On the maximum horizontal distance of a matrix’s eigenvalues from the imaginary axis: A new criterion for control system stability

This paper presents a rigorously proven theorem for determining the maximum horizontal distance of a matrix’s eigenvalues from the imaginary axis, based on the matrix exponential function and spectral radius. By applying the theorem twice, we formulate an efficient criterion for assessing the Hurwitz stability of linear time-invariant (LTI) systems, without requiring the characteristic polynomial. The utility of the proposed method is demonstrated through detailed analysis of practical systems, including a sixth-order high-pass filter and a network of three interconnected buck-boost converters. Comparative robustness tests under perturbations show that the proposed method significantly outperforms the classical Routh-Hurwitz criterion, correctly classifying 96.8 % of stable systems (versus 44.9 %) while maintaining 100 % accuracy for unstable systems across 5000 randomly generated 50th-order systems. Furthermore, CPU time analysis indicates sublinear computational complexity, with empirical scaling of approximately O ( n 0.13 ) , in stark contrast to the polynomial growth observed in traditional methods. These results demonstrate the method’s potential for robust and highly efficient stability analysis, especially in high-order and real-time control systems.