Electrical and Computer Engineering ETDs

Publication Date

1-22-1975

Abstract

This dissertation is divided into two parts. The first part presents a method for the design of time-optimal feedback controllers for linear discrete stable systems with a single bounded input. In the second part, a method is presented for the design of time sub-optimal feedback controllers in which a tight upper bound on the time sub-optimal feedback controller's performance for state points in a convex polyhedron can be guaranteed. Two sets related to the reachable set--the zero control set and the cylindrical set--are formulated. Both are helpful in the realization of the time-optimal feedback controller. The cylindrical set is composed of subsets which are called sub-cylindrical regions. The sub-cylindrical regions can be obtained easily by matrix calculations. The time-optimal feedback controller operates by first determining which sub-cylindrical region contains the present state, and then determining the corresponding optimal control which is a fixed function of the state on the sub-cylindrical region. The cost of implementation is a function of the dimension of the system and of the number of steps needed to transfer the state to the origin. Implementation for higher-order systems is possible. If implementation of the optimal control is too expensive, then sub-optimal control must be used. The sub-optimal feedback controller is realized with a single hyperplane outside a given reachable set. The sub-optimal feedback control policy is defined as follows: The optimal feedback control policy is applied when the state belongs to a given reachable set and a stable control policy is applied when the state is outside of that reachable set. The stable control policy is determined using a Liapunov function approximation to the reachable set. The stable control policy used is defined so that it maximizes the rate of shrink­age of the Liapunov function. The stable control policy for a state point is a saturation function of the distance from that state to the specific single hyperplane. The design and realization of the sub-optimal feedback control is much easier than the design and realization of the optimal feedback controller. A complete test algorithm is devised for testing the quality of the sub-optimal feedback controller on a convex polyhedron in the state space. The test algorithm uses the concept of mapping the convex polyhedron with the sub-optimal feedback control policy. The results of the complete test are compared with the corresponding optimal performances. A practical design procedure for the sub-optimal feedback controller is developed. The design procedure is tested for two second-order systems, two third-order systems, and a fourth-order system. When the modulus of the eigenvalues are near 1, the performance of the sub-optimal feedback controller is not as good as in the case in which the modulus of the eigenvalues are small. The performance of the sub-optimally controlled system is, however, much better than the performance obtained with zero control outside of the reachable set. When the modulus of the eigenvalues are less than .9, experimental evidence shows that the sub-optimal control is very close to optimal. A design procedure is also given by which the controller performance can be improved.

Project Sponsors

The University of New Mexico and The Sandia Corporation

Document Type

Dissertation

Language

English

Degree Name

Electrical Engineering

Level of Degree

Doctoral

Department Name

Electrical and Computer Engineering

First Committee Member (Chair)

Harold Knud Knudsen

Second Committee Member

James Vernon Lewis

Third Committee Member

Arnold Herman Koschmann

Fourth Committee Member

Joseph Thomas Cordaro Jr.

Download

Share

COinS