Constrained State Estimation Research Articles

Range-Parameterized Range and Velocity Constrained State Estimation

Range-Parameterized Range and Velocity Constrained State Estimation

Data-driven moving horizon state estimation of nonlinear processes using Koopman operator

In this paper, a data-driven constrained state estimation method is proposed for nonlinear processes. Within the Koopman operator framework, we propose a data-driven model identification procedure for state estimation based on the algorithm of extended dynamic mode decomposition, which seeks an optimal approximation of the Koopman operator for a nonlinear process in a higher-dimensional space that correlates with the original process state-space via a prescribed nonlinear coordinate transformation. By implementing the proposed procedure, a linear state-space model can be established based on historic process data to describe the dynamics of a nonlinear process and the nonlinear dependence of the sensor measurements on process states. Based on the identified Koopman operator, a linear moving horizon estimation (MHE) algorithm that explicitly addresses constraints on the original process states is formulated to efficiently estimate the states in the higher-dimensional space. The states of the treated nonlinear process are recovered based on the state estimates provided by the MHE estimator designed in the higher-dimensional space. Two process examples are utilized to demonstrate the effectiveness and superiority of the proposed framework.

Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation

Several industrial processes fall under the category of Distributed parameter systems (DPS) which are represented as coupled partial differential equations (PDEs). Obtaining accurate and physically meaningful profile estimates of states despite only a limited number of spatially distributed measurements, is a pre-requisite for adequately controlling such systems. In this work, we propose an online constrained state profile estimation approach for DPS. The approach is based on reduced-dimension model represented by differential and algebraic equations (DAE) which in turn is developed by discretization of the PDEs along the spatial coordinate using the orthogonal collocation (OC) method. A profile prediction error based approach is proposed for systematically fixing the dimension of the OC model to be used. Further, to compensate for the inevitable model plant mismatch (MPM) arising in the process of the dimensionality reduction, additive zero mean Gaussian white noise processes are introduced in both state dynamics as well as measurement model. Statistics of these noises are obtained based on simulation based realizations of approximation errors at collocation points and measurement locations. To incorporate the identified correlation between the resulting state and measurement noises in the estimation method, we propose a constrained DAE-EKF (extended Kalman Filter) extension to deal with such correlations. The estimation approach also involves an appropriately formulated optimization based update step to ensure constraint satisfaction over the entire spatial profile leading to physically meaningful state profile estimates. The constrained profile estimator is also integrated with an online moving window based parameter estimation, which enables tracking of both time varying parameters and state profiles. The efficacy of the proposed approach is demonstrated by conducting stochastic simulation studies using two tubular reactor systems. The results show that the proposed constrained online state estimation approach is able to obtain accurate profile estimates at low computational cost, while ensuring that the profile estimates satisfy the given constraints.

Multiple Active Material Lithium-Ion Batteries: Finite-Dimensional Modeling and Constrained State Estimation

Lithium-ion batteries with composite electrodes are being developed in the industry for their improved power, energy, and durability properties over classical single active material accumulators. To make the best out of multiple active material batteries, their management system needs to be fed with reliable information on the internal states, which are not available by measurements. It appears that tools developed for this purpose for single active material batteries are not suitable for multiple active material ones due to their distinctive electrical coupling rules within the electrodes. In this context, we first present a new finite-dimensional model of lithium-ion batteries for which the cathode is represented by two particles made of two different materials, and the anode is described by two graphite particles. We then use the model to design a nonlinear observer based on polytopic techniques that estimate the internal battery variables and thus the state of charge (SOC). The convergence of the observer is guaranteed under a linear matrix inequality (LMI) condition. We then exploit a technique recently proposed in the literature to modify the observer so that its state estimates remain in given plausible range at all times, while preserving its original convergence properties. This allows avoiding aberrant estimated values in the transient and may also favor the future implementation of the observer on embedded devices. Simulation results using literature data illustrate the efficiency of the observer to estimate internal battery variables.

Robust State and Unknown Input Estimator and Its Application to Robot Localization

The robot localization problem is typically solved using state estimation techniques, where process and sensing inaccuracies are invariably present. Moreover, disturbances in the sensing and actuating mechanisms add to the uncertainties. Any system may degrade over time, and its parameter values may be ambivalently known. Cumulatively, all these sources of errors and uncertainties are considered as unknown inputs. This work aims to address the unknown inputs using a robust state (pose in the robot localization problem) estimator. The proposed robust state estimator deals with the unknown inputs such that the solution of the estimator is constrained in a way that warrants unbiased state estimates in the presence of the unknown inputs. This article explores the formulation of these constraints and the development of a constrained state estimator for a system, where the unknown inputs appear in both the state transition map (i.e., system model) and the state-output map (i.e., measurement model). The theoretical development of such a strategy stems from the localization problem of a wheeled mobile robot. The residuals of the constrained state estimator developed contain information about the unknown inputs. We conceive a recursive least squares strategy to estimate the unknown inputs simultaneously with the states using this information. Using simulations and experimental studies, we demonstrate the adequacy of our strategy for a differential drive robot.

