Nonstationary Response of Optimal Controlled Stochastic Van Der Pol Oscillator

2014 ◽  
Vol 875-877 ◽  
pp. 2000-2005
Author(s):  
Lu Yuan Qi ◽  
Wei Xu ◽  
Wei Ting Gao
Keyword(s):  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Van Der Pol Oscillator ◽  
Kolmogorov Equation ◽  
System Response ◽  
Control Law ◽  
Dynamical Programming ◽  
Control Constraint ◽  
Van Der Pol ◽  
Ito Equation

A procedure to calculate the transient response of optimal controlled stochastic Van Der Pol oscillator is proposed. The stochastic averaging method is employed to obtain a partially averaged Itô equation for the amplitude process. The dynamical programming equation is adopted to minimize the system response. An optimal control law with a control constraint is established. The completed averaged Itô equation is obtained. The transient probability density function is solved from Fokker-Planck-Kolmogorov equation by Galerkin method. Results obtained show the proposed method is accurate. The effective of the control strategy is significant.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

scholarly journals Applying higher order stochastic averaging method to the Van der Pol system with timedelay under random excitation

2019 ◽  
Vol 2 (2) ◽  
pp. 102-109
Author(s):  
Hao Ngoc Duong ◽  
Anh Dong Nguyen ◽  
Dung Quang Nguyen
Keyword(s):  
Time Delay ◽  
Averaging Method ◽  
Order Approximation ◽  
Original System ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Van Der Pol ◽  
System With Time Delay

The paper investigated the Van der Pol system with time-delay under random excitation by the higher stochastic averaging method. The original system was expressed in terms without time-delay under the assumption that the state variabled of the system were slowly varying processed. Then the higher stochastic averaging method was applied on the approximation system. By this technique, the analytical expression of the stationary probability density function for the Van der Pol system with time-delay under random excitation was showed in higher order approximation for the first time. Effects of the parameter time-delay on the system’s response were investigated. The analytical results were suited well to numerical ones obtained by Monte-Carlo method. It was also showed that the higher order averaging solution was better than the one obtained by the traditional stochastic averaging method.

scholarly journals Optimal Vibration Control of a Class of Nonlinear Stochastic Systems with Markovian Jump

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
R. H. Huan ◽  
R. C. Hu ◽  
D. Pu ◽  
W. Q. Zhu
Keyword(s):  
Optimal Control ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Control Force ◽  
Dynamical Programming ◽  
Markovian Jump ◽  
Finite Set ◽  
Time Optimal ◽  
Set Up ◽  
Ito Equation

The semi-infinite time optimal control for a class of stochastically excited Markovian jump nonlinear system is investigated. Using stochastic averaging, each form of the system is reduced to a one-dimensional partially averaged Itô equation of total energy. A finite set of coupled dynamical programming equations is then set up based on the stochastic dynamical programming principle and Markovian jump rules, from which the optimal control force is obtained. The stationary response of the optimally controlled system is predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Itô equation. Two examples are worked out in detail to illustrate the application and effectiveness of the proposed control strategy.

Stochastic Stability of Some Hamiltonian Systems

2002 ◽  
Author(s):  
F. Jedrzejewski
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Stability ◽  
Stochastic Modelling ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Van Der Pol Oscillator ◽  
Stochastic Fluctuations ◽  
The Stability ◽  
Averaged Hamiltonian

Stochastic stability plays an important role in modern theories of nonlinear structural dynamics. Recently, more realistic models based on stochastic modelling and Itoˆ calculus, like flow induced vibrations and seismic excitations have been proposed. In this paper, the almost-sure asymptotic stability of some hamiltonian systems subjected to stochastic fluctuations is investigated. Dynamical systems are reduced to Itoˆ stochastic differential equations for the averaged hamiltonian by using a new stochastic averaging method. The stability of the original system is determined approximately by examining the behavior of the averaged hamiltonian. Analytical expressions for the stochastic stability exponents are obtained. The proposed procedure is illustrated on the Rayleigh Van der Pol Oscillator.

scholarly journals Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

2021 ◽  
pp. 125-125
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Qixun Lan ◽  
Yujie Cai ◽  
Huafeng Xu ◽  
...  
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Original System ◽  
Stochastic Averaging ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Damping ◽  
Transition Set ◽  
Fractional Order Controller

