Plank Kolmogorov Equation Research Articles

The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

AbstractConsidering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Dynamics of a Nonlinear Energy Harvesting System in Time-Delayed Feedback Control under Stochastic Excitations

A time-delayed feedback control is applied to a nonlinear piezoelectric energy harvesting system excited by additive and multiplicative Gaussian white noises to improve its energy harvesting performance. An equivalent decoupling system can be obtained by using a variable transformation. Based on the standard stochastic averaging method, the Fokker–Plank–Kolmogorov equation and the stationary probability density functions of the amplitude, displacement, and velocity of the harvester are obtained, respectively. In addition, the approximate expressions of mean square electric voltage and the mean extracted output power are derived. Finally, the paper explores the influences of parameters on the mean square electric voltage. The results show that noise intensity, time delay, feedback strength, time constant ratio, and coupling coefficients have great influences on the mean square electric voltage. The accuracy of the theoretical method is verified by the Monte Carlo simulation.

Probabilistic analysis and ghost-stochastic resonance of a hybrid energy harvester under Gaussian White noise

In this present work, a hybrid energy scavenger using two mechanisms of transduction namely piezoelectric and electromagnetic and subjected to the Gaussian white noise is investigated. The stochastic averaging method is used here in other to construct the Fokker–Plank–Kolmogorov equation of the system whose the statistic response in the stationary state is the probability density. The mean square voltage and current are obtained for different value of white noise intensities as the output power generated by piezoelectric circuit and electromagnetic circuit. In addition, combining the Gaussian white noise and coherence excitation, the Ghost-Stochastic resonance is observed through the mean residence time and improve the amount of energy harvested by the scavenger. The agreement between the analytical method and those obtained numerically validates the effectiveness of analytical investigations. The results obtained in this manuscript reveal that, while the natural frequency is absent in the external coherent force, the amount of energy harvested by the energy scavenging device can be improved for certain value of noise intensity.

Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations

Hertz contact model has not included the impact of inelastic, while the modified Hertz contact models are too complicated to handle. The classical impact model only uses a material-related constant to describe the relationship between the velocity before and after a collision, without considering the energy dissipation mechanism in the impact process. On the other hand, many recent experiments have shown that the restitution factor was a function of impact velocity. Such impact velocity dependent restitution factor offers a comprehensive and reasonable impact model. In the present paper, the random vibration of single-degree-of-freedom (SDOF) vibro-impact oscillators with the restitution factor related to velocity under wide-band noise excitations is investigated. The given stochastic vibro-impact oscillator is firstly transformed into a nonlinear system without impact by a non-smooth transformation. Then, a Markov approximation is applied to obtain the Fokker–Plank-Kolmogorov (FPK) equation governing the evolution of the probability density function (PDF) of the energy envelope. The stationary PDFs of the energy envelope and amplitude envelope are solvable in a closed-form. Finally, two examples are given to verify the effectiveness of the proposed method. The results showed that the classical impact model can simplify the calculations to some extent. However, the error caused by this method cannot be ignored. The revised model presented in this paper is more in line with the actual working conditions. For complex dynamical system, especially for multi-point impact vibration systems, the work of this paper is helpful to the optimization of the system.

Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

An optimal bounded control strategy for strongly nonlinear vibro-impact systems under stochastic excitations with actuator saturation is proposed. First, the impact effect is incorporated in an equivalent equation by using a nonsmooth transformation. Under the assumption of light damping and weak random perturbation, the system energy is a slowly varying process. By using the stochastic averaging of envelope for strongly nonlinear systems, the partially averaged Itô stochastic differential equation for system energy can be derived. The optimal control problem is transformed from the original optimal control problem for the state variables to an equivalent optimal control problem for the system energy, which decreases the dimensions of the optimal control problem. Then, based on stochastic maximum principle, an adjoint equation for the adjoint variable and the maximum condition of partially averaged control problem are established. For infinite time-interval ergodic control, the adjoint variable is assumed to be a stationary process and the adjoint equation can be further simplified. Finally, the probability density function of the system energy and other statistics of the optimally controlled system are derived by calculating the associated Fokker–Plank–Kolmogorov equation. For comparison, the bang–bang control is also investigated and the control results are compared to show the advantages of the developed control strategy.

