System Of Stochastic Equations Research Articles

<inline-formula> <tex-math notation="LaTeX">$H_{\infty}$ </tex-math> </inline-formula> Fault Estimation for 2-D Linear Discrete Time-Varying Systems Based on Krein Space Method

This paper addresses the finite horizon ${H_\infty }$ fault estimation problem for 2-D linear discrete time-varying systems with bounded unknown input and measurement noise. The main contribution of this paper is the ${H_\infty }$ fault estimator for 2-D systems with a necessary and sufficient existence condition. By introducing a partially equivalent stochastic dynamic system in Krein space, the necessary and sufficient condition for the existence of the ${H_\infty }$ fault estimator is derived based on innovation analysis and projection formula in Krein space. Then, the solution of the estimator is achieved by means of a Riccati-like difference equation for 2-D systems. Finally, a thermal process example is given to demonstrate the effectiveness of the proposed method.

Stability of Random Processes with the 1/f α Spectrum

Extreme fluctuations have been modeled based on point nonpotential and potential systems of stochastic equations, in which white noise induces random processes with a power-law frequency dependence of the power spectra. The distribution of extreme fluctuations corresponds to the maximum of statistical entropy, which indicates their stability. The stability of fluctuation processes with 1/f α power spectra has been analyzed based on the principle of maximum information entropy.

Stochastic models for forward systems of nonlinear parabolic equations

Stochastic differential equations give a constructive way to obtain a large number of stochastic processes. They can be used as well to simulate these processes numerically. As a by-product we can construct numerically solutions of the Cauchy problem for some parabolic equations and systems. In this paper we derive systems of stochastic equations for stochastic processes underlying systems of coupled parabolic equations which are particular cases of cross-diffusion systems of PDEs. Stochastic processes satisfying these stochastic systems allow to obtain probabilistic representations of classical and weak solutions of the Cauchy problem for original PDE systems and can be used to simulate the PDE solutions.

Set stability and synchronization of logical networks with probabilistic time delays

Using the algebraic state space representation (ASSR) method, this paper investigates the set stability and synchronization of Boolean networks with probabilistic time delays (PTDs). Firstly, an equivalent stochastic system is established for the Boolean network with PTDs by using the ASSR method. Secondly, based on the probabilistic state transition matrix of equivalent stochastic system, a necessary and sufficient condition is proposed for the set stability of Boolean networks with PTDs. Thirdly, as an application of set stability, the synchronization of coupled Boolean networks with PTDs is studied, and a necessary and sufficient condition is presented. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained new results.

Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay

Dynamics of a stochastic Logistic model with distributed delay are considered. We first transfer a scalar stochastic Logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique. Then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one. Moreover, in the case of persistence, we prove that there exists a unique stationary distribution by the Markov semigroups theory. The results show that, the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution, though for the deterministic model with strong kernel or weak kernel, the average delay may induce the existence of a group of small amplitude periodic solutions.

Determination of preferential exponent [formula omitted] in random processes with a [formula omitted] power spectrum

The stability of fluctuation processes with 1∕fα power spectra in systems of stochastic equations simulating coupled phase transitions has been analyzed with the use of the maximum information entropy principle. It is shown that in the class of processes described by a nonpotential system of model equations fluctuations with a 1/f power spectrum, i.e. with the exponent α=1, are preferential, Preferential for the class of potential systems under consideration are fluctuations with a 1∕fα spectrum at α=1.3. The spectral entropy of the random processes has been calculated. Calculation of the spectral entropy makes it possible to investigate the stability of random processes directly from the power spectra, without calculating the probability density functions. The dependence of the spectral entropies on the amplitude of white noise has a minimum whose position corresponds to the critical behavior.