A Framework for Constrained Static State Estimation in Unbalanced Distribution Networks

State estimation plays a key role in the transition from the passive to the active operation of distribution systems, as it allows to monitor the networks and provides the necessary information to perform control actions. However, designing state estimators for distribution systems is challenging, due to the characteristics of the networks, such as limited measurement availability. Furthermore, the features of the distribution system present significant local variations, e.g., voltage level and number and type of customers, which makes it hard to design a “one-size-fits-all” state estimator. This paper introduces a unifying framework that allows to easily implement and compare diverse unbalanced static state estimation models. This is achieved by formulating state estimation as a general constrained optimization problem. The advantages of this approach are described and supported by numerical illustration on a large set of real distribution feeders. The framework is also implemented in software and made available open-source.

Robust Filter-Based Visual Navigation Solution with Miscalibrated Bi-Monocular or Stereo Cameras

This work addressed the problem of miscalibration or decalibration of mobile stereo/bi-monocular camera setups. We especially focused on the context of autonomous vehicles. In real-world conditions, any optical system is subject to various mechanical stresses, caused by vibration, rough handling, collisions, or even thermal expansion. Such mechanical stresses change the stereo pair geometry, and as a consequence, the pre-calculated epipolar geometry or any geometric-based approach is no longer valid. The standard method, which consists of estimating the calibration online, fails in such harsh conditions. The proposed method was based on a robust linearly constrained state estimation technique able to mitigate the model mismatch without estimating the model parameters. Therefore, our solution was able to mitigate the errors with negligible use of additional computing resources. We propose to use a linearly constrained extended Kalman filter for a stereo-based visual odometry or simultaneous localization and mapping approach. Simulations confirmed that the method kept the system (and objects of the map) localized in real-time even with huge miscalibration errors and parameter variations. The results confirmed that the method was robust to a miscalibration of all the extrinsic calibration parameters even when the standard online calibration procedure failed.

State Estimation With a Destination Constraint Imposed by Proportional Navigation Guidance Law

Proportional navigation (PN) guidance laws have been commonly applied to homing missile systems guiding the missile to its target. When the target is stationary and its location is known a priori, the state of the missile is subjected to a destination constraint, which contains extra information on missile state that can help increase tracking performance. This article concerns with the constraint modeling and the state estimation with a destination constraint imposed by PN guidance law in 3-D space. Constraint models corresponding to three representative PN guidance laws: pure PN, true PN, and generalized PN, are presented using the state augmentation method, which stacks the states at the previous and current time steps in one state vector to express the constraint relationships. For cases where the destination information is noisy and the parameters of the guidance law, i.e., the navigation constant and the bias angle, are not known a priori, the unknown parameters are estimated along with the missile state in the state vector in an augmentation way. Based on the formulations of constraint models, the construction of pseudomeasurements is carried out. A constrained state estimation method is proposed by incorporating the pseudomeasurements into the nonlinear filtering process. In this method, a sequential process framework is employed and the cubature Kalman filter is used to handle the high-dimensional filtering problem and the strong nonlinearity involved in pseudomeasurements. Numerical experiments in three scenarios are performed to demonstrate the validity of the proposed models along with the estimation method.

Minimum variance constrained estimator

This paper is concerned with the problem of state estimation for discrete-time linear systems in the presence of additional (equality or inequality) constraints on the state (or estimate). By use of the minimum variance duality, the estimation problem is converted into an optimal control problem. Two algorithmic solutions are described: the full information estimator (FIE) and the moving horizon estimator (MHE). The main result is to show that the proposed estimator is stable in the sense of an observer. The proposed algorithm is distinct from the standard algorithm for constrained state estimation based upon the use of the minimum energy duality. The two are compared numerically on the benchmark batch reactor process model.

Constrained State Estimation for Nonlinear Systems: A Redesign Approach Based on Convexity

Given a plant whose trajectories of interest remain in a known compact set and an associated observer, we propose a general framework to modify this observer so that its state remains in a given convex set for all times, without altering the observer guaranteed performances in terms of convergence and robustness to external disturbances. The methodology can be applied to any time-varying continuous-time, discrete-time, and hybrid system/observer, for which a quadratic Lyapunov function is used for the analysis. The proposed approach is relevant, for instance, to remove the peaking phenomenon, to attenuate the effect of impulsive outliers in the measurement, to avoid aberrant estimates during transients, or to guarantee a given range for variables in embedded systems.

