scholarly journals The Gamma Stein equation and noncentral de Jong theorems

Bernoulli ◽  
2018 ◽  
Vol 24 (4B) ◽  
pp. 3384-3421 ◽  
Author(s):  
Christian Döbler ◽  
Giovanni Peccati
Keyword(s):  
Stein Equation

Distributed optimisation design for solving the Stein equation with constraints

2019 ◽  
Vol 13 (15) ◽  
pp. 2492-2499 ◽  
Author(s):  
Guanpu Chen ◽  
Xianlin Zeng ◽  
Yiguang Hong
Keyword(s):  
Stein Equation

scholarly journals Approximate Design of Optimal Disturbance Rejection for Discrete-Time Systems with Multiple Delayed Inputs: Application to a Jacket-Type Offshore Structure

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Shi-Yuan Han ◽  
Dong Wang ◽  
Yue-Hui Chen ◽  
Gong-You Tang ◽  
Xi-Xin Yang
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Performance Index ◽  
Disturbance Rejection ◽  
Offshore Structure ◽  
Approximate Design ◽  
Free System ◽  
Discrete Time Systems ◽  
Stein Equation ◽  
Time Systems

The study is concerned with problem of optimal disturbance rejection for a class of discrete-time systems with multiple delayed inputs. In order to avoid the two-point boundary value (TPBV) problem with items of time-delay and time-advance caused by multiple delayed inputs, the discrete-time system with multiple delayed inputs is transformed into a delay-free system by introducing a variable transformation, and the original performance index is reformulated as a corresponding form without the explicit appearance of time-delay items. Then, the approximate optimal disturbance rejection controller (AODRC) is derived from Riccati equation and Stein equation based on the reduced system and reformulated performance index, which is combined with feedback item of system state, feedforward item of disturbances, and items of delayed inputs. Also, the existence and uniqueness of AODRC are proved, and the stability of the closed-loop system is analysed. Finally, numerical examples of disturbance rejection for jacket-type offshore structure and pure mathematical model are illustrated to validate the feasibility and effectiveness of the proposed approach.

scholarly journals Solving the Stein Equation in compound poisson approximation

1998 ◽  
Vol 30 (02) ◽  
pp. 449-475 ◽  
Author(s):  
A. D. Barbour ◽  
Sergey Utev
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Restricted Range ◽  
Stein's Method ◽  
Kolmogorov Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation ◽  
The Mean ◽  
Accuracy Of Approximation

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

scholarly journals New error bounds for Laplace approximation via Stein's method

2021 ◽  
Author(s):  
Robert Gaunt
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Laplace Approximation ◽  
Stein’S Method ◽  
Independent Random Variables ◽  
Stein's Method ◽  
Smooth Test ◽  
Stein Equation ◽  
Technical Advances ◽  
Derivatives Of

We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren \cite {pike} for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and its first two derivatives, of the Rayleigh Stein equation.

Optimal Vibration Control for Uncertain Nonlinear Sampled-Data Systems With Actuator and Sensor Delays: Application to a Vehicle Suspension

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Jing Lei
Keyword(s):  
Vibration Control ◽  
Time Delays ◽  
Design Procedure ◽  
Data Systems ◽  
Sampled Data ◽  
Car Suspension ◽  
Stein Equation ◽  
Vector Difference ◽  
Suspension Model ◽  
Original Time

The problem of optimal sampled-data vibration control for nonlinear systems with time delays and uncertainties is considered. With the purpose of simplifying the nonlinear optimal vibration control (NOVC) design, the original time-delay sampled-data system is converted into a discrete-time nondelayed system first, as well as the nonlinear and uncertain terms are treated as external excitations. Therefore, the design procedure for NOVC law is reduced and the successive approximation approach is sequentially developed in it. The obtained NOVC law is derived from a Riccati equation, a Stein equation, and sequences of adjoint vector difference equations. It is combined with a feedforward term, the nonlinearity and uncertainty compensator terms, and some control memory terms, which compensate for the effects produced by the disturbance, the nonlinearity and uncertainties, and the time delays. Moreover, the existence and uniqueness of NOVC law are proved and the stability of the closed-loop system is analyzed. In order to make the controller physically realizable, an observer is constructed and the corresponding dynamical control law is given. Furthermore, by this means, the NOVC law for a sampled-data quarter-car suspension model with actuator and sensor delays is designed. The results of numerical simulations illustrate that the NOVC gives satisfactory conclusions in effectiveness of suspension performance responses and feasibility of the proposed design approach.

scholarly journals Solving the Stein Equation in compound poisson approximation

1998 ◽  
Vol 30 (2) ◽  
pp. 449-475 ◽  
Author(s):  
A. D. Barbour ◽  
Sergey Utev
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Restricted Range ◽  
Stein's Method ◽  
Kolmogorov Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation ◽  
The Mean ◽  
Accuracy Of Approximation

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

Sign in / Sign up

Export Citation Format

Share Document