Monotone One-Port Circuits

Abstract

Maximal monotonicity is explored as a generalization of the linear theory of passivity, aiming at an algorithmic input/output analysis of physical models. The theory is developed for maximal monotone one-port circuits, formed by the series and parallel interconnection of basic elements. These circuits generalize passive LTI transfer functions. Periodic input signals are shown to be mapped to periodic output signals, and these input-output behaviors can be efficiently computed using a maximal monotone splitting algorithm, which decomposes the computation according to the circuit topology. A new splitting algorithm is presented, which applies to any monotone circuit defined as a port interconnection of monotone elements.The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n. 670645.

Hilbert Direct Integrals of Monotone Operators

“…Similar examples arise in probability theory [4], circuit theory [15], approximation theory [18], calculus of variations [21], partial differential equations [22], variational analysis [32], convex analysis [35], and evolution systems [37]. In terms of modeling, (1.2) is limited to a finite number of operators, (1.3) requires that all the factor operators be identical to A, and (1.4) imposes that all the factor spaces be identical to H and operates with the standard Lebesgue measure space [0, ].…”

Synchronization in Electrical Circuits With Memristors and Grounded Capacitors

“…Substitution of the above in (17), together with the dynamics of the capacitors given in (6), leads to…”

“…Examples range from the study of bifurcations [14] to their use in solving optimisation problems [15]. We note that electrical circuits with memristors are necessarily nonlinear and that the study of nonlinear circuits has a long history [16], see also [17] for a recent contribution.…”

Port-Hamiltonian Systems Theory and Monotonicity

“…Of course, the assumption of a quadratic-affine Hamiltonian in order to let the monotone port-Hamiltonian system be incrementally and differentially passive is restrictive. On the other hand, it is known from the literature [4,34] that for ``unconditional"" incremental properties such an assumption may be necessary as well. For example, we can formulate the following simple result.…”

“…With regard to the current paper the following two aspects of monotonicity are most relevant. First, monotonicity has been a key concept in the study of nonlinear electrical circuits and general nonlinear network dynamics; see, e.g., the recent paper [4] for a historical context and references. From a systems and control point of view this view on monotonicity is intimately related to notions of incremental passivity [5].…”