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 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

Robust stability and stabilization for singular systems with state delay and parameter uncertainty

This note considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method.published_or_final_versio

Infinite horizon control and minimax observer design for linear DAEs

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.Comment: This is an extended version of the paper which is to appear in the proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10-13, 201

The Optimal Linear Quadratic Feedback State Regulator Problem for Index One Descriptor Systems

In this note we present both necessary and sufficient conditions for the existence of a linear static state feedback controller if the system is described by an index one descriptor system. A priori no definiteness restrictions are made w.r.t. the quadratic performance criterium. It is shown that in general the set of solutions that solve the problem constitutes a manifold. This feedback formulation of the optimization problem is natural in the context of differential games and we provide a characterization of feedback Nash equilibria in a deterministic context.linear quadratic optimal control;descriptor systems;static stabilizing state feedback control

The Open-Loop Discounted Linear Quadratic Differential Game for Regular Higher Order Index Descriptor Systems

In this paper we consider the discounted linear quadratic differential game for descriptor systems that have an index larger than one. We derive both necessary and sufficient conditions for existence of an open-loop Nash (OLN) equilibrium. In a small macro-economic stabilization game we illustrate that the corresponding optimal response is generically cyclic.linear quadratic differential games;open-loop information structure;descriptor systems

The Optimal Linear Quadratic Feedback State Regulator Problem for Index One Descriptor Systems

In this note we present both necessary and sufficient conditions for the existence of a linear static state feedback controller if the system is described by an index one descriptor system. A priori no definiteness restrictions are made w.r.t. the quadratic performance criterium. It is shown that in general the set of solutions that solve the problem constitutes a manifold. This feedback formulation of the optimization problem is natural in the context of differential games and we provide a characterization of feedback Nash equilibria in a deterministic context.