White Measurement Noise Research Articles

Global dynamics for the stochastic KdV equation with white noise as initial data

We study the stochastic Korteweg-de Vries equation (SKdV) with an additive space-time white noise forcing, posed on the one-dimensional torus. In particular, we construct global-in-time solutions to SKdV with spatial white noise initial data. Due to the lack of an invariant measure, Bourgain’s invariant measure argument is not applicable to this problem. In order to overcome this difficulty, we implement a variant of Bourgain’s argument in the context of an evolution system of measures and construct global-in-time dynamics. Moreover, we show that the white noise measure with variance 1 + t 1+t is an evolution system of measures for SKdV with the white noise initial data.

How Useful is Learning in Mitigating Mismatch Between Digital Twins and Physical Systems?

In the control of complex systems, we observe two diametrical trends: model-based control derived from digital twins, and model-free control through AI. There are also attempts to bridge the gap between the two by incorporating learning-based AI algorithms into digital twins to mitigate mismatches between the digital twin model and the physical system. One of the most straightforward approaches to this is direct input adaptation. In this paper, we ask whether it is useful to employ a generic learning algorithm in such a setting, and our conclusion is “not very”. We denote an algorithm to be more useful than another algorithm based on three aspects: 1) it requires fewer data samples to reach a desired minimal performance, 2) it achieves better performance for a reasonable number of data samples, and 3) it accumulates less regret. In our evaluation, we randomly sample problems from an industrially relevant geometry assurance context and measure the aforementioned performance indicators of 16 different algorithms. Our conclusion is that blackbox optimization algorithms, designed to leverage specific properties of the problem, generally perform better than generic learning algorithms, once again finding that “there is no free lunch”. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Digital twins have the potential to improve productivity and quality in complex systems such as manufacturing systems. Their impact on system performance hinges on the accuracy of their digital models around the system’s operating points. Difficult to measure phenomena, such as wear and tear of equipment, however, may cause a mismatch between the digital twin model and the physical system. In this paper, we formalize this problem and compare 16 potential solution strategies under practical aspects. We argue that readily available off-the-shelf blackbox optimization algorithms may prove more useful for this problem, than more recent learning-based approaches. Specifically, gradient-based algorithms will perform best in systems with high-dimensional, continuous, and non-linear performance functions – even in the presence of white measurement noise.

Fish Swarmed Kalman Filter for State Observer Feedback of Two-Wheeled Mobile Robot Stabilization

Over the past few decades, there have been significant technological advancements in the field of robots, particularly in the area of mobile robots. The performance standards of speed, accuracy, and stability have become key indicators of progress in robotic technology. Self-balancing robots are designed to maintain an upright position without toppling over. By continuously adjusting their center of mass, they can maintain stability even when disturbed by external forces. This research aims to achieving and maintaining balance is a complex task. Self-balancing robots must accurately sense their orientation, calculate corrective actions, and execute precise movements to stay upright. Eliminating disturbances and measurement noise in self-balancing robot can enhance the accuracy of their output. One common technique for achieving this is by using Kalman filters, which are effective in addressing non-stationary linear plants with unknown input signal strengths that can be optimized through filter poles and process covariances. Additionally, advanced Kalman filter methods have been developed to account for white measurement noise. In this research, state estimation was conducted using the Fish Swarm Optimization Algorithm (FSOA) to provide feedback to the controller to overcome the effects of disturbances and noise in the measurements through the designed filter. FSOA mimics the social interactions and coordinated movements observed in fish groups to solve optimization problems. FSOA is primarily used for optimization tasks where finding the global optimal solution is desired. The results show that the use of an optimized Kalman filter with FSOA on a two-wheeled mobile robot to handle system stability reduces noise values by 38.37%, and the system reaches a steady state value of 3.8 s with a steady error of 0.2%. In addition, by using the proposed method, filtering disturbances and measurement noise in self-balancing robot can help improve the accuracy of the self balancing robot’s output. System response becomes faster towards stability compared to other methods which are also applied to two-wheeled mobile robots.

