Immersion and Invariance Stabilization of Nonlinear Systems Via Virtual and Horizontal Contraction

Abstract

The main objective of this paper is to revisit one of the key steps of immersion and invariance stabilizing controller design. Namely, the one that ensures attractivity of the manifold whose internal dynamics contains a copy of the desired system behavior. Towards this end we invoke contraction theory principles and propose two alternative procedures to carry out this step: (i) to replace attractivity of the manifold by virtual contraction of the off-themanifold coordinate and (ii) to ensure the attractivity of the manifold rendering it horizontally contractive. This makes more systematic the design with more explicit degrees of freedom to accomplish the task. Several examples, including the classical case of systems in feedback form, are used to illustrate the proposed design.

Orbital stabilization of nonlinear systems via the immersion and invariance technique

“…Remark As indicated in Reference , the constraint condition is absent in (Theorem 2.1). In addition, to reduce the number of mappings to be found, we have expressed the FBI Equation projecting it into the null space of the input matrix g ( x ).…”

“… In addition, to reduce the number of mappings to be found, we have expressed the FBI Equation projecting it into the null space of the input matrix g ( x ). As shown in (Propositions 2 and 3], the stability condition A4 can be replaced by a contraction condition.…”

“…Although, the second step of the design involves the solution of a partial differential equation—corresponding to the Francis‐Byrnes‐Isidori (FBI) equations—the third step is a stabilization problem where it is desired to drive to zero the rest of the state, that is, the off‐the‐manifold coordinates, while preserving bounded trajectories. As shown in Reference , this latter step can also be translated into a contraction problem.…”

Contraction Analysis of Monotone Systems via Separable Functions

“…The concept of partial stability [43] has been extended towards contraction analysis in the notions of partial contraction [33], [34] or horizontal contraction [17]. For instance, partial contraction is applied to observer design [19], [21], [44] and tracking control [45], [46]. Propositions 4.5 and 4.6 can be extended for partial contraction analysis.…”

Voice control in mixed reality

“…It is only necessary to turn off the camera and set up goggles to be able to move into a fully artificially created world. Studies on immersion are dispelled in increasing the feeling of detachment from the environment [5]. Enabling and capturing movements is quite a significant problem.…”