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

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