The aim of this paper is to define an extension of the controllability and observability for linear quaternion-valued systems (QVS). Some criteria for controllability and observability are derived, and the minimum norm control and duality theorem are also investigated. Compared with real-valued or complex-valued linear systems, it is shown that the classical Caylay-Hamilton Theorem as well as Popov-Belevitch-Hautus (PBH) type controllability and observability test do not hold for linear QVS. Hence, a modified PBH type necessary condition is studied for the controllability and observability, respectively. Finally, some examples are given to illustrate the effectiveness of the obtained results.

本文的目的是定义线性四元数值系统（QVS）的可控制性和可观察性的扩展。推导了一些可控性和可观察性的准则，并研究了最小范数控制和对偶定理。与实值或复值线性系统相比，表明经典的Caylay-Hamilton定理以及Popov-Belevitch-Hautus（PBH）类型的可控性和可观察性测试不适用于线性QVS。因此，针对可控性和可观察性分别研究了改进的PBH型必要条件。最后，通过一些例子说明了所得结果的有效性。