State Estimation Of Dynamic Systems Research Articles

Symbolic Representation and Toolkit Development of Iterated Error-State Extended Kalman Filters on Manifolds

Error-state extended Kalman filter (ESEKF) is one of extensively used filtering techniques in robot systems. There are many works that cast ESEKF on manifolds to improve consistency. However, most of these works are designed case by case, which makes it difficult to extend to new manifolds. In this paper, we propose a generic method to formulate the iterated error-state extended Kalman filter (IESEKF) on manifolds, which aims to facilitate the deployment of IESEKF for on-manifold systems (e.g., lidar-inertial and visual-inertial systems). Firstly, a canonical on-manifold representation of the robot system is proposed, based on which, an on-manifold IESEKF framework is formulated and solved by linearization at each estimation point. The proposed framework has two main advantages, one is that an equivalent error-state system is derived from linearization, which is minimally-parameterized without any singularities in practice. And the other is that in each step of IESEKF, the manifold constraints are decoupled from the system behaviors, ultimately leading to a generic and symbolic IESEKF framework that naturally evolving on manifolds. Based on the separation of manifold constraints from the system behaviors, the on-manifold IESEKF is implemented as a toolkit in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C$</tex-math></inline-formula> ++ packages, with which the user needs only to provide the system-specific descriptions, and then call the respective filter steps (e.g., predict, update) without dealing with any manifold constraints. The existing implementation supports full iterated Kalman filtering for versatile systems on manifold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M} = \mathbb {R}^{m}\!\times SO(3)\!\times \!\cdots \!\times \! SO(3)\!\times \! SE_{N}(3)\!\times \!\cdots \!\times \! SE_{N}(3)\!\times \mathbb {S}^{2} \times \cdots \times \mathbb {S}^{2}$</tex-math></inline-formula> or any of its sub-manifolds, and is extendable to other types of manifold when necessary. The proposed symbolic IESEKF and the developed toolkit are verified by implementing two filter-based tightly-coupled lidar-inertial navigation systems. Results show that, while greatly facilitating the EKF deployment the developed toolkit leads to estimation performances and computation efficiency comparable to hand-engineered counterparts. Finally, the toolkit is open-sourced at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/hku-mars/IKFoM</uri> . The aimed application is the real-time state estimation of dynamic systems (e.g., robots) whose states are evolving on manifolds.

Gaussian Mixture Filtering with Nonlinear Measurements Minimizing Forward Kullback-Leibler Divergence

A Gaussian mixture filter is proposed for the state estimation of dynamical systems with nonlinear measurements. The filter is derived by solving an assumed density filtering problem where Kullback-Leibler (KL) divergence from the assumed posterior, which is a Gaussian mixture, to the true posterior (which we call forward KL divergence) is minimized. The approximate solution to this problem gives an iterative measurement update by which the weights, means and covariances of the assumed posterior are optimized. The resulting Gaussian mixture filter is shown to be a generalization of the (damped) posterior linearization filter to Gaussian mixture posteriors. The performance of the proposed filter is illustrated and compared to alternatives on a challenging example of target tracking in a sensor network and on an example of tracking with a radar. The results show that the proposed filter can outperform Gaussian filters as well as the Gaussian sum filter and the Gaussian sum particle filter yielding results very close to a bootstrap particle filter when the number of components in the assumed posterior is sufficiently large.

PASE: An autonomous sequential framework for the state estimation of dynamical systems

