On the Use of Resultant-Based Frequency-Sweeping Approaches to Construct Stability Charts for Systems With Two Delays

The resultant-based frequency-sweeping approach is appealing for creating a stability map to study stability robustness against delay uncertainties. It relies on converting the system’s transcendental characteristic function into a polynomial form such that the crossing frequency set and the corresponding stability switching curves in the delay parameter plane can be exactly and exhaustively determined using only algebraic computations. In the existing literature, the equation conversion has been achieved by using Rekasius substitution, bilinear substitution, or half-angle tangent substitution, and variable eliminations are carried out by using Sylvester resultant or Dixon resultant. In this paper, we first point out that since the computation of Dixon resultant involves a polynomial division, it could be beneficial in a symbolic programming environment. Then, we provide numerical evidence to compare the computational complexity of using different equation conversion methods and the subsequent resultant elimination to obtain the critical crossing frequencies from which the exact crossing frequency set can be determined. Next, based on the resultant theory, we reveal the roles of the critical crossing frequencies that play in the characterization of the delay map. Finally, we provide an efficient procedure for constructing stability charts in the delay parameter plane for linear time-invariant systems with two delays in states. To demonstrate our contributions, explanatory examples are provided along with a complete illustrative example.