An novel treatment of robust stability of continuous-time systems

The four-vertices concept of Kharitonov´s theorem is one of the most important results in the area of robust stability of linear time-invariant continuous-time systems. If the coefficients of the characteristic polynomial of a given system are dependent, the vertex concept cannot be applied in general. If the coefficients are linearly dependent, various results can be achieved. The case where the coefficients are partly dependent is considered, and more general vertex theorems on robust stability are given where the set considered in the coefficient space does not need to be convex. The notion of the vertex polynomial is generalized. The results are based on a modified Hermite-Biehler theorem, which presents an irredundant characterization of a reactance function. A new geometrical interpretation of stability conditions is also given. Further results are presented for low-order polynomials