Indefinite LQ Research Articles

Pareto Optimal Strategy under H∞ Constraint for Discrete-Time Stochastic Systems

This paper investigates the Pareto optimal strategy of discrete-time stochastic systems under H∞ constraint, in which the weighting matrices of the weighted sum cost function can be indefinite. Combining the H∞ control theory with the indefinite LQ control theory, the generalized difference Riccati equations (GDREs) are obtained. By means of the solution of the GDREs, the Pareto optimal strategy with H∞ constraint is derived, and the necessary and sufficient conditions for the existence of the strategy are presented. Then the Pareto optimal solution under the worst-case disturbance is solved. Finally, the efficiency of the obtained results is illustrated by a numerical example.

Indefinite LQ Optimal Control with Terminal State Constraint for Discrete-Time Uncertain Systems

Uncertainty theory is a branch of mathematics for modeling human uncertainty based on the normality, duality, subadditivity, and product axioms. This paper studies a discrete-time LQ optimal control with terminal state constraint, whereas the weighting matrices in the cost function are indefinite and the system states are disturbed by uncertain noises. We first transform the uncertain LQ problem into an equivalent deterministic LQ problem. Then, the main result given in this paper is the necessary condition for the constrained indefinite LQ optimal control problem by means of the Lagrangian multiplier method. Moreover, in order to guarantee the well-posedness of the indefinite LQ problem and the existence of an optimal control, a sufficient condition is presented in the paper. Finally, a numerical example is presented at the end of the paper.

Indefinite LQ Problem for Irregular Singular Systems

The indefinite LQ problem for irregular singular systems is investigated. Under some general conditions, the optimal control-state pair is obtained by solving an algebraic Riccati equation. The optimal control is synthesized as state feedback. All the finite poles of the closed-loop system are located on the left-half complex plane. An example is given to show the validity of the proposed conclusion.

Study on Stochastic Linear Quadratic Optimal Control with Quadratic and Mixed Terminal State Constraints

This paper studies the indefinite stochastic LQ control problem with quadratic and mixed terminal state equality constraints, which can be transformed into a mathematical programming problem. By means of the Lagrangian multiplier theorem and Riesz representation theorem, the main result given in this paper is the necessary condition for indefinite stochastic LQ control with quadratic and mixed terminal equality constraints. The result shows that the different terminal state constraints will cause the endpoint condition of the differential Riccati equation to be changed. It coincides with the indefinite stochastic LQ problem with linear terminal state constraint, so the result given in this paper can be viewed as the extension of the indefinite stochastic LQ problem with the linear terminal state equality constraint. In order to guarantee the existence and the uniqueness of the linear feedback control, a sufficient condition is also presented in the paper. A numerical example is presented at the end of the paper.

Indefinite LQ Control for Discrete‐Time Stochastic Systems via Semidefinite Programming

This paper is concerned with a discrete‐time indefinite stochastic LQ problem in an infinite‐time horizon. A generalized stochastic algebraic Riccati equation (GSARE) that involves the Moore‐Penrose inverse of a matrix and a positive semidefinite constraint is introduced. We mainly use a semidefinite‐programming‐ (SDP‐) based approach to study corresponding problems. Several relations among SDP complementary duality, the GSARE, and the optimality of LQ problem are established.

Discrete-time Indefinite Stochastic LQ Optimal Control: Infinite Horizon Case

The asymptotic analysis method is applied to the infinite horizon discrete-time indefinite stochastic LQ problem. This paper carries out the research on the basis of the results of its finite horizon counterpart and the mean-square stabilizing assumption. Properties of the solutions to the generalized algebraic Riccati equation (GARE) are also considered. Finally, two examples are provided to justify the developed theory.

Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon

Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon

Discrete-time Indefinite LQ Control with State and Control Dependent Noises

This paper deals with the discrete-time stochastic LQ problem involving state and control dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. In this general setting, it is shown that the well-posedness and the attainability of the LQ problem are equivalent. Moreover, a generalized difference Riccati equation is introduced and it is proved that its solvability is necessary and sufficient for the existence of an optimal control which can be either of state feedback or open-loop form. Furthermore, the set of all optimal controls is identified in terms of the solution to the proposed difference Riccati equation.

Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls

The optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential Riccati equation. In this paper, we provide general necessary and sufficient conditions for the solvability of this generalized differential Riccati equation. Furthermore, its asymptotic behavior is investigated along with its connection to the generalized algebraic Riccati equation associated with the linear quadratic control problem in finite time horizon. Examples are presented to illustrate the results established.

Input/output norms in general linear systems

Upper bounds on the induced norm of the input/output operators in linear systems with L2 input and output signals, are provided. The main idea is to transform the original problem into a parametized set of indefinite LQ optimization problems. Using the classical maximum principle, the existence of solutions to the latter is equivalent to the solvability of a boundary value hamiltonian system or, equivalenty, of an ‘indefinite’ Riccati equation. Our time domain approach enables the treatment of a very general linear set-up, including time-varying, infinite-dimensional, finite- and infinite-horizon systems.

Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness