  This report presents the results of a simulation study of the linear model
and bilinear model approximations of the Koopman system model of the nonlinear
utility functions in optimal control of a 3-robot system. In such a control
problem, the nonlinear utility functions are maximized to achieve the control
objective of moving the robots to their target positions and avoiding
collisions. With the linear and bilinear model approximations of the utility
functions, the optimal control problem is solved, based on the approximate
model state variables rather than the original nonlinear utility functions.
This transforms the original nonlinear game theory problem to a linear
optimization problem. This report studies both the centralized and
decentralized implementations of the approximation model based control signals
for the 3-robot system control problem. The simulation results show that the
maximum value of the posteriori estimation error of the bilinear approximation
model is several thousand times less than the linear approxiamtion model. This
indicates that the bilinear model has more capacity to approximate the
nonlinear utility functions. Both the centralized and decentralized bilinear
approximation model based control signals can achieve the control objective of
moving the robots to their target positions. Based on the analysis of the
simulation time, the bilinear model based optimal control solution is fast
enough for real-time control implementation.
