Additive Measurement Noise Research Articles

System identification in a networked environment using second order statistical properties

System identification for networked control is considered. Due to the time-delays in the network, it can be difficult to work with a discrete-time model and a continuous-time model is therefore chosen. A covariance function based method that relies on the second order statistical properties of the output signal, where it is assumed that the input signal samples are from a discrete-time white noise sequence, is proposed for estimating the parameters. The method is easy to use since the actual time instants when new input signal levels are applied at the actuator do not have to be known. An analysis of the networked system and the effects of the time-delays is made, and the results of the analysis motivate and support the chosen approach. Numerical studies indicate that the method is robust to randomly distributed time-delays, packet drop-outs, and additive measurement noise.

A patchy approximation of explicit model predictive control

The explicit solution of multi-parametric optimisation problems (MPOP) has been used to construct an off-line solution to relatively small- and medium-sized constrained control problems. The control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics. In this context, it can be shown that the optimal control law is a piecewise linear (PWL) state feedback defined over polytopic cells of the state space. However, as the complexity of the related optimisation problems increases, the memory footprint and implementation of such explicit optimal solution may be burdensome for the available hardware, principally due to the high number of polytopic cells in the state-space partition. In this article we provide a solution to this problem by proposing a patchy PWL feedback control law, which intend to approximate the optimal control law. The construction is based on the linear interpolation of the exact solution at the vertices of a feasible set and the solution of an unconstrained linear quadratic regulator (LQR) problem. With a hybrid patchy control implementation, we show that closed-loop stability is preserved in the presence of additive measurement noise despite the existence of discontinuities at the switch between the overlapping regions in the state-space partition.

Modeling of Multipath Environment Using Copulas for Particle Filtering Based GPS Navigation

Today in GPS navigation, an accuracy from 5 to 10 m can be achieved, but performance can be strongly degraded in a multipath environment. Multipath can introduce large errors when measuring the distance between the satellites and the GPS receiver. They are commonly modeled by additive-measurement noise variance jumps affecting GPS measurements if there is a direct path between the satellites and the receiver and by additive-measurement noise mean-value jumps otherwise. If two signals from satellites have close directions of arrival, they are very likely to be simultaneously degraded by multipath. Therefore, in this letter we suggest taking into account the spatial dependencies between GPS measurements when modeling multipath occurrence/disappearance. For that purpose, we use a probabilistic tool, namely copulas. Then, as the proposed model is strongly nonlinear and non-Gaussian, we jointly estimate the mobile location and perform the multipath detection/estimation by using particle filtering.

Distributed Seeking of Nash Equilibria With Applications to Mobile Sensor Networks

We consider the problem of distributed convergence to a Nash equilibrium in a noncooperative game where the players generate their actions based only on online measurements of their individual cost functions, corrupted with additive measurement noise. Exact analytical forms and/or parameters of the cost functions, as well as the current actions of the players may be unknown. Additionally, the players' actions are subject to linear dynamic constraints. We propose an algorithm based on discrete-time stochastic extremum seeking using sinusoidal perturbations and prove its almost sure convergence to a Nash equilibrium. We show how the proposed algorithm can be applied to solving coordination problems in mobile sensor networks, where motion dynamics of the players can be modeled as: 1) single integrators (velocity-actuated vehicles), 2) double integrators (force-actuated vehicles), and 3) unicycles (a kinematic model with nonholonomic constraints). Examples are given in which the cost functions are selected such that the problems of connectivity control, formation control, rendezvous and coverage control are solved in an adaptive and distributed way. The methodology is illustrated through simulations.

A statistical approach to target detection and localization in the presence of noise

The problem addressed in this paper is the combined detection and localization of a point reflector embedded in a medium by sensor array imaging when the array response matrix is obtained in a noisy environment. We consider additive measurement noise or a clutter noise in the multiple scattering regime. We study a detection test based on reverse-time migration of the array response matrix that is the most powerful for a given false alarm rate and compare it with a test based on the singular values of the response matrix. Moreover, we show that reflector localization should be performed with reverse-time migration rather than any other form of weighted-subspace migration and we give the standard deviation of the localization error.

