Koopman System Approximation-Based Optimal Control of Multiple Mobile Robots

This article presents a study of the Koopman operator theory and its application to optimal control of a multiple-mobile-robot system. The operator, while operating on a set of observation functions of the state vector of a nonlinear system, produces a set of dynamic equations that, through a dynamic transformation, form a new dynamic system. The Koopman system technique is then applied to the development of a linear or bilinear model approximation of nonlinear utility functions for optimal control of a system of multiple (mobile) robots, by selecting the utility functions as the Koopman system state variables and expressing the set of Koopman variables as the state variables of a linear or bilinear system whose parameters are determined through optimization. An iterative algorithm is developed to estimate the parameters adaptively. Finally, the optimal control problems based on a linear or bilinear approximation model are formulated, by transforming the nonlinear programming problem to a linear programming problem. The above models are simulated on a three-mobile-robot system to verify their performance in both centralized and decentralized versions. From the simulation results, 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 and are generated fast enough for real-time control.