Numerical Nonlinear and Optimal Control Using Wavelets

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PI's Name: Panagiotis TsiotrasInstitution: School of Aerospace Engineering, Georgia Institute of TechnologyProposal Number: 0084954Proposal Title: Numerical Nonlinear and Optimal Control Using WaveletsAbstract: The development of optimal feedback control laws for nonlinear systems has been the topic of intense research over the past half a century. The cornerstone of this theory is the so-called Hamilton-Jacobi-Bellman partial differential equation, the solution of which provides an analytic expression for the optimal feedback control. Unfortunately, the analytic solution of the Hamilton-Jacobi-Bellman equation is a formidable task. Solutions can be obtained only for some special cases of very low dimension. Even traditional numerical methods for solving the Hamilton-Jacobi-Bellman equation are often inadequate due to the ``curse of dimensionality''. That is, the number of computations involved increases tremendously with the state of the system. In this project, we propose to use wavelets for solving the Hamilton-Jacobi-Bellman equation numerically. Wavelets are orthogonal basis functions that use the concept of multiresolution. Namely, they are able to capture the local behaviour of signals both in frequency and time. Using translations and dilations, they decompose the solution space into finer and finer subspaces and the final solution is computed as an aggregation of the solutions in these finer subspaces. Only the subspaces which significantly contribute to the solution have non-zero coefficients in the final Fourier/Wavelet series expansion. Therefore, given a level of accuracy for the solution, only few non-zero Fourier coefficients are needed to capture the exact solution. As a result, solutions can be obtained much more efficiently than with other methods. This research will have an immediate impact on the area of embedded control systems. Dedicated computer chips running the developed algorithms will be able to calculate optimal feedback control laws on-line, thus achieving unprecedented levels of intelligence, autonomy and versatility for several applications such as, automobile control, aircraft navigation, autonomous mobile robot control, etc. In addition, the multiresolution property of wavelet expansions implies a decoupling of the solutions in the different resolution levels. Therefore, wavelet-based solutions are uniquely suited to parallel processing for computer implementation.