Marginalised iterated unscented Kalman filter

In this study, the authors investigate the role of iteration in the unscented Kalman filter (UKF) with additive measurement noise and propose a novel filter referred to as the marginalised iterated unscented Kalman filter (MIUKF). In each iteration of the MIUKF, the new sigma points are regenerated and propagated through the same measurement update strategy as the UKF. In order to guarantee the state to be statistically independent with the measurement noise, the state variables are augmented with the measurement noise. Since the measurement noise is additive, the measurement function is conditionally linear of the augmented state, then the marginalised unscented transformation is investigated to reduce the computational burden. Compared with the traditional iterated UKF, the MIUKF is more rigorous in terms of efficiency and accuracy. The simulation results agree well with the theoretical analyses.

State Estimation for Linear Scalar Dynamic Systems with Additive Cauchy Noises: Characteristic Function Approach

Uncertainties in many practical systems, such as radar glint and sonar noise, have impulsive character and are better described by heavy-tailed non-Gaussian densities, for example, the Cauchy probability density function (pdf). The Cauchy pdf does not have a well defined mean and its second moment is infinite. Nonetheless, the conditional density of a Cauchy random variable, given a scalar linear measurement with an additive Cauchy noise, has a conditional mean and a finite conditional variance. In particular, for scalar discrete linear systems with additive process and measurement noises described by Cauchy pdfs, the unnormalized characteristic function of the conditional pdf is considered. It is expressed as a growing sum of terms that at each measurement update increases by one term, constructed from four new measurement-dependant parameters. The dynamics of these parameters is linear. These parameters are shown to decay, allowing an approximate finite dimensional recursion. From the first two differen...

Deterministic regression methods for unbiased estimation of time-varying autoregressive parameters from noisy observations

A great deal of interest has been paid to autoregressive parameter estimation in the noise-free case or when the observation data are disturbed by random noise. Tracking time-varying autoregressive (TVAR) parameters has been also discussed, but few papers deal with this issue when there is an additive zero-mean white Gaussian measurement noise. In this paper, one considers deterministic regression methods (or evolutive methods) where the TVAR parameters are assumed to be weighted combinations of basis functions. However, the additive white measurement noise leads to a weight-estimation bias when standard least squares methods are used. Therefore, we propose two alternative blind off-line methods that allow both the variance of the additive noise and the weights to be estimated. The first one is based on the errors-in-variable issue whereas the second consists in viewing the estimation issue as a generalized eigenvalue problem. A comparative study with other existing methods confirms the effectiveness of the proposed methods.

Parameter estimation and change-point detection from Dynamic Contrast Enhanced MRI data using stochastic differential equations

Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular–extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients’ anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results.

An Optimum Land Cover Mapping Algorithm in the Presence of Shadows

Occurrence of shadowy pixels in remote sensing images is a common phenomenon particularly with passive sensors. In these cases, analysts may mistakenly treat these pixels as a separate land cover class. This may result in the loss of information present in the shadow pixels A better approach may be to correct light intensity values in shadowy pixels and use the light-corrected image to produce the land cover map. Most light intensity correction algorithms are not designed to optimize classification performance. Consequently, the accuracy of the resulting land cover map may be degraded. To address this problem, this paper proposes an algorithm that employs the maximum a posteriori criterion for classifying a multispectral image in the presence of shadows. The observed image is assumed to be the product of a shadow-free image with a light intensity image along with an additive measurement noise. The main purpose of this algorithm is to find the most likely land cover map along with the shadow-free image and light intensity image as byproducts. Our results show that a large number of misclassified pixels can be corrected. Furthermore, in the shadow-free image, the materials in the shadowy regions can also be successfully reconstructed.