Kalman filter (KF) and its variants are commonly used in estimating states of dynamical systems. For accurate estimation of states, researchers have developed intelligent KFs by combining them with machine learning algorithms. The inherent variations in the vehicle dynamics and uncertainties in the environment require the development of an autonomous sequential framework for continuous estimation. Further, the need for real-time estimation of states in certain applications requires low computer memory usage and low computational cost while implementing autonomous-structured learning algorithms. In this paper, a parsimonious autonomous sequential estimator (PASE) is proposed, which combines the KF-based estimator and autonomous-structured recurrent parsimonious learning machine (rPALM) in a sequential manner. The rPALM overcomes the dependency on target variables while point-to-hyperplane distance calculation. The performance of PASE has been evaluated extensively by comparing it with various batch-learning algorithms and single-pass learning-based intelligent estimators. The results clearly indicate that PASE provides better estimation accuracy with a compact architecture for both linear and nonlinear dynamical systems. Finally, the performance of PASE has been evaluated with experimental data for the state estimation of an unmanned ground vehicle while the training of the learning machine is performed with the simulated data. The estimation accuracy in such a scenario is justifying its appropriateness in real-world applications.

State estimation for dynamic systems with higher-order autoregressive moving average non-Gaussian noise

The classical Kalman filter is a very important state estimation approach, which has been widely used in many engineering applications. The Kalman filter is optimal for linear dynamic systems with independent Gaussian noises. However, the independence and Gaussian assumptions may not be satisfied in practice. On the one hand, modeling physical systems usually results in discrete-time state-space models with correlated process and measurement noises. On the other hand, the noise is non-Gaussian when the system is disturbed by heavy-tailed noise. In this case, the performance of the Kalman filter will deteriorate, or even diverge. This paper is devoted to addressing the state estimation problem of linear dynamic systems with high-order autoregressive moving average (ARMA) non-Gaussian noise. First, a triplet Markov model is introduced to model the system with high-order ARMA noise, since this model relaxes the independence assumption of the hidden Markov model. Then, a new filter is derived based on correntropy, instead of the commonly used minimum mean square error (MMSE), to deal with non-Gaussian noise. Unlike the MMSE, which uses only second-order statistics of error, correntropy can capture second-order and higher-order statistics. Finally, simulation results verify the effectiveness of the proposed algorithm.

Variational-Based Nonlinear Bayesian Filtering With Biased Observations

State estimation of dynamical systems is crucial for providing new decision-making and system automation information in different applications. However, the assumptions on the standard computational models for sensor measurements can be violated in practice due to different types of data abnormalities such as outliers and biases. In this work, we focus on the occurrence of measurement biases and propose a robust filter for their detection and mitigation during state estimation of nonlinear dynamical systems. We model the presence of bias in each dimension within the generative structure of the state-space models. Subsequently, employing the theory of Variational Bayes and general Gaussian filtering, we devise a recursive filter which we call the Bias Detecting and Mitigating (BDM) filter. As the error detection mechanism is embedded within the filter structure its dependence on any external detector is obviated. Simulations verify the performance gains of the proposed BDM filter compared to similar Kalman filtering-based approaches in terms of robustness to temporary and persistent bias presence.

KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics

State estimation of dynamical systems in real-time is a fundamental task in signal processing. For systems that are well-represented by a fully known linear Gaussian state space (SS) model, the celebrated Kalman filter (KF) is a low complexity optimal solution. However, both linearity of the underlying SS model and accurate knowledge of it are often not encountered in practice. Here, we present KalmanNet, a real-time state estimator that learns from data to carry out Kalman filtering under non-linear dynamics with partial information. By incorporating the structural SS model with a dedicated recurrent neural network module in the flow of the KF, we retain data efficiency and interpretability of the classic algorithm while implicitly learning complex dynamics from data. We demonstrate numerically that KalmanNet overcomes non-linearities and model mismatch, outperforming classic filtering methods operating with both mismatched and accurate domain knowledge.

A distributed algorithm for state estimation of dynamic coupled and networked systems with cross‐correlated noises

AbstractWe study the distributed problem for state estimation of dynamic systems, in which the system and measurement noises of each subsystem are cross‐correlated, and the local state of each subsystem is also coupled with that of the neighboring subsystems. By newly constructing measurements, the noises are uncorrelated, and we derive an algorithm only using the measurements and low‐dimensional information from neighboring subsystems. Furthermore, the obtained error covariance matrices have lower and upper bounds and the proposed algorithm enjoys local stability properties. In addition, two examples are shown the effectiveness of our algorithm when compared to the centralized Kalman filter, which is the optimal algorithm for entire information.

