Finite dimensional smoothers for MAP state estimation of bilinear systems

The ability to theoretically model the propagation of photon noise through PET and SPECT tomographic reconstruction algorithms is crucial in evaluating the reconstructed image quality as a function of parameters of the algorithm. In a previous approach for the important case of the iterative ML - EM (maximum-likelihood - expectation-maximization) algorithm, judicious linearizations were used to model theoretically the propagation of a mean image and a covariance matrix from one iteration to the next. Our analysis extends this approach to the case of MAP (maximum a posteriori) - EM algorithms, where the EM approach incorporates prior terms. We analyse in detail two cases: a MAP - EM algorithm incorporating an independent gamma prior, and a one-step-late (OSL) version of a MAP - EM algorithm incorporating a multivariate Gaussian prior, for which familiar smoothing priors are special cases. To validate our theoretical analyses, we use a Monte Carlo methodology to compare, at each iteration, theoretical estimates of mean and covariance with sample estimates, and show that the theory works well in practical situations where the noise and bias in the reconstructed images do not assume extreme values.

The application of maximum a posteriori (MAP) detection to DS-CDMA systems in time-selective fading channels is investigated. By sampling the outputs of a matched filter and combining antenna array elements, strong multiple-access interference (MAI) is estimated and suppressed. By incorporating channel approximations, Kalman filtering, and the probability density of the prediction error, a joint channel estimation and MAP detection algorithm is developed. Unknown channel parameters are identified by using an algorithm similar to the extended Kalman filter (EKF). Also, differential modulation is applied to eliminate the need for pilot insertion. Through computer simulations, a near-optimum bit-error-rate (BER) performance is found.

ABSTRACTWe consider a Bayesian framework for estimating a high-dimensional sparse precision matrix, in which adaptive shrinkage and sparsity are induced by a mixture of Laplace priors. Besides discussing our formulation from the Bayesian standpoint, we investigate the MAP (maximum a posteriori) estimator from a penalized likelihood perspective that gives rise to a new nonconvex penalty approximating the ℓ0 penalty. Optimal error rates for estimation consistency in terms of various matrix norms along with selection consistency for sparse structure recovery are shown for the unique MAP estimator under mild conditions. For fast and efficient computation, an EM algorithm is proposed to compute the MAP estimator of the precision matrix and (approximate) posterior probabilities on the edges of the underlying sparse structure. Through extensive simulation studies and a real application to a call center data, we have demonstrated the fine performance of our method compared with existing alternatives. Supplementary materials for this article are available online.

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts.

This paper studies three-dimensional multi-robot Cooperative Localization (CL) problem. Most of existing CL strategies adopt Extended Kalman Filter (EKF) or Maximum a Posteriori (MAP). In this paper, a novel approach based on Moving Horizon Estimation (MHE) is proposed. The main contribution of this paper is twofold: 1) MHE is integrated with EKF for three-dimensional CL using single mobile beacon, which can bound localization error, impose various constraints on states and noises, and make use of previous range mea- surements for current estimation. 2) A sufficient condition on observability of multi-robot CL is derived by using Fisher Information Matrix. Simulation is conducted to verify that the proposed MHE based CL algorithm outperforms EKF based method in terms of localization accuracy, and two scenarios where our algorithm is superior to EKF are discussed.

Motivated by the many applications associated with estimation of sparse multivariate models, the estimation of sparse directional connectivity between the imperfectly measured nodes of a network is studied. Node dynamics and interactions are assumed to follow a multivariate autoregressive model driven by noise, and the observations are a noisy linear combination of the underlying node activities. The corresponding maximum a posteriori (MAP) problem is derived to estimate system parameters. Due to the intractability of the MAP problem, the expectation maximization (EM) framework is used to iteratively implement the MAP estimation. To impose sparsity, the EM algorithm is augmented with an $\ell _1$ regularization of the connectivity matrix. Multiple techniques have been used to lower the computational complexity. Importantly, an efficient coordinate descent algorithm utilizing a closed-form solution is designed to solve the $\ell _1$ -regularized sub-problem of the EM. Cholesky factors of the unknown covariance matrices are used directly in the optimization process in order to impose positive definiteness and guarantee the functionality of the $\ell _1$ optimization. The algorithm is first applied to synthetic data to evaluate the estimation accuracy. A comparison with the previous work over an extensive set of configurations shows that our method is superior under moderate to high sparsity. Finally, the algorithm is applied to real temperature data obtained from 98 stations across the U.S. mainland to identify the predictive interactions between the time series.

The state estimation problem for the nonlinear system is a most important issue for industrial control application. It is also preferable that the estimation technique should be accurate and easily implementable. In the present work, we compare the performance of two estimation algorithm for state estimation of the Twin Rotor MIMO system (TRMS). Here, the Kalman filter (KF)and extended Kalman filter (EKF) are used for realtime state estimation for the system. It is observed that in both simulation and experimental result, that the performance of EKF is better than the KF in terms of robustness and root mean square error (RMSE) in state estimation.

