输入矩阵定常型切换线性系统的能控性探讨

Discussion on Controllability of Switched Linear System with Constant Input Matrix

摘要:

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：该判据只适用于二阶系统，对于三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．最后通过一个反例对结论予以验证．

Abstract:

This paper re-examines the controllability of a class of switched linear systems which has different systematic matrices and the same input matrix. Investigations reveal that the necessary and sufficient condition derived in literatures is a false proposition: it only holds for second-order systems, while the necessity is not always true for systems of third and higher order. We also prove that the first invariant subspace is a proper subset of controllable state set rather than the whole of it. Finally, a counterexample is presented to illustrate the conclusion.