Some new algebraic and geometric analysis for local stability crossing curves

Abstract The algebraic and geometric properties for the local stability crossing curves (SCCs) of systems with two free parameters have attracted considerable interest, but some fundamental issues still remain unsolved. In this paper, we will develop a systematic approach for addressing such SCCs. First, we will parametrize the local SCCs through proposing an algorithm, with which the parametrization in the general case can be obtained. It will be seen that the parametrization in the general case is subject to some systems of Puiseux series, a new notion introduced in this paper. With the systems of Puiseux series, we can further explore the algebraic as well as the geometric properties of local SCCs. Next, all possible cases regarding the systems of Puiseux series and the topological structure will be appropriately classified. The resultant topological classifications, which take into account the distribution of the characteristic roots (from the stability standpoint), are insightful for the stability and stabilization studies. Finally, the asymptotic behavior issue when the parameters move along any given curve is addressed. We will show that the asymptotic behavior, in this case, corresponds to a group of Puiseux series, from which the detailed information on the local root loci is available. Moreover, this approach helps to verify the approaches proposed previously, such that all the approaches presented in our paper can be cross-validated.