Second-order Stochastic Process Research Articles

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

A New Method for Determining the Gravity Disturbance Vector in Strapdown Airborne Gravimetry

A two-step method for determining the gravity disturbance vector based on strapdown airborne gravimeter (SAG) and Global navigation satellite system (GNSS) is proposed. First, a strapdown navigation algorithm and a Kalman filter are used to estimate and compensate for the biases of the accelerometers by the maneuvering flight of SAG. Second, the gravity disturbance vector equations of SAG are established, and the forward and smoothing Kalman filter tuned by the formula of the gravity parameter in the Markov stochastic process are used to estimate the gravity disturbance vector, the model of which is described by the second-order Markov stochastic process to separated it from the biases of the accelerometers. The validity of the above method is verified by processing the data of the repeated line flight test of SAG in the South China Sea, and the results reveal that the internal coincidence accuracy of gravity anomaly is better than 0.4 mGal, and that of the horizontal gravity disturbance is better than 2 mGal.

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

• A system of linear stochastic integro-differential equations is under consideration. • Gaussian nonstationary noises and random model uncertainties excite the system. • The problem is to find the mean and covariance functions of the state vector process. • Green matrix-valued functions are used to solve the problem. • As example, a response of a thermoviscoelastic system is estimated by our scheme. The paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations. It is suggested that these systems are subjected to Gaussian nonstationary stochastic noises in the presence of model parameter uncertainties that are described in the framework of the probability theory. If the uncertain model parameter is fixed to a given value, then a time-history of the system will be fully represented by a second-order Gaussian vector stochastic process whose properties are completely defined by its conditional vector-valued mean function and matrix-valued covariance function. The scheme that is proposed is constituted of a combination of two subschemes. The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by the Monte Carlo method. A full scheme realized on the base of Wolfram Mathematica and Intel Fortran software programs, is demonstrated by an example devoted to an estimation of a nonstationary stochastic response of a mechanical system with a thermoviscoelastic component. Results obtained by using the proposed scheme are compared with a reference solution constructed by using a direct Monte Carlo simulation.

A Mean Square Chain Rule and its Application in Solving the Random Chebyshev Differential Equation

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.

A New Wavelet-based Expansion of a Random Process

There has been obtained an expansion of a second-order stochastic process into a system of wavelet-based functions. This expansion is used for simulation of a sub-Gaussian stochastic process with given accuracy and reliability.

Data-driven stochastic models for spatial uncertainties in micromechanical systems

Accurate uncertainty quantification in engineering systems requires the use of proper data-driven stochastic models that bear a good fidelity with respect to experimentally observed variations. This paper looks at a variety of modeling techniques to represent spatially varying uncertainties in a form that can be incorporated into numerical simulations. In the context of microelectromechanical systems, we consider spatial uncertainties at the device level in the form of surface roughness and at the wafer level in the form of non-uniformities that arise as a result of various microfabrication steps. We discuss methods to obtain roughness characterization data ranging from the use of a simple profilometer probe to imaging-based techniques for the extraction of digitized data from images. We model spatial uncertainties as second-order stochastic process and use Bayesian inference to estimate the model parameters from the input data. We apply the data-driven stochastic models generated from this process to micromechanical actuators and sensors in which these spatial uncertainties are likely to cause significant variation. These include an electrostatically-actuated torsion-spring micromirror, an electromechanical comb-drive actuator and a pressure sensor with a piezoresistive strain gauge. We show that the performance of these devices is sensitive to the presence of spatial uncertainties and a proper modeling of these uncertainties helps us make reliable predictions about the variation in device performance. Where data is available, we even show that the predicted variation can be validated against experimental observations, highlighting the significance of proper stochastic modeling in the analysis of such devices.

On estimation of the extended Orey index for Gaussian processes

Orey suggested the definition of an index for a Gaussian process with stationary increments which determines various properties of the sample paths of this process. We provide an extension of the definition of the Orey index towards a second-order stochastic process which may not have stationary increments and estimate the Orey index towards a Gaussian process from discrete observations of its sample paths.