The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the nonlinear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary Probability Density Function (PDF) is obtained by stochastic averaging. And then according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary PDF curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems

1997 ◽  
Vol 64 (1) ◽  
pp. 157-164 ◽  
Author(s):  
W. Q. Zhu ◽  
Y. Q. Yang
Keyword(s):  
Nonlinear System ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Present Method ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
System Method

A stochastic averaging method is proposed to predict approximately the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable Hamiltonian systems with lightly linear and (or) nonlinear dampings and subject to weakly external and (or) parametric excitations of Gaussian white noises). According to the present method, a one-dimensional approximate Fokker-Planck-Kolmogorov equation for the transition probability density of the Hamiltonian can be constructed and the probability density and statistics of the stationary response of the system can be readily obtained. The method is compared with the equivalent nonlinear system method for stochastically excited and dissipated nonintegrable Hamiltonian systems and extended to a more general class of systems. An example is given to illustrate the application and validity of the present method and the consistency of the present method and the equivalent nonlinear system method.

scholarly journals Van Der Pol-Duffing oscillator under combined harmonic and random excitations

2014 ◽  
Vol 36 (3) ◽  
pp. 161-172 ◽  
Author(s):  
N. D. Anh ◽  
V. L. Zakovorotny ◽  
D. N. Hao
Keyword(s):  
Duffing Oscillator ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Linearization Method ◽  
Auxiliary Function ◽  
Equivalent Linearization ◽  
Van Der Pol ◽  
Averaged Equations ◽  
Equivalent Linearization Method ◽  
A New Technique

A new technique is proposed to investigate the response of Van der Pol-Duffing (V-D for short) oscillator to a combination of harmonic and random excitations in the primary resonant frequency region. The analytical approach is based on the stochastic averaging method and equivalent linearization method. The stochastic averaging is applied to the original equation transformed into Cartesian coordinates. Then the resulting nonlinear averaged equations are linearized by the equivalent linearization method so that the equations obtained can be solved exactly by the technique of auxiliary function. Numerical results show that the proposed approximate technique is an effective approach to solving the V-D equation. Although the technique has been used for the V-D equation in the paper, however, it can also be used to solve many other nonlinear oscillators.

Stochastic optimal semi-active control of stay cables by using magneto-rheological damper

2010 ◽  
Vol 17 (13) ◽  
pp. 1921-1929 ◽  
Author(s):  
M Zhao ◽  
WQ Zhu
Keyword(s):  
Active Control ◽  
Control Strategy ◽  
Mr Damper ◽  
Stochastic Averaging ◽  
Control Law ◽  
Control Force ◽  
Stay Cable ◽  
Dynamical Programming ◽  
Magneto Rheological ◽  
Bang Bang Control

Stochastic optimal semi-active control for stay cable multi-mode vibration attenuation by using magneto-rheological (MR) damper is developed. The Bingham model for an MR damper is used. The force produced by an MR damper is split into passive and active parts. The passive part is combined with structural damping forces into effective damping forces. The partially averaged Itô stochastic differential equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable Hamiltonian systems. Then the dynamical programming equation for controlled modal energies with an index involving control force is established by applying the stochastic dynamical programming principle, and a stochastic optimal semi-active control law is obtained by solving the dynamical programming equation. For controlled modal energies with an index not involving control force, bang-bang control law is obtained without solving a dynamical programming equation. A comparison between the two control laws shows that the stochastic optimal semi-active control strategy is superior to the bang-bang control strategy in the sense of higher control effectiveness and efficiency and less chattering.

Stochastic Averaging of Quasi-Nonintegrable-Hamiltonian Systems Under Poisson White Noise Excitation

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Y. Zeng ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Degree Of Freedom ◽  
Kolmogorov Equation ◽  
Transition Probability Density ◽  
Poisson White Noise ◽  
Van Der Pol Oscillators

A stochastic averaging method for predicting the response of multi-degree-of-freedom quasi-nonintegrable-Hamiltonian systems (nonintegrable-Hamiltonian systems with lightly linear and (or) nonlinear dampings subject to weakly external and (or) parametric excitations of Poisson white noises) is proposed. A one-dimensional averaged generalized Fokker–Planck–Kolmogorov equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and two-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.

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