State and Parameter Estimation of Non-Uniform Transmission Line Using Kronecker Product Based Modeling

The distributed parameters of a non-uniform transmission line (NTL) are assumed to depend on a finite set of unknown parameters, for example, their spatial Fourier series coefficients. The line equations are set up taking random voltage and current line loading into account with the random loading being assumed to be white Gaussian noise in the time and spatial variables. Bypassing over to the spatial Fourier series domain, these line stochastic differential equations (SDEs) are expressed as a truncated sequence of coupled stochastic differential equations for the spatial Fourier series components of the line voltage and current. The coefficients that appear in these stochastic differential equations are the spatial Fourier series coefficients of the distributed line parameters and these coefficients in turn are linear functions of the unknown parameters. By separating these complex stochastic differential equations into their real and imaginary part, the line equations become a finite set of SDEs for the extended state defined to be the real and imaginary components of the line voltage and current as well as the parameters on which the distributed parameters depend. The measurement model is assumed to be white Gaussian noise corrupted versions of the line voltage and current sampled at a discrete set of spatial points on the line and this measurement model can be expressed as a linear transformation of the state variables, i.e, of the spatial Fourier series components of the line voltage and current plus a white measurement noise vector. This work uses sparse matrix factorization using Kronecker product to implement the Fourier unitary transform. This work implements perturbation theory for non-uniform transmission line. The extended Kalman filter (EKF) equations are derived for this state and measurement model and our real timeline voltage and current and distributed parameter estimates show excellent agreement with the true values based on MATLAB simulations. The use of sparse matrix factorization using Kronecker product ease the mathematical complexity and provides compact representation.

Pathwise large deviations for white noise chaos expansions

We consider a family of continuous processes $\{X^\varepsilon\}_{\varepsilon>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C([0,1]^d:\mathbb{R})$, and have the Wiener chaos expansion \[ X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). \] We provide sufficient conditions for the large deviations principle of $\{X^\varepsilon\}_{\varepsilon>0}$ to hold in $C([0,1]^d:\mathbb{R})$, thereby refreshing a problem left open by P\'erez-Abreu (1993) in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.

IDENTIFICATION OF WHITE NOISE AND 1/f IN CURRENT MIRROR CONFIGURATION BASED ON VDS MOSFET

Identifying noise in the Current Mirror (CM) circuit is essential to locate noise signals in biosensor applications so that measurements become more accurate and precise. There are two dominant types of noise: white noise, which consists of thermal noise and shot noise, and also low-frequency noise (1/f noise). The main component of the CM circuit is the BS250 type MOSFET, which works by varying the width of the charge carrier channelcontrolled by the voltage at the gate. When the drain is given a voltage, electrons will flow from the source to the drain which generates the noise.This study was carried out to identify the noise in the CM configuration by varying the reference voltage of MOSFET using the PCI-6221 card data integrated with the LabVIEW program. The reference voltage values ​​used are 1 mV, 10 mV, and 100 mV to determine the effect of the input voltage on the CM circuit noise signal, while the measurement frequency is varied from 0.1 Hz to 100 kHz with a resolution of 0.1 Hz. The results show that the noise characteristics vary with the applied voltage, which will increase at a higher voltage. Analysis of 1/f noise at frequencies up to 0.2 Hz has a gradient increase of up to 10 times for each given voltage value. Based on the value of the data distribution on the white noise measurement, it shows that a voltage of 100 mV produces the highest noise with an average of 3.62 × 10-7 Vrms/Hz1/2. The results of this study are used in the design of CM circuits with minimal noise.

Poisson multi‐Bernoulli mixture filters with coloured measurement noise

To solve multitarget tracking (MTT) problems with coloured measurement noise, this study proposes a Poisson multi-Bernoulli mixture filter with coloured measurement noise (PMBM-CMN) and a robust PMBM-CMN filter. By using the measurement differencing method and state augmentation approach, the proposed PMBM-CMN filter transforms a state estimation problem with coloured measurement noise into a problem with white measurement noise. However, covariances of the true process and measurement noise in the proposed PMBM-CMN filter are time-varying and unknown, which may degrade the filtering performance. Therefore, a robust PMBM-CMN filter is proposed for estimating the augmented state, including the kinematic state, the predicted state covariance, and the white measurement noise covariance. For linear Gaussian systems, the augmented state is modelled as a Gaussian inverse Wishart inverse Wishart (GIWIW) distribution. The variational Bayesian method is also employed to guarantee the conjugacy of the GIWIW density. Simulation results demonstrate the ability of the PMBM-CMN filter to solve MTT problems with coloured measurement noise and show that the robust PMBM-CMN filter based on the GIWIW model (GIWIW-PMBM-CMN) has the best overall performance in comparison with existing state-of-the-art filters.