For land-based navigation, Euler angles are often used in INS/GPS integrated navigation systems. However, the trigonometric operations required in the updates and forming of the rotation matrices for transforming the INS measurements from the body frame to the navigation frame turns the system model to be highly nonlinear. Besides, using low-cost MEMS-based IMUs, the gyroscope bias errors must be correctly estimated and compensated, which makes the nonlinearity problem a critical one. In this contribution, three Kalman filtering methods (i.e., Extended Kalman filter with simplified system model, Extended Kalman filter with linearized system model and Unscented Kalman filter with nonlinear system model) are utilized in INS/GPS tightly-coupled integration. Simulations and field experiments are conducted. Numerical results are compared in terms of both estimation accuracy and processing time.

This paper addresses the problem of estimating time varying phase noise caused by imperfect oscillators in multiple-input multiple-output (MIMO) systems. The estimation problem is parameterized in detail and based on an equivalent signal model its dimensionality is reduced to minimize the overhead associated with phase noise estimation. New exact and closed-form expressions for the Bayesian Cramer-Rao lower bounds (BCRLBs) and soft-input maximum a posteriori (MAP) estimators for online, i.e., filtering, and offline, i.e., smoothing, estimation of phase noise over the length of a frame are derived. Simulations demonstrate that the proposed MAP estimators' mean-square error (MSE) performances are very close to the derived BCRLBs at moderate-to-high signal-to-noise ratios. To reduce the overhead and complexity associated with tracking the phase noise processes over the length of a frame, a novel soft-input extended Kalman filter (EKF) and extended Kalman smoother (EKS) that use soft statistics of the transmitted symbols given the current observations are proposed. Numerical results indicate that by employing the proposed phase tracking approach, the bit-error rate performance of a MIMO system affected by phase noise can be significantly improved. In addition, simulation results indicate that the proposed phase noise estimation scheme allows for application of higher order modulations and larger numbers of antennas in MIMO systems that employ imperfect oscillators.

SummaryThis paper considers the state estimation problem of bilinear systems in the presence of disturbances. The standard Kalman filter is recognized as the best state estimator for linear systems, but it is not applicable for bilinear systems. It is well known that the extended Kalman filter (EKF) is proposed based on the Taylor expansion to linearize the nonlinear model. In this paper, we show that the EKF method is not suitable for bilinear systems because the linearization method for bilinear systems cannot describe the behavior of the considered system. Therefore, this paper proposes a state filtering method for the single‐input–single‐output bilinear systems by minimizing the covariance matrix of the state estimation errors. Moreover, the state estimation algorithm is extended to multiple‐input–multiple‐output bilinear systems. The performance analysis indicates that the state estimates can track the true states. Finally, the numerical examples illustrate the specific performance of the proposed method.

This paper presents a new two-step pseudo maximum a posteriori (MAP) segmentation method for the Markov random field (MRF)-modeled image because the exact MAP estimation is hard to implement due to intractable complexity. The expectation maximization (EM) and Markov Chain Monte Carlo (MCMC) methods are adopted to estimate the parameters for the MRF model due to their comparatively good performance. Although the image segmentation algorithms via graph cuts have become very popular nowadays, our proposed algorithm still performs significantly better in automatic identification and segmentation of fuzzy images than them, which is shown by the quantitative results on synthesized images. In practical applications, the proposed two-step pseudo MAP method is superior in segmenting the fuzzy laser images reflected from the weld pool surfaces during the P-GMAW welding process.

Maximum A Posteriori estimation for AUV localization with USBL measurements

This paper investigates the problem of nonlinear state estimation for control moment gyroscope (CMG) inverted pendulum. The dynamics of CMG inverted pendulum result into stiff nonlinearities and are included here. Both Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) algorithms are utilized for state estimation to investigate their effectiveness. The computer simulations are included for comparison of the two algorithms. There are inherent difficulties in state estimation for systems with stiff nonlinearities. In this research, the results show that EKF performs far better than UKF in most cases for this system. However, this is an attempt to explore the nonlinear estimation problem for this challenging system.

Robust unscented Kalman filter with adaptation of process and measurement noise covariances

We consider the identification problem for a linear system when the variances of the system and/or the observation noises take on two unknown values. It is assumed that the system and observation noises can be represented in terms of Bernoulli sequences and two random variables with specified but unknown variances. We propose an algorithm to simultaneously estimate the unknown variances and the switching parameters by applying the EM (expectation-maximization) algorithm, avoiding non-linear optimizations that arise in the Maximum Likelihood (ML) and the Maximum A Posteriori (MAP) estimations. Several numerical examples are given to illustrate the effectiveness of the present algorithm.