Suboptimal State Estimation Research Articles

Reconstructing tephra fall deposits via ensemble-based data assimilation techniques

Abstract. In recent years, there has been a growing interest in ensemble approaches for modelling the atmospheric transport of volcanic aerosol, ash, and lapilli (tephra). The development of such techniques enables the exploration of novel methods for incorporating real observations into tephra dispersal models. However, traditional data assimilation algorithms, including ensemble Kalman filter (EnKF) methods, can yield suboptimal state estimates for positive-definite variables such as those related to volcanic aerosols and tephra deposits. This study proposes two new ensemble-based data assimilation techniques for semi-positive-definite variables with highly skewed uncertainty distributions, including aerosol concentrations and tephra deposit mass loading: the Gaussian with non-negative constraints (GNC) and gamma inverse-gamma (GIG) methods. The proposed methods are applied to reconstruct the tephra fallout deposit resulting from the 2015 Calbuco eruption using an ensemble of 256 runs performed with the FALL3D dispersal model. An assessment of the methodologies is conducted considering two independent datasets of deposit thickness measurements: an assimilation dataset and a validation dataset. Different evaluation metrics (e.g. RMSE, MBE, and SMAPE) are computed for the validation dataset, and the results are compared to two references: the ensemble prior mean and the EnKF analysis. Results show that the assimilation leads to a significant improvement over the first-guess results obtained from the simple ensemble forecast. The evidence from this study suggests that the GNC method was the most skilful approach and represents a promising alternative for assimilation of volcanic fallout data. The spatial distributions of the tephra fallout deposit thickness and volume according to the GNC analysis are in good agreement with estimations based on field measurements and isopach maps reported in previous studies. On the other hand, although it is an interesting approach, the GIG method failed to improve the EnKF analysis.

Predicting the output error of the suboptimal state estimator to improve the performance of the MPC-based artificial pancreas

The error of single step-ahead output prediction is the information traditionally used to correct the state estimate while exploiting the new measurement of the system output. However, its dynamics and statistical properties can be further studied and exploited in other ways. It is known that in the case of suboptimal state estimation, this output prediction error forms a correlated sequence, hence it can be effectively predicted in real time. Such a suboptimal scenario is typical in applications where the process noise model is not known or it is uncertain. Therefore, the paper deals with the problems of analytical and empirical modeling, identification, and prediction of the output error of the suboptimal state estimator for the sake of improving the output prediction accuracy and ultimately the performance of the model predictive control. The improvements are validated on an empirical model of type 1 diabetes within an in-silico experiment focused on glycemia prediction and implementation of the MPC-based artificial pancreas.

Ensemble Kalman filter based data assimilation for tropical waves in the MJO skeleton model

AbstractThe Madden–Julian oscillation (MJO) is the dominant component of tropical intraseasonal variability, with wide‐reaching impacts even on extratropical weather and climate patterns. However, predicting the MJO is challenging. One reason is the suboptimal state estimates obtained with standard data assimilation (DA) approaches. These are typically based on filtering methods with Gaussian approximations and do not take into account physical properties that are important specifically for the MJO. In this article, a constrained ensemble DA method is applied to study the impact of different physical constraints on the state estimation and prediction of the MJO. The quadratic programming ensemble (QPEns) algorithm utilized extends the standard stochastic ensemble Kalman filter (EnKF) with specifiable constraints on the updates of all ensemble members. This allows us to recover physically more consistent states and to respect possible associated non‐Gaussian statistics. The study is based on identical twin experiments with an adopted nonlinear model for tropical intraseasonal variability. This so‐called skeleton model succeeds in reproducing the main large‐scale features of the MJO and closely related tropical waves, while keeping adequate simplicity for fast experiments on intraseasonal time‐scales. Conservation laws and other crucial physical properties from the model are examined as constraints in the QPEns. Our results demonstrate an overall improvement in the filtering and forecast skill when the model's total energy is conserved in the initial conditions. The degree of benefit is found to be dependent on the observational setup and the strength of the model's nonlinear dynamics. It is also shown that, even in cases where the statistical error in some waves remains comparable with the stochastic EnKF during the DA stage, their prediction is improved remarkably when using the initial state resulting from the QPEns.