DYNAMIC SYSTEMS STATE, DISTURBANCES AND NOISES SET-VALUED ESTIMATION UNDER CONDITIONS OF INCOMPLETE INFORMATION

The paper considers the problem of set-valued dynamic systems state estimation under conditions of uncertainty, when the sets of disturbances and noises possible values are known and statistical information about them is absent or cannot be obtained. An algorithm for feasible set polyhedral approximation is described, when the sets of possible values of disturbances and noises are polyhedra. The algorithm is based on the implicit description of the information set with linear equations and inequalities systems and solving a number of linear programming problems. Methods for increasing the estimation accuracy by taking into account additional information about disturbances and noises models are considered. Set-valued estimation of the dynamical system state vector is described when the disturbances are given as a system of functions with unknown coefficients. In this case, due to the use of information that the coefficients are constant, the dynamic system state estimates are more accurate than in the case when the disturbances are known up to a set of possible values. A numerical example is presented to demonstrate the algorithm performance. Aim. The aim of the research is to develop dynamic system state, disturbance and noises set-valued estimation algorithms. Research methods. Methods of optimization theory, filtering, linear algebra, MATLAB software package were used in the work. Results. Dynamic system state estimation algorithm was described. The algorithm takes into account additional information about disturbances and noises models. A method of feasible set polyhedral approximation is described, which makes it possible to obtain a set-valued estimate of a state vector, a vector of disturbances and noises, and an evolution of reachable sets. It can be used in the adaptive estimation and control algorithms development. The algorithm for set-valued estimation of the system state vector and coefficients in the disturbance decomposition as a system of given functions is developed. Conclusion. An algorithm for feasible set polyhedral approximation was described.The numerical example was performed and the analysis of the estimateswas presented.

Power Test System Development and Dynamic Performance State Estimation Based on Hub Motor Vehicle

Background: The power performance of an electric vehicle is the basic parameter. Traditional test equipment, such as the expensive chassis dynamometer, not only increases the cost of testing but also makes it impossible to measure all the performance parameters of an electric vehicle. Objective: A set of convenient, efficient and sensitive power measurement system for electric vehicles is developed to obtain the real-time power changes of hub-motor vehicles under various operating conditions, and the dynamic performance parameters of hub-motor vehicles are obtained through the system. Methods: Firstly, a set of on-board power test system is developed by using virtual instrument (Lab- VIEW). This test system can obtain the power changes of hub-motor vehicles under various operating conditions in real-time and save data in real-time. Then, the driving resistance of hub-motor vehicles is analyzed, and the power performance of hub-motor vehicles is studied in depth. The power testing system is proposed to test the input power of both ends of the driving motor, and the chassis dynamometer is combined to test so that the output efficiency of the driving motor can be easily obtained without disassembly. Finally, this method is used to carry out the road test and obtain the vehicle dynamic performance parameters. Results: The real-time current, voltage and power, maximum power, acceleration time and maximum speed of the vehicle can be obtained accurately by using the power test system in the real road experiment. Conclusion: The maximum power required by the two motors reaches about 9KW, and it takes about 20 seconds to reach the maximum speed. The total power required to maintain the maximum speed is about 7.8kw, and the maximum speed is 62km/h. In this article, various patents have been discussed.

A Novel Smooth Variable Structure Smoother for Robust Estimation.

