Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique

A practical method of improving the accuracy of the Gaussian statistical linearization technique is presented. The method uses a series expansion of the unknown probability density function which includes up to fourth order terms. It is shown that by the use of the Gram-Charlier expansion a simple generating function can be derived to evaluate analytically the integrals required. It is also shown how simplifying assumptions can be used to substantially reduce the required extra equations, e.g. symmetric or assymetric and single input nonlinearities. It is also shown that the eigenvalues of the statistically linearized system can be used to estimate the stability and speed of response of the nonlinear system. The reduced expansion technique is applied to first and second order nonlinear systems and the predicted mean square response is compared to the Gaussian statistical linearization and the exact solution. The prediction of the time response of the mean of a nonlinear first order system by the use of the statistically linearized eigenvalues is compared to a 300 run Monte Carlo digital solution.

Download Full-text

Analysis of Robust Stability for a Class of Stochastic Systems via Output Feedback: The LMI Approach

Journal of Function Spaces and Applications ◽

10.1155/2013/873578 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Xin-rong Cong ◽

Long-suo Li

Keyword(s):

Linear Matrix Inequality ◽

Robust Stability ◽

Output Feedback ◽

Stochastic Systems ◽

Linear Matrix ◽

Matrix Inequality ◽

Convex Optimization Problem ◽

Control Laws ◽

The Stability ◽

And Control

This paper investigates the robust stability for a class of stochastic systems with both state and control inputs. The problem of the robust stability is solved via static output feedback, and we convert the problem to a constrained convex optimization problem involving linear matrix inequality (LMI). We show how the proposed linear matrix inequality framework can be used to select a quadratic Lyapunov function. The control laws can be produced by assuming the stability of the systems. We verify that all controllers can robustly stabilize the corresponding system. Further, the numerical simulation results verify the theoretical analysis results.

Download Full-text

The stability and control of fractional-order system with distributed time delay

IECON 2017 - 43rd Annual Conference of the IEEE Industrial Electronics Society ◽

10.1109/iecon.2017.8217098 ◽

2017 ◽

Author(s):

Keyong Shao ◽

Xiaofeng Gu ◽

Guangyuan Yang

Keyword(s):

Time Delay ◽

Fractional Order ◽

Fractional Order System ◽

Order System ◽

Stability And Control ◽

Distributed Time Delay ◽

The Stability ◽

And Control

Download Full-text

Dynamics Analysis and Control of a Five-Term Fractional-Order System

Discrete Dynamics in Nature and Society ◽

10.1155/2018/8284121 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Li-xin Yang ◽

Xiao-jun Liu

Keyword(s):

Fractional Order ◽

Equilibrium Points ◽

Fractional Order System ◽

Phase Portraits ◽

Order System ◽

Bifurcation Diagrams ◽

Compound Structure ◽

Dynamical Properties ◽

The Stability ◽

And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

Download Full-text

Recent Developments on the Stability and Control of Stochastic Systems

Mathematical Problems in Engineering ◽

10.1155/2015/454951 ◽

2015 ◽

Vol 2015 ◽

pp. 1-3

Author(s):

Quanxin Zhu ◽

Son Nguyen ◽

Ruihua Liu ◽

Leonid Shaikhet

Keyword(s):

Stochastic Systems ◽

Stability And Control ◽

Recent Developments ◽

The Stability ◽

And Control

Download Full-text

Completeness on the stability criterion of fractional order LTI systems

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2017-0008 ◽

2017 ◽

Vol 20 (1) ◽

Cited By ~ 25

Author(s):

Yiheng Wei ◽

Yuquan Chen ◽

Songsong Cheng ◽

Yong Wang

Keyword(s):

Fractional Order ◽

Stability Criterion ◽

Positive Definite Matrix ◽

Order System ◽

Linear Matrix ◽

Matrix Inequalities ◽

Stability Criteria ◽

The Stability ◽

Definition Of ◽

And Control

AbstractThe importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that many conclusions were drawn, but little consensus was reached. Consequently, there is an urgent need for a much deeper understanding of such a concept. With the definition of fractional order positive definite matrix, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before. All the results are formed in terms of linear matrix inequalities. Afterwards, a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc. Eventually, a simulation study is provided to validate the effectiveness of the obtained results.

