Energy shaping, interconnection and damping assignment, and integral control in the bond graph domain

This paper presents a methodology to perform energy shaping and interconnection and damping assignment in the bond graph domain, and addresses a new result on integral action control which improves the robustness of these passivity based control methods, which were first introduced on the port-controlled Hamiltonian systems with dissipation formalism. The methods perform expressing the desired closed-loop energy, interconnection and damping properties on a so-called target bond graph, which is built as follows on the plant bond graph: (i) adding virtual storage elements enforces a minimum on the target bond graph energy at a prespecified stable closed-loop equilibrium state, (ii) changing the R-field and its interconnection with the rest of the graph assigns a new dissipation function, and (iii) suitably inserting of power conserving elements (bonds and other structural BG-elements) among junctions yields the desired power conserving interconnection structure. The control law is then determined developing physically based heuristics and formal techniques on the target and plant bond graphs, which deliver a set of partial differential equations to be solved. The method to achieve integral control consists in adding to the target bond graph virtual elements representing the integral action and a change of variables, and then computing the integral control law with standard BG equation-reading procedures. These BG heuristics and prototyping allow to provide integral action on outputs of relative degree greater than one, a contribution of this work not previously available in the literature. This shows that the physical properties of bond graphs are beneficial not only to expediently perform methods contributed by control theory, but also to derive new theoretical results.