Classifications of Single-input Lower Triangular Forms

The purpose of this paper is twofold. The first is to classify lower triangular forms such that, under any lower triangular coordinate transformation and any state feedback, the type a given lower triangular form belongs to is invariant. The second is to determine under what conditions a nonlinear system is equivalent to a specific type of lower triangular form. According to the least multi-indices and the largest 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. Each of the two classifications induces a classification of all systems that are equivalent to lower triangular forms. 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 presented to illustrate how to determine which type of lower triangular form a given nonlinear system is equivalent to without performing an equivalence transformation.