Optimal Synchronization for Networks of Noisy Double Integrators

In this technical note, we present a novel synchronization protocol to synchronize a network of controlled discrete-time double integrators which are nonidentical, with unknown model parameters and subject to additive measurement and process noise. This framework is motivated by the typical problem of synchronizing a network of clocks whose speeds are nonidentical and are subject to variations. This synchronization protocol is formally studied in its synchronous implementation. In particular, we provide a completely distributed strategy that guarantees convergence for any undirected connected communication graph and we also propose an optimal design strategy when the underlaying communication graph is known. Moreover, this protocol can be readily used to study the effect of noise and external disturbances on the steady-state performance. Finally, some simulations including also randomized implementation of the proposed algorithm are presented.

Evaluating strong measurement noise in data series with simulated annealing method

Many stochastic time series can be described by a Langevin equation composed of a deterministic and a stochastic dynamical part. Such a stochastic process can be reconstructed by means of a recently introduced nonparametric method, thus increasing the predictability, i.e. knowledge of the macroscopic drift and the microscopic diffusion functions. If the measurement of a stochastic process is affected by additional strong measurement noise, the reconstruction process cannot be applied. Here, we present a method for the reconstruction of stochastic processes in the presence of strong measurement noise, based on a suitably parametrized ansatz. At the core of the process is the minimization of the functional distance between terms containing the conditional moments taken from measurement data, and the corresponding ansatz functions. It is shown that a minimization of the distance by means of a simulated annealing procedure yields better results than a previously used Levenberg-Marquardt algorithm, which permits a rapid and reliable reconstruction of the stochastic process.

A-Posteriori Identifiability of the Maxwell Slip Model of Hysteresis

AbstractThe a–posteriori identifiability of the Maxwell Slip model of hysteresis is addressed. The necessary and sufficient conditions that guarantee that the available data are informative enough are provided. An Output Error type estimator is subsequently postulated and its consistency is established. It is specifically shown that the estimates converge in probability to their actual counterparts under easily verifiable conditions on the Maxwell Slip model structure, the excitation, and mild assumptions on the additive measurement noise. The analytical results are verified through Monte Carlo simulations.

An iterative Kalman smoother/least-squares algorithm for the identification of delta-ARX models

Additive measurement noise on the output signal is a significant problem in the δ-domain and disrupts parameter estimation of auto-regressive exogenous (ARX) models. This article deals with the identification of δ-domain linear time-invariant models of ARX structure (i.e. driven by known input signals and additive process noise) by using an iterative identification scheme, where the output is also corrupted by additive measurement noise. The identification proceeds by mapping the ARX model into a canonical state-space framework, where the states are the measurement noise-free values of the underlying variables. A consequence of this mapping is that the original parameter estimation task becomes one of both a state and parameter estimation problem. The algorithm steps between state estimation using a Kalman smoother and parameter estimation using least squares. This approach is advantageous as it avoids directly differencing the noise-corrupted ‘raw’ signals for use in the estimation phase and uses different techniques to the common parametric low-pass filters in the literature. Results of the algorithm applied to a simulation test problem as well as a real-world problem are given, and show that the algorithm converges quite rapidly and with accurate results.

Non-parametric methods for [formula omitted]-gain estimation using iterative experiments

In this paper we develop non-parametric methods to estimate the L 2 -gain ( H ∞ -norm) of a linear dynamical system from iterative experiments. This work is mainly motivated by model error modeling, where the error dynamics are more complex than can be captured by a low order parametric model. The standard system identification approach to the gain estimation problem is to estimate a parametric model of the system, which is then used to calculate the gain. If it is possible to update the input signal during the experiment, an alternative way is to iteratively optimize the input signal in order to maximize the estimated input to output gain. A key observation is that the gradient of the gain with respect to the input signal can, without knowing a model, be found from two experiments. Iterative numerical methods for calculation of eigenvalues of matrices, e.g., the Power Method or the Lanczos Method, can then be applied to update the input signal sequence between experiments in order to find the maximum gain. The main difficulty compared to the corresponding eigenvalue problem in numerical analysis is the effects of additive measurement noise, which require modified schemes that avoid bias errors. Three such related methods are derived and evaluated by a numerical example. Partial results on convergence and statistical properties of the gain estimator are given. A constrained stochastic gradient method with local optimization of step-length gives the best numerical results in the case of noisy data.

