  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.