Influence of restricted operating space on the performance of random vibration energy harvesting

The ambition to create self-powered microscale electrical devices motivates scientific and industrial communities to investigate the energy harvesting technique, especially working in random vibration circumstances. The mechanical response of the random vibration system may approach infinity with small probability, and then the restricted operating space of the energy harvesting system will unavoidably induce the occurrence of collision interaction. Here, the random mechanical vibration and electrical output of the vibration energy harvesting system including inelastic collision are investigated, in which the random excitation is described by Gaussian white noise, while the collision interactions are described by the transient impact model and inelastic contact model, respectively. Introducing the generalized harmonic transformation of mechanical states and adopting a slow-varying process assumption of amplitude and averaged frequency, the output voltage can be explicitly expressed as the function of displacement, velocity, and system total energy by directly integrating the linear electrical equation. The transient impact interaction is equivalent to an effective damping with energy-dependent damping coefficient, while the inelastic contact interaction is equivalent to an effective damping and an affiliated potential energy. The averaged equations with respect to mechanical energy are then derived through the stochastic averaging technique. The stationary probabilistic density function of mechanical states is established by solving the reduced Fokker–Plank–Kolmogorov equation, and then the statistical quantities of electrical voltage are obtained by the relation between voltage and mechanical states. The effectiveness and precision of the analytical procedure are validated through the results from Monte Carlo simulations, and the influence of collision interaction on the performance of energy harvesting is discussed in detail. Also, for the energy harvesting system excited by colored noise, the influence of collision interaction on the performance is evaluated through Monte Carlo simulations.

Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise

In this paper, the stochastic bifurcations and the performance analysis of a strongly nonlinear tri-stable energy harvesting system with colored noise are investigated. Using the stochastic averaging method, the averaged Fokker–Plank–Kolmogorov equation and the stationary probability density (SPD) of the amplitude are obtained, respectively. Meanwhile, the Monte Carlo simulations are performed to verify the effectiveness of the theoretical results. D-bifurcation is studied through the largest Lyapunov exponent calculations, which implies the system undergoes D-bifurcation twice with increasing the nonlinear stiffness coefficients. The effects of the nonlinear stiffness coefficients, noise intensity and correlation time on P-bifurcation are discussed by the qualitative changes of the SPD. Moreover, the relationship between D- and P-bifurcation is explored. If the strength of stochastic jump has obvious gap with respect to the two statuses before and after the occurrence of P-bifurcation, the D-bifurcation will happen, too. Finally, the performance and the capability of harvesting energy from ambient random excitation are analyzed.

Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Ito equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.

Probabilistic behavior analysis of a sandwiched buckled beam under Gaussian white noise with energy harvesting perspectives

In this paper, a sandwiched buckled beam with axial compressive force under Gaussian white noise is considered as a piezoelectric energy harvester. A stochastic averaging method is proposed to analytically predict the system’s response, the stability and the estimation of system’s reliability. By using the generalized harmonic transformation, the Itô differential equations with respect to the mechanical and electrical amplitude are derived through this technique. From these differential equations, we construct the Fokker–Plank–Kolmogorov equation for the electrical and mechanical subsystem where the solution of each equation in the stationary state is a probability density. The mean first passage time (MFPT) is numerically provided in order to study the attractor stability(stable equilibrium point observed in the effective potential) which give rise to the noise-enhanced stability(NES) phenomenon. The mean square response and voltage are obtained for different white noise intensities and others system parameters. The effects of linear damping and noise intensity on the mean square voltage are investigated. We notice that harvested energy can be enhanced by suitable choice of noise intensity and others system parameters. In additional, by combining the random signal with harmonic excitation, the stochastic resonance(SR) phenomenon is observed via the mean residence time(TMR) which give rise to the large amplitude of vibrations and consequently, an optimization of harvested energy. The agreements between the analytical method and those obtained numerically validate the effectiveness of analytical investigations.

Stochastic averaging for nonlinear vibration energy harvesting system

A stochastic averaging technique for the nonlinear vibration energy harvesting system to Gaussian white noise excitation is developed to analytically evaluate the mean-square electric voltage and mean output power. By introducing the generalized harmonic transformation, the influence of the external circuit on the mechanical system is equivalent to a quasi-linear stiffness and a quasi-linear damping with energy-dependent coefficients, and then the equivalent nonlinear system with respect to the mechanical states is completely established. The Ito stochastic differential equation with respect to the mechanical energy of the equivalent nonlinear system is derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the stationary probability density of the mechanical states, and then the mean-square electric voltage and mean output power are analytically obtained through the approximate relation between the electric quantity and the mechanical states. The agreements between the analytical results and those from the moment method and from Monte Carlo simulations validate the effectiveness of the proposed technique.

Constructing transient response probability density of non-linear system through complex fractional moments

The probability density function for transient response of non-linear stochastic system is investigated through the stochastic averaging and Mellin transform. The stochastic averaging based on the generalized harmonic functions is adopted to reduce the system dimension and derive the one-dimensional Itô stochastic differential equation with respect to amplitude response. To solve the Fokker–Plank–Kolmogorov equation governing the amplitude response probability density, the Mellin transform is first implemented to obtain the differential relation of complex fractional moments. Combining the expansion form of transient probability density with respect to complex fractional moments and the differential relations at different transform parameters yields a set of closed-form first-order ordinary differential equations. The complex fractional moments which are determined by the solution of the above equations can be used to directly construct the probability density function of system response. Numerical results for a van der Pol oscillator subject to stochastically external and parametric excitations are given to illustrate the application, the convergence and the precision of the proposed procedure.

