Gaussian Noise Excitation Research Articles

Control and stochastic dynamic behavior of Fractional Gaussian noise-excited time-delayed inverted pendulum system

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H∈(1/2,1), we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic Itoˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

Model-Free Optimal Vibration Control of a Nonlinear System Based on Deep Reinforcement Learning

Optimal control of nonlinear vibration requires precise knowledge of the system and the solution to Hamilton–Jacobi–Bellman (HJB) equation. However, in practical engineering applications, acquiring precise system parameters poses challenges, and the analytical solutions for the HJB equation are difficult to obtain. In this paper, two reinforcement learning algorithms, Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm and Soft Actor-Critic (SAC) algorithm, are employed to train neural network-based optimal controllers for the van der Pol vibration system in the presence of unknown system parameters. To validate their performance, the controllers undergo testing in a series of experiments, including assessments of free vibration, frequency sweep excitation, and Gaussian noise excitation. The results indicate that both the TD3-trained and SAC-trained neural network-based controller are capable of proficiently suppress the vibration of the van der Pol oscillator. Additionally, these two model-free controllers can approximate the optimal control law which solved based on the dynamic model of the nonlinear system.

Dynamical responses of a Gaussian colored noise-driven shape memory alloy oscillator with a periodic force

For shape memory alloy (SMA) oscillator, the stochastic transitions between low-amplitude and undesirable high-amplitude oscillations will affect the safety and normal use of the system. Therefore, the nonlinear dynamical responses of a SMA oscillator with both periodic and Gaussian colored noise excitations are investigated in this paper. Firstly, to reveal the influences of random factors, a stochastic SMA model with a Gaussian colored noise is introduced. Subsequently, the approximate analytical solution of the steady-state amplitude response under the deterministic case is given by the averaging method, and the time-averaging mean-square response of the stochastic SMA system is obtained by the stochastic averaging method. The deduced approximated analytical solutions are verified by Monte Carlo numerical results, and a good consistency is observed. Finally, the effects of the random excitation and temperature on the steady-state responses of the SMA system are explored. We uncover that the temperature influences the frequency of bifurcation points, which can disrupt the stable structure of the SMA oscillator. In addition, noise intensity and correlation time of the colored noise can induce the occurrence of stochastic transitions between low-amplitude and high-amplitude oscillations, which are visible through the steady-state probability density and time history diagrams. All these obtained results may provide theoretical guidance and valuable insights for avoiding catastrophic stochastic transitions in SMA materials.

Stochastic dynamic analysis of nonlinear MDOF systems under combined Gaussian and Poisson noise excitation based on DPIM

Random vibration analysis of structures subjected to combined Gaussian and Poisson white noise excitation is a challenging issue. In this paper, a novel direct probability integral method (DPIM) is suggested to address stochastic responses and dynamic reliability of nonlinear multi-degree-of-freedom (MDOF) systems under combined excitation. Firstly, probability density integral equation (PDIE) of MDOF system under combined excitation is derived based on the principle of probability conservation. Then, DPIM is proposed to achieve the probability density function of stochastic response by solving deterministic motion equation of MDOF system and PDIE in sequence. To solve PDIE, two techniques, i.e., partition of input probability space and smoothing of Dirac delta function, are introduced. From the perspective of probability conservation, furthermore, the equivalent relationship between the PDIE and the corresponding probability density differential equation of a Markov system under combined Gaussian and Poisson noise is established. Since Dirac delta function is analytically integrated as Heaviside function, the first-passage dynamic reliability of MDOF system under combined excitation is readily evaluated by introducing the extreme value mapping of stochastic response. Finally, two nonlinear MDOF systems, including multiple-span bridge under combined vehicle load and non-stationary seismic excitation, are solved. Results demonstrate that the proposed DPIM is effective for random vibration and dynamic reliability analyses of MDOF structures excited by combined Gaussian and Poisson noise, and considering the randomness of vehicle load and seismic excitation simultaneously in the bridge design benefits the safe operation of bridge.

Response of Cantilever Model with Inertia Nonlinearity under Transverse Basal Gaussian Colored Noise Excitation

Considering the curvature nonlinearity and longitudinal inertia nonlinearity caused by geometrical deformations, a slender inextensible cantilever beam model under transverse pedestal motion in the form of Gaussian colored noise excitation was studied. Present stochastic averaging methods cannot solve the equations of random excited oscillators that included both inertia nonlinearity and curvature nonlinearity. In order to solve this kind of equations, a modified stochastic averaging method was proposed. This method can simplify the equation to an Itô differential equation about amplitude and energy. Based on the Itô differential equation, the stationary probability density function (PDF) of the amplitude and energy and the joint PDF of the displacement and velocity were studied. The effectiveness of the proposed method was verified by numerical simulation.

Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Ito stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.

Research on Output Characteristics of Double-Ended Fixed Beam Piezoelectric Energy Harvester Under Random Excitation

By use of the concept of impedance, this paper establishes a random energy harvester output model of double-ended fixed beam piezoelectric energy harvester under random White Gaussian noise excitation, which is based on the discussion of the output characteristics influenced by changing vibration beam shape and piezoelectric sheet length under random excitation, while improving piezoelectric structure of double-ended fixed beam piezoelectric energy harvester. It also deduces the output voltage, current and power of energy harvester with Fokker–Planck equation, which are verified by Monte Carlo simulation and experimental testing. It can be found that improving vibration beam shape can reduce resonant frequency and increase average output power of the system. The optimal value of piezoelectric sheet length can be different corresponding to different structures, which proves that double-ended fixed beam piezoelectric energy harvester under random White Gaussian noise excitation can achieve maximum output with optimal piezoelectric sheet length. In addition, in order to further improve the output performance of piezoelectric energy harvester under random excitation, it is necessary to increase the acceleration spectral density under random excitation and set an optimal value of the piezoelectric loading resistance, which is more suitable for energy harvester applied under random low-frequency environment.

A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitation

Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This non-locality is treated by applying an extension of the Novikov–Furutsu theorem and a novel approximation, employing a stochastic Volterra–Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of non-locality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious.

The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations

• A new stochastic two-degree-freedom airfoil (TDOFA) model was established. • A stochastic jump phenomenon was observed and effects of the colored noise were included. • A sliding mode control of the stochastic TDOFA system was investigated. • Based on the practical stability, the stability of the controlled stochastic TDOFA system was analyzed. This paper addresses a sliding mode control (SMC) for an airfoil model excited by a combination of harmonic force and colored Gaussian noise . Firstly, to reveal effects of random factors, the airfoil model with colored Gaussian noise is established. Next, via a perturbation technique and the stochastic averaging method, an analytical expression for the time-averaging mean square response is derived, which agrees well with results by Monte Carlo simulations . Additionally, we uncover that colored noise can induce a stochastic jump phenomenon, which can cause a catastrophic structural failure of the airfoil or even a disintegration of the aircraft. Subsequently, the SMC strategy is employed to design an effective controller for suppressing such a jump phenomenon of the stochastic airfoil system. In the case of the proposed stochastic airfoil system, we introduce concepts of ultimately reachability with an arbitrary small bound and a mean square practical stability to realize the reachability of the sliding mode and the stability of the system state. Finally, several numerical results are presented to demonstrate the effectiveness of the proposed SMC algorithm. We show that the jump phenomenon can be suppressed efficiently to avoid a catastrophic failure of the wing structure due to large deformation/deflection, and the energy cost is discussed to analyze the SMC approach.

Probabilistic solutions of nonlinear oscillators to subject random colored noise excitations

Random colored noise excitations, more close to real environmental excitations, are considered. In this paper, probabilistic solutions of nonlinear SDOF systems excited by colored noise are analyzed. With the aid of filtering method, the initial equation of motion is transferred to four coupled first-order SDEs. As a result, a Markov process of enlarged system responses comes into being, and traditional FPK equation method can be employed. However, this way makes the corresponding FPK equation become four-dimensional. In order to solve this problem, the developed and improved exponential polynomial closure (EPC) method is employed. In the solution procedure, an approximate exponential function is extended to include four state variables. Then additional unknown coefficients associated with extended state variables are produced, and the solution procedure becomes more complicated. On the other hand, since there is no exact solution, Monte Carlo simulation (MCS) is performed to verify the efficiency of the developed EPC method. Four examples associated with Gaussian and non-Gaussian colored noise excitations are considered. Numerical results show that the developed EPC method provides good agreement with the MCS method. Moreover, the effect of bandwidth for colored noise on system responses is taken into account.

Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations

In this paper, single-degree-of-freedom (SDOF) systems combined to Gaussian white noise and Gaussian/non-Gaussian colored noise excitations are investigated. By expressing colored noise excitation as a second-order filtered white noise process and introducing colored noise as an additional state variable, the equation of motion for SDOF system under colored noise is then transferred artificially to multi-degree-of-freedom (MDOF) system under white noise excitations with four-coupled first-order differential equations. As a consequence, corresponding Fokker-Planck-Kolmogorov (FPK) equation governing the joint probabilistic density function (PDF) of state variables increases to 4-dimension (4-D). Solution procedure and computer programme become much more sophisticated. The exponential-polynomial closure (EPC) method, widely applied for cases of SDOF systems under white noise excitations, is developed and improved for cases of systems under colored noise excitations and for solving the complex 4-D FPK equation. On the other hand, Monte Carlo simulation (MCS) method is performed to test the approximate EPC solutions. Two examples associated with Gaussian and non-Gaussian colored noise excitations are considered. Corresponding band-limited power spectral densities (PSDs) for colored noise excitations are separately given. Numerical studies show that the developed EPC method provides relatively accurate estimates of the stationary probabilistic solutions, especially the ones in the tail regions of the PDFs. Moreover, statistical parameter of mean-up crossing rate (MCR) is taken into account, which is important for reliability and failure analysis. Hopefully, our present work could provide insights into the investigation of structures under random loadings.

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.

Effects of distance-dependent delay on small-world neuronal networks.

We study firing behaviors and the transitions among them in small-world noisy neuronal networks with electrical synapses and information transmission delay. Each neuron is modeled by a two-dimensional Rulkov map neuron. The distance between neurons, which is a main source of the time delay, is taken into consideration. Through spatiotemporal patterns and interspike intervals as well as the interburst intervals, the collective behaviors are revealed. It is found that the networks switch from resting state into intermittent firing state under Gaussian noise excitation. Initially, noise-induced firing behaviors are disturbed by small time delays. Periodic firing behaviors with irregular zigzag patterns emerge with an increase of the delay and become progressively regular after a critical value is exceeded. More interestingly, in accordance with regular patterns, the spiking frequency doubles compared with the former stage for the spiking neuronal network. A growth of frequency persists for a larger delay and a transition to antiphase synchronization is observed. Furthermore, it is proved that these transitions are generic also for the bursting neuronal network and the FitzHugh-Nagumo neuronal network. We show these transitions due to the increase of time delay are robust to the noise strength, coupling strength, network size, and rewiring probability.

Stochastic averaging of quasi-non-integrable Hamiltonian systems under fractional Gaussian noise excitation

A stochastic averaging method for quasi-non-integrable Hamiltonian systems under excitation of fractional Gaussian noise (fGn) with Hurst index $$1/2<H<1$$ is proposed. First, the definitions and the basic properties of fGn and fractional Brownian motion are briefly introduced. Then, the averaged stochastic differential equation for the total energy of the quasi-non-integrable Hamiltonian system under fGn excitation is derived. The simulation and asymptotic analysis for two examples are conducted to illustrate the proposed stochastic averaging method. It is shown that the simulation results obtained from averaged equation and from original systems agree well and that the displacement response process of linear system to fGn has the same long-range dependence index and the same H self-similarity as those of the fGn excitation.

Simple nonlinearity evaluation and modeling of low-noise amplifiers with application to radio astronomy receivers

This paper describes a comparative nonlinear analysis of low-noise amplifiers (LNAs) under different stimuli for use in astronomical applications. Wide-band Gaussian-noise input signals, together with the high values of gain required, make that figures of merit, such as the 1 dB compression (1 dBc) point of amplifiers, become crucial in the design process of radiometric receivers in order to guarantee the linearity in their nominal operation. The typical method to obtain the 1 dBc point is by using single-tone excitation signals to get the nonlinear amplitude to amplitude (AM-AM) characteristic but, as will be shown in the paper, in radiometers, the nature of the wide-band Gaussian-noise excitation signals makes the amplifiers present higher nonlinearity than when using single tone excitation signals. Therefore, in order to analyze the suitability of the LNA's nominal operation, the 1 dBc point has to be obtained, but using realistic excitation signals. In this work, an analytical study of compression effects in amplifiers due to excitation signals composed of several tones is reported. Moreover, LNA nonlinear characteristics, as AM-AM, total distortion, and power to distortion ratio, have been obtained by simulation and measurement with wide-band Gaussian-noise excitation signals. This kind of signal can be considered as a limit case of a multitone signal, when the number of tones is very high. The work is illustrated by means of the extraction of realistic nonlinear characteristics, through simulation and measurement, of a 31 GHz back-end module LNA used in the radiometer of the QUIJOTE (Q U I JOint TEnerife) CMB experiment.

