Approximate bond graph models for linear singularly perturbed systems

A method for obtaining approximate bond graph models for linear time invariant (LTI) Multi-Input Multi-Output (MIMO) systems with singular perturbations is presented. The basic idea of using time-scale analysis in obtaining low-order models is to decouple the slow and fast models. This is achieved by using two-stage linear transformations. Hence, a procedure to construct decoupled bond graph models based on $$R$$R -fields representing each dynamic of the singularly perturbed system is proposed.When the linear transformations are applied to the system with singular perturbations, non-linear and linear equations have to be solved for separating the subsystems. In many cases, the exact solutions of these equations are complicated, but approximate solutions can be determined and approximate models can be obtained.Thus, zeroth- and first-order solutions in a bond graph approach are proposed. The key to finding the approximate solutions is to obtain the relations of the original bond graph with a predefined integral causality of the system and another bond graph called the Singularly Perturbed Bond Graph whose storage elements of the fast dynamics have derivative causality and for the slow dynamics they maintain an integral causality assignment.Finally, the proposed method is applied to an illustrative example where the simulation results show the exact solutions and zeroth- and first-order approximations.