Cauchy Estimation for Linear Scalar Systems

An estimation paradigm is presented for scalar discrete linear systems entailing additive process and measurement noises that have Cauchy probability density functions (pdf). For systems with Gaussian noises, the Kalman filter has been the main estimation paradigm. However, many practical system uncertainties that have impulsive character, such as radar glint, are better described by stable non-Gaussian densities, for example, the Cauchy pdf. Although the Cauchy pdf does not have a well defined mean and does have an infinite second moment, the conditional density of a Cauchy random variable, given its linear measurements with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. For a single measurement, simple expressions are obtained for the conditional mean and variance, by deriving closed form expressions for the infinite integrals associated with the minimum variance estimation problem. To alleviate the complexity of the multi-stage estimator, the conditional pdf is represented in a special factored form. A recursion scheme is then developed based on this factored form and closed form integrations, allowing for the propagation of the conditional mean and variance over an arbitrary number of time stages. In simulations, the performance of the newly developed scalar discrete-time Cauchy estimator is significantly superior to a Kalman filter in the presence of Cauchy noise, whereas the Cauchy estimator deteriorates only slightly compared to the Kalman filter in the presence of Gaussian noise. Remarkably, this new recursive Cauchy conditional mean estimator has parameters that are generated by linear difference equations with stochastic coefficients, providing computational efficiency.

Unscented Kalman filtering in the additive noise case

The unscented Kalman filter (UKF) has four implementations in the additive noise case, according to whether the state is augmented with noise vectors and whether a new set of sigma points is redrawn from the predicted state (which is so-called resampling) for the observation prediction. This paper concerns the differences of performances for those implementations, such as accuracy, adaptability, computational complexity, etc. The conditionally equivalent relationships between the augmented and non-augmented unscented transforms (UTs) are proved for several sampling strategies that are commonly used. Then, we find that the augmented and non-augmented UKFs have the same filter results with the additive measurement noise, but only have the same state predictions with the additive process noise. Resampling is not believed to be necessary in some researches. However, we find out that resampling can be helpful for an adaptive Kalman gain. This will improve the convergence and accuracy of the filter when the large scale state modeling bias or unknown maneuvers occur. Finally, some universal designing principles for a practical UKF are given as follows: 1) for the additive observation noise case, it’s better to use the non-augmented UKF; 2) for the additive process noise case, when the small state modeling bias or maneuvers are involved, the non-resampling algorithms with state whether augmented or not are candidates for filters; 3) the resampling and non-augmented algorithm is the only choice while the large state modeling bias or maneuvers are latent.

A Covariance Function Based Approach to Networked System Identification

System identification for networked control is considered. Due to the random time-delays in the network, it can be difficult to work with a discrete-time model and a continuous-time model is therefore chosen here. A covariance function based method is proposed for estimating the parameters, which relies on the second order statistical properties of the output signal and that the input signal samples can be taken from a discrete-time white noise sequence. Numerical studies indicate that the method is robust to additive measurement noise and packet dropouts.

Linear invariant estimation of clock phase and trend from noisy phase data

We show how to solve two problems of optimal linear estimation from a set of noisy measurements of clock phase. The phase is modelled as a process with stationary dth increments, e.g. a sum of processes with power-law spectra. The additive measurement noise can be any mean-zero process with a known autocovariance function. The estimation targets are the phase at a given time and the coefficient of the overall trend, e.g. the frequency drift. A feasibility condition called ‘invariance’ is imposed according to the target and the degree of non-stationarity of the phase model. The solution of a set of linear equations gives the regression coefficients and the mean squared error of the best linear invariant estimate. Proofs of these results are available from the online version of this journal.

Blind maximum likelihood identification of Hammerstein systems

This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér–Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss–Newton based optimization scheme. The paper includes the illustration of the method on a simulation example. A theoretical analysis demonstrates that additive output measurement noise introduces a bias that is proportional to the variance of that additive, unmodeled noise source. The simulations support this result, and show that this bias is insignificant beyond a certain Signal-to-Noise Ratio (40 dB in the example).