Causality in dual faceted linearization of nonlinear dynamical systems

Abstract

Dual Faceted Linearization (DFL) is an emerging methodology for recasting a nonlinear dynamical system in an augmented state space where the system behaves more linearly. While the original system is highly nonlinear when represented with independent state variables, the system can be described as a linear dynamical system in a larger space with additional variables, termed Auxiliary Variables. A caveat of DFL, however, is that Auxiliary Variables may depend on input variables, leading to non-causal state equations containing the time derivative of input variables. Here, we a) examine when an anti-causal problem occurs, and b) how the problem can be mitigated. Analysis based on graphical physical modeling theory, Bond Graphs, reveals that all the Auxiliary Variables associated with nonlinear energy storage elements are causal. Auxiliary Variables become anti-causal only for systems containing nonlinear damping elements through which inputs propagate. To eliminate the influence of inputs, revised Auxiliary Variables are introduced, and their causal state equations are obtained. A few numerical examples verify the results and demonstrate the effectiveness of the method.