中文摘要：

能控性和能观性是控制系统的两个重要概念．利用非线性控制系统的微分几何理论，研究了双线性系统和它的延伸系统的能控性之间的关系以及它们的能观测性之间的关系．通过计算双线性系统的输入向量场争系统向量场之间的李括号，以及输出函数沿着这些向量场的李导数，证明了双线性系统和它的延伸系统的能控性是等价的，且它们的能观测性也是等价的．

英文摘要：

Controllability and observability are two important concepts of control systems. The relations between controllability and observability of bilinear systems and controllability and observability of prolongation system are studied by using differential geometry theory of nonlinear control systems. At the same time, the equivalence between controllability of bilinear systems and controllability of prolongation system is proved by discussed Lie bracket between input vector fields and systems vector fields. The equivalence between observability of bilinear systems and observability of prolongation system is proved by discussed the Lie derivative of output function along these vector fields.