Explicit Port-Hamiltonian Representation of Feedthrough-Systems with Nonlinear Dissipation

In this technical note, we present an explicit port-Hamiltonian formulation of feedthrough systems subject to nonlinear energy-dissipating effects. To this end, we merge the Dirac structure which describes the system\u27s internal interconnection structure with the constitutive relations of energy-storing and energy-dissipating elements. The resulting port-Hamiltonian system (PHS) is proven to be passive and generalizes an existing nonlinear-dissipative port-Hamiltonian formulation from the literature by feedthrough

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

Interconnection of Discrete-Time Dissipative Systems

Strictly proper discrete-time systems cannot be passive. 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 local 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. To demonstrate the validity of our methods, we provide numerical examples in the context of islanded microgrids

Linear Matrix Inequality Design of Exponentially Stabilizing Observer-Based State Feedback Port-Hamiltonian Controllers

The design of an observer-based state feedback (OBSF) controller with guaranteed passivity properties for port-Hamiltonian systems (PHS) is addressed using linear matrix inequalities (LMIs). The observer gain is freely chosen and the LMIs conditions such that the state feedback is equivalent to control by interconnection with an input strictly passive (ISP) and/or an output strictly passive (OSP) and zero state detectable (ZSD) port-Hamiltonian controller are established. It is shown that the proposed controller exponentially stabilizes a class of infinite-dimensional PHS and asymptotically stabilizes a class of finite-dimensional non-linear PHS. A Timoshenko beam model and a microelectromechanical system are used to illustrate the proposed approach