Enhanced stochastic averaging of non-integrable nonlinear systems subjected to stochastic excitations

Stochastic averaging method based on energy envelope for a non-integrable system reduces the dimension of dynamic systems, while preserving the nonlinearity in system behavior and its stochasticity arising from input excitations. However, this approach for a general system neglects effects of off-diagonal damping components, considers a lumped effect of diagonal damping entries, and assumes independent stochastic excitations for involved degrees of freedom. These constraints may limit the application of stochastic averaging as nonlinear systems often do not satisfy these conditions. This paper addresses these shortcomings and proposes a new strategy to derive an equivalent nonlinear stochastic system. For this purpose, an equivalent excitation intensity and modified damping parameters are derived by equating drift and diffusion components of the modified and the original system through the method of weighted residuals using high order moments of system velocities. The proposed approach, called enhanced stochastic averaging, is demonstrated for a one-story building on a raft foundation in loose sand subjected to lateral stochastic excitations. Results, such as probability density function at different intensity levels, indicate that the enhanced stochastic averaging method significantly improves the analytical predictions of the probabilistic properties of system responses compared to the conventional stochastic averaging method.

Stochastic analysis of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation

To the best of authors’ knowledge, the dynamical behaviors of vibration energy harvesters with fractional derivative damping have not been discussed by researchers with the help of the stochastic averaging method. As the fractional-order models are more accurate than the classical integer-order models, so it is necessary to investigate the dynamical behaviors of fractional vibration energy harvesters. This paper aims to investigate the stochastic response of monostable vibration energy harvesters with fractional derivative damping under Gaussian white noise excitation. First, we can get the equivalent stochastic system with the help of variable transformation. Then, the approximately analytical solutions of the equivalent stochastic system can be obtained by the stochastic averaging method. Third, the numerical results are considered as the benchmark to prove the effectiveness of the proposed method. The results indicate that the proposed method has a satisfactory level of accuracy. We also discuss the effect of system parameters on the mean square voltage.

An entropy-based stability analysis of extreme fluctuations in a system featuring a 1/f spectrum

Extreme fluctuations are modeled by a point system of stochastic equations, in which power spectra inversely proportional to the frequency are produced under the effect of white noise. The distribution of extreme fluctuations corresponds to the maximum of statistical entropy, which points to their stability in nature. By calculating the spectral entropy of random processes, it becomes possible to investigate their stability directly from power spectra without the need to calculate the amplitude distribution functions. The spectral entropy as a function of white noise amplitude has a minimum. The position of the spectral entropy minimum corresponds to the critical state of the system in which the spectra of fluctuating quantities are inversely proportional to the frequency.

Quantized/Saturated Control for Sampled-Data Systems Under Noisy Sampling Intervals: A Confluent Vandermonde Matrix Approach

In this paper, a unified framework is established to investigate both the quantized and the saturated control problems for a class of sampled-data systems under noisy sampling intervals. A random variable obeying the Erlang distribution is used to describe the noisy sampling intervals. In virtue of the matrix exponential, the sampled-data control system is transformed into an equivalent discrete-time stochastic system, and the aim of this paper is to design a quantized/saturated sampled-data controller such that the resulting discrete-time stochastic system is stochastically stable when the sampling error follows the Erlang distribution. In order to deal with the case of multiple control inputs, a confluent Vandermonde matrix approach is proposed in the design process. By using the Kronecker product operation and the matrix inequality techniques, the desired quantized/saturated controller gains are designed in terms of the solution to certain matrix inequalities. Finally, a simulation example is exploited to verify the effectiveness of the proposed design approach.

Stochastic Interpretation of Quasilinear Parabolic Systems with Cross Diffusion

We construct a stochastic interpretation of the generalized solution of the Cauchy problem for a strongly coupled system of quasilinear parabolic equations. To this end we derive a system of stochastic equations and prove that its solution allows us to construct a probabilistic representation of the original Cauchy problem solution and investigate its properties. Our construction of a probabilistic representation of the generalized solution of the Cauchy problem is crucially based on results of stochastic flow theory.