The smooth variable structure filter (SVSF) is a new-type filter based on the sliding-mode concepts and has good stability and robustness in overcoming the modeling uncertainties and errors. However, SVSF is insufficient to suppress Gaussian noise. A novel smooth variable structure smoother (SVSS) based on SVSF is presented here, which mainly focuses on this drawback and improves the SVSF estimation accuracy of the system. The estimation of the linear Gaussian system state based on SVSS is divided into two steps: Firstly, the SVSF state estimate and covariance are computed during the forward pass in time. Then, the smoothed state estimate is computed during the backward pass by using the innovation of the measured values and covariance estimate matrix. According to the simulation results with respect to the maneuvering target tracking, SVSS has a better performance compared with another smoother based on SVSF and the Kalman smoother in different tracking scenarios. Therefore, the SVSS proposed in this paper could be widely applied in the field of state estimation in dynamic system.

A Polytopic Box Particle Filter for state estimation of Non Linear Discrete-Time Systems

The development of Particle Filters has made possible state estimation of dynamic systems presenting non-linear dynamics and potential multi-modalities. However, the efficiency of these approaches depends tightly of the required number of particles which may prove very high to approximate large range of uncertainty on the process or the measurements. To overcome this issue, the Box-Particle Filter (BPF) combines the versatility of the Particle Filter and the robustness of set-membership algorithms. The particles are replaced by boxes which represent in a compact way large variations of the estimates. Although this filter presents various advantages and requires a small number of boxes to estimate the state, the resulting estimates may prove pessimistic, as the uncertainty description as unions of axis-aligned intervals can be rather rough and doesn’t account for potential dependencies between the resulting estimate components. In the proposed paper, a new version of the BPF is proposed. Boxes are replaced by polytopes (multidimensional polygons) in the filter algorithm, so that they can adapt to represent state components dependency. This modification tends to ameliorate the estimation precision (i.e. the size of the final set that includes the true state decreases) while keeping the number of required polyhedrons small. Several examples illustrate the benefits of such an approach.

State estimation of dynamic systems with sandwich structure and hysteresis

In this paper, a non-smooth observer is proposed to estimate the states of a sandwich system with hysteresis. Based on the analysis of observability of this system, a non-smooth observer is constructed. The non-smooth observer has a switchable transition matrix and contains an additional item which can switch according to the working zone of the system. By considering the suppression of model uncertainty and disturbance, the robust design of the non-smooth observer for the sandwich system with hysteresis is proposed. In the robust observer, the estimation error caused by the switched error is included as an extended disturbance and the selection of the feedback matrices of the robust observer has been converted into an optimal problem. Finally, the proposed methods are implemented to estimate the states of a micro-positioning stage with piezoelectric actuator which can be described as a non-smooth sandwich system with hysteresis. The estimations results by the non-smooth and the robust observer are superior to the conventional and the non-robust observer in the experimental case, respectively.

Double hybrid Kalman filtering for state estimation of dynamical systems

In this paper authors present a new approaches to the hybrid Kalman filtering and modified hybrid Kalman filtering, with the changed order of methods inside (Unscented Kalman Filter and Extended Kalman Filter). For these algorithms, the modification based on double use of Hybrid Kalman Filters (Excented and Unscented) has been proposed. This new modification has been checked for Hybrid Kalman Particle Filters too, for the variable number of particles. Based on the obtained results, one can see that duplication of hybrid filters can improve the estimation quality.

Comparison of Auxiliary and Likelihood Particle Filters for State Estimation of Dynamical Systems

Comparison of Auxiliary and Likelihood Particle Filters for State Estimation of Dynamical Systems

Globally Optimal Distributed Kalman Filtering for Multisensor Systems with Unknown Inputs.

In this paper, the state estimation for dynamic system with unknown inputs modeled as an autoregressive AR (1) process is considered. We propose an optimal algorithm in mean square error sense by using difference method to eliminate the unknown inputs. Moreover, we consider the state estimation for multisensor dynamic systems with unknown inputs. It is proved that the distributed fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurement; therefore, it achieves the best performance. The computation complexity of the traditional augmented state algorithm increases with the augmented state dimension. While, the new algorithm shows good performance with much less computations compared to that of the traditional augmented state algorithms. Moreover, numerical examples show that the performances of the traditional algorithms greatly depend on the initial value of the unknown inputs, if the estimation of initial value of the unknown input is largely biased, the performances of the traditional algorithms become quite worse. However, the new algorithm still works well because it is independent of the initial value of the unknown input.

