On the classification and canonical forms of nonlinear controllable systems

A differential geometrical approach to classification of nonlinear systems to be controlled is proposed. For systems of the form $\mathbf y=\mathbf f_0(\mathbf y)+\sum_{\alpha=1}^r\mathbf f_\alpha(\mathbf y)\mathbf u^\alpha$, $\mathbf y\in R^n$, $\mathbf u\in R^r$, results in reducing the problem to classification of Pfaff equation sets which result from system equations when the variables $\mathbf u$ are eliminated. Canonical forms are given for two cases: 1) $\operatorname{rank}\|f_\alpha^i(\mathbf y)\|_{\alpha=1,\dots,r}^{i=1,\dots,n}=n-1$, 2) $\operatorname{rank}\|f_\alpha^i(\mathbf y)\|_{\alpha=1,\dots,r}^{i=1,\dots,n}= \operatorname{rank}\|f_\alpha^i(\mathbf y)\|_{\alpha=0,1,\dots,r}^{i=1,\dots,n}$, $n\leqslant4$. The number of canonical forms in finite in these cases.