Optimal Adaptive Filtering Algorithm by Using the Fractional-Order Derivative

The previous work for the filter design considers uncorrelated white measurement noise disturbance. For more complex correlated noise disturbance, the conventional adaptive filter results in biased estimates. To overcome this problem, we introduce a linear prefilter to whiten the correlated noise (i.e., colored noise) for obtaining the unbiased estimate of the filter weight. Moreover, the design of some adaptive filters mainly focuses on the integer-order optimization methods. However, compared with the integer-order-based adaptive algorithms, the fractional-order-based algorithms show better performance. Thus, this letter develops a new gradient approach for the adaptive filter design based on the fractional-order derivative and a linear filter. Finally, the simulation results are provided from the system identification perspective for demonstrating the performance analysis of the proposed algorithms.

Stochastic mSQG equations with multiplicative transport noises: White noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T2, perturbed by multiplicative transport noise. The equation admits the white noise measure on T2 as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space–time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski’s approach in Gubinelli and Perkowski (2020).

System identification approach for inverse optimal control of finite-horizon linear quadratic regulators

The inverse optimal control for finite-horizon discrete-time linear quadratic regulators is investigated in this paper, which is to estimate the parameters in the objective function using noisy measurements of partial optimal states only. By the Pontryagin’s minimum principle, the concerned inverse optimal control problem is recast as the identification of a parameterized causal-and-anticausal mixed system excited by boundary conditions. Sufficient identifiability conditions for the unknown parameters are provided in terms of the system model itself, rather than relying on the exact values of optimal states or control inputs. In addition, an elegant algebraic solution is provided for the concerned identification problem that is inherently a challenging optimization problem with trilinear equality constraints, and it can recover the true parameters (up to a scalar ambiguity) in the absence of measurement noise or can consistently identify the parameters (up to a scalar ambiguity) in the presence of white measurement noise. The presented algebraic solution relies on recursive matrix calculations so that its computational burden is much less than directly solving a high-dimensional non-convex optimization problem as done in many existing works. The effectiveness of the proposed method as well as its noise sensitivity issue is illustrated by simulation examples.

Contraction-Based Nonlinear Controller for a Laser Beam Stabilization System Using a Variable Gain

In this letter, we propose a contraction-based variable gain nonlinear control scheme for the laser-beam stabilizing (LBS) servo-system, which guarantees that the closed-loop system is convergent. With the variable gain acting on the velocity error, the well known waterbed effect of the low-frequency/bandwidth trade-off can be overcome. Moreover, the contraction-based framework allows us to extend the linear control performance metrics for analyzing the closed-loop nonlinear system behavior. The closed-loop system’s performance is evaluated in numerical simulations under input disturbances and/or white noise measurements and its efficacy is compared to that using PID and LQG controllers.

Recursive Identification of Errors-in-Variables Systems Based on the Correlation Analysis

This paper considers a single-input single-output linear dynamic system, whose input and output are corrupted by Gaussian white measurement noises with zero means and unknown variances; the parameter estimation of such a system is a typical errors-in-variables (EIV) system identification problem. This paper proposes the correlation function-based two-step identification methods for the EIV systems. In order to obtain the unbiased parameter estimates of the EIV system, we derive the correlation function equation by using the correlation analysis method and adopt the least squares method and the instrumental variable method to recursively compute the parameter estimates of the model, resulting in the unbiased parameter estimates of the EIV systems. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithms.

Invariance of white noise for KdV on the line

We consider the Korteweg–de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

A Novel Student’s t-based Kalman Filter with Colored Measurement Noise

In this paper, a novel robust Student’s t-based Kalman filter (RSTKF) is proposed to solve the problem of a linear system with heavy-tailed process and measurement noises (HPMN) and colored measurement noise (CMN). The above problem is transformed into the filtering problem of a linear system with HPMN and white measurement noise after using the measurement differencing method and state augmentation approach. The augmentation state vector, the scale matrix and the auxiliary random variables are jointly estimated based on the variational Bayesian approach. Simulation results are provided to demonstrate the superiority of the proposed RSTKF by comparing with the existing filtering algorithms for systems with HPMN and CMN.

Solving the 4NLS with white noise initial data

AbstractWe construct global-in-time singular dynamics for the (renormalized) cubic fourth-order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the ‘random-resonant / nonlinear decomposition’, which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work, and we instead establish the convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.

