Classifications of Single-input Lower Triangular Forms

TL;DR

This paper introduces new classification schemes for lower triangular forms of nonlinear systems, establishing invariance properties and providing methods to determine system equivalence to these forms using differential geometric control theory.

Contribution

It develops two novel classification schemes based on multi-indices, proves invariance under coordinate transformations, and offers a new method to identify the form type of nonlinear systems.

Findings

Classification schemes are invariant under transformations.

A new method to determine the form type of a system.

Necessary and sufficient conditions for feedback equivalence.

Abstract

The purposes of this paper are to classify lower triangular forms and to determine under what conditions a nonlinear system is equivalent to a specific type of lower triangular forms. According to the least multi-indices and the greatest essential multi-index sets, which are introduced as new notions and can be obtained from the system equations, two classification schemes of lower triangular forms are constructed. It is verified that the type that a given lower triangular form belongs to is invariant under any lower triangular coordinate transformation. Therefore, although a nonlinear system equivalent to a lower triangular form is also equivalent to many other appropriate lower triangular forms, there is only one type that the system can be transformed into. Each of the two classifications induces a classification of all the systems that are equivalent to lower triangular forms. A new method for transforming a nonlinear system into a lower triangular form, if it is possible, is provided to find what type the system belongs to. Additionally, by using the differential geometric control theory, several necessary and sufficient conditions under which a nonlinear system is locally feedback equivalent to a given type of lower triangular form are established. An example is given to illustrate how to determine which type of lower triangular form a given nonlinear system is equivalent to without performing an equivalent transformation.