Integrating bond graph with Port-Hamiltonian formulation for memristor non-linear circuit elements

The discovery of flux controlled memristors (Memory Resistor) by Leon Chua in 1971 as the missing element relating flux to charge, opens up possibilities for the development of a novel class of dielectrics over the coming years. With memristive components there is a departure from linearity; and components exhibit nonlinear characteristics. These properties enable the memristive elements to be used for the successful modeling of a number of physical devices and systems. The Bond Graph is one of graph theory modeling techniques, whose graphical description directly reveals the allocation and management of energy in the system (storage and dissipation) as well as the interconnection structure through which internal and external power exchange occurs via power ports. The graphical expansion of bond graph with the causal relationships among the system variables leads into a formulation of different types of mathematical models such as Port-Hamiltonian Systems. Incorporation of memory based elements leads to circuits with far more complex behaviour than normal dielectrics display. System dynamics may be studied using differential algebraic models arising from descriptor representations of the derived Port-Hamiltonian systems through Bond graph analysis. A derivation of unique generic Input-State-Output Port-Hamiltonian (ISO PHS) formulation from Bond graph representation of memristive circuits is proposed, which is suitable for simulation as well as providing engineering insight through analysis. In the proposed framework, the dissipation field splits into resistive and memristive parts in order to derive the Input-State-Output Port-Hamiltonian expressions and discuss different classes of systems of the proposed framework. Applications of the generic bond graph ISO PHS formulation using case studies with a memristive element are presented as examples of the proposed analysis. Consistency of the formulation is shown with transfer function formulations as well as with hybrid systems modelling. The nonlinear bond graph port-Hamiltonian methodology has applications in nonlinear network analysis and enables the formulation of input-output models of complex components embedded in non-linear circuits and systems