Application Proposal of Sequential Probability Ratio to Dynamic System State Estimation

Application Proposal of Sequential Probability Ratio to Dynamic System State Estimation

Use of uninformative priors to initialize state estimation for dynamical systems

The admissible region must be expressed probabilistically in order to be used in Bayesian estimation schemes. When treated as a probability density function (PDF), a uniform admissible region can be shown to have non-uniform probability density after a transformation. An alternative approach can be used to express the admissible region probabilistically according to the Principle of Transformation Groups. This paper uses a fundamental multivariate probability transformation theorem to show that regardless of which state space an admissible region is expressed in, the probability density must remain the same under the Principle of Transformation Groups. The admissible region can be shown to be analogous to an uninformative prior with a probability density that remains constant under reparameterization. This paper introduces requirements on how these uninformative priors may be transformed and used for state estimation and the difference in results when initializing an estimation scheme via a traditional transformation versus the alternative approach.

An Indoor Pedestrian Localization Algorithm Based on Multi-Sensor Information Fusion

For existing indoor localization algorithm has low accuracy, high cost in deployment and maintenance, lack of robustness, and low sensor utilization, this paper proposes a particle filter algorithm based on multi-sensor fusion. The pedestrian’s localization in indoor environment is described as dynamic system state estimation problem. The algorithm combines the smart mobile terminal with indoor localization, and filters the result of localization with the particle filter. In this paper, a dynamic interval particle filter algorithm based on pedestrian dead reckoning (PDR) information and RSSI localization information have been used to improve the filtering precision and the stability. Moreover, the localization results will be uploaded to the server in time, and the location fingerprint database will be built incrementally, which can adapt the dynamic changes of the indoor environment. Experimental results show that the algorithm based on multi-sensor improves the localization accuracy and robustness compared with the location algorithm based on Wi-Fi.

State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems

State estimation of dynamic systems is quite significant in many research areas, such as state-based control and stabilization, state-based monitoring and fault detection. This paper is concerned with the problem of observer-based state estimation for a class of fractional-order (FO) nonlinear dynamic systems. The objective is to consider the state estimation of FO nonlinear system with the sliding mode control (SMC) technique, and conduct the asymptotic stability analysis for the estimate error dynamic system. Firstly, the considered FO nonlinear system models with constant and uncertain parameters are both presented. Fractional order sliding mode observer (FOSMO) structures are established for the FO nonlinear system models. Then, the asymptotic stability of the estimate error dynamic systems are analyzed via employing the Lyapunov stability analysis method for FO systems, and the sufficient conditions of asymptotic stability are derived. FOSMOs design for FO nonlinear systems of Caputo׳s and Riemann–Liouville׳s differential operators are both investigated, and the corresponding asymptotic stability sufficient conditions of the error dynamic systems are presented to insure the estimation accuracy of FOSMOs. Finally, multiple simulation examples are provided to demonstrate the effectiveness of the presented FOSMOs.

A Limited-Memory Multiple Shooting Method for Weakly Constrained Variational Data Assimilation

Maximum-likelihood-based state estimation for dynamical systems with model error raises computational challenges in memory usage due to the much larger number of free variables when compared to the perfect model case. To address this challenge, we present a limited-memory method for maximum-likelihood-based estimation of state space models. We reduce the memory storage requirements by expressing the optimal states as a function of checkpoints bounding a shooting interval. All states can then be recomputed as needed from a recursion stemming from the optimality conditions. The matching of states at checkpoints is imposed, in a multiple shooting fashion, as constraints on the optimization problem, which is solved with an augmented Lagrangian method. We prove that for nonlinear systems under certain assumptions the condition number of the Hessian matrix of the augmented Lagrangian function is bounded above with respect to the number of shooting intervals. Hence the method is stable for increasing time horizo...