输入矩阵定常型切换线性系统的能控性

Controllability of Switched Linear System with Constant Input Matrix

重新考察一类切换线性系统的能控性,这类系统具有不同的系统矩阵和相同的输入矩阵.研究发现文献导出的必要充分条件为假命题,该判据只适用于2阶系统,对于3阶及以上系统其必要性不一定成立.并证明第一不变子空间只是能控状态集的真子集,而非其全体.最后通过1个反例对结论予以验证.

The paper presents a reexamination of the controllability of a class of switched linear systems with different systematic matrices but the same input matrix. Investigations reveal that the necessary and sufficient condition derived in the former literatures is a false proposition, which only holds for the second-order systems, while the necessity is not always true for systems of the third or a higher order. The first invariant subspace proves to be a proper subset of controllable state set rather than the whole of it. Finally, a counterexample is presented to illustrate the conclusion.