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The purposes of this paper are to classify lower triangular forms and todetermine under what conditions a nonlinear system is equivalent to a specifictype of lower triangular forms. According to the least multi-indices and thegreatest essential multi-index sets, which are introduced as new notions andcan be obtained from the system equations, two classification schemes of lowertriangular forms are constructed. It is verified that the type that a givenlower triangular form belongs to is invariant under any lower triangularcoordinate transformation. Therefore, although a nonlinear system equivalent toa lower triangular form is also equivalent to many other appropriate lowertriangular forms, there is only one type that the system can be transformedinto. Each of the two classifications induces a classification of all thesystems that are equivalent to lower triangular forms. A new method fortransforming a nonlinear system into a lower triangular form, if it ispossible, is provided to find what type the system belongs to. Additionally, byusing the differential geometric control theory, several necessary andsufficient conditions under which a nonlinear system is locally feedbackequivalent to a given type of lower triangular form are established. An exampleis given to illustrate how to determine which type of lower triangular form agiven nonlinear system is equivalent to without performing an equivalenttransformation.

Classifications of Single-input Lower Triangular Forms