based on the work of Astrom and Furuta; (5) Stabilizing local control, via full state feedback; (6) Hybrid control, which sums up the previous approaches; (7) Development project, which describes the work realized in the FabLab and (8) Conclusion, discussing the knowledge extracted from the development of this thesis