  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.