Probabilistic Modeling and Bayesian Filtering for Improved State Estimation for Soft Robots

State estimation is one of the key requirements in robot control, which has been achieved by kinematic and dynamic models combined with motion sensors in traditional robotics. However, it is challenging to acquire accurate proprioceptive information in soft robots due to relatively high noise levels and hysteretic responses of soft actuators and sensors. In this article, we propose a method of estimating real-time states of soft robots by filtering noisy output signals and including hysteresis in the models using a Bayesian network. This approach is useful in constructing a state observer for soft robot control when both the kinematic model of the actuator and the model of the sensor are used. In our method, we regard a hysteresis function as a conditional random process model. We then introduce a dynamic Bayesian network composed of the actuator and the sensor models of the target system using distribution hysteresis mapping. Finally, we show that solving a Bayesian filtering problem is equivalent to suboptimal state estimation of the soft system. This article describes two ways for defining modeling and filtering; one is by Gaussian process regression combined with an extended Kalman filter, and the other is based on variational inference with a particle filter. While the first approach relaxes the uncertainty level in modeling to Gaussian, the second approach illustrates a general probability distribution. We experimentally validate the proposed methods through real-time state estimation of a sensor-integrated soft robotic gripper. The result shows significant improvement in state estimation compared to conventional estimation methods.

Adaptive State Estimation for Power Systems Measured by PMUs With Unknown and Time-Varying Error Statistics

Measurement error is a crucial factor that determines the accuracy of state estimation (SE). Conventional estimators have fixed models, and can yield optimal performance only when the measurement error statistics exactly meet the assumptions. In reality, however, the error distribution is usually unknown and time-varying, resulting in suboptimal state estimates in most cases. This paper develops the concept of adaptive SE for power systems measured by phasor measurement units (PMUs). First, the Gaussian-Laplacian Mixture (GLM) model is developed to fit the body and tail of unknown measurement error distributions. Then, an adaptive estimation framework is proposed based on the Expectation-Maximization (EM) algorithm. It is capable of tracking the actual error statistics online, and adjusting the parameters of SE to maintain near-optimality of state estimates under complex measurement error conditions. Simulation results demonstrate that the proposed adaptive estimator yields more accurate state estimates than the well-known Weighted Least Squares (WLS) and Weighted Least Absolute Value (WLAV) estimators by adapting itself to the variations of measurement error statistics.

State estimation of CPSs with deception attacks: Stability analysis and approximate computation

This paper is concerned with the state estimation problems of cyber-physical systems (CPSs) under unobservable deception attacks. First, the optimal state estimator is provided based on the derived state probability density function, which consists of an exponentially increasing number of linear Gaussian hypotheses. The exponentially growing number of components will lead to high computational cost. Therefore, a suboptimal state estimator based on the IMM algorithm is proposed, which is computationally more efficient than the optimal estimator. Finally, numerical results are given to verify the effectiveness and superiority of the proposed suboptimal estimator, rendering an efficient and stable state estimation when the privacy of sensor measurements is attacked.

On stability and convergence of suboptimal estimation for systems over lossy networks without acknowledgement

This paper concentrates on the suboptimal state estimation problem for systems without acknowledgement (ACK). The ACK signal is used for informing the estimator of whether control-input packets have been lost during the transmission or not. For such systems, which are usually regarded as NACK systems, the optimal estimator has been proved to be beset by exponentially increasing computational burden so that it is scarcely possible to be implemented in practice and then the suboptimal estimation becomes a necessary need. Inspired by the idea of multiple-model approaches, we first design an efficient suboptimal estimator for NACK systems to significantly improve computational efficiency. More importantly, we obtained a sufficient and necessary condition to characterise the stability of this proposed suboptimal estimator, which depends only on the packet loss rate (PLR) of the observation regardless of the control-input PLR. We also prove that its estimation error will converge to that of the optimal estimator under a certain condition. A distinguishing feature of the suboptimal estimator proposed in this paper is that not only its stability and convergence can be determined theoretically, but also they are proved to be the same as those of the optimal one for NACK systems.