Optimal execution strategy in liquidity framework

AbstractA trader wishes to execute a given number of shares of an illiquid asset. Since the asset price also depends on the trading behaviour, the trader main aim is to find the execution strategy that minimizes the related expected costs. We solve this problem in a discrete time framework, by modeling the asset price dynamic as an arithmetic random walk with drift and volatility both modeled as Markov stochastic processes. The market impact is assumed to follow a Markov process. We found the unique execution strategy minimizing the implementation shortfall when short selling is allowed. This optimal strategy is given as solution of a forward-backward system of stochastic equations depending on conditional expectations of future values of model parameters. In the opposite case, namely when short selling is prohibited, we numerically obtain the solution for the associated Bellman equation that an optimal trading strategy must satisfy.

Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations

This paper is mainly dealing with the stochastic responses of nonlinear vibro-impact (VI) system coupled with viscoelastic force excited by colored noise. By the aid of approximate conversion for the viscoelastic force, the original stochastic VI system is transformed into an equivalent stochastic system without viscoelastic term. Then, the equations of the converted system are simplified by non-smooth transformation, and the stochastic averaging method is employed to solve the above simplified system. A Van der Pol VI oscillator coupled with viscoelastic force is worked out in detail to illustrate the application of the mentioned method, and therewith the analytical solutions fit the numerical simulation results based on the original system. Therefore, the present analytical means of investigating this system is proved to be feasible. Additionally, the exploration of stochastic P-bifurcation by two different ways is also demonstrated in this paper through varying the value of the certain system parameters. Besides, it shows a noteworthy fact that assigning zero or a positive value to the magnitude of viscoelastic force can also lead to the bimodal shape of different degrees in the process of stochastic bifurcations.

Response analysis of a class of quasi-linear systems with fractional derivative excited by Poisson white noise

The Poisson white noise, as a typical non-Gaussian excitation, has attracted much attention recently. However, little work was referred to the study of stochastic systems with fractional derivative under Poisson white noise excitation. This paper investigates the stationary response of a class of quasi-linear systems with fractional derivative excited by Poisson white noise. The equivalent stochastic system of the original stochastic system is obtained. Then, approximate stationary solutions are obtained with the help of the perturbation method. Finally, two typical examples are discussed in detail to demonstrate the effectiveness of the proposed method. The analysis also shows that the fractional order and the fractional coefficient significantly affect the responses of the stochastic systems with fractional derivative.

Stationary response analysis of vibro-impact system with a unilateral nonzero offset barrier and viscoelastic damping under random excitations

The stationary response analysis of stochastic vibro-impact (VI) dynamical system with a unilateral nonzero offset barrier and viscoelastic damping under additive and multiplicative random excitations is explored. First, the conversion of the original stochastic VI system coupled with viscoelastic damping into an equivalent stochastic VI system without viscoelastic damping terms is performed by adding the equivalent stiffness and damping. Then, the equivalent stochastic VI system is studied according to the level of system energy instead of non-smooth transformation of the previous literature; thus, description on effects of impacts draws support from the loss of the system energy. The energy loss at each impact is found to be related to the restitution factor and the energy level before impact. Furthermore, the probability density functions of the equivalent VI system based on assumption of the weak damping and random perturbation is produced by the application of the stochastic averaging of energy envelope. The effectiveness of the suggested method is verified by comparing the analytical results with those from Monte Carlo simulations.

Stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation

This paper aims to investigate the stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation. Firstly, the original stochastic vibroimpact system with fractional derivative is transformed into equivalent stochastic vibroimpact system without fractional derivative. Then, the non-smooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system. At last, in order to verify the effectiveness of the above mentioned approach, the van der Pol vibroimpact system with fractional derivative is worked out in detail. A very satisfactory agreement can be found between the analytical results and the numerical results. An interesting phenomenon we found in this paper is that the fractional order and fractional coefficient of the stochastic van der Pol vibroimpact system can induce the occurrence of stochastic P-bifurcation. To the best of authors’ knowledge, the stochastic P-bifurcation phenomena induced by fractional order and fractional coefficient have not been found in the present available literature which studies the dynamical behaviors of stochastic system with fractional derivative under Gaussian white noise excitation.