Optimal bounded control of quasi-nonintegrable Hamiltonian systems using stochastic maximum principle

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation associated with the fully averaged Ito equation of the controlled system. For comparison, the bang–bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang–bang control at the price of slightly less control effectiveness.

Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

Energy harvesting of monostable Duffing oscillator under Gaussian white noise excitation

Energy harvesting of monostable Duffing oscillator with piezoelectric coupling under Gaussian white noise excitation is investigated. Based on the Fokker–Plank–Kolmogorov equation of piezoelectric coupling systems, the statistical moments of the response are derived from the Van Kampen expansion. The effects of the spectral density of the random excitation and the coefficient of cubic nonlinearity on the expected response moments are analyzed. Some numerical examples are presented to demonstrate the effects of excitation spectral density, coefficient of cubic nonlinearity and initial conditions on the output voltage.

Random response of vibro-impact systems with inelastic contact

The probability density and statistics of random response are investigated for vibro-impact systems with inelastic contact in this paper. The inelastic contact is described through an empirical model incorporating highly non-linear and inelastic mechanical property of the impact interface, and the system response is determined through the stochastic averaging and energy dissipation balance technique. The inelastic contact force is separated into an elastic restoring component and a dissipative component; the former is expressed as the gradient of the potential energy preserved in the system while the latter is approximated by a damping force with energy-dependent damping coefficient through energy dissipation balance technique. The averaged Itô stochastic differential equation with respect to system total energy is then derived through stochastic averaging. The solution of the associated Fokker–Plank–Kolmogorov equation yields stationary probability density of system total energy, joint probability density of system displacement and velocity, as well as statistics of system response. The agreement between analytical results and Monte-Carlo simulations validates effectiveness of the proposed technique.

Response of Duffing system with delayed feedback control under bounded noise excitation

This paper presents a procedure for predicting the response of Duffing system with time-delayed feedback control under bounded noise excitation by using stochastic averaging method. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Ito stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker–Plank–Kolmogorov equation associated with the averaged Ito equations. It is shown that the time delay in feedback control will deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.

On intentional introduction of stiffness nonlinearities for energy harvesting under white Gaussian excitations

A significant body of the open literature on vibratory energy harvesting is currently focused on the concept of purposeful inclusion of stiffness nonlinearities for broadband transduction. When compared to their linear resonant counterparts, nonlinear energy harvesters have a wider steady-state frequency bandwidth, leading to the idea that they can be utilized to improve performance especially in random and non-stationary vibratory environments. To further investigate this common belief, this paper studies the response of vibratory energy harvesters to white Gaussian excitations. Both mono- and bi-stable piezoelectric Duffing-type harvesters are considered. The Fokker–Plank–Kolmogorov equation governing the evolution of the system’s transition probability density function is formulated and used to generate the moment differential equations governing the response statistics. The moment equations are then closed using a fourth-order cumulant-neglect closure scheme and the relevant steady-state response statistics are obtained. It is demonstrated that the energy harvester’s time constant ratio, i.e., the ratio between the nominal period of the mechanical subsystem and the time constant of the harvesting circuit, plays a critical role in characterizing the performance of nonlinear harvesters in a random environment. When the time constant ratio is large, stiffness-type nonlinearities have very little influence on the voltage response. In such a case, no matter how the potential function of the harvester is altered, it does not affect the average output power of the device. When the time constant ratio is small, the influence of the nonlinearity on the voltage output becomes more prevalent. In this case, a Duffing-type mono-stable harvester can never outperform its linear counterpart. A bi-stable harvester, on the other hand, can outperform a linear harvester only when it is designed with the proper potential energy function based on the known noise intensity of the excitation. Such conclusions hold for harvesters with nonlinearities appearing in the restoring force.

Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree

The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.

Time-delayed feedback control optimization for quasi linear systems under random excitations

A strategy for time-delayed feedback control optimization of quasi linear systems with random excitation is proposed. First, the stochastic averaging method is used to reduce the dimension of the state space and to derive the stationary response of the system. Secondly, the control law is assumed to be velocity feedback control with time delay and the unknown control gains are determined by the performance indices. The response of the controlled system is predicted through solving the Fokker–Plank–Kolmogorov equation associated with the averaged It? equation. Finally, numerical examples are used to illustrate the proposed control method, and the numerical results are confirmed by Monte Carlo simulation.

State Analysis of Hidden Markov Models Governed by Special Jump Processes

This paper investigates a class of stochastic differential systems with random structure, the transitions of which are generated by the special Markov jump processes. A statement concerning the Markovian property of the couple “jump process–governed diffusion" is presented. An analogue of the Fokker–Plank–Kolmogorov equation in the form of a system of partial integro-differential equations, describing the evolution of the mutual transition probability of this couple, is derived.