Cell Mapping Methods
Nonlinear Stochastic Optimal Control
The project proposes a new numerical solution method for nonlinear stochastic optimal control problems. Such problems can be found in civil engineering structures subject to earthquake, ocean waves and wind loading excitations, in precision machines, manufacturing processes and robots, and in armed vehicles (land or air), target tracking systems and guidance systems operated in combat or rough environments. At present, few effective methods exist for obtaining optimal control solutions of complex nonlinear systems under random excitations.The proposed method is called the short-time Gaussian cell mapping method. The method has been applied to some very challenging nonlinear oscillators under Gaussian white noise excitations and has been proven to be very effective and accurate in analyzing very complex nonlinear stochastic systems. This project will further develop the method for nonlinear stochastic optimal control problems. Research tasks include 1) development of solutions for general nonlinear stochastic optimal control problems, 2) study of state constrained stochastic systems, and 3) the methodology refinement. The first two tasks are concerned with the further development and extensions of the short-time Gaussian cell mapping method to the new application areas. The last task deals with the issues related to the computational algorithms such as nonuniform state space partition and parallel computing. The results of this project will enable us to more efficiently and accurately predict and control the response of nonlinear systems under stochastic excitations, to better assess the reliability of the system under harsh random environments, and perhaps to develop new strategies for controlling the system.
This project was supported by National Science Foundation.
Bifurcation Analysis of Fuzzy Dynamical Systems
When a nonlinear dynamical system has random or fuzzy uncertainties, the definition of bifurcation is still open to discussion. Furthermore, the bifurcation analysis of such systems often requires knowledge of the global properties of the response. The cell mapping methods were specifically developed for the global analysis of nonlinear dynamical systems.
In this research, we apply the generalized cell mapping method to study bifurcation of nonlinear dynamical systems whose response is a fuzzy process. We have found some very interesting bifurcation scenarios when a system is subject to fuzzy disturbances or parametric uncertainties. For more information on this study, please refer to our recent publications.