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: | |
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| Format: | Thesis |
| Language: | English English |
| Published: |
2007
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/5037/1/FS_2007_37.pdf |
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| Summary: | 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. |
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