Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems

The linear quadratic (LQ) control problem is a widely studied field in the area of control and optimization, particularly, in the area of optimal control. This problem is, in general, concerned with determining a controller such that the controller satisfies the dynamic constraint. On the other h...

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Main Author: Muhafzan
Format: Thesis
Language:English
English
Published: 2007
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Online Access:http://psasir.upm.edu.my/id/eprint/5037/1/FS_2007_37.pdf
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spelling my-upm-ir.50372013-05-27T07:19:56Z Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems 2007 Muhafzan, The linear quadratic (LQ) control problem is a widely studied field in the area of control and optimization, particularly, in the area of optimal control. This problem is, in general, concerned with determining a controller such that the controller satisfies the dynamic constraint. On the other hand, the descriptor systems have received considerable interest over the last decade because it has some specificity in the structure of its solution. It is a natural mathematical model of many types of physical systems of which it appears frequently in the fields such as circuit systems, economics, power systems, robots and electric network. Therefore, the LQ control problem subject to the descriptor system has a great potential for the system modeling, because it can preserve the structure of physical systems and can include the non dynamic constraint and impulsive element. In this thesis we consider the most general class of the LQ control problem subject to descriptor system of which the constraint is of the class of the nonregular descriptor system and rectangular descriptor system in the infinite horizon time. In addition, we allow the control weighting matrix in the cost functional to be positive semidefinite, but our results, however, hold for the constraint to be regular descriptor system as well as the control weighting matrix in the cost functional to be positive definite. Our main aims are to find the optimal smooth solution of the LQ control problem subject to both nonregular and rectangular descriptor systems, respectively. For these purposes, we create the sufficient conditions that guarantee the existence, or existence and uniqueness if possible, of the smooth optimal solution of the problems. In order to solve the considered problem we transform the LQ control problem subject to both nonregular and rectangular descriptor systems, respectively, into the standard LQ control problem. In fact, by utilizing the means of the restricted system equivalent of two descriptor systems and the equivalence principle of two optimal control problems, we can construct some bijections which show that there are equivalent relationship between the considered problem and the standard LQ control problem. As a result of the transformation process, we have two kinds of standard LQ control problem, that are, the cases of which the control weighting matrix in the cost functional is positive definite and positive semidefinite. In the positive definite case, we utilize the available results of the standard LQ control problem. Otherwise, in the positive semidefinite case, the semidefinite programming approach is used in order to obtain the smooth solutions. As the ultimate results, the conditions that guarantee the existence of the smooth optimal solution are presented formally in several theorems. Some testing problems are presented. The graphs of the trajectories are plotted to visualize the behavior of the optimal control-state pairs. The Maple 9 and Matlab 6.5 softwares are used for calculation and plotting the trajectories of the optimal solution. Linear control systems. 2007 Thesis http://psasir.upm.edu.my/id/eprint/5037/ http://psasir.upm.edu.my/id/eprint/5037/1/FS_2007_37.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Linear control systems. Science English
institution Universiti Putra Malaysia
collection PSAS Institutional Repository
language English
English
topic Linear control systems.


spellingShingle Linear control systems.


Muhafzan,
Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
description The linear quadratic (LQ) control problem is a widely studied field in the area of control and optimization, particularly, in the area of optimal control. This problem is, in general, concerned with determining a controller such that the controller satisfies the dynamic constraint. On the other hand, the descriptor systems have received considerable interest over the last decade because it has some specificity in the structure of its solution. It is a natural mathematical model of many types of physical systems of which it appears frequently in the fields such as circuit systems, economics, power systems, robots and electric network. Therefore, the LQ control problem subject to the descriptor system has a great potential for the system modeling, because it can preserve the structure of physical systems and can include the non dynamic constraint and impulsive element. In this thesis we consider the most general class of the LQ control problem subject to descriptor system of which the constraint is of the class of the nonregular descriptor system and rectangular descriptor system in the infinite horizon time. In addition, we allow the control weighting matrix in the cost functional to be positive semidefinite, but our results, however, hold for the constraint to be regular descriptor system as well as the control weighting matrix in the cost functional to be positive definite. Our main aims are to find the optimal smooth solution of the LQ control problem subject to both nonregular and rectangular descriptor systems, respectively. For these purposes, we create the sufficient conditions that guarantee the existence, or existence and uniqueness if possible, of the smooth optimal solution of the problems. In order to solve the considered problem we transform the LQ control problem subject to both nonregular and rectangular descriptor systems, respectively, into the standard LQ control problem. In fact, by utilizing the means of the restricted system equivalent of two descriptor systems and the equivalence principle of two optimal control problems, we can construct some bijections which show that there are equivalent relationship between the considered problem and the standard LQ control problem. As a result of the transformation process, we have two kinds of standard LQ control problem, that are, the cases of which the control weighting matrix in the cost functional is positive definite and positive semidefinite. In the positive definite case, we utilize the available results of the standard LQ control problem. Otherwise, in the positive semidefinite case, the semidefinite programming approach is used in order to obtain the smooth solutions. As the ultimate results, the conditions that guarantee the existence of the smooth optimal solution are presented formally in several theorems. Some testing problems are presented. The graphs of the trajectories are plotted to visualize the behavior of the optimal control-state pairs. The Maple 9 and Matlab 6.5 softwares are used for calculation and plotting the trajectories of the optimal solution.
format Thesis
qualification_name Doctor of Philosophy (PhD.)
qualification_level Doctorate
author Muhafzan,
author_facet Muhafzan,
author_sort Muhafzan,
title Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
title_short Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
title_full Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
title_fullStr Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
title_full_unstemmed Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems
title_sort linear quadratic control problem subject to nonregular and rectangular descriptor systems
granting_institution Universiti Putra Malaysia
granting_department Science
publishDate 2007
url http://psasir.upm.edu.my/id/eprint/5037/1/FS_2007_37.pdf
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