Discussion on: “Stabilization of the Experimental Cart-Pendulum System with Proven Domain of Attraction”

Abstract

The article presents an experimental implementation of an Immersion & Invariance (I&I), [2], controller design to an experimental cart-pendulum system. The I&I controller was developed in [1], and in the paper under discussion the domain of attraction is proven to be the open upper half plane. The article serves merely as an experimental proof of principle for the I&I method, where in addition to the I&I controller, extra control actions are taken in order to deal with practical issues such as friction, and the fact that the velocities are not measured. It is interesting to compare these results with another controller implementation on the same set up. In [5] an IDA-PBC (Interconnection and Damping Assignment Passivity Based Control) controller was developed and implemented on the same set up. Even though it is argued in the introduction of the article under discussion that the controller of [5] is rather complicated in comparison to the I&I control, the level of complexity or simplicity of the I&I controller given by expression (19) is not discussed later on in the article. For that, we can consider the IDA-PBC design of [5]. The controller expression is derived by construction of a newmass inertia matrix (Md), interconnection matrix (J2), and damping matrix (Kv), along with a new energy function (Hd), and the overall expression for the control law is obtained by substituting these new matrices and functions into the control law given by