INVITED PAPER Port-Hamiltonian Formulation of Systems With Memory This paper investigates extending the properties of memory elements to model, analyze, and simulate mechanical, electromechanical, hydromechanical, and thermodynamic systems. By Dimitri Jeltsema and Arnau Do `ria-Cerezo ABSTRACT | In this paper, we consider memristors, mem- inductors, and memcapacitors and their properties as port- Hamiltonian systems. The port-Hamiltonian formalism naturally arises from network modeling of physical systems in a variety of domains. Exposing the relation between the energy storage, dissipation, and interconnection structure, this framework underscores the physics of the system. One of the strong aspects of the port-Hamiltonian formalism is that a power-preserving interconnection between port-Hamiltonian systems results in another port-Hamiltonian system with com- posite energy, dissipation, and interconnection structure. This feature can advantageously be used to model, analyze, and simulate networks consisting of complex interconnections of both conventional and memory circuit elements. Furthermore, the port-Hamiltonian formalism naturally extends the funda- mental properties of the memory elements beyond the realm of electrical circuits. KEYWORDS | Memory elements; memristive systems; memris- tor; port-based modeling; port-Hamiltonian systems I. INTRODUCTION AND MOTIVATION In the early 1970s, Chua [1] postulated the existence of a new basic electrical circuit element, called the memris- tor. The memristor, a contraction of memory and resis- tance that refers to a resistor with memory, completes the family of the well-known existing fundamental circuit elements: the resistor, the inductor, and the capacitor. Although a variety of physical devices, including thermis- tors, discharge tubes, Josephson junctions, and even ionic systems like the Hodgkin–Huxley model of a neuron, were shown to exhibit memristive effects [2], [3], a phy- sical passive two-terminal memristive prototype could not be constructed until very recently scientists of Hewlett- Packard Laboratories announced its realization in [14]. Strukov et al. show that memristance naturally arises in nanoscale systems when electronic and atomic trans- ports are coupled under an external bias voltage. In [6], the concept of memristance is extended to inductive and capacitive elements, called meminductors and memcapacitors. One of the main reasons why the concept of memris- tance has not yet played a major role in modeling problems can most likely be explained from the fact that so far the majority of practical devices are reasonably well modeled by some (though often artificial) combination of standard modeling building blocks, like resistive, inductive, and ca- pacitive elements, and their nonlinear and multiport ver- sions. However, as nanoscale devices become more and more important and complex [2], it might be beneficial, and on the longer term even necessary, to enlarge our repertoire of modeling building blocks that establishes a closer connection between the mathematics and the ob- served physics. In this contribution, we study memristive, meminduc- tive, and memcapacitive behavior in the port-Hamiltonian modeling framework. Based on the mechanics formula- tion introduced by Sir W. R. Hamilton in the 19th cen- tury, the port-Hamiltonian formalism naturally arises from network modeling of physical systems in a variety of domains (e.g., electrical, mechanical, electromechanical, Manuscript received September 15, 2010; revised May 3, 2011; accepted July 25, 2011. Date of publication September 22, 2011; date of current version May 10, 2012. The work of A. Do `ria-Cerezo was supported in part by the Spanish Government under Research Project DPI2010-15110. D. Jeltsema is with the Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail: d.jeltsema@tudelft.nl). A. Do `ria-Cerezo is with the Department of Electrical Engineering and the Institute of Industrial and Control Engineering, Universitat Polite `cnica de Catalunya, 08800 Vilanova i la Geltru ´, Spain (e-mail: arnau.doria@upc.edu). Digital Object Identifier: 10.1109/JPROC.2011.2164169 1928 Proceedings of the IEEE | Vol. 100, No. 6, June 2012 0018-9219/$26.00 Ó2011 IEEE