The converse of the passivity and small-gain theorems for input-output maps

We prove the following converse of the passivity theorem. Consider a causal system given by a sum of a linear time-invariant and a passive linear time-varying input-output map. Then, in order to guarantee stability (in the sense of finite L2-gain) of the feedback interconnection of the system with an arbitrary nonlinear output strictly passive system, the given system must itself be output strictly passive. The proof is based on the S-procedure lossless theorem. We discuss the importance of this result for the control of systems interacting with an output strictly passive, but otherwise completely unknown, environment. Similarly, we prove the necessity of the small-gain condition for closed-loop stability of certain time-varying systems, extending the well-known necessity result in linear robust control.Comment: 15 pages, 3 figure

Compositional properties of passivity

The classical passivity theorem states that the negative feedback interconnection of passive systems is again passive. The converse statement, - passivity of the interconnected system implies passivity of the subsystems -, turns out to be equally valid. This result implies that among all feasible storage functions of a passive interconnected system there is always one that is the sum of storage functions of the subsystems. Sufficient conditions guaranteeing that all storage functions are of this type are derived. Closely related is the question when and how the stability of the closed-loop interconnected system implies passivity of the subsystems. We recall a folklore theorem which was proved for SISO linear systems, and derive some preliminary results towards a more general result, using the theory of simulation relations.