A matrix-exponential-based stability test for high-dimensional dynamical systems across low-to-high sparsity regimes

Abstract

A matrix-exponential-based criterion for stability analysis of continuous-time linear time-invariant (LTI) systems is presented. The proposed formulation establishes that a system matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{A}}$$\end{document} is Hurwitz stable if and only if the spectral radius of its matrix exponential satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho \left( {\exp _{{{\text{mat}}}} ({\mathbf{A}})} \right) < 1 $$\end{document}. Unlike classical polynomial-based approaches such as the Routh–Hurwitz criterion, the proposed method operates directly on the system matrix without explicit construction of the characteristic polynomial. The matrix exponential is approximated using a truncated Taylor expansion following normalization of the system matrix, which improves the convergence of the approximation while preserving the original stability classification. The spectral radius is then estimated using an iterative power method. The influence of the truncation order is discussed through theoretical error bounds. The applicability of the proposed criterion is demonstrated using two representative chemical engineering case studies, including a high-order absorption–desorption process and a nonlinear biotechnology fermentation network. In the latter case, the nonlinear model is linearized about an operating equilibrium, and local asymptotic stability is assessed using Lyapunov’s indirect method. Numerical experiments involving randomly generated LTI systems demonstrate favorable computational performance relative to the classical Routh–Hurwitz approach. This assessment is further complemented by a large-scale validation involving 160 randomly generated LTI systems with matrix dimensions up to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=4000$$\end{document}, which produced complete agreement with direct eigenvalue computation while maintaining competitive computational performance. Additional Monte Carlo studies on ill-conditioned and sparse nonnormal systems further demonstrate the numerical robustness of the proposed criterion under varying perturbation levels and sparsity ratios. Overall, the results indicate that the proposed matrix-exponential-based criterion provides a practical and numerically reliable alternative for stability assessment of high-order dynamical systems arising in chemical engineering and related applications.