Advanced stability assessment for control systems: Gelfand’s method with iterated shanks transformation

In this paper, we present an efficient method for evaluating the stability of both linear time-invariant (LTI) systems and continuous-time nonlinear systems. Our approach centres around the modulus of the spectral radius derived from a specific matrix transformation of the original system state matrix in the case of LTI systems, or the Jacobian of the nonlinear system. We enhance computational speed by employing the iterated Shanks transformation on Gelfand's formula, eliminating the need to rely on the system's characteristic equation. Notably, our stability criterion avoids the computation of all eigenvalues of the system matrix. Illustrative examples, including a fluidised bed reactor, a continuous stirred tank reactor and a hypothetical 20th-order dynamical system, are provided. Additionally, we conduct a CPU-time analysis, revealing that our method demonstrates several orders of magnitude faster computational speed compared to the classical Routh-Hurwitz stability test for systems ranging from two to 80 dimensions.