Passive and reciprocal linear time-and-space-invariant systems
Brayan M. Shali, Rodolphe Sepulchre
TL;DR
This work extends the classical notion of reciprocity from finite-dimensional LTI systems to linear time-and-space-invariant (LTSI) systems, addressing the ill-posedness that often arises for infinite-dimensional models. By decomposing LTSI dynamics into a family of LTI subsystems via the Fourier transform, reciprocity is enforced at every frequency $\omega$, and a self-adjoint, translation-invariant operator $\mathcal{S}$ with Fourier symbol $\hat{S}_\omega$ is constructed to realize internal reciprocity and self-duality. Coupled with a weak form of impedance passivity, this framework yields a well-posed, internally impedance-passive, self-dual LTSI realization, which can be partitioned into a port-Hamiltonian interconnection. The Timoshenko beam example motivates the approach and demonstrates how external reciprocity and passivity can guide the selection of physically meaningful internal states to overcome unboundedness and domain issues in infinite-dimensional systems.
Abstract
Reciprocity is a fundamental symmetry property observed across many physical domains, including acoustics, elasticity, electromagnetics, and thermodynamics. In systems and control theory, it provides key insights into the internal structure of linear time-invariant (LTI) systems and is closely linked to properties such as passivity, relaxation, and time-reversibility. This paper extends the concept of reciprocity to linear time-and-space-invariant (LTSI) systems, a class of infinite-dimensional systems with spatio-temporal dynamics. It is suggested that, analogously to the LTI case, combining the internal properties of reciprocity and (impedance) passivity entails physical state-space realizations. This is of particular relevance for infinite-dimensional systems, where issues of unboundedness can be detrimental to the well-posedness of the system. The results are motivated and illustrated with a physical example.