Variable-State-Dimension Kalman-Based Filter for Orientation Determination Using Inertial and Magnetic Sensors

In this paper a quaternion-based Variable-State-Dimension Extended Kalman Filter (VSD-EKF) is developed for estimating the three-dimensional orientation of a rigid body using the measurements from an Inertial Measurement Unit (IMU) integrated with a triaxial magnetic sensor. Gyro bias and magnetic disturbances are modeled and compensated by including them in the filter state vector. The VSD-EKF switches between a quiescent EKF, where the magnetic disturbance is modeled as a first-order Gauss-Markov stochastic process (GM-1), and a higher-order EKF where extra state components are introduced to model the time-rate of change of the magnetic field as a GM-1 stochastic process, namely the magnetic disturbance is modeled as a second-order Gauss-Markov stochastic process (GM-2). Experimental validation tests show the effectiveness of the VSD-EKF, as compared to either the quiescent EKF or the higher-order EKF when they run separately.

A note on the cross-covariance operator and on congruence relations for Hilbert space valued stochastic processes

Explicit formulas are derived for the congruence mappings that connect three Hilbert spaces associated with a second-order stochastic process. In particular, an insightful expression is obtained for the mapping that connects a process to its corresponding reproducing kernel Hilbert space. In addition, a useful infinite dimensional extension of a result from Khatri (1976) which pertains to cross-covariance operators is provided.

An RKHS formulation of the inverse regression dimension-reduction problem

Suppose that Y is a scalar and X is a second-order stochastic process, where Y and X are conditionally independent given the random variables ξ1, …, ξp which belong to the closed span LX2 of X. This paper investigates a unified framework for the inverse regression dimension-reduction problem. It is found that the identification of LX2 with the reproducing kernel Hilbert space of X provides a platform for a seamless extension from the finite- to infinite-dimensional settings. It also facilitates convenient computational algorithms that can be applied to a variety of models.

Generalized Levinson–Durbin Sequences and Binomial Coefficients

AbstractThe Levinson–Durbin recursion is used to construct the coefficients which define the minimum mean square error predictor of a new observation for a discrete time, second-order stationary stochastic process. As the sample size varies, the coefficients determine what is called a Levinson–Durbin sequence. A generalized Levinson–Durbin sequence is also defined, and we note that binomial coefficients constitute a special case of such a sequence. Generalized Levinson–Durbin sequences obey formulas which generalize relations satisfied by binomial coefficients. Some of these results are extended to vector stationary processes.

Gabor discretization of the Weyl product for modulation spaces and filtering of nonstationary stochastic processes

We discretize the Weyl product acting on symbols of modulation spaces, using a Gabor frame defined by a Gaussian function. With one factor fixed, the Weyl product is equivalent to a matrix multiplication on the Gabor coefficient level. If the fixed factor belongs to the weighted Sjöstrand space M ω ∞ , 1 , then the matrix has polynomial or exponential off-diagonal decay, depending on the weight ω. Moreover, if its operator is invertible on L 2 , the inverse matrix has similar decay properties. The results are applied to the equation for the linear minimum mean square error filter for estimation of a nonstationary second-order stochastic process from a noisy observation. The resulting formula for the Gabor coefficients of the Weyl symbol for the optimal filter may be interpreted as a time–frequency version of the filter for wide-sense stationary processes, known as the noncausal Wiener filter.

Approximate series representations of linear operations on second-order stochastic processes: application to Simulation

Series representations of the more usual linear operations in weak sense on a second-order stochastic process are studied. The starting point of this analysis is the optimal Cambanis expansion of the stochastic process considered. Likewise, the extensions of the approximate series expansions based on the Rayleigh-Ritz method are presented for such linear operations on the process. The main advantages of these extensions are that they are computationally feasible and entail a significant reduction in the computational burden. Finally, their applicability as a practical simulation tool is examined.

