Abstract. For a polynomial linearly dependent on two parameters, several methods are proposed to approximate its stability region with respect to a given root localization region (also called a root clustering set in the literature). The first method is to apply a sufficiently uniform grid to the stability region boundary that ensures its complete coverage with a given accuracy. The second (semi-grid) method yields an internal approximation of the stability region using line segments or curve arcs bounded by the stability region. The third method is to cover the stability region boundary with simple sets (cells) in order to obtain piecewise linear internal and external approximations of the stability region. All methods are based on the constructive D-partition (constructive D-decomposition) method, which describes the stability region boundary as a set of line segments and rational curve arcs. The exact stability radius and its simple estimate are derived in the parameter plane. Implementation of all methods and algorithms is reduced to finding the real roots of polynomials.
Keywords: constructive D-partition, rational curves, approximation of the stability region, sufficiently uniform grid, grid methods, semi-grid methods, support function, stability radius.
Acknowledgments. The research presented in Sections 2 and 3 was supported by the Russian Science Foundation, project no. 21-71-30005-П, https://rscf.ru/project/21-71-30005/.
Constructive D-Partition for Two Parameters Entering a Polynomial Linearly. Part II: Approximation of Stability Regions and Robustness Analysis