The Bolza-form of the finite-time constrained optimal control problem leads to
the Hamilton-Jacobi-Bellman (HJB) equation with terminal boundary conditions and tobe-
determined parameters. In general, it is a formidable task to obtain analytical and/or
numerical solutions to the HJB equation. This dissertation presents two novel
polynomial expansion methodologies for solving optimal feedback control problems for
a class of polynomial nonlinear dynamical systems with terminal constraints. The first
approach uses the concept of higher-order series expansion methods. Specifically, the
Series Solution Method (SSM) utilizes a polynomial series expansion of the cost-to-go
function with time-dependent coefficient gains that operate on the state variables and
constraint Lagrange multipliers. A significant accomplishment of the dissertation is that
the new approach allows for a systematic procedure to generate optimal feedback control
laws that exactly satisfy various types of nonlinear terminal constraints.
The second approach, based on modified Galerkin techniques for the solution of
terminally constrained optimal control problems, is also developed in this dissertation. Depending on the time-interval, nonlinearity of the system, and the terminal
constraints, the accuracy and the domain of convergence of the algorithm can be related
to the order of truncation of the functional form of the optimal cost function. In order to
limit the order of the expansion and still retain improved midcourse performance, a
waypoint scheme is developed. The waypoint scheme has the dual advantages of
reducing computational efforts and gain-storage requirements. This is especially true for
autonomous systems. To illustrate the theoretical developments, several aerospace
application-oriented examples are presented, including a minimum-fuel orbit transfer
problem.
Finally, the series solution method is applied to the solution of a class of partial
differential equations that arise in robust control and differential games. Generally, these
problems lead to the Hamilton-Jacobi-Isaacs (HJI) equation. A method is presented that
allows this partial differential equation to be solved using the structured series solution
approach. A detailed investigation, with several numerical examples, is presented on the
Nash and Pareto-optimal nonlinear feedback solutions with a general terminal payoff.
Other significant applications are also discussed for one-dimensional problems with
control inequality constraints and parametric optimization.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2008-12-156 |
| Date | 14 January 2010 |
| Creators | Sharma, Rajnish |
| Contributors | Vadali, Dr. Srinivas R. |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation |
| Format | application/pdf |
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