Maximal feedback linearization and its internal dynamics with applications to mechanical systems on

Abstract

Summary For systems that are not feedback linearizable, a natural question is: how to find the largest feedback linearizable subsystem and, if the partial linearization is not unique, what are the control‐theoretic properties of various partial linearizations. In this paper, we will consider the problem of how to choose a partially linearizing output that renders the zero dynamics asymptotically stable and when such an output exists. We will state general results solving completely the problem for systems whose linearizability defect is one by identifying and describing two classes of systems. For the first class, all maximal partial linearizations lead to the same zero dynamics. For the second class, any asymptotic behavior of the zero dynamics can be achieved by a suitable choice of a partially linearizing output. In the second part, we apply our results to mechanical systems with two‐degrees‐of‐freedom and provide a detailed study of their partial linearizations. We illustrate the obtained results by examples of Acrobot (which belongs to the second class) and Pendubot (which belongs to the first class).

Evaluation of Linearization Methods for Control of the Pendubot

“…Recently, in [15] a new insight to control of under-actuated systems is proposed. The presented approach is based on finding the largest feedback linearizable subsystem for mechanical systems that are not fully feedback linearizable.…”

“…An important issue of the linearization approach using state and feedback transformations discussed in [15] is the existence of singular points. Due to this obstruction, a feasible state space is constrained and a feedback cannot be designed in a global way.…”

Anti‐Swing control of the Pendubot using damper and spring with positive or negative stiffness

“…The Pendubot is a two‐link planar robot with a single actuator at the base joint of the first link, and the joint between two links is unactuated and allowed to swing freely 1,2 . It, together with other mechanical systems such as the inverted pendulum on a cart 3 and the Acrobot, 4 is used for control and robot education and for research as one of typical examples of underactuated mechanical systems 5,6 …”

Robust global stabilization of a class of underactuated mechanical systems of two degrees of freedom

“…Another alternative to control a class of these underactuated mechanical systems of two DoF is to find a partially linearizing output for which the zero dynamics is asymptotically stable, 15 or that can be asymptotically stabilized using a virtual control 16 . In these cases, it is possible to obtain global asymptotic stabilization.…”

“…In these cases, it is possible to obtain global asymptotic stabilization. In these papers, 15,16 subactuated mechanical systems are classified according to the structure of their normal forms. The control design is then proposed corresponding to the class they belong to.…”