Linear Quadratic Control Problem subject To Nonregular and Rectangular Descriptor Systems - Muhafzan, (2007)
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.