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

Abstract

Please check your proof carefully and mark all corrections at the appropriate place in the proof (e.g., by using on-screen annotation in the PDF file) or compile them in a separate list. Note: if you opt to annotate the file with software other than Adobe Reader then please also highlight the appropriate place in the PDF file. To ensure fast publication of your paper please return your corrections within 48 hours.For correction or revision of any artwork, please consult http://www.elsevier.com/artworkinstructions.Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by flags in the proof. Click on the Q link to go to the location in the proof.Your article is registered as a regular item and is being processed for inclusion in a regular issue of the journal. If this is NOT correct and your article belongs to a Special Issue/Collection please contact p.atterbury@elsevier.com immediately prior to returning your corrections. Location in articleQuery / Remark: click on the Q link to go Please insert your reply or correction at the corresponding line in the proof Q1Please confirm that given names and surnames have been identified correctly and are presented in the desired order. Q2Corresponding author has not been indicated in the author group. Kindly provide the corresponding author's footnote. Q3Please provide the city of publication for Refs. [10,12,15].Thank you for your assistance. 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 portcontrolled 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 verified.

IDA-PBC with adaptive friction compensation for underactuated mechanical systems

“…The effect of viscous friction was studied within the CL approach by (Chang, 2010;Woolsey et al, 2004), while continuous and linear friction forces were considered in IDA-PBC by (Delgado & Kotyczka, 2014;Gómez-Estern & Van der Schaft, 2004). Furthermore, the compensation of nonlinear and discontinuous friction forces on actuated joints was investigated in (Ryalat & Laila, 2016;Sandoval, Kelly, & Santibáñez, 2011;Zhang & Liu, 2016). Beyond energyshaping control, a technical solution for dry friction compensation on both actuated and unactuated joints based on sliding-mode-control was proposed in (Martinez, Alvarez, & Orlov, 2008) considering the case of small displacements and under specific assumptions on the magnitude of the friction forces.…”

Self-generated limit cycle tracking of the underactuated inertia wheel inverted pendulum under IDA-PBC

“…Moreover, in [35], Khraief et al enhanced the IDA-PBC by combining it with an adaptive control technique to estimate controller gains. Ryalat and Laila [36] used a simplified IDA-PBC by considering friction parameters. They estimated the friction parameters in order to calculate the friction compensation term.…”

Robust dynamic state feedback for underactuated systems with linearly parameterized disturbances

“…As one of its main drawbacks, IDA‐PBC requires solving a set of partial differential equations (PDE), termed matching equations, which in general can be a challenging task. Considerable effort has been directed toward addressing this issue and notable results include the following works: a detailed analysis of the matching equations in Reference ; a new parameterization leading to a simplification of the PDE for a class of mechanical systems in Reference ; sufficient conditions for PDE solvability in Reference . More recently, in Reference the closed‐loop energy was permitted to depend on the control input warranting more flexibility in the definition of the PDE, while in Reference the PDE were avoided altogether introducing the notion of algebraic solution.…”