On the robust stability of some parameter-dependent linear systems: solutions via matrix pencil techniques

This note focuses on deriving stability conditions for a class of linear parameter-dependent systems in a state-space representation. More precisely, we will compute the set of parameters for which the characteristic roots are located on the imaginary axis, and next we will give the characterization of the way such critical roots are crossing the imaginary axis. The methodology considered makes use of the computation of the generalized eigenvalues of an appropriate matrix pencil combined with an operator perturbation approach for deriving the crossing direction. Finally, the particular case of parameter-dependent polynomials will be also considered, and the stability analysis of time-delay systems is also revisited in this perspective.