Constrained State Estimation - A Review
---
paper_title: Nonlinear constrained tracking of targets on roads

paper_content:

Ground targets are constrained to move on the Earth's surface and are most likely to travel along a road network. For targets on road, their interaction with the environment and with each other particularly at intersections is more structured, thus useful to tracking algorithms. Indeed, the knowledge of terrain database and road maps can be used as constraints and incorporated into the tracking algorithms. In this paper, we set forth a nonlinear formulation of tracking multiple interacting targets along road networks and apply nonlinear filtering techniques (namely, the extended Kalman filter, the unscented Kalman filter, the particle filter, and a hybrid Kalman particle filter) to the problem for such scenarios as air-to-ground surveillance. Simulation results are present to illustrate the performance of the nonlinear filters with the nonlinear formulation as compared to conventional ones.
---
paper_title: Robust and reliable estimation via Unscented Recursive Nonlinear Dynamic Data Reconciliation

paper_content:

Abstract The quality of process data in a chemical plant significantly affects the performance and benefits gained from activities like performance monitoring, online optimization and control. Since many chemical processes often exhibit nonlinear dynamics, techniques like Extended Kalman Filter (EKF) and Nonlinear Dynamic Data Reconciliation (NDDR) have been developed to improve the data quality. There are various issues that arise with the use of either of these techniques: EKF cannot handle inequality or equality constraints, while the NDDR has high computational cost. Recently a recursive estimation technique for nonlinear dynamic processes has been proposed which combines the merits of EKF and NDDR techniques. This technique, named as Recursive Nonlinear Dynamic Data Reconciliation (RNDDR), provides state and parameter estimates that satisfy bounds and other constraints imposed on them. However, the estimate error covariance matrix in RNDDR is computed in the same manner as in EKF, that is, the effects of both nonlinearity and constraints are neglected in the computation of the estimate error covariance matrix. A relatively new method known as the Unscented Kalman Filter has been developed for nonlinear processes, in which the statistical properties of the estimates are computed without resorting to linearization of the nonlinear equations. This leads to improved accuracy of the estimates. In this paper, we combine the merits of the Unscented Kalman Filter and the RNDDR to obtain the Unscented Recursive Nonlinear Dynamic Data Reconciliation (URNDDR) technique. This technique addresses all concerns arising due to the presence of nonlinearity and constraints within a recursive estimation framework, resulting in an efficient, accurate and stable method for real-time state and parameter estimation for nonlinear dynamic processes.
---
paper_title: Exploitation of a priori information for tracking maritime intermittent data sources

paper_content:

This paper presents results achieved by the Authors within the IPP1 Multi Sensorics program. The addressed research deals with the exploitation of context information, denoted as “Knowledge Base” within the different steps of multi-sensor data processing for surface surveillance. An overview on the research motivations and the application field (e.g., multi-target tracking in the Maritime Scenario) is given with focus on intermittent information sources (i.e., non-continuous sensor reports, due to low-SNR signals, sensor blind areas, communication loss, etc). The Knowledge Base is described by means of mathematical models and the inclusion of the a priori information into the Extended Kalman Filter is presented. Different target motion models are herein considered. Finally, the achievements are summarised in terms of track continuity performance over a live maritime traffic scenario.
---
paper_title: Recursive parameter identification of constrained systems: an application to electrically stimulated muscle

paper_content:

A method of incorporating constraints on model parameters is developed. This method is applicable to most recursive parameter-identification algorithms. It enforces linear equality constraints on identified parameters. The use of this method for the real-time identification of autoregressive-moving-average time series models, subject to parameter constraints, is described. These constraints may be time varying. The use of this algorithm is demonstrated in the identification of electrically stimulated quadriceps muscles in paraplegic human subjects, using percutaneous intramuscular electrodes. The nonlinear steady-state force versus pulsewidth recruitment characteristic of the electrode-muscle system is identified simultaneously with the input-output muscle response dynamics, using a Hammerstein-type model. Knowledge of the recruitment curve's shape is translated into constraints on the identified parameters. This information improves the experimental predictive quality of the identified model. >
---
paper_title: On Kalman Filtering With Nonlinear Equality Constraints

paper_content:

The state space description of some physical systems possess nonlinear equality constraints between some state variables. In this paper, we consider the problem of applying a Kalman filter-type estimator in the presence of such constraints. We categorize previous approaches into pseudo-observation and projection methods and identify two types of constraints-those that act on the entire distribution and those that act on the mean of the distribution. We argue that the pseudo-observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only. We propose a new method that utilizes the projection method twice-once to constrain the entire distribution and once to constrain the statistics of the distribution. We illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation
---
paper_title: Constrained Kalman filtering via density function truncation for turbofan engine health estimation

paper_content:

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter estimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine model. It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).
---
paper_title: Recursive estimation in constrained nonlinear dynamical systems

paper_content:

In any modern chemical plant or refinery, process operation and the quality of product depend on the reliability of data used for process monitoring and control. The task of improving the quality of data to be consistent with material and energy balances is called reconciliation. Because chemical processes often operate dynamically in nonlinear regimes, techniques such as extended-Kalman filter (EKF) and nonlinear dynamic data reconciliation (NDDR) have been developed for reconciliation. There are various issues that arise with the use of either of these techniques. EKF cannot handle inequality or equality constraints, whereas the NDDR has high computational cost. Therefore, a more efficient and robust method is required for reconciling process measurements and estimating parameters involved in nonlinear dynamic processes. Two solution techniques are presented: recursive nonlinear dynamic data reconciliation (RNDDR) and a combined predictor–corrector optimization (CPCO) method for efficient state and parameter estimation in nonlinear systems. The proposed approaches combine the efficiency of EKF and the ability of NDDR to handle algebraic inequality and equality constraints. Moreover, the CPCO technique allows deterministic parameter variation, thus relaxing another restriction of EKF where the parameter changes are modeled through a discrete stochastic equation. The proposed techniques are compared against the EKF and the NDDR formulations through simulation studies on a continuous stirred tank reactor and a polymerization reactor. In general, the RNDDR performs as well as the two traditional approaches, whereas the CPCO formulation provides more accurate results than RNDDR at a marginal increase in computational cost. © 2005 American Institute of Chemical Engineers AIChE J, 51: 946–959, 2005
---
paper_title: Neural drive estimation using the hypothesis of muscle synergies and the state-constrained Kalman filter

paper_content:

We explore the hypothesis of muscle synergies to estimate the neural drive (movement intent) for upper extremity myoelectric prosthesis using the surface myoelectric signals. Commonly employed pattern classification systems have certain limitations, like inherent discrete nature, finite movement classes and limited degrees-of-freedom. We propose a novel framework based on the state space modeling and the hypothesis of muscle synergies. The problem is formulated in the state space framework in a novel way, where the movement intent is modeled as the hidden state of the system. A continuous stream of the movement intent (the hidden state) is estimated using the state-constrained Kalman filter. Preliminary experimental results also confirm the applicability of the proposed framework for estimation of movement intent.
---
paper_title: Aircraft turbofan engine health estimation using constrained Kalman filtering

paper_content:

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This paper develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The incorporation of state variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results obtained from application to a turbofan engine model. This model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering.Copyright © 2003 by ASME
---
paper_title: Filtering method for nonlinear systems with constraints

paper_content:

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is converted to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pseudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.
---
paper_title: Optimal Combination and Constraints for Geometrical Sensor Data

paper_content:

A formalism is described for the statistical combination of geometrical information from multiple sensors. This is illus trated by its applications to stereo vision, including the com bination of multiple stereo views to increase the accuracy of a wire frame model and the consistent imposition of geometri cal constraints on such a model.
---
paper_title: Modeling and State Estimation for Dynamic Systems With Linear Equality Constraints

paper_content:

The problem of modeling and estimation for linear equality constrained (LEC) systems is considered. The exact constrained dynamic model usually is not readily available or is too complicated, and hence in many studies an auxiliary dynamic model is employed in which the state does not necessarily obey the constraint strictly. Based on the understanding that the constraints, as prior information about the state, should be incorporated into the dynamics modeling, an LEC dynamic model (LECDM) is constructed first. The model optimally fuses the linear equality constraint (LEC) and the auxiliary dynamics. Some of its superior properties are presented. Next, the linear minimum mean squared error (LMMSE) estimate of the LEC state is proved to satisfy the constraint. The LMMSE estimator for linear systems, called the LEC Kalman filter (LECKF), and two approximate LMMSE estimators for nonlinear systems are presented. The LECKF is compared with other constrained estimators, and a sufficient condition is also provided under which the estimate projection method mathematically equals the LECKF. Furthermore, extensions of the LECDM for the LEC systems with uncertain or unknown constraint parameters are discussed. Finally, illustrative examples are provided to show the effectiveness and efficiency of the LECKF and to verify the theoretical results given in the paper.
---
paper_title: Robust and reliable estimation via Unscented Recursive Nonlinear Dynamic Data Reconciliation

paper_content:

