Strong Stability of a Coupled System Composed of Impedance-Passive Linear Systems Which May Both Have Imaginary Eigenvalues

Abstract

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 $GGG$, 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 $o_j$, where $jin{1, ldots n}$, which means that its transfer function $gggg$ has poles at the points $io_j$. We also assume that $gggg$ has a feedthrough term $d$ with $Re d>0$. We assume that $ReGGG(io_j)>0$ for all $jin{1,ldots n}$ and the points $io_j$ are not eigenvalues of $A$. We can show that the closed-loop system is well-posed and input-output stable (in particular, $(I+ggggGGG)^{-1}in H^infty$ and also $GGG(I+ggggGGG)^{-1}in H^infty$). 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.