Accelerated BIBO stability criterion for dynamical systems based on matrix hyperbolic tangent function - Fatoorehchi, Hooman; Bohr, Henrik G.; Fosbøl, Philip L. (International journal of dynamics and control, 2025)
In this paper, an efficient technique is developed to examine the bounded-input bounded-output (BIBO) stability of linear time-invariant (LTI) systems. Our method is based on eigenvalue separation, traditionally relying on the matrix sign function. To simplify the intricate calculations, we propose a theorem linking the matrix sign function to the matrix hyperbolic tangent function. This reformulation reduces complexity, requiring at most two matrix exponentials and a matrix inversion. Additionally, we enhance the computation of matrix exponentials using a scaling and squaring technique that features pre-adjusted accuracy up to any desired level. Unlike conventional approaches, our method does not require the characteristic polynomial of the system, avoiding numerical errors that can lead to a misrepresentation of the system’s stability. The proposed criterion approximately achieves a 77-fold reduction in CPU time for a system of order 5, with this improvement increasing progressively, reaching a 1465-fold reduction at order 200, as confirmed by our analysis. Based on an uncertain parameter test conducted on 1000 random systems of order 10 subject to ±20% perturbation, our criterion achieved a Matthews correlation coefficient of unity, whereas the Routh–Hurwitz method attained a value of 0.5827. Several case studies, including a binary distillation column from chemical engineering and an enhanced Sallen-Key filter system from electronics engineering, are presented to illustrate how the criterion can be applied in practice.