scholarly journals Discrete-Time Indefinite Stochastic LQ Control via SDP and LMI Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Shaowei Zhou ◽  
Weihai Zhang
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Algebraic Riccati Equation ◽  
Mean Square ◽  
Infinite Time ◽  
Lq Problem ◽  
The Cost ◽  
And Control ◽  
Indefinite Stochastic Lq Control ◽  
Lq Control

This paper studies a discrete-time stochastic LQ problem over an infinite time horizon with state-and control-dependent noises, whereas the weighting matrices in the cost function are allowed to be indefinite. We mainly use semidefinite programming (SDP) and its duality to treat corresponding problems. Several relations among stability, SDP complementary duality, the existence of the solution to stochastic algebraic Riccati equation (SARE), and the optimality of LQ problem are established. We can test mean square stabilizability and solve SARE via SDP by LMIs method.

scholarly journals Indefinite LQ Control for Discrete-Time Stochastic Systems via Semidefinite Programming

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Shaowei Zhou ◽  
Weihai Zhang
Keyword(s):  
Semidefinite Programming ◽  
Riccati Equation ◽  
Discrete Time ◽  
Time Horizon ◽  
Stochastic Systems ◽  
Positive Semidefinite ◽  
Algebraic Riccati Equation ◽  
Infinite Time ◽  
Lq Problem ◽  
Lq Control

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.

scholarly journals Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments

2004 ◽  
Vol 36 (4) ◽  
pp. 1278-1299 ◽  
Author(s):  
Qihe Tang ◽  
Gurami Tsitsiashvili
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Economic Environment ◽  
Ruin Probabilities ◽  
Insurance Risk ◽  
Infinite Time ◽  
Time Period ◽  
Stochastic Returns ◽  
Regularly Varying

This paper investigates the finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly-varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, and so apply in the case of infinite-time ruin.

scholarly journals Discrete-Time Indefinite Stochastic Linear Quadratic Optimal Control with Second Moment Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Weihai Zhang ◽  
Guiling Li
Keyword(s):  
Discrete Time ◽  
Terminal State ◽  
Linear Quadratic ◽  
Linear Quadratic Optimal Control ◽  
Second Moment ◽  
New Class ◽  
The Matrix ◽  
Moment Constraint ◽  
Lq Problem ◽  
The Cost

This paper studies the discrete-time stochastic linear quadratic (LQ) problem with a second moment constraint on the terminal state, where the weighting matrices in the cost functional are allowed to be indefinite. By means of the matrix Lagrange theorem, a new class of generalized difference Riccati equations (GDREs) is introduced. It is shown that the well-posedness, and the attainability of the LQ problem and the solvability of the GDREs are equivalent to each other.

scholarly journals Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments

2004 ◽  
Vol 36 (04) ◽  
pp. 1278-1299 ◽  
Author(s):  
Qihe Tang ◽  
Gurami Tsitsiashvili
Keyword(s):  
Discrete Time ◽  
Time Horizon ◽  
Economic Environment ◽  
Ruin Probabilities ◽  
Insurance Risk ◽  
Infinite Time ◽  
Time Period ◽  
Stochastic Returns ◽  
Regularly Varying

This paper investigates the finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly-varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, and so apply in the case of infinite-time ruin.

scholarly journals Minimum System Sensitivity Study of Linear Discrete Time Systems for Fault Detection

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Xiaobo Li ◽  
Hugh H. T. Liu
Keyword(s):  
Fault Detection ◽  
Riccati Equation ◽  
Discrete Time ◽  
Time Horizon ◽  
Linear Time ◽  
Sensitivity Study ◽  
Algebraic Riccati Equation ◽  
System Sensitivity ◽  
Discrete Time Systems ◽  
Time Systems

Fault detection is a critical step in the fault diagnosis of modern complex systems. An important notion in fault detection is the smallest gain of system sensitivity, denoted asℋ−index, which measures the worst fault sensitivity. This paper is concerned with characterizingℋ−index for linear discrete time systems. First, a necessary and sufficient condition on the lower bound ofℋ−index in finite time horizon for linear discrete time-varying systems is developed. It is characterized in terms of the existence of solution to a backward difference Riccati equation with an inequality constraint. The result is further extended to systems with unknown initial condition based on a modifiedℋ−index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of theℋ−index in frequency domain, a condition in terms of algebraic Riccati equation is developed. In comparison with the well-known bounded real lemma, it is found thatℋ−index is not completely dual toℋ∞norm. Finally, several numerical examples are given to illustrate the main results.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Hongli
Keyword(s):  
State Constraints ◽  
Terminal State ◽  
Equality Constraints ◽  
Necessary Condition ◽  
Mathematical Programming Problem ◽  
Lagrangian Multiplier ◽  
Lq Problem ◽  
Indefinite Stochastic Lq Control ◽  
Lq Control ◽  
Stochastic Lq Control

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.

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