A Convexity Embedded Necessary and Sufficient Condition for the Hurwitz stability of a Real Matrix using its elemental sign structure and qualitative determinant concept

Abstract

In this paper, we highlight the importance of the ‘elemental sign structure’ of a matrix in its Hurwitz stability/instability assessment. The issue of determining conditions for the Hurwitz stability of a matrix is an age old topic of interest to engineers working with continuous time linear state space systems. However, the reason for revisiting this old issue is that the currently available conditions of stability, namely the Routh-Hurwitz test and the associated Kronecker Lyapunov matrix characteristic equation test (Fuller's condition) do not possess convexity property with respect to stability. In this paper, we present an alternative to these famous conditions by making use of the sign structure of the matrix. The proposed new condition of stability combines the concepts of both Quantitative Determinant (involving eigenvalues) and the Qualitative Determinant (involving only the sign information of the matrix elements) and serves as an alternative way of assessing the stability of a matrix. This new necessary and sufficient condition with convexity property is deemed extremely helpful in solving the related problems of stability including the testing of robust stability of interval parameter matrix families, which has attracted intense attention and scrutiny in the last few decades. The implications and importance of this new condition with convexity property are discussed.