Quasi-poles of linear time-varying systems in an intrinsic algebraic approach

In a previous piece of work it has been shown that the exponential stability of a linear time-varying (LTV) system can be evaluated using new definitions of the poles of such a system. The latter are given by a fundamental set of roots of the skew polynomial P(∂) which defines the autonomous part of the system. Such a set may not exist over the initial field K of definition of the coefficients of the system, but can exist over a suitable field extension K̃⊃K. It is shown here that conditions for stability can also be obtained using linear factors of the polynomial P(∂) over another field extension Ǩ which may be smaller: K̃⊃Ǩ⊃K. The roots of these factors are called the quasi-poles of the system. The necessary condition for system stability, expressed in function of these quasi-poles, is more restrictive than the one involving a fundamental set of roots.