Reduced-Order Model for an Impacting Beam using the Karhunen-Loµeve Expansion

The Karhunen-Loµeve expansion (KLE), also known in the literature as the proper orthogonal decomposition, is a powerful tool for the model reduction of structural systems. Although the method has been used for quite some time in turbulence studies to uncover spatial coherent structures in °ow fields, only recently has it been applied to structural dynamics problems. The KL method is a primarily statistical one where the system dynamics is assumed to be a second-order stochastic process. It consists in obtaining a set of orthogonal eigenfunctions where the dynamics is to be projected. Practically, one constructs a spatial autocorrelation tensor and then performs its spectral decomposition. The resulting eigenfunctions will provide the required proper orthogonal modes (POMs) or empirical eigenmodes and the correspondent empirical eigenvalues (or proper orthogonal values, POVs) represent the mean energy contained in that projection. Finally, one uses a number of the computed modes in Galerkin's method in order to obtain a reduced dimension system (this is sometimes called the KLG method). Although this method can also be applied to linear systems, its main application stems from nonlinear ones. In this present work, such a system is studied, namely a linear clamped beam impacting a °exible barrier. The KLE will be used to look into the dynamics of the system and to generate a reduced-order model (ROM).

Spectral multiplicity of some stochastic processes

In this paper we consider the connection between the canonical and the weak-canonical representations for the given second-order stochastic process in a separable Hilbert space and we extend a well-known theorem of H. Cramer concerning sufficient conditions for a process to be of multiplicity one.

Unimodal spectral windows

The class of all symmetric unimodal probability densities on the line whose Fourier transforms have support in an interval [− T, T] is studied. It is shown that there exists a unique density with minimal variance in this class. The Fourier transform of the minimizing density is everywhere non-negative. These results complement earlier work by Bohman who solved the analogous problem with no unimodality restriction. Motivation for studying this problem is given in the context of estimating the spectral density of a second-order stationary continuous-time stochastic process. Some comments are made regarding analogous problems for measures on the unit circle and discrete-time processes.

Chaos characteristics of tree ring series

The climatic characteristics of an area are reflected in the growth of the trees. Consequently, the characteristics of tree rings are studied to determine the long-term climatic behavior of a region. In the present study, 13 tree ring series from the Salt and Verde River basins in Arizona are analyzed to determine their chaos characteristics. Two methods of analysis are used in the study: the Grassberger-Procaccia algorithm and the method proposed by Scargle. Neither indicated any evidence of chaotic behavior in these tree ring series. The series are also successfully modeled by a second-order stationary stochastic process. These results confirm the earlier result of Grassberger, who analyzed a much longer tree ring series and found no evidence of chaotic behavior.

Kalman filtering on approximate state-space models

Several state-space models for estimating a second-order stochastic process are proposed in this paper on the basis of the approximate Karhunen-Loeve expansion. Properties of these models are studied and then the Kalman filtering method is applied. The accuracy of the models on the basis of two different situations, deterministic or random inputs, is studied by means of a simulation of a Brownian motion.

Ergodic behavior and estimation for periodically correlated processes

A complex second-order stochastic process { X( t); t ∈ R } is periodically correlated (PC) when its mean function m( t) and covariance R( s, t) are periodic with the period T. In the following note the relationship between the hierarchy of chaos and estimation for the stochastic process is investigated. The results are based on the Correspondence Principle for the PC processes.

The Influence of Planetary-Wave Transience on Horizontal Air Motions in the Stratosphere

The influence of transience of the planetary-wave field on the horizontal air motions and tracer distributions in the stratosphere was investigated in equivalent barotropic calculations. Two classes of transience are considered: a monochromatic traveling wave, representative of discrete components such as the 5- and 16-day waves, and a second-order stochastic process representative of broadband variability. The response to each of these forms of unsteady forcing is investigated in terms of the characteristic time scale of the transience. Results are presented, and the implications these results have on stratospheric behavior are discussed.