Numerical Gridding Stability Charts Estimation using Quasi-polynomial Approximation for TDS

The aim of this study is to present and summarize our numerical algorithm for the determination of stability charts in the delay space for linear time-invariant time systems with constant delays (TDS), both retarded and neutral ones. The core of algorithm lies in a successive (iterative) approximation of the infinite-dimensional characteristic quasi-polynomial in each grid node of the delay space. This approximation resulting in a polynomial or an exponential polynomial with commensurate delays is made in the neighborhood of the dominant characteristic value (pole) that has recently been estimated in the closest grid node. Two different approximation techniques are presented; namely, continuous-time and discrete-time ones. A complete numerical example for retarded TDS is presented, whereas the approximation issues are highlighted in another example for neutral TDS.