State estimate for stochastic systems with dual unknown interference inputs

Stochastic system state estimation subject to the unknown interference input widely exists in many fields, such as the control, communication, signal processing, and fault diagnosis. However, the research results are mostly limited to the stochastic system in which only the dynamic state model or the measurement model concerns the individual unknown interference input, and the state model and the measurement model are both with the same unknown interference input. State estimate of the stochastic systems where the state model and the measurement model contain dual Unknown Interference inputs (dual-UI) with different physical meanings and mathematical definitions is concerned here. Firstly, the decoupling condition with the Unknown Interference input in the State model (S-UI) is shown, which introduces the decoupled system with the adjacent Measurement concerned Unknown Interference inputs (M-UI) appearing in the state model and the measurement model. Then, through defining the Differential term of the adjacent M-UI (M-UID), the equivalent system with only M-UID in the state model is obtained. Finally, considering the design freedom of the equivalent system, the decoupling filter in the minimum mean square error sense and the adaptive minimum upper filter with different applicable conditions are represented to obtain the optimal and sub-optimal state estimate, respectively. Two simulation cases verify the effectiveness and superiority compared with the traditional methods.

Non-fragile Suboptimal Set-membership Estimation for Delayed Memristive Neural Networks with Quantization via Maximum-error-first Protocol

This paper is concerned with the non-fragile protocol-based set-membership estimation problem for a class of discrete memristive neural networks (MNNs) with mixed time-delays, quantization and unknown but bounded noises. The nonlinear neural activation function satisfies the sector-bounded condition and the logarithmic quantization error is transformed to the norm-bounded uncertainty. In order to save the networks resources, the maximum-error-first (MEF) protocol is introduced to allocate the utilization order of the network channel. The focus is on the design of non-fragile state estimator to ensure such that, in the simultaneous presence of the mixed time-delays, quantization errors and estimator gain perturbations, real state is confined to the ellipsoid. In particular, a minimization problem is given to determine the radius of the designed ellipsoid and the estimator gain matrix by testifying the feasibility of some recursive matrix inequalities. Finally, some simulations are used to show the feasibility of the developed non-fragile suboptimal state estimation strategy.

Correcting observation model error in data assimilation.

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications, the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models and on a single-column radiative transfer model with multicloud parameterization.

The effects of proportional steering strategies on the behavior of controlled clocks

Two approaches to steering clocks to a reference standard via frequency steers have been explored in the literature, the pole placement (PP) approach and the linear quadratic Gaussian (LQG) approach. The PP method sets the response time of the controlled clock. LQG minimizes a user-selected cost function, which is a linear combination of the expected variances of the phase, frequency, and control effort (frequency steers). This paper unifies these techniques by developing an analysis that quantifies what time constants follow from the LQG-derived gains, and what costs are incurred by the PP-derived gains. With this analysis, one can compute how sensitive the outcomes would be to variations of the steering constants. Another outcome we present is a quantification of the degradation due to a common form of suboptimal state estimation, as well as that due to delays before steering decisions can be implemented into the system.

An upper bound of mean-square error in state estimation with quantized measurements

In this paper, we study the state estimation for a linear time-invariant (LTI) discrete-time system with quantized measurements. The quantization law under consideration has a time-varying data rate. To cope with nonlinearities in quantization laws and to analyse stability in the state estimation problem, a Kalman-filter-based sub-optimal state estimator is developed and an upper bound of its estimation error covariance is minimized. It turns out that, to guarantee the convergence of the upper bound, the averaged data rate of the quantization law must be greater than a minimum rate. This minimum data rate for the quantization law is presented in terms of the poles of the system and design parameters in the state estimator. Numerical examples are presented to illustrate the results in this work.