Static State Estimation with Inequality Constraints in a Complementarity Framework for Detecting Infeasible Operation States

This paper presents a new approach for incorporating inequality constraints into the equality-constrained weighted least squares state estimator, without modifying the way in which this estimator is formulated and solved. In this proposal, the inequality constraints are first represented as complementarity constraints, which are then transformed into equality constraints by using the Fischer-Burmeister merit function. The resulting set of equations are directly included in the set of equality constraints associated with the original constrained state estimation problem; therefore, the infeasible operating state, which is associated with the violation of a system's physical limit, is automatically detected during the iterative solution of the estimation problem. Under this condition, the compliance of the system's physical constraints is also automatically performed during the estimation solution process. The proposed approach also provides information about physical limits that were enforced, which could be employed to analyze the infeasible state estimation's causes. The validity and effectiveness of the proposed approach is fully demonstrated in the IEEE 14-bus benchmark network, the CIGRE test system and in the 7659-bus representation of the Mexican interconnected power system.

Moving Horizon Estimation for Uncertain Networked Control Systems with Packet Loss

This paper studies a moving horizon estimation approach to solve the constrained state estimation problem for uncertain networked systems with random packet loss. The system model error range is known, and the packet loss phenomena are modeled by a binary switching random sequence. Taking the model error, the packet loss, the system constraints, and the network transmission noise into account, a time-varying weight matrix is obtained by solving a least-square problem. Then, a robust moving horizon estimator is designed to estimate the system state by minimizing an optimization problem with an arrival cost function. The proposed estimator ensures that the optimal estimated state can be obtained in the worst case. Furthermore, the asymptotic convergence of the estimator is analyzed and some sufficient conditions for convergence are given. Finally, the validity of the proposed approach can be demonstrated by numerical simulations.

State Estimation With Implicit Constraints of Circular Trajectory Using Pseudomeasurements

In some target tracking scenarios, tracking performance can be improved by the incorporation of a constraint imposed by a preset trajectory. When the complete information about the trajectory is known, the constraint can be formulated in an explicit way. However, in practical applications, only partial information about the trajectory may be available. This article deals with the modeling of implicit constraints, imposed by a circular trajectory, along with the constrained state estimation. First, constraint models for two kinds of representative implicit constraints, the destination constraint and the trajectory shape constraint, are proposed for various cases with different conditions of prior information. In the cases with destination constraints, the destination information and the trajectory shape information are assumed to be known a priori . For each case, two forms of constraint models are derived. One is a direct form model where the constraint relationships are used to eliminate the unknowns in the final model, whereas the other is a state augmentation form model, where the unknowns are augmented into the state vector being estimated along with the target state. In the cases with trajectory shape constraints, only the trajectory shape information is available. For each case, the state augmentation form models are formulated. Second, constrained state estimation methods are proposed by utilizing pseudomeasurement constructed based on the constraint models. Furthermore, an extension to the maneuvering target tracking by integrating the proposed constraint filters into an interacting multiple model estimator is also presented. Numerical simulations in two scenarios are provided to show the effectiveness of the proposed constraint models and state estimation methods.

Variable Splitting Methods for Constrained State Estimation in Partially Observed Markov Processes

In this paper, we propose a class of efficient, accurate, and general methods for solving state-estimation problems with equality and inequality constraints. The methods are based on recent developments in variable splitting and partially observed Markov processes. We first present the generalized framework based on variable splitting, then develop efficient methods to solve the state-estimation subproblems arising in the framework. The solutions to these subproblems can be made efficient by leveraging the Markovian structure of the model as is classically done in so-called Bayesian filtering and smoothing methods. The numerical experiments demonstrate that our methods outperform conventional optimization methods in computation cost as well as the estimation performance.

Real-time equality-constrained hybrid state estimation in complex variables

The hybrid power system state estimation problem requires computing the state of the power network using data from both legacy and phasor measurements. Recent research has shown that the normal equations approach in complex variables is computationally advantageous, particularly in the presence of phasor measurement values, and that its software implementation is best suited to modern processors that employ single instruction multiple data (SIMD) processor extensions. The complex normal equations approach is however not ideal for handling zero injection measurements, as it requires their modeling as virtual measurements with high weights. This paper employs Wirtinger calculus for extending the complex normal equations approach to include equality constraints, and contrasts it with two previously published implementations: the normal equations approach in complex variables and the hybrid equality constrained state estimator in real variables. Numerical results are reported on transmission networks having up to 9241 nodes; they show that the complex variable equality constrained hybrid state estimator exhibits superior performance as compared to the above two techniques in terms of both computational time and accuracy. Moreover, the execution time on the largest network is less than 300 ms, which makes the proposed implementation commensurate with the requirements of real-time applications.

