Interconnection of (Q,S,R)-Dissipative Systems in Discrete Time

Discrete-time systems cannot be passive unless there is a direct feedthrough from the input to the output. For passivity-based control to be exploited nevertheless, some authors introduce virtual outputs, while others rely on continuous-time passivity and then apply discretization techniques that preserve passivity in discrete time. Here we argue that quadratic supply rates incorporate and extend the effect of virtual outputs, allowing one to exploit dissipativity properties directly in discrete time. We derive decentralized (Q,S,R)-dissipativity conditions for a set of nonlinear systems interconnected with arbitrary topology, so that the overall network is guaranteed to be stable. For linear systems, we develop dissipative control conditions that are linear in the supply rate matrices. To demonstrate the validity of our methods, we provide numerical examples in the context of islanded microgrids.