Strong stability of a coupled system composed of impedance-passive linear systems which may both have imaginary eigenvalues

摘要

We consider coupled systems consisting of a well-posed and impedance passive linear system (that may be infinite dimensional), with semigroup generator A and transfer function G, and an internal model controller (IMC), connected in feedback. The IMC is finite dimensional, minimal and impedance passive, and it is tuned to a finite set of known disturbance frequencies ω_j, where j E {1, ... n }, which means that its transfer function g has poles at the points iω_j. We also assume that g has a feedthrough term d with Re d > 0. We assume that Re G(iω_j) > 0 for all j ϵ {1, ... n} and the points iω_j are not eigenvalues of A. We can show that the closed-loop system is well-posed and input-output stable (in particular, (I + gG)^{-1 E H ∞ and also G(1 + gG)-l E H ∞). It is also easily seen that the closed-loop system is impedance passive. We show that if A has at most a countable set of imaginary eigenvalues, that are all observable, and A has no other imaginary spectrum, then the closed-loop system is strongly stable. This result is illustrated with a wind turbine tower model controlled by an IMC.}