Conditions for Hurwitz Stability/Instability of a Real Matrix via its Sign Pattern with a Necessary and Sufficient Condition for Magnitude Independent Stability

Abstract

In this paper, we address the issue of discerning the Hurwitz stability/instability of real matrix directly from its sign pattern.In that connection, in this paper, we first classify all sign patterns of matrices into three categories, namely i) Qualitative Sign Unstable (QLSU) matrices (i.e. matrices with a sign pattern that it is unstable for any magnitudes in the entries, that is, magnitude independent instability), then ii) Qualitative Sign Stable (QLSS) matrices (i.e. matrices with a sign pattern that it is stable for any magnitudes in the entries, that is, magnitude independent stability) and finally iii) MDSU matrices (which require magnitudes to determine its stability/instability). We then propose a necessary and sufficient condition for a sign matrix to be QLSS. The proposed necessary and sufficient condition is derived solely based on the nature (signs) of interactions and interconnections (i.e. based only on the signs of the entries of the matrix), borrowed from ecological principles. The proposed condition in this paper, serves as a better (being much simpler) alternative to the necessary and sufficient condition for QLSS matrices that is available in the ecology literature which uses a complicated test labeled ‘the Color test’. Identifying QLSS/MDSU/QLSU sign structures in a necessary and sufficient way has significant implications in many engineering systems whose dynamics are described by linear state space representation.