Hua Xu

New method for composite optimal control of singularly perturbed systems

Abstract In this paper, a new method, based on a generalized algebraic Riccati equation arising in descriptor systems, is presented to solve the composite optimal control problem of singularly perturbed systems. Contrary to the existing method, the slow subsystem is viewed as a special kind of descriptor system. A new composite optimal controller is obtained which is valid for both standard and non-standard singularly perturbed systems. It is shown that the composite optimal control can be obtained simply by revising the solution of the slow regulator problem. It is proven that the composite optimal control can achieve a performance which is O(ee2) close to the optimal performance. Although this result is well-known for the standard singularly perturbed systems, it is new in the non-standard case. The equivalence between the new composite optimal controller and the existing one is also established for the standard singularly perturbed systems.

Linear-quadratic zero-sum differential games for generalized state space systems

In this note, we consider linear-quadratic zero-sum differential games for generalized state space systems. It is well known that a unique linear feedback saddle-point solution can exist in the game of state space systems. However, for the generalized state space system, we show that the game admits uncountably many linear feedback saddle-point solutions. Sufficient conditions for the existence of Linear feedback saddlepoint solutions are found. A constructive method is given to find these linear feedback saddle-point solutions. A simple example is included to illustrate the nonuniqueness of the linear feedback saddle-point solutions. >

New sufficient conditions for linear feedback closed-loop Stackelberg strategy of descriptor systems

This paper is concerned with the derivation of linear feedback closed-loop Stackelberg (LFCLS) strategies for a class of continuous-time two-person nonzero-sum differential games characterized by linear descriptor systems and the quadratic cost functionals. Compared with existing results, new sufficient conditions for the existence of a LFCLS strategy are obtained under less restrictive conditions. Furthermore, from the results of this paper, the authors also arrive at some conclusions about the linear-quadratic closed-loop Nash game for continuous-time descriptor systems. These conclusions are distinguished from those of the linear-quadratic closed-loop Nash game for state-space systems. >

On linear-quadratic optimal regulator for continuous-time descriptor system

An approach to dealing with the linear-quadratic optimal control problems for the continuous-time descriptor systems is proposed. A Riccati equation with a new form is established through the use of the necessary conditions for the optimal control. Unlike the existing approaches, transformation to the Riccati equation is made directly to find its solution. The sufficient conditions for the existence of the unique solution of the Riccati equation are found. The solution of the Riccati equation is then used in the construction of the optimal control strategies. It is believed that the full-order Riccati equation derived here may have some relation to the dynamic programming theory for the descriptor system.<<ETX>>

DERIVATION OF A MAXIMUM PRINCIPLE FOR DESCRIPTOR SYSTEMS WITHOUT AN ADMISSIBLE INITIAL CONDITION ASSUMPTION

Abstract In this paper, we derive a maximum principle for descriptor systems via the dynamic programming method. By using this method, we can avoid the admissible initial condition assumption in the derivation of the maximum principle, which is required in the other papers. Therefore, the results obtained in this paper can extend the domain of application of the maximum principle for descriptor systems. An illustrative example without the admissible initial condition assumption is given to show the application of the maximum principle for descriptor systems.

Upper and lower bounds of H/sub /spl infin//-optimal performance for a class of continuous-time descriptor systems

In this paper, finite- and infinite-horizon H/sub /spl infin// optimal control problems for a class of linear time-invariant descriptor systems are studied. Using a dynamic game theoretic approach, we provide upper and lower bounds of H/sub /spl infin//-optimal performance, which are characterized in terms of a parameterized reduced-order Riccati-like algebraic equation and a parameterized reduced-order game Riccati differential or algebraic equation. Moreover, we show that, if a prespecified performance level is achievable, the central H/sub /spl infin//-controller to guarantee such a performance level is not unique. A numerical example is included to illustrate the results obtained in the paper.

Team-Optimal Closed-Loop Stackelberg Strategies for Discrete-Time Descriptor Systems

In this paper we investigate the team-optimal closed-loop Stackelberg strategies for discrete-time descriptor systems. We show that the closed-loop no-memory information on the descriptor variables is sufficient for the leader to design the team-optimal feedback closed-loop Stackelberg strategies for a general class of linear-quadratic Stackelberg games. Sufficient conditions for the existence of such strategies are derived. A recursive scheme is presented to determine the team-optimal feedback closed-loop Stackelberg strategies. A numerical example is solved to illustrate the validity of the sufficient conditions.

Recursive Approach of H∞ Control Problems for Nonstandard Singularly Perturbed Systems Under Imperfect State Measurements

Abstract In this paper, we study the H ∞ control problem for singularly perturbed systems under imperfect state measurements by using the recursive approach. We construct a controller that guarantees a disturbance attenuation level larger than a boundary value of the reduced-order slow and fast subsystems when the singular perturbation parameter e approaches zero. In order to obtain the controller, we must solve the generalized algebraic Riccati equations. The main results in this paper is to propose a new recursive algorithm to solve the generalized algebraic Riccati equations and to find sufficient conditions for the convergence of the proposed algorithm.

On the near-optimality of composite optimal control for nonstandard singularly perturbed systems

In this paper, a method based on a generalized algebraic Riccati equation arising in descriptor systems is presented to solve the composite optimal control problem of nonstandard singularly perturbed systems. It is shown that the composite optimal control can be obtained very simply by only revising the solution of the slow regulator problem. It is proven that the composite optimal control can achieve a performance which is O(/spl epsiv//sup 2/) close to the optimal performance. Although this result is well-known for the standard singularly perturbed systems, it is new in the nonstandard case.

Feedback coordinating control for a class of two-level descriptor systems

A Stackelberg optimization problem for two-level systems was formulated by M. Simaan (1977) in which it was assumed that there is a coordinator at the high level who exerts control on N subsystems at the low level. The authors consider a similar problem with two differences. The first one is that the subsystems are described by the equations of the form Ex=Ax+Bv+Cu, that is the descriptor systems. The second one is that the coordinator does not have a cost functional of its own, and the objective of the coordinator is to optimize N different cost functionals simultaneously by using one coordinating strategy. In contrast with the problem for state space systems, it is found that there exists a linear feedback coordinating strategy for the coordinator because of the first difference.<<ETX>>