Control-Coherent Koopman Modeling: A Physical Modeling Approach

Abstract: The modeling of nonlinear dynamics based on Koopman operator theory, which is originally applicable only to autonomous systems with no control, is extended to non-autonomous control system without approximation to input matrix B. Prevailing methods using a least square estimate of the B matrix may result in an erroneous input matrix, misinforming the controller about the structure of the input matrix in a lifted space. Here, a new method for constructing a Koopman model that comprises the exact input matrix B is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact control matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. The proposed method is applied to multi degree-of-freedom robotic arms and multi-cable manipulation systems. Model Predictive Control is applied to the former. It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate control matrix B does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct B matrix.