Download Full-text

Robust stabilization and control using fractional order integrator

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331216645178 ◽

2016 ◽

Vol 39 (10) ◽

pp. 1559-1576 ◽

Cited By ~ 2

Author(s):

Amina Ben Hmed ◽

Messaoud Amairi ◽

Mohamed Aoun

Keyword(s):

Stability Boundary ◽

Robust Stabilization ◽

Smith Predictor ◽

Order System ◽

Stabilization Problem ◽

Stabilizing Controller ◽

First Order ◽

Analytic Expressions ◽

The Stability ◽

And Control

This paper addresses the robust stabilization problem of first-order uncertain systems. To treat the robust stabilization problem, an interval-based stabilization method using stability conditions of the non-commensurate elementary fractional transfer function of the second kind is developed. Some analytic expressions are determined to compute the set of all stabilizing controller parameters and plot the stability boundary. A robust performance control is also developed to fulfil some desired time-domain performances as the iso-overshoot property. The fractional controller can be used combined with the Smith predictor to control a first-order system with time delay and achieve desired specifications. Numerical examples are presented to illustrate the obtained results.

Download Full-text

Improved Statistical Linearization for Analysis and Control of Nonlinear Stochastic Systems: Part II: Application to Control System Design

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3139638 ◽

1981 ◽

Vol 103 (1) ◽

pp. 22-27 ◽

Cited By ~ 1

Author(s):

J. J. Beaman ◽

J. K. Hedrick

Keyword(s):

Stochastic Systems ◽

Pole Placement ◽

Linear Feedback ◽

Statistical Linearization ◽

Feedback Controllers ◽

Nonlinear Stochastic Systems ◽

Input Noise ◽

Gaussian Method ◽

And Control ◽

Digital Simulations

Using the techniques of Gaussian statistical linearization and the extension given in Part I, this paper describes the synthesis of linear feedback controllers for nonlinear stochastic systems. The method used is that of pole placement of the statistically linearized “eigenvalues”. The technique is described in terms of a design example, a position servomechanism with backlash, it is shown that for this type of system the standard Gaussian method works well for large input noise levels but can lead to an unstable design for low input levels. The extended fourth cumulant method is satisfactory for both cases studied. The results of the analysis are compared to Monte Carlo digital simulations to test their accuracy.

Download Full-text

Data-Driven Stability Assessment of Multilayer Long Short-Term Memory Networks

Applied Sciences ◽

10.3390/app11041829 ◽

2021 ◽

Vol 11 (4) ◽

pp. 1829

Author(s):

Davide Grande ◽

Catherine A. Harris ◽

Giles Thomas ◽

Enrico Anderlini

Keyword(s):

Neural Network ◽

Network Architecture ◽

Short Term Memory ◽

Moving Average ◽

Machine Learning Algorithms ◽

Physical Models ◽

Data Driven ◽

The Stability ◽

And Control ◽

Nonlinear Autoregressive

Recurrent Neural Networks (RNNs) are increasingly being used for model identification, forecasting and control. When identifying physical models with unknown mathematical knowledge of the system, Nonlinear AutoRegressive models with eXogenous inputs (NARX) or Nonlinear AutoRegressive Moving-Average models with eXogenous inputs (NARMAX) methods are typically used. In the context of data-driven control, machine learning algorithms are proven to have comparable performances to advanced control techniques, but lack the properties of the traditional stability theory. This paper illustrates a method to prove a posteriori the stability of a generic neural network, showing its application to the state-of-the-art RNN architecture. The presented method relies on identifying the poles associated with the network designed starting from the input/output data. Providing a framework to guarantee the stability of any neural network architecture combined with the generalisability properties and applicability to different fields can significantly broaden their use in dynamic systems modelling and control.

Download Full-text