An intrinsic algebraic setting for poles and zeros of linear time-varying systems

Independent roots of a polynomial Independent solutions of differential equations Galois/Picard-Vessiot extensions a b s t r a c t In this paper, poles and zeros are defined for linear time-varying systems using suitable ground field extensions. The definitions of the system poles, transmission poles, invariant zeros, hidden modes, etc, are given in an intrinsic module-based framework and are consistent in the sense that the poles are connected to the stability of the system and the zeros to the zeroing of the output for non zero inputs. In particular, it is proved that the necessary and sufficient condition for a continuous-time system to be exponentially stable is similar to the well-known condition in the time-invariant case.