Application of Federal Kalman Filter with Neural Networks in the Velocity and Attitude Matching of Transfer Alignment

The centralized Kalman filter is always applied in the velocity and attitude matching of Transfer Alignment (TA). But the centralized Kalman has many disadvantages, such as large amount of calculation, poor real-time performance, and low reliability. In the paper, the federal Kalman filter (FKF) based on neural networks is used in the velocity and attitude matching of TA, the Kalman filter is adjusted by the neural networks in the two subfilters, the federal filter is used to fuse the information of the two subfilters, and the global suboptimal state estimation is obtained. The result of simulation shows that the federal Kalman filter based on neural networks is better in estimating the initial attitude misalignment angle of inertial navigation system (INS) when the system dynamic model and noise statistics characteristics of inertial navigation system are unclear, and the estimation error is smaller and the accuracy is higher.

Discrete-Time LQG Control With Input Delay and Multiplicative Noise

This paper is concerned with linear quadratic Gaussian (LQG) control problem for discrete-time systems with both multiplicative noises and input delay. The main contributions are two-fold. First, when the state variables can be obtained exactly, the optimal LQG controller is derived which consists of the feedback form of the conditional expectation of the state and one additive deterministic term based on the coupled Riccati equations. Through the completing square approach, the optimal controller is obtained based on the complete solution to the forward and backward stochastic difference equations. Second, when the state variables are partially observed, we derive a suboptimal linear state estimate feedback controller for LQG models involving multiplicative noises and input delay. Since the separation principle does not hold and the optimal estimator cannot be obtained, the suboptimal linear state estimate feedback controller is derived by linearizing the linear optimal estimator and using the obtained results of the optimal LQG control.

Robust recovery of myocardial kinematics using dual ℋ ∞ $\mathcal {H}_{\infty }$ criteria

Accurate estimation of myocardial motion can help to better understand the pathophysiological processes of ischemic heart diseases. However, because of partial and noisy image-derived measurements on the cardiac kinematics, the performance of model-based motion estimation relies heavily on the assumption of noise distribution on the measurement data. While existing studies of model-based motion estimation have often adopted the $\mathcal {H}_{2}$ criteria (e.g. least square error) based on fixed model constraints from mathematical or mechanical nature, we present a robust estimation framework with an adaptive biomechanical model constraint using dual $\mathcal {H}_{\infty }$ criteria for the first time. Comparing to the minimization of average gaussian error in $\mathcal {H}_{2}$ criteria, our $\mathcal {H}_{\infty }$ criteria aims to minimize the maximum error without regarding the noise distribution. In this work, our dual estimation framework consists of two iterative $\mathcal {H}_{\infty }$ filters: One filter for the kinematics estimation and another one for the elasticity estimation. At each time step, heart kinematics is estimated with sub-optimal fixed material parameters, followed by an elasticity property recovering given these sub-optimal kinematic state estimates. Such coupled estimation processes are iteratively repeated as necessary until convergence. We evaluate the performance of dual estimation framework on synthetic data, cine image sequences, and human image sequence. Our dual estimation framework shows a higher tolerance of noise than the conventional extended Kalman filter. The results obtained by both synthetic data of varying noises and magnetic resonance image sequences demonstrate the accuracy and robustness of the proposed strategy.

Input Design for Fault Detection Using Extended Kalman Filter and Reinforcement Learning

The paper deals with an active fault detection problem for jump Markov nonlinear systems with Gaussian noises. The problem is formulated as a functional optimization problem over an infinite-time horizon with a general discounted detection criterion. The design of an active fault detector is performed in two steps. First, the original problem is recast as a perfect state information problem by complementing the nonlinear system with a suboptimal state estimator based on a bank of extended Kalman filters. Then, a temporal-difference learning algorithm is used to train the active fault detector such that the criterion is minimized. A simulation example of a differential wheeled robot is used to illustrate the performance of the proposed design.

