Precise linearization of nonlinear, non-autonomous systems based on physical system modeling theory

Abstract

Nonlinear dynamical systems behave linearly when recast in a higher dimensional space. This paper presents a new data-driven approach to precise linearization of a class of nonlinear dynamical systems. State variables are augmented by adding auxiliary variables that sufficiently inform the nonlinear dynamics of the system. A data matrix containing samples taken from the augmented state space is analyzed to extract latent variables that predict state transition in the latent space. First, the sufficiently informing augmented state variables are derived from Bond Graph where the connective structure of networked elements is known, but the constitutive laws of individual elements, which may be nonlinear, are unknown. Second, the rank of the data matrix and its covariance is analyzed based on the system's Bond Graph to determine the number of latent variables that can completely recover the data matrix. Third, using the latent variables, an exact linear state equation in Differential Algebraic Equation form is obtained for the class of nonlinear systems represented in the augmented state space. Furthermore, the latent variables are truncated to obtain a causal state equation that is linear, yet precisely represent the original nonlinear dynamics in the augmented state space. Finally, a practical example verifies and demonstrates the new methodology.