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