Stability Properties of Coupled Impedance Passive LTI Systems

Abstract

We study the stability of the feedback interconnection of two impedance passive linear time-invariant systems, of which one is finite dimensional. The closed-loop system is well known to be impedance passive, but no stability properties follow from this alone. We are interested in two main issues: 1) the strong stability of the operator semigroup associated with the closed-loop system, 2) the input–output stability (meaning transfer function in $H^\infty$ ) of the closed-loop system. Our results are illustrated with the system obtained from the nonuniform SCOLE (NASA Spacecraft Control Laboratory Experiment) model representing a vertical beam clamped at the bottom, with a rigid body having a large mass on top, connected with a trolley mounted on top of the rigid body, via a spring and a damper. Such an arrangement, called a tuned mass damper (TMD), is used to stabilize tall buildings. We show that the SCOLE-TMD system is strongly stable on the energy state space and that the system is input–output stable from the horizontal force input to the horizontal velocity output.