Physics-Based Causal Lifting Linearization of Nonlinear Control Systems Underpinned by the Koopman Operator

Methods for constructing causal linear models from nonlinear dynamical systems through lifting linearization underpinned by Koopman operator and physical system modeling theory are presented. Outputs of a nonlinear control system, called observables, may be functions of state and input, $\phi(x,u)$. These input-dependent observables cannot be used for lifting the system because the state equations in the augmented space contain the time derivatives of input and are therefore anticausal. Here, the mechanism of creating anticausal observables is examined, and two methods for solving the causality problem in lifting linearization are presented. The first method is to replace anticausal observables by their integral variables $\phi^*$, and lift the dynamics with $\phi^*$, so that the time derivative of $\phi^*$ does not include the time derivative of input. The other method is to alter the original physical model by adding a small inertial element, or a small capacitive element, so that the system's causal relationship changes. These augmented dynamics alter the signal path from the input to the anticausal observable so that the observables are not dependent on inputs. Numerical simulations validate the effectiveness of the methods.