An Actuator Pre-Filtering Approach to Control-Coherent Koopman Modeling: Extending Koopman Operators to Systems With Control

The original Koopman operator theory cannot be applied to non-autonomous systems with exogenous input. Linearizing a nonlinear state equation with respect to the control input causes significant error, misinforming the controller. Bilinear approximations produce better accuracy, but the resultant model is no longer linear. This paper presents an alternative method for constructing a Koopman model for non-autonomous systems. Control-Coherent Koopman (CCK) modeling allows us to obtain a linear model with a constant control matrix by incorporating the dynamics of actuators. Here, the principle of CCK modeling is extended to those systems where actuator dynamics do not meet the requirements for the CCK formulation. Physical actuator dynamics are replaced by virtual dynamics, which are analogous to actuator pre-filters. These virtual dynamics possess independent state variables and new input terms that appear linearly in the filter dynamics. Although the original CCK physical modeling approach suffers a stiff equation problem, the actuator pre-filter approach does not. Conditions for implementation and numerical examples are discussed at the end.