Study of the LTI relations between the outputs of two coupled Wiener systems and its application to the generation of initial estimates for Wiener–Hammerstein systems

This paper consists of two parts. In the first, more theoretic part, two Wiener systems driven by the same Gaussian noise excitation are considered. For each of these systems, the best linear approximation (BLA) of the output (in mean square sense) is calculated, and the residuals, defined as the difference between the actual output and the linearly simulated output is considered for both outputs. The paper is focused on the study of the linear relations that exist between these residuals. Explicit expressions are given as a function of the dynamic blocks of both systems, generalizing earlier results obtained by Brillinger [Brillinger, D. R. (1977). The identification of a particular nonlinear time series system. Biometrika, 64(3), 509–515] and Billings and Fakhouri [Billings, S. A., & Fakhouri, S. Y. (1982). Identification of systems containing linear dynamic and static nonlinear elements. Automatica, 18(1), 15–26]. Compared to these earlier results, a much wider class of static nonlinear blocks is allowed, and the efficiency of the estimate of the linear approximation between the residuals is considerably improved. In the second, more practical, part of the paper, this new theoretical result is used to generate initial estimates for the transfer function of the dynamic blocks of a Wiener–Hammerstein system. This method is illustrated on experimental data.

Non-linear dynamics of a stochastically excited beam system with impact

The response of non-linear, dynamic systems to stochastic excitation exhibits many interesting characteristics. In this paper, a strongly non-linear beam-impact system under both broad- and small-banded, Gaussian noise excitations is investigated. The response of this system is investigated both numerically, through a multi-degree-of-freedom model, and experimentally focusing on frequency-domain phenomena such as stochastic equivalents of harmonic and subharmonic solutions. An improved understanding of these stochastic response characteristics is obtained by comparing these to non-linear periodic response features of the system. It will be shown that in modelling such a continuous, linear system with a local non-linearity, the linear part can be effectively reduced to a description based on several modes. Combining this reduced, linear part with the local non-linearity in a reduced, non-linear model is shown to result in a non-linear model, which can be used to accurately predict the stochastic response characteristics of the original, continuous, non-linear system. It is shown that including more modes to the model causes its response to differ significantly from that of a single-degree-of-freedom model and show a better correspondence with experimental results, also in the frequency range of the first mode.

Blind channel estimation and echo cancellation using chaotic coded signals

A novel dynamic-based deconvolution (DBD) algorithm is presented that can be exploited in blind channel estimation as well as in telephony echo cancellation. Chaotic coded signals generated by the logistic map are employed while the channel is represented by an autoregressive model. The applicability is demonstrated in a speech transmission scenario using chaotic coded speech showing a modeling misadjustment improvement of 24 dB/100 iterations which is sixfold that obtained by the LMS for a 128 tap digital adaptive filter driven by ideal white Gaussian noise excitation.

Experimental and Numerical Analysis of Nonlinear Phenomena in a Stochastically Excited Beam System with Impact

The response of strongly nonlinear dynamic systems to stochastic excitation exhibits many interesting characteristics. In this paper, a strongly nonlinear beam-impact system under broad and small banded, Gaussian noise excitations is investigated. The response of this system is investigated numerically as well as experimentally. The emphasis lies on frequency domain characteristics. Phenomena like multiple resonance frequencies and stochastic equivalents of harmonic and subharmonic solutions are found. Improved understanding of these stochastic response characteristics is obtained by comparing them to nonlinear periodic response features of the system. It is shown that these stochastic response phenomena can provide information on the periodic response characteristics of the system. The observed stochastic characteristics have more general value and meaning than mere application to this system suggests.

Envelope correlation function of the narrow-band non-Gaussian process

The stationary response of oscillators possessing linear damping and non-linear stiffness to the white Gaussian noise excitation is considered. The original stochastic differential equation (SDE) is used to establish relations between the system parameters and the envelope correlation function. The approximation approach is based on the moment approximation and replacement of the envelope generating SDE by the new one with analytical solution of the corresponding Fokker-Planck equation. The usefulness of the method is illustrated by application to the randomly driven Duffing oscillator.