An efficient stability criterion for process control systems based on numerical computation of matrix square root with predefined accuracy

In this paper, we present a rigorous criterion for bounded input, bounded output (BIBO) stability of linear time‐invariant dynamical systems and for local stability analysis of nonlinear continuous‐time systems. Our approach involves computing the principal matrix square root of a relevant matrix from an improper matrix‐valued integral to ultimately determine the matrix sign of the state matrix. It allows the computation of the principal matrix square root, and consequently the matrix sign of the state matrix, with a pre‐specified error margin. By exploiting the concept of eigenvalue separation, we can assess the stability of the system without the need for eigendecomposition. Our method is easily extended to nonlinear systems using the indirect Lyapunov method. We provide proofs of convergence and reliable error estimates for the formulas involved. The effectiveness of the method is demonstrated through several case studies from chemical engineering, including a single input, single output (SISO) continuous stirred tank reactor (CSTR) model and a multiple input, multiple output (MIMO) system of two CSTRs in series. Computational performance analysis shows that our method is extremely superior to Routh's array test, achieving an average speed improvement of about 90 times for systems with state variable numbers ranging from 10 to 1035, based on CPU time analyses.