The GIGG‐EnKF: ensemble Kalman filtering for highly skewed non‐negative uncertainty distributions

Observations and predictions of near‐zero non‐negative variables such as aerosol, water vapour, cloud, precipitation and plankton concentrations have uncertainty distributions that are skewed and better approximated by gamma and inverse‐gamma probability distribution functions (pdfs) than Gaussian pdfs. Current Ensemble Kalman Filters (EnKFs) yield suboptimal state estimates for these variables. Here, we introduce a variation on the EnKF that accurately solves Bayes' theorem in univariate cases where the prior forecasts and error‐prone observations given truth come in (gamma, inverse‐gamma) or (inverse‐gamma, gamma) or (Gaussian, Gaussian) distribution pairs. Since its multivariate extension is similar to an EnKF, we refer to it as the GIGG‐EnKF or GIGG where GIGG stands for Gamma, Inverse‐Gamma and Gaussian. The GIGG‐EnKF enables near‐zero semi‐positive‐definite variables with highly skewed uncertainty distributions to be assimilated without the need for observation bias inducing log‐normal or Gaussian anamorphosis nonlinear transformations. In the special case that all observations are treated as Gaussian, the GIGG‐EnKF gives identical results to the original EnKF. A multi‐grid‐point and multi‐variable idealized system was used to compare and contrast the data assimilation performance of the GIGG with that of both the perturbed observation and deterministic forms of the EnKF. This test system featured variables and observation types whose uncertainty distributions approximate Gaussian, gamma and inverse‐gamma distributions. The normalized analysis error variance of the GIGG ensemble mean was found to be significantly smaller than that of the EnKFs. The higher moments of the analysed ensemble distributions were tested by subjecting the ensemble members to nonlinear ‘forecast’ mappings. The normalized mean square error of the mean of the corresponding GIGG forecast ensemble was found to be less than a 3rd of that obtained from either form of the original EnKF.

Almost sure state estimation with [formula omitted]-type performance constraints for nonlinear hybrid stochastic systems

This paper is concerned with the problem of almost sure state estimation for general nonlinear hybrid stochastic systems whose coefficients only satisfy local Lipschitz conditions. By utilizing the stopping time method combined with martingale inequalities, a theoretical framework is established for analyzing the so-called almost surely asymptotic stability of the addressed system. Within such a theoretical framework, some sufficient conditions are derived under which the estimation dynamics is almost sure asymptotically stable and the upper bound of estimation error is also determined. Furthermore, a suboptimal state estimator is obtained by solving an optimization problem in the H2 sense. According to the obtained results, for a class of special nonlinear hybrid stochastic systems, the corresponding conditions reduce to a set of matrix inequalities for the purpose of easy implementation. Finally, two numerical simulation examples are used to demonstrate the effectiveness of the results derived.

White noise estimation for discrete-time systems with random delay and packet dropout

This paper is concerned with the optimal and suboptimal deconvolution problems for discrete-time systems with random delayed observations. When the random delay is known online, i.e., time stamped, the random delayed system is reconstructed as an equivalent delay-free one by using measurement reorganization technique, and then an optimal input white noise estimator is presented based on the stochastic Kalman filtering theory. However, the optimal white-noise estimator is time- varying, stochastic, and doesn't converge to a steady state in general. Then an alternative suboptimal input white-noise estimator with deterministic gains is developed under a new criteria. The estimator gain and its respective error covariance-matrix information are derived based on a new suboptimal state estimator. It can be shown that the suboptimal input white-noise estimator converges to a steady-state one under appropriate assumptions.

State estimation for discrete-time Markov jump linear systems with multiplicative noises and delayed mode measurements

In this paper, the state estimation problem for discrete-time Markov jump linear systems affected by multiplicative noises is considered. The available measurements for the system under consideration have two components: the first is the model measurement and the second is the output measurement, where the model measurement is affected by a fixed amount of delay. Using Bayes' rule and some results obtained in this paper, a novel suboptimal state estimation algorithm is proposed in the sense of minimum mean-square error under a lot of Gaussian hypotheses. The proposed algorithm is recursive and does not increase computational and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed algorithm.