Optimal-control Theoretic Methods For Optimization And Regulation Of Distributed Parameter Systems

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Title: Optimal-control Theoretic Methods For Optimization And Regulation Of Distributed Parameter Systems
Author: Goss, Jennifer Dawn
Abstract: Optimal control and optimization of distributed parameter systems are discussed in the context of a common control framework . The adjoint method of optimization and the traditional linear quadratic regulator implementation of optimal control both employ adjoint or costate variables in the determination of control variable progression . As well both theories benefit from a reduced order model approximation in their execution . This research aims to draw clear parallels between optimization and optimal control utilizing these similarities . Several applications are presented showing the use of adjoint /costate variables and reduced order models in optimization and optimal control problems . The adjoint method for shape optimization is derived and implemented for the quasi -one -dimensional duct and two variations of a two -dimensional double ramp inlet . All applications are governed by the Euler equations . The quasi -one -dimensional duct is solved first to test the adjoint method and to verify the results against an analytical solution . The method is then adapted to solve the shape optimization of the double ramp inlet . A finite volume solver is tested on the flow equations and then implemented for the corresponding adjoint equations . The gradient of the cost function with respect to the shape parameters is derived based on the computed adjoint variables .The same inlet shape optimization problem is then solved using a reduced order model . The basis functions in the reduced order model are computed using the method of snapshots form of proper orthogonal decomposition . The corresponding weights are derived using an optimization in the design parameter space to match the reduced order model to the original snapshots . A continuous map of these weights in terms of the design variables is obtained via a response surface approximations and artificial neural networks . This map is then utilized in an optimization problem to determine the optimal inlet shape . As in the adjoint method of optimization , the methodology for a reduced order model is validated using the quasi -one -dimensional duct . The reduced order model is tested for efficiency and accuracy by performing an inverse optimization to match the pressure along the duct to a desired pressure profile . The method is then extended to generate a reduced order model for the two dimensional double ramp inlet . In this case , we optimize the inlet shape to minimize the mass weighted total pressure loss .The optimal control problem addressed is a two -dimensional channel flow governed by the Burgers equation . An obstacle in the flow is utilized for the implementation of boundary control to influence the flow . The Burgers equation is written in the abstract Cauchy form to allow for the implementation of linear control routines . The Riccati and Chandrasekhar equations are used to solve for the optimal control input to influence a region downstream of the obstacle . The results of both the controlled and uncontrolled scenarios are presented , and the Riccati and Chandrasekhar methods of gain calculation are compared . Reduced order modelling of the channel flow is performed using proper orthogonal decomposition and standard projection techniques . The reduced order model is then used for feedback control of the system in both set point and time -varying tracking problems .
URI: http : / /hdl .handle .net /10106 /1653
Date: 2009-09-16


Optimal-control Theoretic Methods For Optimization And Regulation Of Distributed Parameter Systems. Available electronically from http : / /hdl .handle .net /10106 /1653 .

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