A Jacobian‐based frequency‐sweeping approach for constructing stability crossing curves of systems with two delays

Abstract This paper introduces a novel Jacobian‐based frequency‐sweeping approach for constructing a stability map on a delay plane for linear time‐invariant (LTI) systems with two delays. It is demonstrated that by imposing a zero Jacobian condition on the Rekasius‐transformed polynomial equations defining the stability crossing curves (SCCs), the set of stability crossing frequencies (SCFs) can be determined exactly and exhaustively. Compared to existing frameworks that use a discrimination criterion to identify the upper bound of SCFs, our proposed framework offers the following three merits: (i) it establishes a direct link between the zero Jacobian criterion and the end points of the SCF intervals; (ii) it reveals the invariance property of the curve‐wise crossing direction for each SCC; (iii) it provides a simple Jacobian calculation to determine the root crossing directions relative to the imaginary axis. These merits significantly enhance the cluster treatment of the characteristic root (CTCR) paradigm for investigating stability robustness against delay uncertainties. A computational procedure is provided for the implementation of the proposed framework. Additionally, the role of Jacobian in constructing stability crossing curves is demonstrated through two examples.