Advances on the analysis of the LMS algorithm with a colored measurement noise

Due to the inherent feedback feature of adaptive filtering algorithms, a comprehensive theoretical understanding of their learning process is still challenging. In order to make the mathematics tractable, most stochastic analyses adopt simplifications. One of these is the independence assumption, which presumes statistical independence between adaptive weights and input signal samples. Furthermore, the additive noise is usually assumed to be white, although it may not be the case in practice. This paper advances a novel theoretical model of the least-mean-square adaptive filter that, under the independence assumption, does not presume a white noise signal. Additionally, a theoretical result is established which implies that the stability properties of such an adaptive filter are not influenced by noise coloring. Such a result does not employ the above-mentioned assumptions, i.e., it is valid for both colored noise and non-infinitesimally small step size values. An optimal step-size sequence that minimizes the mean-square deviation and takes into account the stochastic coupling of the excitation data and adaptive weights is proposed. It is noteworthy that such a design does not induce divergence in practice and does not assume, for the first time, a white measurement noise. The theoretical contributions are confirmed by simulations.

A New Insight Into GAMP and AMP

A concise expectation propagation (EP) based message passing algorithm (MPA) is derived for the general measurement channel. By neglecting some high-order infinitesimal terms, the EP-MPA is proven to be equivalent to the Generalized Approximate Message Passing (GAMP), which exploits central limit theorem and Taylor expansion to simplify the belief propagation process. Furthermore, for additive white gaussian noise measurement channels, EP-MPA is proven to be equivalent to the AMP. Such intrinsic equivalence between EP and GAMP/AMP offers a new insight into GAMP and AMP via a unified message passing rule for non-linear processing, and may provide clues towards building new MPAs in solving more general non-linear problems.

Overlay Cognitive Radio Based on OFDM with Channel Estimation Issues

Cognitive radio (CR) has been proposed as a technology to improve the spectrum efficiency by giving an opportunistic access of the licensed-user spectra to unlicensed users. We consider an overlay CR consisting of a primary macro-cell and cognitive small cells of cooperative secondary base stations (SBS). We suggest studying a CR where an orthogonal frequency division multiplexing is used for both the primary users (PU) and the secondary users (SU). In order to cancel the interferences, a precoding is required at the SBS. Therefore, we first derive the interferences expression due to SU at the PU receiver. Then, zero forcing beamforming (ZFBF) is considered to cancel the interferences. However, applying ZFBF depends on the channels between the SBS and the PU. A channel estimation is hence necessary. For this purpose, we propose to approximate the channel by an autoregressive process (AR) and to consider the channel estimation issue by using a training sequence. The received signals, also called the observations, are considered to be disturbed by an additive white measurement noise. In that case, the AR parameters and the channel can be jointly estimated from the received noisy signal by using a recursive approach. Nevertheless, the corresponding state space representation of the system is non-linear. Then, we propose to carry out a complementary study by compare non-linear Kalman filter based approaches.

Consensus-Based Filter for Distributed Sensor Networks with Colored Measurement Noise.

Consensus filtering is an effective method for distributed state estimation of distributed sensor networks and the assumption of white measurement noise is widely used. However, when the measurement noise is colored, the traditional consensus filter cannot work well. In this paper, we first propose a consensus-based distributed filter for colored measurement noise by augmenting the state to include the colored measurement noise. To improve the efficiency of the filter, only local colored measurement noise is integrated into the augmented state for each local filter. Furthermore, another consensus-based distributed filter based on measurement differencing scheme is developed to eliminate the ill-conditioned computations of the augmented state approach. In addition, this method does not need to augment the state and thus has lower dimension than the augmented state filter. Simulation results demonstrate the superiority of the proposed methods.

Distributed adaptive model-free cooperative control for a network of generic unknown nonlinear systems

In this article, a distributed model-free consensus control is proposed for a network of nonlinear agents with unknown nonlinear dynamics, unknown process disturbances, and white noise measurement disturbances. Here, the purpose of the control protocol is to first synchronize the states of all follower agents in the network to a leader and then track a reference trajectory in the state space. The leader has at least one information connection with one of the follower agents in the network. The design procedure includes adaptive laws for estimating the unknown linear and nonlinear terms of each agent’s dynamics. The salient feature of the proposed control scheme is that each agent’s estimation is a model-free adaptive law, that is, the need for regressor or linear-in-parameter basis is alleviated. In addition, without requiring direct connection to the leader, the leader’s control input can still be reconstructed by virtue of a robust observer which can be defined in a distributed manner in the network. The entire design procedure is analyzed successfully for the stability using Lyapunov stability theorem. In addition, it is shown that the proposed distributed controller includes an optimal term. Besides, a modified Kalman filter is added to eliminate the measurement noise. Finally, the simulation results on three networks of unknown nonlinear systems are presented. Moreover, a comparative study is presented to evaluate the proposed algorithm against a model-based cooperative control algorithm.