Abstract The quality of process data in a chemical plant significantly affects the performance and benefits gained from activities like performance monitoring, online optimization and control. Since many chemical processes often exhibit nonlinear dynamics, techniques like Extended Kalman Filter (EKF) and Nonlinear Dynamic Data Reconciliation (NDDR) have been developed to improve the data quality. There are various issues that arise with the use of either of these techniques: EKF cannot handle inequality or equality constraints, while the NDDR has high computational cost. Recently a recursive estimation technique for nonlinear dynamic processes has been proposed which combines the merits of EKF and NDDR techniques. This technique, named as Recursive Nonlinear Dynamic Data Reconciliation (RNDDR), provides state and parameter estimates that satisfy bounds and other constraints imposed on them. However, the estimate error covariance matrix in RNDDR is computed in the same manner as in EKF, that is, the effects of both nonlinearity and constraints are neglected in the computation of the estimate error covariance matrix. A relatively new method known as the Unscented Kalman Filter has been developed for nonlinear processes, in which the statistical properties of the estimates are computed without resorting to linearization of the nonlinear equations. This leads to improved accuracy of the estimates. In this paper, we combine the merits of the Unscented Kalman Filter and the RNDDR to obtain the Unscented Recursive Nonlinear Dynamic Data Reconciliation (URNDDR) technique. This technique addresses all concerns arising due to the presence of nonlinearity and constraints within a recursive estimation framework, resulting in an efficient, accurate and stable method for real-time state and parameter estimation for nonlinear dynamic processes.
---
paper_title: Filtering method for nonlinear systems with constraints

paper_content:

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is converted to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pseudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---
paper_title: Unscented filtering and nonlinear estimation

paper_content:

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---
paper_title: New extension of the Kalman filter to nonlinear systems

paper_content:

The Kalman Filter (KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearizes all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterize mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact, is directly comparable to that of the second order Gauss filter. The method is not restricted to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and more accurate estimation features of the new filter recommend its use over the EKF in virtually all applications.
---
paper_title: Unscented filtering and nonlinear estimation

paper_content:

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.
---
paper_title: A new method for the nonlinear transformation of means and covariances in filters and estimators

paper_content:

This paper describes a new approach for generalizing the Kalman filter to nonlinear systems. A set of samples are used to parametrize the mean and covariance of a (not necessarily Gaussian) probability distribution. The method yields a filter that is more accurate than an extended Kalman filter (EKF) and easier to implement than an EKF or a Gauss second-order filter. Its effectiveness is demonstrated using an example.
---
paper_title: Filtering method for nonlinear systems with constraints

paper_content:

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is converted to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pseudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.
---
paper_title: Constrained process monitoring: Moving‐horizon approach

paper_content:

Moving-horizon estimation (MHE) is an optimization-based strategy for process monitoring and state estimation. One may view MHE as an extension for Kalman filtering for constrained and nonlinear processes. MHE, therefore, subsumes both Kalman and extended Kalman filtering. In addition, MHE allows one to include constraints in the estimation problem. One can significantly improve the quality of state estimates for certain problems by incorporating prior knowledge in the form of inequality constraints. Inequality constraints provide a flexible tool for complementing process knowledge. One also may use inequality constraints as a strategy for model simplification. The ability to include constraints and nonlinear dynamics is what distinguishes MHE from other estimation strategies. Both the practical and theoretical issues related to MHE are discussed. Using a series of example monitoring problems, the practical advantages of MHE are illustrated by demonstrating how the addition of constraints can improve and simplify the process monitoring problem.
---
paper_title: Constrained Bayesian state estimation – A comparative study and a new particle filter based approach

paper_content:

Abstract This paper investigates constrained Bayesian state estimation problems by using a Particle Filter (PF) approach. Constrained systems with nonlinear model and non-Gaussian uncertainty are commonly encountered in practice. However, most of the existing Bayesian methods are unable to take constraints into account and require some simplifications. In this paper, a novel constrained PF algorithm based on acceptance/rejection and optimization strategies is proposed. The proposed method retains the ability of PF in nonlinear and non-Gaussian state estimation, while take advantage of optimization techniques in constraints handling. The performance of the proposed method is compared with other accepted Bayesian estimators. Extensive simulation results from three examples show the efficacy of the proposed method in constraints handling and its robustness against poor prior information.
---
paper_title: Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations

paper_content:

State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.
---
paper_title: Nonlinear Bayesian state estimation: A review of recent developments

paper_content:

Abstract Online estimation of the internal states is a perquisite for monitoring, control, and fault diagnosis of many engineering processes. A cost effective approach to monitor these variables in real time is to employ model-based state estimation techniques. Dynamic model-based state estimation is a rich and highly active area of research and many novel approaches have emerged over the last few years. In this paper, we review various recent developments in the area of nonlinear state estimators from a Bayesian perspective. In particular, we focus on the constrained state estimation (including systems modeled using differential-algebraic equations), the handling of multi-rate and delayed measurements and recent advances in model parameter estimation. Recent advances on the stability analysis of the estimation error dynamics are also briefly discussed. The review aims to provide an integrated view of important ideas, from the authors' perspective that have driven the research in this area in recent years.
---
paper_title: Constrained linear state estimation - a moving horizon approach

paper_content:

This article considers moving horizon strategies for constrained linear state estimation. Additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states, noise, and other variables. Formulating a linear state estimation problem with inequality constraints, however, prevents recursive solutions such as Kalman filtering, and, consequently, the estimation problem grows with time as more measurements become available. To bound the problem size, we explore moving horizon strategies for constrained linear state estimation. In this work we discuss some practical and theoretical properties of moving horizon estimation. We derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate. We also discuss smoothing strategies for moving horizon estimation. Our framework is solely deterministic.
---
paper_title: Constrained Nonlinear State Estimation Using Ensemble Kalman Filters

paper_content:

Recursive estimation of states of constrained nonlinear dynamic systems has attracted the attention of many researchers in recent years. In this work, we propose a constrained recursive formulation of the ensemble Kalman filter (EnKF) that retains the advantages of the unconstrained EnKF while systematically dealing with bounds on the estimated states. The EnKF belongs to the class of particle filters that are increasingly being used for solving state estimation problems associated with nonlinear systems. A highlight of our approach is the use of truncated multivariate distributions for systematically solving the estimation problem in the presence of state constraints. The efficacy of the proposed constrained state estimation algorithm using the EnKF is illustrated by application on two benchmark problems in the literature (a simulated gas-phase reactor and an isothermal batch reactor) involving constraints on estimated state variables and another example problem, which involves constraints on the process...
---
paper_title: Analysis Scheme in the Ensemble Kalman Filter

paper_content:

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter. It is shown that the observations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the observations and generate an ensemble of observations that then is used in updating the ensemble of model states. Traditionally, this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low. This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach.
---
paper_title: Data Assimilation: The Ensemble Kalman Filter

paper_content:

Statistical definitions.- Analysis scheme.- Sequential data assimilation.- Variational inverse problems.- Nonlinear variational inverse problems.- Probabilistic formulation.- Generalized Inverse.- Ensemble methods.- Statistical optimization.- Sampling strategies for the EnKF.- Model errors.- Square Root Analysis schemes.- Rank issues.- Spurious correlations, localization, and inflation.- An ocean prediction system.- Estimation in an oil reservoir simulator.
---
paper_title: What is the ensemble Kalman filter and how well does it work?

paper_content:

In this paper we described the ensemble Kalman filter algorithm. This approach to nonlinear Kalman filtering is a Monte Carlo procedure, which has been widely used in weather forecasting applications. Our goal was to apply the ensemble Kalman filter to representative examples to quantify the tradeoff between estimation accuracy and ensemble size. For all of the linear and nonlinear examples that we considered, the ensemble Kalman filter worked successfully once a threshold ensemble size was reached. In future work we will investigate the factors that determine this threshold value.
---
paper_title: Resampling Algorithms for Particle Filters: A Computational Complexity Perspective

paper_content:

Newly developed resampling algorithms for particle filters suitable for real-time implementation are described and their analysis is presented. The new algorithms reduce the complexity of both hardware and DSP realization through addressing common issues such as decreasing the number of operations and memory access. Moreover, the algorithms allow for use of higher sampling frequencies by overlapping in time the resampling step with the other particle filtering steps. Since resampling is not dependent on any particular application, the analysis is appropriate for all types of particle filters that use resampling. The performance of the algorithms is evaluated on particle filters applied to bearings-only tracking and joint detection and estimation in wireless communications. We have demonstrated that the proposed algorithms reduce the complexity without performance degradation.
---
paper_title: The Unscented Particle Filter

paper_content:

In this paper, we propose a new particle filter based on sequential importance sampling. The algorithm uses a bank of unscented filters to obtain the importance proposal distribution. This proposal has two very "nice" properties. Firstly, it makes efficient use of the latest available information and, secondly, it can have heavy tails. As a result, we find that the algorithm outperforms standard particle filtering and other nonlinear filtering methods very substantially. This experimental finding is in agreement with the theoretical convergence proof for the algorithm. The algorithm also includes resampling and (possibly) Markov chain Monte Carlo (MCMC) steps.
---
paper_title: On sequential Monte Carlo sampling methods for Bayesian filtering

paper_content:

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literatures these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.
---
paper_title: Sampling Densities of Particle Filter: A Survey and Comparison

paper_content:

The paper deals with the particle filter in discrete-time nonlinear non-Gaussian system state estimation. One of the key parameters affecting estimation quality of the particle filter is the sampling density (also called importance function or proposal density). In the literature, there are many sampling density proposals based on various ideas. The goal of the paper is to provide a survey of sampling densities, to classify them and to compare estimation quality of the particle filter with various sampling densities in an illustration example.
---
paper_title: Novel approach to nonlinear/non-Gaussian Bayesian state estimation

paper_content:

An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linear- ity or Gaussian noise: it may be applied to any state transition or measurement model. A simula- tion example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter.
---
paper_title: Sigma-Point Kalman Filters for Nonlinear Estimation and Sensor-Fusion: Applications to Integrated Navigation

paper_content:

A probabilistic framework, called Sigma-point Kalman Filters (SPKF) was applied to the problem domain addressed by the extended Kalman Filter (EKF). SPKF methods are superior to the standard EKF based estimation approaches, as an SPKF achieves second-order or higher accuracy. The SPKF has also been applied to the integrated navigation problem as it relates to unmanned aerial vehicle (UAV) autonomy. The SPKF-based sensor latency compensation technique is used to demonstrate the lagged GPS measurements.
---
paper_title: On Resampling Algorithms for Particle Filters

paper_content:

In this paper a comparison is made between four frequently encountered resampling algorithms for particle filters. A theoretical framework is introduced to be able to understand and explain the differences between the resampling algorithms. This facilitates a comparison of the algorithms with respect to their resampling quality and computational complexity. Using extensive Monte Carlo simulations the theoretical results are verified. It is found that systematic resampling is favourable, both in terms of resampling quality and computational complexity.
---
paper_title: Sequential Monte Carlo Methods for Dynamic Systems

paper_content:

Abstract We provide a general framework for using Monte Carlo methods in dynamic systems and discuss its wide applications. Under this framework, several currently available techniques are studied and generalized to accommodate more complex features. All of these methods are partial combinations of three ingredients: importance sampling and resampling, rejection sampling, and Markov chain iterations. We provide guidelines on how they should be used and under what circumstance each method is most suitable. Through the analysis of differences and connections, we consolidate these methods into a generic algorithm by combining desirable features. In addition, we propose a general use of Rao-Blackwellization to improve performance. Examples from econometrics and engineering are presented to demonstrate the importance of Rao–Blackwellization and to compare different Monte Carlo procedures.
---
paper_title: Kalman Filtering with State Constraints : A Survey of Linear and Nonlinear Algorithms

paper_content:

The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.
---
paper_title: Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations

paper_content:

State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.
---
paper_title: A gain modification form for Kalman filtering under state inequality constraints

paper_content:

This paper presents a method for dealing with Kalman filtering under particular classes of nonlinear state model and state inequality constraints. This method is based on a modification of the optimal Kalman gain in order to enforce the state constraints if necessary. The proposed method is applied to a nonlinear hydraulic system in order to highlight the efficiency of this method. Copyright © 2010 John Wiley & Sons, Ltd.
---
paper_title: Best linear unbiased state estimation with noisy and noise-free measurements

paper_content:

Numerous state estimation problems (e.g., under linear or nonlinear equality constraints, with correlated or singular measurement noise) can be formulated as the problem with noisy and noise-free measurements. Under the assumption that the initial state and the noise are jointly Gaussian distributed and the coefficient matrix for linear equality constraint has full row rank, some optimal state estimation algorithms were obtained in the literature. Given only the first two moments and without any assumption on the rank of the measurement matrix for noise-free measurement, optimal batch and two sequential forms of the state estimation algorithms in the sense of best linear unbiased estimation (BLUE) are obtained in this paper. Extension to nonlinear measurements is also discussed.
---
paper_title: A Robust Null Space Method for Linear Equality Constrained State Estimation

paper_content:

We present a robust null space method for linear equality constrained state space estimation. Exploiting a degeneracy in the estimator statistics, an orthogonal factorization is used to decompose the problem into stochastic and deterministic components, which are then solved separately. The resulting dimension reduction algorithm has enhanced numerical stability, solves the constrained problem completely, and can reduce computational load by reducing the problem size. The new method addresses deficiencies in commonly used pseudo-observation or projection methods, which either do not solve the constrained problem completely or have unstable numerical implementations, due in part to the degeneracy in the estimator statistics. We present a numerical example demonstrating the effectiveness of the new method compared to other current methods.
---
paper_title: Some results on Kalman filtering with linear equality state constraints

paper_content:

This paper addresses the estimation problem for linear system with linear equality state constraints. We review the existing state projection method and present a simple way to find the optimal filter gain of the constrained Kalman filter. Then, we transform the constrained system into a reduced-order model and construct a reduced-order Kalman filter for this estimation problem. Next, we discuss the properties and error covariance matrices of different estimates. Finally, a vehicle tracking example is provided to compare the effectiveness of these estimators.
---
paper_title: Target tracking problems subject to kinematic constraints

paper_content:

Filtering problems with kinematic constraints which may arise in target tracking problems are considered. A novel approach which treats kinematic constraints as additional fictitious or pseudomeasurements is proposed. A numerical example is provided to show the technical feasibility of the proposed idea for target tracking problems. This example shows that the proposed method can improve estimation accuracy significantly for velocity and acceleration states in the tracking problem. However, it is noted that the tracking performance may be deteriorated if the constraints do not properly represent the target characteristics and a small R/sub c/ is chosen. >
---
paper_title: Kalman Filtering with State Constraints : A Survey of Linear and Nonlinear Algorithms

paper_content:

The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.
---
paper_title: Model-based multi-sensor data fusion

paper_content:

The authors describe an algorithm for implementing a multisensor system in a model-based environment with consideration of the constraints. Based on an environment model, geometric features and constraints are generated from a CAD model database. Sensor models are used to predict sensor response to certain features and to interpret raw sensor data. A constrained MMS (minimum mean squared) estimator is used to recursively predict, match, and update feature location. The effects of applying various constraints in estimation were shown by simulation system mounted on a robot arm for localization of known object features. >
---
paper_title: Constraining Kalman Filter and Smoothing Estimates to Satisfy Time-Varying Restrictions

paper_content:

It sometimes happens that the unobservable state vector of a linear dynamic model expressed in the state space is subject to known restrictions. Incorporation of this information into the Kalman filter procedure will increase the efficiency of estimation. It is shown that a simple augmentation of the measurement equation constrains the estimated state vector to obey the restrictions. The method applies whether the restrictions are time-invariant, time-varying, linear, or nonlinear. Copyright 1992 by MIT Press.
---
paper_title: State estimation for linear systems with state equality constraints

paper_content:

This paper deals with the state estimation problem for linear systems with linear state equality constraints. Using noisy measurements which are available from the observable system, we construct the optimal estimate which also satisfies linear equality constraints. For this purpose, after reviewing modeling problems in linear stochastic systems with state equality constraints, we formulate a projected system representation. By using the constrained Kalman filter for the projected system and comparing its filter Riccati equation with those of the unconstrained and the projected Kalman filters, we clearly show, without using optimality, that the constrained estimator outperforms the other filters for estimating the constrained system state. Finally, a numerical example is presented, which demonstrates performance differences among those filters.
---
paper_title: Use of a kinematic constraint in tracking constant speed, maneuvering targets

paper_content:

The use of a kinematic constraint as a pseudomeasurement in the tracking of constant-speed, maneuvering targets is considered. The kinematic constraint provides additional information about the target motion that can be processed as a pseudomeasurement to improve tracking performances. A new formulation of the constraint equation is presented, and the rationale for the new formulation is discussed. The filter using the kinematic constraint as a pseudomeasurement is shown to be unbiased, and sufficient conditions for stochastic stability of the filter are given. Simulated tracking results are given to demonstrate the potential that the new formulation provides for improving tracking performance. >
---
paper_title: Constrained Kalman Filtering: Additional Results

paper_content:

This paper deals with linear state space modelling subject to general linear constraints on the state vector. The discussion concentrates on four topics: the constrained Kalman filtering versus the recursive restricted least squares estimator; a new proof of the constrained Kalman filtering under a conditional expectation framework; linear constraints under a reduced state space modelling; and state vector prediction under linear constraints. The techniques proposed are illustrated in two real problems. The first problem is related to investment analysis under a dynamic factor model, whereas the second is about making constrained predictions within a GDP benchmarking estimation. Copyright (c) 2010 The Author. Journal compilation (c) 2010 International Statistical Institute.
---
paper_title: Reduced-order Kalman filtering for state constrained linear systems

paper_content:

This paper aims at solving the state filtering problem for linear systems with state constraints. Three classes of typical state constraints, i.e., linear equality, quadratic equality and inequality, are discussed. By using the linear relationships among different state variables, a reduced-order Kalman filter is derived for the system with linear equality constraints. Afterwards, such a solution is applied to the cases of the quadratic equality constraint and inequality constraints and the two constrained state filtering prob- lems are transformed into two relative constrained optimization problems. Then they are solved by the Lagrangian multiplier and linear matrix inequality techniques, respectively. Finally, two simple tracking examples are provided to illustrate the effectiveness of the reduced-order filters.
---
paper_title: Design and analysis of linear equality constrained dynamic systems

paper_content:

The state of many dynamic systems evolves subject to some equality constraints. Under the assumption that the equality constrained dynamic systems are already available, most existing work focuses on developing state estimation algorithms for these systems. However, how to design and analyze such a system is rarely addressed, even though it is critically important for performance evaluation and application. In this paper, unlike the existing system conversion-based design techniques, we propose a new systematic way to design and analyze linear equality constrained linear dynamic systems through a direct elimination technique, where the desired model class is given and only the distributions of the initial state and driving process noise need to be determined. Comparatively, it is much easier. It is also found that the existing formulations of linear equality constrained linear dynamic systems only cover a small part of the whole class. Numerical examples are provided to illustrate the effectiveness of the proposed way of design and analysis.
---
paper_title: Kalman Filtering with Equality and Inequality State Constraints

paper_content:

Both constrained and unconstrained optimization problems regularly appear in recursive tracking problems engineers currently address -- however, constraints are rarely exploited for these applications. We define the Kalman Filter and discuss two different approaches to incorporating constraints. Each of these approaches are first applied to equality constraints and then extended to inequality constraints. We discuss methods for dealing with nonlinear constraints and for constraining the state prediction. Finally, some experiments are provided to indicate the usefulness of such methods.
---
paper_title: Track Fusion with Road Constraints

paper_content:

This paper is concerned with tracking of ground targets on roads and investigates possible ways to improve target state estimation via fusing a target's track with information about a road along which the target is believed to be traveling. A target track is estimated by a surveillance radar whereas a digital map provides the road network of a region under surveillance. When the information about roads is as accurate as (or even better than) radar measurements, it is desired naturally to incorporate such information (fusion) into target state estimation. In this paper, roads are modeled with analytic functions and its fusion with a target track is cast as linear or nonlinear state constraints in an optimization procedure. The constrained optimization is then solved with the Lagrangian multiplier, leading to a closed-form solution for linear constraints and an iterative solution for nonlinear constraints. Geometric interpretations of the solutions are provided for simple cases. Computer simulation results are presented to illustrate the algorithms.
---
paper_title: Constrained Kalman filtering via density function truncation for turbofan engine health estimation

paper_content:

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter estimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine model. It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).
---
paper_title: Aircraft turbofan engine health estimation using constrained Kalman filtering

paper_content:

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This paper develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter is a combination of a standard Kalman filter and a quadratic programming problem. The incorporation of state variable constraints increases the computational effort of the filter but significantly improves its estimation accuracy. The improvement is proven theoretically and shown via simulation results obtained from application to a turbofan engine model. This model contains 16 state variables, 12 measurements, and 8 component health parameters. It is shown that the new algorithms provide improved performance in this example over unconstrained Kalman filtering.Copyright © 2003 by ASME
---
paper_title: Mathematically equivalent approaches for equality constrained Kalman Filtering

paper_content:

Kalman Filtering problems often have inherent and known constraints in the physical dynamics that are not exploited despite potentially significant gains (e.g., fixed speed of a motor). In this paper, we review existing methods and propose some new ideas for filtering in the presence of equality constraints. We then show that three methods for incorporating state space equality constraints are mathematically equivalent to the more general "Projection" method, which allows different weighting matrices when projecting the estimate. Still, the different approaches have advantages in implementations that may make one better suited than another for a given application.
---
paper_title: Kalman Filtering in the Presence of State Space Equality Constraints

paper_content:

We discuss two separate techniques for Kalman filtering in the presence of state space equality constraints. We discuss these methods and then prove that despite the lack of similarity in their formulations, under certain conditions, the two methods result in mathematically equivalent constrained estimate structures. We then discuss numerical stability for implementations. We conclude that the potential benefits of using equality constraints in Kalman filtering often outweigh the computational costs, and as such, equality constraints, when present, should be enforced by way of one of these two methods.
---
paper_title: Filtering method for nonlinear systems with constraints

paper_content:

A constrained filtering method is proposed to deal with the filtering problems for nonlinear systems with constraints. The problem is converted to a sequence of recursive estimation problems in which the system equations and constraint conditions are treated as pseudo-measurements. To resolve the singularity problem arising from the constraints, a modified maximum-likelihood method for nonlinear systems is developed. The simulation results from the application of the proposed scheme to the target tracking problem shows that the constrained filtering method can enhance the performance of filter design significantly.
---
paper_title: Optimal Combination and Constraints for Geometrical Sensor Data

paper_content:

A formalism is described for the statistical combination of geometrical information from multiple sensors. This is illus trated by its applications to stereo vision, including the com bination of multiple stereo views to increase the accuracy of a wire frame model and the consistent imposition of geometri cal constraints on such a model.
---
paper_title: The Linear Minimum Mean-Square Error Estimation with Constraints and Its Applications

paper_content:

The minimum mean-square error (MMSE) is an important optimization criterion, which is widely applied to many fields in signal processing and others, such as waveforms estimation, signal detection and system identification. In many practical scenarios, the optimal solutions are usually expected to locate in some especial subspace. So, it is significant to study the MMSE detector or estimator with various constraints. A rigorous analytic method to incorporate the optimal estimation with linear equality constraints is developed in the most general case. It can be directly applied to the Kalman filtering with state equality constraints, in which the assumption of the states and noises to be Gaussian can be avoided.
---
paper_title: The Role of Pseudo Measurements in Equality-Constrained State Estimation

paper_content:

The pseudo measurement method is a main approach to equality-constrained state estimation due to its simplicity. It is, however, not popular due to possible numerical problems and increased computational complexity. The work presented here further develops the pseudo measurement method. To avoid numerical problems resulting from singular measurement noise when a matrix inverse is used, the Moore-Penrose (MP) inverse is used instead. Also, to reduce the computational load without performance loss and to simplify the analysis of this type of estimation problem, two sequential forms are obtained. They differ only in the processing order of the physical measurement and the pseudo measurement (i.e., the equality constraint). Although form 1 is the same as some existing results, form 2 is new. This motivates the discussion of processing order for this type of estimation problem, especially in the extension to the nonlinear case. It is found that under certain conditions, the use of the pseudo measurement for filtering is redundant. This differs in effect from update by the physically error-free measurement. However, if there exists model mismatch, update by the pseudo measurement is necessary and helpful. Supporting numerical examples are provided.
---
paper_title: State estimation for linear and non-linear equality-constrained systems

paper_content:

This article addresses the state-estimation problem for linear and non-linear systems for the case in which prior knowledge is available in the form of an equality constraint. The equality-constrained Kalman filter (KF) is derived as the maximum-a-posteriori solution to the equality-constrained state-estimation problem for linear and Gaussian systems and is compared to alternative algorithms. Then, four novel algorithms for non-linear equality-constrained state estimation based on the unscented KF are presented, namely, the equality-constrained unscented KF, the projected unscented KF, the measurement-augmentation unscented KF, and the constrained unscented KF. Finally, these methods are compared on linear and non-linear examples.
---
paper_title: State Estimation with Probability Constraints

paper_content:

This paper considers a state estimation problem for a discrete-time linear system driven by a Gaussian random process. The second order statistics of the input process and state initial condition are uncertain. However, the probability that the state and input satisfy linear constraints during the estimation interval is known. A minimax estimation problem is formulated to determine an estimator that minimizes the worst-case mean square error criterion, over the uncertain second order statistics, subject to the probability constraints. It is shown that a solution to this constrained state estimation problem is given by a Kalman filter for appropriately chosen input and initial condition models. These models are obtained from a finite dimensional convex optimization problem. The application of this estimator to an aircraft tracking problem quantifies the improvement in estimation accuracy obtained from the inclusion of probability constraints in the minimax formulation.
---
paper_title: Attitude Error Representations for Kalman Filtering

paper_content:

The quaternion has the lowest dimensionality possible for a globally nonsingular attitude representation. The quaternionmustobeyaunitnormconstraint,though,whichhasledtothedevelopmentofanextendedKalmane lter usingaquaternionfortheglobalattitudeestimateandathree-componentrepresentationforattitudeerrors.Various attitude error representations are considered for this multiplicative extended Kalman e lter, which incorporates a nonlinear, norm preserving quaternion reset operation. Second-order bias corrections are computed in this framework.
---
paper_title: Kalman Filtering with Nonlinear State Constraints

paper_content:

In [Simon and Chia, 2002], an analytic method was developed to incorporate linear state equality constraints into the Kalman filter. When the state constraint is nonlinear, linearization was employed to obtain an approximately linear constraint around the current state estimate. This linearized constrained Kalman filter is subject to approximation errors and may suffer from a lack of convergence. In this paper, we present a method that allows exact use of second-order nonlinear state constraints. It is based on a computational algorithm that iteratively finds the Lagrangian multiplier for the nonlinear constraints. The method therefore provides better approximation when higher order nonlinearities are encountered. Computer simulation results are presented to illustrate the algorithm.
---
paper_title: State estimation under nonlinear state inequality constraints. A tracking application

paper_content:

This paper proposes a general method for dealing with state estimation under nonlinear state soft inequality constraints. This method is based on the projection approach, and then has the advantage to be compatible with any kind of state estimator. In order to be taken into account, the nonlinear constraints are linearized about the constrained state using an iterated approach. The proposed algorithm is tested on a three- dimension tracking application with nonlinear constraints on the moving body acceleration. The results are compared with those of an unconstrained Kalman filter.
---
paper_title: Kalman Filtering with Equality and Inequality State Constraints

paper_content:

Both constrained and unconstrained optimization problems regularly appear in recursive tracking problems engineers currently address -- however, constraints are rarely exploited for these applications. We define the Kalman Filter and discuss two different approaches to incorporating constraints. Each of these approaches are first applied to equality constraints and then extended to inequality constraints. We discuss methods for dealing with nonlinear constraints and for constraining the state prediction. Finally, some experiments are provided to indicate the usefulness of such methods.
---
paper_title: Track Fusion with Road Constraints

paper_content:

This paper is concerned with tracking of ground targets on roads and investigates possible ways to improve target state estimation via fusing a target's track with information about a road along which the target is believed to be traveling. A target track is estimated by a surveillance radar whereas a digital map provides the road network of a region under surveillance. When the information about roads is as accurate as (or even better than) radar measurements, it is desired naturally to incorporate such information (fusion) into target state estimation. In this paper, roads are modeled with analytic functions and its fusion with a target track is cast as linear or nonlinear state constraints in an optimization procedure. The constrained optimization is then solved with the Lagrangian multiplier, leading to a closed-form solution for linear constraints and an iterative solution for nonlinear constraints. Geometric interpretations of the solutions are provided for simple cases. Computer simulation results are presented to illustrate the algorithms.
---
paper_title: Mathematically equivalent approaches for equality constrained Kalman Filtering

paper_content:

Kalman Filtering problems often have inherent and known constraints in the physical dynamics that are not exploited despite potentially significant gains (e.g., fixed speed of a motor). In this paper, we review existing methods and propose some new ideas for filtering in the presence of equality constraints. We then show that three methods for incorporating state space equality constraints are mathematically equivalent to the more general "Projection" method, which allows different weighting matrices when projecting the estimate. Still, the different approaches have advantages in implementations that may make one better suited than another for a given application.
---
paper_title: Kalman Filtering with State Constraints : A Survey of Linear and Nonlinear Algorithms

paper_content:

The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.
---
paper_title: A smoothly constrained Kalman filter

paper_content:

This paper presents the Smoothly Constrained Kalman Filter (SCKF) for nonlinear constraints. A constraint is any relation that exists between the state variables. Constraints can be treated as perfect observations. But, linearization errors can prevent the estimate from converging to the true value. Therefore, the SCKF iteratively applies nonlinear constraints as nearly perfect observations, or, equivalently, weakened constraints. Integration of new measurements is interlaced with these iterations, which reduces linearization errors and, hence, improves convergence compared to other iterative methods. The weakening is achieved by artificially increasing the variance of the nonlinear constraint. The paper explains how to choose the weakening values, and when to start and stop the iterative application of the constraint.
---
paper_title: Constrained linear state estimation - a moving horizon approach

paper_content:

This article considers moving horizon strategies for constrained linear state estimation. Additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states, noise, and other variables. Formulating a linear state estimation problem with inequality constraints, however, prevents recursive solutions such as Kalman filtering, and, consequently, the estimation problem grows with time as more measurements become available. To bound the problem size, we explore moving horizon strategies for constrained linear state estimation. In this work we discuss some practical and theoretical properties of moving horizon estimation. We derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate. We also discuss smoothing strategies for moving horizon estimation. Our framework is solely deterministic.
---
paper_title: Adaptive beamforming using the constrained Kalman filter

paper_content:

From the viewpoint of achieving rapid convergence, application of a Kalman filter to an adaptive array is considered. Compared with the Frost's (1972) constrained least-mean-square algorithm, the constrained Kalman algorithm for adaptive beamforming is proposed to overcome the problem of signal distortion along the look direction which occurs in the unconstrained Kalman beamformer of Baird (1974). A constraint on the array response along the look direction is added to the measurement equation of the Kalman filter. The weight vector of the constrained Kalman beamformer is derived and shown to converge to that of the minimum-variance distortionless-response beamformer. The convergence rate of the proposed algorithm is also analyzed. Compared to Baird's algorithm and the sidelobe canceller with one-step Kalman predictor, simulation results show the effectiveness of the proposed algorithm. >
---
paper_title: A smoothly constrained Kalman filter

paper_content:

This paper presents the Smoothly Constrained Kalman Filter (SCKF) for nonlinear constraints. A constraint is any relation that exists between the state variables. Constraints can be treated as perfect observations. But, linearization errors can prevent the estimate from converging to the true value. Therefore, the SCKF iteratively applies nonlinear constraints as nearly perfect observations, or, equivalently, weakened constraints. Integration of new measurements is interlaced with these iterations, which reduces linearization errors and, hence, improves convergence compared to other iterative methods. The weakening is achieved by artificially increasing the variance of the nonlinear constraint. The paper explains how to choose the weakening values, and when to start and stop the iterative application of the constraint.
---
paper_title: Use of a kinematic constraint in tracking constant speed, maneuvering targets

paper_content:

The use of a kinematic constraint as a pseudomeasurement in the tracking of constant-speed, maneuvering targets is considered. The kinematic constraint provides additional information about the target motion that can be processed as a pseudomeasurement to improve tracking performances. A new formulation of the constraint equation is presented, and the rationale for the new formulation is discussed. The filter using the kinematic constraint as a pseudomeasurement is shown to be unbiased, and sufficient conditions for stochastic stability of the filter are given. Simulated tracking results are given to demonstrate the potential that the new formulation provides for improving tracking performance. >
---
paper_title: PARAMETER ESTIMATION USING KALMAN FILTERS WITH CONSTRAINTS

paper_content:

We suggest incorporating dynamical information such as locations of unstable fixed points into parameter estimation algorithms in order to improve the method of reconstructing dynamics from time series data. We show how the process of reconstruction using nonlinear filters such as the extended Kalman filter can be easily modified to take advantage of the additional information. We demonstrate the methods using data from two systems exhibiting chaotic dynamics — the Chua circuit and Chen's equations. In both cases we find the models reconstructed using constraints that better approximate the unstable fixed point structure of the underlying systems.
---
paper_title: Constrained State Estimation for Nonlinear Systems with Unknown Input

paper_content:

This paper extends the problem of state estimation for linear discrete-time systems with unknown input to the nonlinear systems. Based on physical consideration, the constraints of state are also considered. And the constraints which can improve the quality of estimation are imposed on individual updated sigma points as well as the updated state. The advantage of algorithm is that it is able to deal with arbitrary constraints on the states during the estimation procedure, Least-squares unbiased estimation algorithm can be used to obtain unknown input, and the unknown input which can be any signal affects both the system and the outputs. The state estimation problem is transformed into a standard Unscented Kalman filter problem which can easily be solved. Simulations are provided to demonstrate the effectiveness of the theoretical results.
---
paper_title: Constrained state estimation using the Unscented Kalman Filter

paper_content:

A simple procedure to include state inequality constraints in the unscented Kalman filter is proposed. With this procedure, the information of active state constraints influences the state covariance matrix, resulting in better estimates. In a numerical example, the approach outperforms the extended Kalman filter implemented with constraint handling via ldquoclippingrdquo.
---
paper_title: Applying the unscented Kalman filter for nonlinear state estimation

paper_content:

Based on presentation of the principles of the EKF and UKF for state estimation, we discuss the differences of the two approaches. Four rather different simulation cases are considered to compare the performance. A simple procedure to include state constraints in the UKF is proposed and tested. The overall impression is that the performance of the UKF is better than the EKF in terms of robustness and speed of convergence. The computational load in applying the UKF is comparable to the EKF. 2007 Elsevier Ltd. All rights reserved.
---
paper_title: Constrained nonlinear state estimation based on the UKF approach

paper_content:

In this paper we investigate the use of an alternative to the extended Kalman filter (EKF), the unscented Kalman filter (UKF). First we give a broad overview of different UKF algorithms, then present an extension to the ensemble of UKF algorithms, and finally address the issue of how to add constraints using the UKF approach. The performance of the constrained approach is compared with EKF and a selection of UKF algorithms on nonlinear process systems with multimodal probability density functions. The conclusion is that with an algebraic reformulation of the correction part, the reformulated UKF shows strong performance on our selection of nonlinear constrained process systems.
---
paper_title: Constrained unscented recursive estimator for nonlinear dynamic systems

paper_content:

Abstract Nonlinear constrained state estimation is an important task in performance monitoring, online optimization and control. There has been recent interest in developing estimators based on the idea of unscented transformation for constrained nonlinear systems. One of these approaches is the unscented recursive nonlinear dynamic data reconciliation (URNDDR) method. The URNDDR approach follows the traditional predictor-corrector framework. Constraints are handled in the prediction step through a projection algorithm and in the correction step through an optimization formulation. It has been shown that URNDDR produces very accurate estimates at the cost of computational expense. However, there are two issues that need to be addressed in the URNDDR framework: (i) URNDDR approach was primarily developed to handle bound constraints and needs to be enhanced to handle general nonlinear equality and inequality constraints, and (ii) computational concerns in the application of the URNDDR approach needs to be addressed. In this paper, a new estimation technique named constrained unscented recursive estimator (CURE) is proposed, which eliminates these disadvantages of URNDDR, while providing estimates with almost the same accuracy.
---
paper_title: Constrained Nonlinear State Estimation Using Ensemble Kalman Filters

paper_content:

Recursive estimation of states of constrained nonlinear dynamic systems has attracted the attention of many researchers in recent years. In this work, we propose a constrained recursive formulation of the ensemble Kalman filter (EnKF) that retains the advantages of the unconstrained EnKF while systematically dealing with bounds on the estimated states. The EnKF belongs to the class of particle filters that are increasingly being used for solving state estimation problems associated with nonlinear systems. A highlight of our approach is the use of truncated multivariate distributions for systematically solving the estimation problem in the presence of state constraints. The efficacy of the proposed constrained state estimation algorithm using the EnKF is illustrated by application on two benchmark problems in the literature (a simulated gas-phase reactor and an isothermal batch reactor) involving constraints on estimated state variables and another example problem, which involves constraints on the process...
---
paper_title: Robust and reliable estimation via Unscented Recursive Nonlinear Dynamic Data Reconciliation

paper_content:

Abstract The quality of process data in a chemical plant significantly affects the performance and benefits gained from activities like performance monitoring, online optimization and control. Since many chemical processes often exhibit nonlinear dynamics, techniques like Extended Kalman Filter (EKF) and Nonlinear Dynamic Data Reconciliation (NDDR) have been developed to improve the data quality. There are various issues that arise with the use of either of these techniques: EKF cannot handle inequality or equality constraints, while the NDDR has high computational cost. Recently a recursive estimation technique for nonlinear dynamic processes has been proposed which combines the merits of EKF and NDDR techniques. This technique, named as Recursive Nonlinear Dynamic Data Reconciliation (RNDDR), provides state and parameter estimates that satisfy bounds and other constraints imposed on them. However, the estimate error covariance matrix in RNDDR is computed in the same manner as in EKF, that is, the effects of both nonlinearity and constraints are neglected in the computation of the estimate error covariance matrix. A relatively new method known as the Unscented Kalman Filter has been developed for nonlinear processes, in which the statistical properties of the estimates are computed without resorting to linearization of the nonlinear equations. This leads to improved accuracy of the estimates. In this paper, we combine the merits of the Unscented Kalman Filter and the RNDDR to obtain the Unscented Recursive Nonlinear Dynamic Data Reconciliation (URNDDR) technique. This technique addresses all concerns arising due to the presence of nonlinearity and constraints within a recursive estimation framework, resulting in an efficient, accurate and stable method for real-time state and parameter estimation for nonlinear dynamic processes.
---
paper_title: Optimization-Based Sigma Points Selection for Constrained State Estimation

paper_content:

Various deterministic sampling-based constrained state estimation techniques for nonlinear dynamical systems have been proposed in the literature. These algorithms mainly focus on correcting the estimated (or filtered) states such that they satisfy all of the given constraints. However, the aspect of correcting predicted states (after model propagation) such that they satisfy all of the constraints has not been given much attention in the literature. These states are obtained by propagating deterministically chosen points (called sigma points) through the process model. Most approaches either do not incorporate constraints during the generation of these sigma points or use arbitrary strategies for constraint incorporation. Further, the weights associated with these points are not systematically updated after incorporation of any state constraint. In this article, we propose an optimization-based sigma points selection approach that overcomes these limitations. Additionally, we incorporate the effect of co...
---
paper_title: Unscented filtering for interval-constrained nonlinear systems

paper_content:

This paper addresses the state-estimation problem for nonlinear systems with an interval constraint on the state vector. Approximate solutions to this problem are reviewed and compared with new algorithms, which are based on the unscented Kalman filter. An illustrative example is discussed.
---
paper_title: On Kalman Filtering With Nonlinear Equality Constraints

paper_content:

The state space description of some physical systems possess nonlinear equality constraints between some state variables. In this paper, we consider the problem of applying a Kalman filter-type estimator in the presence of such constraints. We categorize previous approaches into pseudo-observation and projection methods and identify two types of constraints-those that act on the entire distribution and those that act on the mean of the distribution. We argue that the pseudo-observation approach enforces neither type of constraint and that the projection method enforces the first type of constraint only. We propose a new method that utilizes the projection method twice-once to constrain the entire distribution and once to constrain the statistics of the distribution. We illustrate these algorithms in a tracking system that uses unit quaternions to encode orientation
---
paper_title: State estimation for equality-constrained linear systems

paper_content:

We address the state-estimation problem for linear systems in a context where prior knowledge, in addition to the model and the measurements, is available in the form of an equality constraint. First, we investigate from where an equality constraint arises in a dynamic system. Then, the equality-constrained Kalman filter (ECKF) is derived as the solution to the equality-constrained state-estimation problem and compared to alternative algorithms. These methods are investigated in an example. In addition to exactly satisfying an equality constraint on the system, ECKF produce more accurate and more informative estimates than the unconstrained estimates.
---
paper_title: On unscented Kalman filtering with state interval constraints

paper_content:

Abstract This paper addresses the state-estimation problem for nonlinear systems for the case in which prior knowledge is available in the form of interval constraints on the states. Approximate solutions to this problem are reviewed and compared with new algorithms. All the algorithms investigated are based on the unscented Kalman filter. Two illustrative examples of chemical processes are discussed. Numerical results suggest that the use of constrained unscented filtering algorithms improves the accuracy of the state estimates compared to the unconstrained unscented filter, especially when a poor initialization is set. Moreover, it is shown that the constrained filters that enforce the state interval constraint on both the state estimate and error covariance yield more accurate state estimates than the methods that enforce such constraint only on the state estimates.
---
paper_title: Constrained state estimation using the ensemble Kalman filter

paper_content:

Recursive estimation of constrained nonlinear dynamical systems has attracted the attention of many researchers in recent years. In this work, we propose a constrained recursive formulation of the ensemble Kalman filter (CEnKF) that retains the advantages of unconstrained Ensemble Kalman Filter while systematically dealing with bounds on the estimated states. The efficacy of the proposed constrained state estimation algorithm using the EnKF is illustrated by application on a simulated gas-phase reactor problem.
---
paper_title: Modified Unscented Recursive Nonlinear Dynamic Data Reconciliation for Constrained State Estimation

paper_content:

In state estimation problems, often, the true states satisfy certain constraints that need to be incorporated during the estimation procedure. Amongst various constrained nonlinear state estimation algorithms proposed in literature, the unscented recursive nonlinear dynamic data reconciliation (URNDDR) proposed by Vachhani et al. (2006) seems to be promising since it is able to incorporate constraints while maintaining the recursive nature of estimation. In this article, we propose a modified URNDDR algorithm that gives superior performance compared to basic URNDDR. The improvements are obtained via better constraint handling and are demonstrated via a representative case study.
---
paper_title: State estimation for linear and non-linear equality-constrained systems

paper_content:

This article addresses the state-estimation problem for linear and non-linear systems for the case in which prior knowledge is available in the form of an equality constraint. The equality-constrained Kalman filter (KF) is derived as the maximum-a-posteriori solution to the equality-constrained state-estimation problem for linear and Gaussian systems and is compared to alternative algorithms. Then, four novel algorithms for non-linear equality-constrained state estimation based on the unscented KF are presented, namely, the equality-constrained unscented KF, the projected unscented KF, the measurement-augmentation unscented KF, and the constrained unscented KF. Finally, these methods are compared on linear and non-linear examples.
---
paper_title: Critical Evaluation of Extended Kalman Filtering and Moving-Horizon Estimation

paper_content:

State estimators for physical processes often must address different challenges, including nonlinear dynamics, states subject to hard constraints (e.g., nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving-horizon estimation (MHE). We outline conditions that lead to the formation of multiple optima in the estimator for systems tending to a steady state and propose tests that determine when these conditions hold for chemical reaction networks. Several simulation examples demonstrate estimation failure in the EKF, even in the absence of plant−model mismatch. We then examine the role that constraints play in determining the performance on these examples of MHE employing local optimization and a “smoothing” update for the arrival cost. This implementation of MHE represents a feasible, on-line alternative to the EKF for industrial practitioners. In each example, the two estimators are given exactly the sa...
---
paper_title: Constrained process monitoring: Moving‐horizon approach

paper_content:

Moving-horizon estimation (MHE) is an optimization-based strategy for process monitoring and state estimation. One may view MHE as an extension for Kalman filtering for constrained and nonlinear processes. MHE, therefore, subsumes both Kalman and extended Kalman filtering. In addition, MHE allows one to include constraints in the estimation problem. One can significantly improve the quality of state estimates for certain problems by incorporating prior knowledge in the form of inequality constraints. Inequality constraints provide a flexible tool for complementing process knowledge. One also may use inequality constraints as a strategy for model simplification. The ability to include constraints and nonlinear dynamics is what distinguishes MHE from other estimation strategies. Both the practical and theoretical issues related to MHE are discussed. Using a series of example monitoring problems, the practical advantages of MHE are illustrated by demonstrating how the addition of constraints can improve and simplify the process monitoring problem.
---
paper_title: A direct sampling particle filter from approximate conditional density function supported on constrained state space

paper_content:

Constraints on the state vector must be taken into account in the state estimation problem. Recently, acceptance/rejection and projection methods are proposed in the particle filter framework for constraining the particles. A weighted least squares formulation is used for constraining samples in unscented and ensemble Kalman filters. In this paper, direct sampling from an approximate conditional probability density function (pdf) is proposed. It is obtained by approximating the a priori pdf as a Gaussian. The support of the conditional density is a subset of the intersection of two supports, the 3-sigma bounds of the priori Gaussian and the constrained state space. A direct sampling algorithm is proposed for handling linear and nonlinear equality and inequality constraints. The algorithm uses the constrained mode for nonlinear constraints.
---
paper_title: Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations

paper_content:

State estimator design for a nonlinear discrete-time system is a challenging problem, further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances. One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained moving horizon estimation. Sufficient conditions for asymptotic and bounded stability are established. We apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation. Examples are used to illustrate the benefits of constrained state estimation. Our framework is deterministic.
---
paper_title: Constrained linear state estimation - a moving horizon approach

paper_content:

This article considers moving horizon strategies for constrained linear state estimation. Additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states, noise, and other variables. Formulating a linear state estimation problem with inequality constraints, however, prevents recursive solutions such as Kalman filtering, and, consequently, the estimation problem grows with time as more measurements become available. To bound the problem size, we explore moving horizon strategies for constrained linear state estimation. In this work we discuss some practical and theoretical properties of moving horizon estimation. We derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate. We also discuss smoothing strategies for moving horizon estimation. Our framework is solely deterministic.
---
paper_title: Constrained Bayesian State Estimation Using a Cell Filter

paper_content:

Constrained state estimation in nonlinear/non-Gaussian processes has been the domain of optimization based methods such as moving horizon estimation (MHE). MHE has a Bayesian interpretation, but it is not practical to implement a recursive MHE without assumptions of Gaussianity and linearized dynamics at various stages. This paper presents the constrained cell filter (CCF) as an alternative to MHE, requiring no linearization, jacobians, or nonlinear program. The CCF computes a piecewise constant approximation of the state probability density function with support defined by constraints; thus, all point estimates are constrained. The CCF can be more accurate and orders of magnitude faster than MHE for problems of a size as investigated in this work.
---
paper_title: A fast moving horizon estimation algorithm based on nonlinear programming sensitivity

paper_content:

Moving horizon estimation (MHE) is an efficient optimization-based strategy for state estimation. Despite the attractiveness of this method, its application in industrial settings has been rather limited. This has been mainly due to the difficulty to solve, in real-time, the associated dynamic optimization problems. In this work, a fast MHE algorithm able to overcome this bottleneck is proposed. The strategy exploits recent advances in nonlinear programming algorithms and sensitivity concepts. A detailed analysis of the optimality conditions of MHE problems is presented. As a result, strategies for fast covariance information extraction from general nonlinear programming algorithms are derived. It is shown that highly accurate state estimates can be obtained in large-scale MHE applications with negligible on-line computational costs.
---
paper_title: Particle filtering and moving horizon estimation

paper_content:

This paper provides an overview of currently available methods for state estimation of linear, constrained and nonlinear systems. The following methods are discussed: Kalman filtering, extended Kalman filtering, unscented Kalman filtering, particle filtering, and moving horizon estimation. The current research literature on particle filtering and moving horizon estimation is reviewed, and the advantages and disadvantages of these methods are presented. Topics for new research are suggested that address combining the best features of moving horizon estimation and particle filters.
---
paper_title: On sequential Monte Carlo sampling methods for Bayesian filtering

paper_content:

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literatures these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.
---
paper_title: A direct sampling particle filter from approximate conditional density function supported on constrained state space

paper_content:

Constraints on the state vector must be taken into account in the state estimation problem. Recently, acceptance/rejection and projection methods are proposed in the particle filter framework for constraining the particles. A weighted least squares formulation is used for constraining samples in unscented and ensemble Kalman filters. In this paper, direct sampling from an approximate conditional probability density function (pdf) is proposed. It is obtained by approximating the a priori pdf as a Gaussian. The support of the conditional density is a subset of the intersection of two supports, the 3-sigma bounds of the priori Gaussian and the constrained state space. A direct sampling algorithm is proposed for handling linear and nonlinear equality and inequality constraints. The algorithm uses the constrained mode for nonlinear constraints.
---
paper_title: Constrained Bayesian state estimation – A comparative study and a new particle filter based approach

paper_content:

Abstract This paper investigates constrained Bayesian state estimation problems by using a Particle Filter (PF) approach. Constrained systems with nonlinear model and non-Gaussian uncertainty are commonly encountered in practice. However, most of the existing Bayesian methods are unable to take constraints into account and require some simplifications. In this paper, a novel constrained PF algorithm based on acceptance/rejection and optimization strategies is proposed. The proposed method retains the ability of PF in nonlinear and non-Gaussian state estimation, while take advantage of optimization techniques in constraints handling. The performance of the proposed method is compared with other accepted Bayesian estimators. Extensive simulation results from three examples show the efficacy of the proposed method in constraints handling and its robustness against poor prior information.
---
paper_title: Bayesian estimation via sequential Monte Carlo sampling - Constrained dynamic systems

paper_content:

Nonlinear and non-Gaussian processes with constraints are commonly encountered in dynamic estimation problems. Methods for solving such problems either ignore the constraints or rely on crude approximations of the model or probability distributions. Such approximations may reduce the accuracy of the estimates since they often fail to capture the variety of probability distributions encountered in constrained linear and nonlinear dynamic systems. This article describes a practical approach that overcomes these shortcomings via a novel extension of sequential Monte Carlo (SMC) sampling or particle filtering. Inequality constraints are imposed by accept/reject steps in the algorithm. The proposed approach provides samples representing the posterior distribution at each time point, and is shown to satisfy the same theoretical properties as unconstrained SMC. Illustrative examples show that results of the proposed approach are at least as accurate as moving horizon estimation, but computationally more efficient and in addition, the approach indicates the uncertainty associated with these estimates.
---
paper_title: Data Assimilation: The Ensemble Kalman Filter

paper_content:

Statistical definitions.- Analysis scheme.- Sequential data assimilation.- Variational inverse problems.- Nonlinear variational inverse problems.- Probabilistic formulation.- Generalized Inverse.- Ensemble methods.- Statistical optimization.- Sampling strategies for the EnKF.- Model errors.- Square Root Analysis schemes.- Rank issues.- Spurious correlations, localization, and inflation.- An ocean prediction system.- Estimation in an oil reservoir simulator.
---
paper_title: Constrained Bayesian state estimation – A comparative study and a new particle filter based approach

paper_content:

Abstract This paper investigates constrained Bayesian state estimation problems by using a Particle Filter (PF) approach. Constrained systems with nonlinear model and non-Gaussian uncertainty are commonly encountered in practice. However, most of the existing Bayesian methods are unable to take constraints into account and require some simplifications. In this paper, a novel constrained PF algorithm based on acceptance/rejection and optimization strategies is proposed. The proposed method retains the ability of PF in nonlinear and non-Gaussian state estimation, while take advantage of optimization techniques in constraints handling. The performance of the proposed method is compared with other accepted Bayesian estimators. Extensive simulation results from three examples show the efficacy of the proposed method in constraints handling and its robustness against poor prior information.
---
paper_title: Nonlinear Bayesian state estimation: A review of recent developments

paper_content:

Abstract Online estimation of the internal states is a perquisite for monitoring, control, and fault diagnosis of many engineering processes. A cost effective approach to monitor these variables in real time is to employ model-based state estimation techniques. Dynamic model-based state estimation is a rich and highly active area of research and many novel approaches have emerged over the last few years. In this paper, we review various recent developments in the area of nonlinear state estimators from a Bayesian perspective. In particular, we focus on the constrained state estimation (including systems modeled using differential-algebraic equations), the handling of multi-rate and delayed measurements and recent advances in model parameter estimation. Recent advances on the stability analysis of the estimation error dynamics are also briefly discussed. The review aims to provide an integrated view of important ideas, from the authors' perspective that have driven the research in this area in recent years.
---
paper_title: Stochastic Processes and Filtering Theory

paper_content:

This book presents a unified treatment of linear and nonlinear filtering theory for engineers, with sufficient emphasis on applications to enable the reader to use the theory. The need for this book is twofold. First, although linear estimation theory is relatively well known, it is largely scattered in the journal literature and has not been collected in a single source. Second, available literature on the continuous nonlinear theory is quite esoteric and controversial, and thus inaccessible to engineers uninitiated in measure theory and stochastic differential equations. Furthermore, it is not clear from the available literature whether the nonlinear theory can be applied to practical engineering problems. In attempting to fill the stated needs, the author has retained as much mathematical rigor as he felt was consistent with the prime objective" to explain the theory to engineers. Thus, the author has avoided measure theory in this book by using mean square convergence, on the premise that everyone knows how to average. As a result, the author only requires of the reader background in advanced calculus, theory of ordinary differential equations, and matrix analysis.
---
paper_title: Constrained State Estimation Using Particle Filters

paper_content:

Abstract Recursive estimation of constrained nonlinear dynamical systems has attracted the attention of many researchers in recent years. For nonlinear/non-Gaussian state estimation problems, particle filters have been widely used. As pointed out by Daum (2005), particle filters require a proposal distribution and the choice of proposal distribution is the key design issue. In this paper, a novel approach for generating the proposal distribution based on a Constrained Unscented Kalman filter is proposed. The efficacy of the proposed constrained state estimation algorithm using a particle filter (CUPF) is illustrated via a successful implementation on a simulated gas-phase reactor.
---
paper_title: An Efficient Constrained Gaussian Particle Filter

paper_content:

Abstract The paper deals with a state estimation of nonlinear stochastic dynamic systems subject to a nonlinear inequality constraint. A special focus is paid to particle filters, which provide an estimate of the whole probability density as opposed to the local filters, such as the extended Kalman filter or the unscented Kalman filter, which provide a point estimate only. Within the particle filtering framework, there are several approaches to the constrained state estimation, mostly based on discarding samples violating the constraint with a possible increase of their number to improve the estimate quality. The paper aims at proposing a modification to an importance function of the particle filter in order to increase efficiency of sampling while keeping the computational complexity low. The proposed modification is utilized within the Gaussian particle filter which is advantageous for its low computational complexity. Complexity and estimation quality of the proposed constrained Gaussian particle filter is compared to other constrained particle filters in a numerical example.
---
paper_title: On the choice of importance distributions for unconstrained and constrained state estimation using particle filter

paper_content:

Abstract Recursive state estimation of constrained nonlinear dynamical system has attracted the attention of many researchers in recent years. For nonlinear/non-Gaussian state estimation problems, particle filters have been widely used (Arulampalam et al. [1] ). As pointed out by Daum [2] , particle filters require a proposal distribution and the choice of proposal distribution is the key design issue. In this paper, a novel approach for generating the proposal distribution based on a constrained Extended Kalman filter (C-EKF), Constrained Unscented Kalman filter (C-UKF) and constrained Ensemble Kalman filter (C-EnkF) has been proposed. The efficacy of the proposed state estimation algorithms using a particle filter is illustrated via a successful implementation on a simulated gas-phase reactor, involving constraints on estimated state variables and another example problem, which involves constraints on the process noise (Rao et al. [10] ). We also propose a state estimation scheme for estimating state variables in an autonomous hybrid system using particle filter with Unscented Kalman filter as a proposal and unconstrained Ensemble Kalman filter (EnKF) as a proposal. The efficacy of the proposed state estimation scheme for an autonomous hybrid system is demonstrated by conducting simulation studies on a three-tank hybrid system. The simulation studies underline the crucial role played by the choice of proposal distribution in formulation of particle filters.
---
paper_title: A direct sampling particle filter from approximate conditional density function supported on constrained state space

paper_content:

Constraints on the state vector must be taken into account in the state estimation problem. Recently, acceptance/rejection and projection methods are proposed in the particle filter framework for constraining the particles. A weighted least squares formulation is used for constraining samples in unscented and ensemble Kalman filters. In this paper, direct sampling from an approximate conditional probability density function (pdf) is proposed. It is obtained by approximating the a priori pdf as a Gaussian. The support of the conditional density is a subset of the intersection of two supports, the 3-sigma bounds of the priori Gaussian and the constrained state space. A direct sampling algorithm is proposed for handling linear and nonlinear equality and inequality constraints. The algorithm uses the constrained mode for nonlinear constraints.
---
paper_title: Truncation nonlinear filters for state estimation with nonlinear inequality constraints

paper_content:

The paper focuses on the state estimation problem of nonlinear non-Gaussian systems with state subject to a nonlinear inequality constraint. Taking into account the available additional information about the state given by the constraint increases the estimate quality compared to classical state estimation methods which cannot utilize the information. Considering the constraint in the form of an inequality involving a nonlinear function of the state makes the state estimation problem difficult and hence treated only marginally. In this paper, a generic local filter for the inequality constrained estimation problem is proposed. It covers the extended Kalman filter, unscented Kalman filter, and divided difference filter as special cases and enforces the constraint by truncating the conditional density of the state. The truncation is computationally cheap, yet it provides high estimate quality of the constrained estimate. The same idea is then utilized in a truncation Gaussian mixture filter which is also proposed in the paper to increase the estimate quality further by providing a global constrained estimate. Superior estimate quality and computational efficiency of the proposed filters are illustrated in two numerical examples.
---
paper_title: A Mata Geweke–Hajivassiliou–Keane multivariate normal simulator

paper_content:

An accurate and efficient numerical approximation of the multivariate normal (MVN) distribution function is necessary for obtaining maximum likeli- hood estimates for models involving the MVN distribution. Numerical integration through simulation (Monte Carlo) or number-theoretic (quasi-Monte Carlo) tech- niques is one way to accomplish this task. One popular simulation technique is the Geweke-Hajivassiliou-Keane MVN simulator. This paper reviews this technique and introduces a Mata function that implements it. It also computes analytical first-order derivatives of the simulated probability with respect to the variables and the variance – covariance parameters. Copyright 2006 by StataCorp LP.
---
paper_title: Truncated unscented particle filter

paper_content:

The problem of state estimation of nonlinear stochastic dynamic systems with nonlinear inequality constraints is treated. The paper focuses on a particle filtering approach, which provides an estimate of the state in the form of a probability density function. A new computationally efficient particle filter for the constrained estimation problem is proposed. The importance function of the particle filter is generated by the unscented Kalman filter that is supplemented with a designed truncation technique to accommodate the constraint. The proposed filter is illustrated in a numerical example.
---
paper_title: Constrained Kalman filtering via density function truncation for turbofan engine health estimation

paper_content:

Kalman filters are often used to estimate the state variables of a dynamic system. However, in the application of Kalman filters some known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not fit easily into the structure of the Kalman filter. This article develops an analytic method of incorporating state variable inequality constraints in the Kalman filter. The resultant filter truncates the probability density function (PDF) of the Kalman filter estimate at the known constraints and then computes the constrained filter estimate as the mean of the truncated PDF. The incorporation of state variable constraints increases the computational effort of the filter but also improves its estimation accuracy. The improvement is demonstrated via simulation results obtained from a turbofan engine model. It is also shown that the truncated Kalman filter may provide a more accurate way of incorporating inequality constraints than other constrained filters (e.g. the projection approach to constrained filtering).
---
paper_title: Gaussian particle filtering

paper_content:

Sequential Bayesian estimation for nonlinear dynamic state-space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations. This paper introduces a new filter called the Gaussian particle filter. It is based on the particle filtering concept, and it approximates the posterior distributions by single Gaussians, similar to Gaussian filters like the extended Kalman filter and its variants. It is shown that under the Gaussianity assumption, the Gaussian particle filter is asymptotically optimal in the number of particles and, hence, has much-improved performance and versatility over other Gaussian filters, especially when nontrivial nonlinearities are present. Simulation results are presented to demonstrate the versatility and improved performance of the Gaussian particle filter over conventional Gaussian filters and the lower complexity than known particle filters.
---
paper_title: A particle filter based on a constrained sampling method for state estimation

paper_content:

Increasingly in practical applications, nonlinearity, non-Gaussianity, and constraint are considered when dealing with state estimation problems. This paper proposes a novel constrained particle filter (PF) approach for state estimation, where three constraint strategies are implemented: First, to ensure the validity of prior, prior particles are restrictedly sampled in the constraint region by a constrained inverse transform sampling method. Second, if constraints are imposed on the posterior, a constrained re-sampling method, similar to the existing acceptance/rejection constrained PF method, is proposed to restrict the posterior particles to be generated from the valid prior particles. Third, the validity of state estimation is ensured through adjustment of part of posterior particles according to the posterior density function of states, which is accomplished by deleting uniformly selected violated posterior particle and uniformly selected valid posterior particle for reproduction. Compared with the existing methods, the proposed method implements constraints with better physical interpretation, and involves no numerical optimization procedure and no restrictive assumptions about the distributions. Simulation results demonstrate its effectiveness.
---

Title: Constrained State Estimation - A Review

Section 1: INTRODUCTION
Description 1: Introduce the concept of constrained state estimation, its relevance, typical applications, and the benefits of using explicit constraints.

Section 2: System Definition
Description 2: Define the system including state transition and measurement models, noise assumptions, and standard assumptions for state estimation.

Section 3: Bayesian Recursion
Description 3: Present the Bayesian recursion framework used for estimating posterior conditional density, including time-update and measurement-update steps.

Section 4: UNCONSTRAINED STATE ESTIMATION
Description 4: Describe various Bayesian recursive schemes for state estimation in linear and nonlinear systems without constraints.

Section 5: CONSTRAINED STATE ESTIMATION
Description 5: Review different methods for state estimation with constraints, including definitions of constraints and specific approaches for linear and nonlinear systems.

Section 6: Constrained State Estimation Using Sequential Monte Carlo Filters
Description 6: Discuss how constraints can be incorporated into particle filtering algorithms at different levels, such as sampling, rejection, and projection methods.

Section 7: Acceptance/Rejection Approach
Description 7: Explain the acceptance/rejection approach for enforcing constraints in particle filters, and its pros and cons.

Section 8: Optimization/Projection Based Approaches
Description 8: Detail the optimization-based approaches that incorporate constraints into the particle filtering algorithm, including different formulations and strategies.

Section 9: Constrained Importance Distributions
Description 9: Describe the methods for drawing particles from a proposal distribution within the constrained region, such as density truncation and perfect Monte Carlo simulations.

Section 10: Miscellaneous Approaches
Description 10: Summarize various alternative strategies for constrained state estimation using particle filters, highlighting unique methods and their challenges.