Multidimensional washboard ratchet potentials for frustrated two-dimensional Josephson junctions arrays on square lattices

In this work, we derive an analytical procedure that allows us to write the multidimensional washboard ratchet potential (MDWBP) Uf for a two-dimensional Josephson junction array. The array has an applied perpendicular magnetic field. The magnetic field is given in units of the quantum flux per plaquette or frustration of the form , where Φ0 is the flux quantum. The derivation is done under the assumption that the checkerboard pattern ground state or unit cell of a two-dimensional Josephson junction array is preserved under current biasing. The resistively and capacitively shunted Josephson junction model with a white noise term describes the dynamics for each junction in the array. The multidimensional potential is the unique expression of the collective effects that emerge from the array in contrast to the single junction. The first step in the procedure is to write the equation for the phases for the unit cell. In doing this, one takes into account the constraints imposed for the gauge invariant phases due to frustration. Second, and the key idea of the procedure, is to perform a variable transformation from the original systems of stochastic equations to a system of variables where the condition for the equality of mixed second partial happens. This is achieved via Poincaré's theorem for differential forms. In this way, we find to a nonlinear matrix equation (equation () in the text), that permits us to find the new coordinate variables xf where the potential exists. The transformation matrix also permits the correct transformation of the original white noise terms of each junction to the intensities in the xf variables. The commensurate symmetries of the ground state pinned vortex lattice leads to discrete symmetries to the part of the washboard potential that does not contain a tilt due to the external bias current (equation () in the text). In this work we apply the procedure for the important cases . For , we show that previously efforts for finding the potential are restricted, leading to a reduced dimension of the potential. The correct potential is given in equation (). We examine this issue in detail. New physics emerge when currents are applied in the x and y directions, in particular, we confirm analytically previous numerical work for , concerning the border of stationary states, a landmark of the potential. For , we give a generalization of previous work, in which we include both the currents in the x and y directions as well the noise terms. We find the MDWBP realizes tilted ratchets analogous to a combustion motor.

Stationary responses of a Rayleigh viscoelastic system with zero barrier impacts under external random excitation.

This paper aims to deal with the stationary responses of a Rayleigh viscoelastic system with zero barrier impacts under external random excitation. First, the original stochastic viscoelastic system is converted to an equivalent stochastic system without viscoelastic terms by approximately adding the equivalent stiffness and damping. Relying on the means of non-smooth transformation of state variables, the above system is replaced by a new system without an impact term. Then, the stationary probability density functions of the system are observed analytically through stochastic averaging method. By considering the effects of the biquadratic nonlinear damping coefficient and the noise intensity on the system responses, the effectiveness of the theoretical method is tested by comparing the analytical results with those generated from Monte Carlo simulations. Additionally, it does deserve attention that some system parameters can induce the occurrence of stochastic P-bifurcation.

Analytical Estimation of Velocity and Temperature Fields in a Circular Pipe on the Basis of Stochastic Equations and Equivalence of Measures

A stream of nonisothermal Newtonian liquid in a circular smooth pipe is considered on the basis of systems of stochastic equations and of the physical law of equivalence of measures between laminar and turbulent motions. Analytical expressions were previously obtained for isothermal flows for the first and second critical Reynolds numbers, critical point, indices of velocity profiles, second-order correlation moments, correlation functions, and spectral functions depending on the parameters of initial turbulence. Analytical expressions, obtained with the use of the earlier derived formulas for the critical Reynolds numbers and the critical points, are presented for the indices of velocity and temperature profiles as functions of the initial turbulence parameters as well as of the Eckert and Prandtl numbers.

Uncertainty quantification for linear elastic bodies with two fluctuating input parameters

AbstractIn this work, we present a numerical method for solving partial differential equations (PDEs) with stochastic coefficients for a linear elastic body. To this end, a stochastic finite element method is applied. We distinguish two different cases for an isotropic material with two fluctuating input parameters in order to analyse the optimal choice of input parameters. Using the GALERKIN projection, the final stochastic equation system is reduced to a system of deterministic PDEs. Subsequently, the solution is determined iteratively. Finally, a numerical example for a plate with a ring hole is presented. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)