Rank reduction for matrix pair and its application in singular systems

In this paper, the rank reduction problem for a rectangle matrix pair is investigated. First, the rank reduction problem is defined and it is solved via an algebraic approach. In fact, the proposed method is a procedure for getting the maximal value for the uncertain parameter such that the rank of the perturbed matrix will remain the same. Based on the results, the maximal robust stability radius problem of singular systems has been solved completely. Finally, three examples are used to illustrate the effectiveness of the proposed approach.