Projection‐based state estimation using noisy destination

The problem of state estimation with a destination constraint using the noisy prior information of the destination is investigated. With the utilisation of constraint information in estimation system, the estimation accuracy can be significantly enhanced. A projection-based constrained state estimation method is proposed to address this problem. In this method, the unscented Kalman filter is employed to obtain the unconstrained estimation. The Taylor series expansion is adopted to deal with the non-linearity of the destination constraint and the projection method is used to project the unconstraint estimate onto the constraint surface. Monte Carlo simulation results are presented to illustrate the effectiveness of the new approach.

State Estimation for Systems With Packet Dropping and State Equality Constraints

We consider the linear estimation problem for systems with packet dropping and equality constraints in this brief, where two kinds of constrained estimators will be designed. First, the constrained state estimator is derived based on the projection method and the unconstrained linear minimum mean square error estimator, and the estimator gains can be yielded in terms of the solutions to a Ricatti equation and a Lyapunov equation. However, when referring to the infinite horizon state estimation, there is a need to consider the convergence of the Lyapunov equation. To meet the requirement of the convergence analysis, the condition that the system matrix is stable should be imposed. Then, in order to remove the restrict condition that the system matrix is stable, another constrained state estimator is designed based on the projection method and time-stamp technique, where the estimator gains can be yielded only in terms of the solutions to a Riccati equation. Finally, we give two numerical examples to show that the second constraint estimator is more effective compared with the first constraint estimator.

Projection-Based Constrained State Estimation and Fusion for Tracking Over Long-Haul Links

In many mission-critical systems, a number of sensors are deployed to track ground targets with linear or nonlinear constraints on their motion dynamics. These sensors generate state estimates to be sent over long-haul links to a remote fusion center. We propose closed-form projection-based methods to incorporate known constraints into the estimation and fusion process. The extensive performance evaluation results demonstrate the extent to which projection-based fusion can improve the overall tracking accuracy performance in the presence of communication loss and delay.

Hybrid State Estimation in Complex Variables

Power system state estimation is classically formulated as a weighted least squares (WLS) optimization problem over the real domain, with the state variables being the voltage magnitudes and angles or the voltage real and imaginary parts. Although the voltages in ac networks are complex variables, a solution in the complex domain has not been previously reported; this is because a real function of complex variables is nonanalytic in its arguments, i.e., the Taylor series expansion in its arguments alone does not exist. By making use of the Wirtinger calculus, this paper presents a new implementation of the WLS state estimation problem in complex variables. The partial derivatives employed in the method give rise to a structure that is very simple to implement. The method can straightforwardly handle both legacy and phasor measurements; particularly, the processing of phasor current measurements does not require any special provisions, unlike previously reported hybrid state estimator implementations over the real domain. Numerical results are reported on networks with up to 9241 nodes, and they demonstrate that the accuracy of the complex variable implementation is competitive with that of the real variable constrained hybrid state estimator, while being significantly faster.

Performances Analysis of GNSS NLOS Bias Correction in Urban Environment Using a Three-Dimensional City Model and GNSS Simulator

The well-known conventional least squares (LS) and extended Kalman filter (EKF) are ones of the most widely used algorithms in science and particularly in localization with global navigation satellites systems (GNSS) measurements. However, these estimators are not optimal when the GNSS measurements become contaminated by nonGaussian errors including multipath (MP) and nonline-of-sight (NLOS) biases. On the other hand, this kind of ranging measurements errors occurs generally in urban areas where GNSS-based positioning applications require more accuracy and reliability. In this paper, we use additional information of the environment consisting of bias prediction from a three-dimensional (3-D) model and a GNSS simulator to exploit constructively NLOS measurements. We use this 3-D GNSS simulator to predict lower and upper bounds of these biases. Then, we integrate this information in the position estimation problem by considering these biases as additive error and exploiting the bounds to end-up with a constrained state estimation problem that we resolve with existing constrained least squares (CLS) and constrained EKF (CEFK) algorithms. Experimental results using real GPS signals in down-town Toulouse show that the proposed estimator is capable of improving the positioning accuracy compared with conventional algorithms. Theoretical conditions have been established to determine the acceptable bias prediction error allowing better positioning performance than conventional estimators. Tests are conducted then to validate these conditions and investigate the influence of the bias prediction error on the localization performance by proposing new accuracy metrics.