Stochastic Averaging for Asymptotic Stability

2012 ◽  
pp. 21-55
Author(s):  
Shu-Jun Liu ◽  
Miroslav Krstic
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Averaging

Asymptotic Stability of Controlled Nonlinear Stochastic Systems Considering the Dynamics of Sensors and Actuators

2021 ◽  
pp. 2150180
Author(s):  
Jiaojiao Sun ◽  
Zuguang Ying ◽  
Ronghua Huan ◽  
Weiqiu Zhu
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stable Region ◽  
High Dimensional ◽  
Nonlinear Stochastic System ◽  
Dimensional System ◽  
Sensors And Actuators ◽  
Controlled System

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. In this paper, asymptotic stability of trivial solutions of a controlled nonlinear stochastic system considering the dynamics of sensors and actuators is investigated. Considering the inherent and intentional nonlinearities and random loadings, the coupled dynamic equations of the controlled system with sensors and actuators are given, which are further formulated by a controlled, randomly excited, dissipated Hamiltonian system. The Hamiltonian of the controlled system is introduced, and, based on the stochastic averaging method, the original high-dimensional system is reduced to a one-dimensional averaged system. The analytical expression of Lyapunov exponent of the averaged system is derived, which gives the approximately necessary and sufficient condition of the asymptotic stability of trivial solutions of the original high-dimensional system. The validation of the proposed method is demonstrated by a four-degree-of-freedom controlled system under pure stochastically parametric excitations in detail. A comparative analysis, which is related to the stochastic asymptotic stability of the system with and without considering the dynamics of sensors and actuators, is carried out to investigate the effect of their dynamics on the motion of the controlled system. Results show that ignoring the dynamics of sensors and actuators will get a shrink stable region of the controlled system.

Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation

1992 ◽  
Vol 59 (3) ◽  
pp. 664-673 ◽  
Author(s):  
S. T. Ariaratnam ◽  
Wei-Chau Xie
Keyword(s):  
Asymptotic Stability ◽  
Degrees Of Freedom ◽  
Elastic Beam ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
System Parameters ◽  
Asymptotic Expressions ◽  
Small Intensity ◽  
Central Cross Section ◽  
Torsional Instability

The almost-sure asymptotic stability of a class of coupled multi-degrees-of-freedom systems subjected to parametric excitation by an ergodic stochastic process of small intensity is studied. Explicit asymptotic expressions for the largest Lyapunov exponent for various values of the system parameters are obtained by using a combination of the method of stochastic averaging and a well-known procedure due to Khas’minskii, from which the asymptotic stability boundaries are determined. As an application, the example of the flexural-torsional instability of a thin elastic beam acted upon by a stochastically fluctuating load at the central cross-section of the beam is investigated.

Stochastic Response and Stability Analysis of Single Leg Articulated Tower

2003 ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Asymptotic Stability ◽  
Probability Density ◽  
Parametric Excitation ◽  
Stochastic Averaging ◽  
Stochastic Response ◽  
Sea State ◽  
Stationary Response ◽  
Stability In Probability ◽  
Hydrodynamic Damping ◽  
Asymptotic Stability In Probability

The stationary response and asymptotic stability in probability of an articulated tower under random wave excitation are investigated. The articulated tower is modelled as a SDOF system having stiffness nonlinearity, damping nonlinearity and parametric excitation. Using a stochastic averaging procedure and Fokker-Plank-Kolomogorov equation (FPK), the probability density function of the stationary solution is obtained for random sea state represented by a P-M sea spectrum. The method involves a Van-Der-Pol transformation of the nonlinear equation of motion to convert it to the Ito’s stochastic differential equation with averaged drift and diffusion coefficients. The asymptotic stability in probability of the system is investigated by obtaining the averaged Ito’s equation for the Hamiltonian of the system. The asymptotic stability is examined approximately by investigating the asymptotic behaviour of the diffusion process Y(t) at its two boundaries Y = 0 and ∞. As an illustrative example, an articulated tower in a sea depth of 150 m is considered. The tower consists of hollow cylinder of varying diameter along the height, providing the required buoyancy of the system. Wave forces on the structure are calculated using Morrison’s equation. The stochastic response and the stability conditions are obtained for a sea state represented by P-M spectrum with 16m significant wave height. The results of the study indicate that the probability density of the stationary response obtained by the stochastic averaging procedure is in very good agreement with that obtained from digital simulation. Further, the articulated tower is found to be asymptotically stable under the parametric excitation arising due to hydrodynamic damping.

scholarly journals Asymptotic Stability of Delay-Controlled Nonlinear Stochastic Systems with Actuator Failures

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
N. Zhou ◽  
R. H. Huan
Keyword(s):  
Time Delay ◽  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Stochastic Averaging ◽  
Largest Lyapunov Exponent ◽  
Control Force ◽  
Nonlinear Stochastic Systems ◽  
Actuator Failures

The problem of asymptotic stability of delay-controlled nonlinear stochastic systems with actuator failures is investigated in this paper. Such a system is formulated as a continuous-discrete hybrid system based on the random switch model of failure-prone actuator. Time delay control force is converted into delay-free one by randomly periodic characteristic of the system. Using limit theorem and stochastic averaging, an approximate formula for the largest Lyapunov exponent of the original system is then derived, from which necessary and sufficient conditions for asymptotic stability are obtained. The validity and utility of the proposed procedure are demonstrated by using a stochastically driven nonlinear two-degree system with time delay feedback and actuator failure.

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