This dissertation introduces novel methods for solving highly challenging model-ing and control problems, motivated by advanced aerospace systems. Adaptable, ro-bust and computationally effcient, multi-resolution approximation algorithms basedon Radial Basis Function Network and Global-Local Orthogonal Mapping approachesare developed to address various problems associated with the design of large scaledynamical systems. The main feature of the Radial Basis Function Network approachis the unique direction dependent scaling and rotation of the radial basis function viaa novel Directed Connectivity Graph approach. The learning of shaping and rota-tion parameters for the Radial Basis Functions led to a broadly useful approximationapproach that leads to global approximations capable of good local approximationfor many moderate dimensioned applications. However, even with these refinements,many applications with many high frequency local input/output variations and ahigh dimensional input space remain a challenge and motivate us to investigate anentirely new approach. The Global-Local Orthogonal Mapping method is based upona novel averaging process that allows construction of a piecewise continuous globalfamily of local least-squares approximations, while retaining the freedom to vary ina general way the resolution (e.g., degrees of freedom) of the local approximations.These approximation methodologies are compatible with a wide variety of disciplinessuch as continuous function approximation, dynamic system modeling, nonlinear sig-nal processing and time series prediction. Further, related methods are developedfor the modeling of dynamical systems nominally described by nonlinear differentialequations and to solve for static and dynamic response of Distributed Parameter Sys-tems in an effcient manner. Finally, a hierarchical control allocation algorithm ispresented to solve the control allocation problem for highly over-actuated systemsthat might arise with the development of embedded systems. The control allocationalgorithm makes use of the concept of distribution functions to keep in check the"curse of dimensionality". The studies in the dissertation focus on demonstrating,through analysis, simulation, and design, the applicability and feasibility of these ap-proximation algorithms to a variety of examples. The results from these studies areof direct utility in addressing the "curse of dimensionality" and frequent redundancyof neural network approximation.
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