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Nonlinear Bayesian Estimation: From Kalman Filtering to a Broader Horizon

Huazhen Fang, Ning Tian, Yebin Wang, MengChu Zhou, Mulugeta A. Haile

arXiv:1712.01406v2eess.SY

TL;DR

Nonlinear state estimation requires methods for inferring unmeasured states under stochastic dynamics and nonlinear transformations. This review develops the Bayesian filtering perspective and surveys Kalman-filter and other Bayesian approaches, concluding with their applications, extensions, and open scope boundaries.

  • Problem

    Nonlinear state estimation is challenging because complete states are usually unmeasured and nonlinear transformations can prevent exact or closed-form conditional-density tracking.

  • Method

    The article systematically reviews Bayesian state estimation, emphasizing Gaussian mean-covariance tracking and KF variants including EKF, UKF, and EnKF.

  • Results

    The review reports that UKF provides the most accurate estimation overall in the speed-sensorless induction-motor study, while EnKF accuracy improves with ensemble size.

  • Takeaways & Limitations

    Bayesian filtering offers a unified framework spanning standard and nonlinear Kalman filters, other filtering approaches, and simultaneous state and parameter estimation.

  • Takeaways & Limitations

    Nonlinear Bayesian estimation for cyber-physical systems remains particularly rare, despite challenges from communication constraints, computing limitations, and cyber data attacks.

Abstract

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This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on the system's measurement data. This article offers a systematic introduction of the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including the Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation forward to more complicated problems such as simultaneous state and parameter/input estimation.

I. INTRODUCTION

The article frames Bayesian analysis as a leading approach to nonlinear stochastic state estimation and provides a systematic review centered on the Kalman-filter family and related estimators.

  • Nonlinear state estimation has attracted sustained research because of applications in signal processing, navigation and guidance, and econometrics.
  • Bayesian analysis provides a probability-based framework for updating beliefs about unknown states using new evidence.
  • The review introduces Bayesian state estimation from first principles, emphasizing model-based prediction and measurement-based belief updates.
  • It focuses on extended, unscented, and ensemble Kalman filters as three primary members of the KF family.
  • The article also surveys other important Bayesian estimators and estimation problems to introduce the field’s state of the art.

II. A BAYESIAN VIEW OF STATE ESTIMATION

Bayesian state estimation treats unknown system quantities probabilistically and recursively updates the conditional state distribution as measurements arrive. The framework alternates prediction from prior information with measurement-based update.

  • The nonlinear system is modeled with state dynamics f and measurement mapping h, while process and measurement noises are independent zero-mean white Gaussian sequences.
  • State estimation infers the unmeasured state xk from output measurements because complete state measurement is unrealistic in most practical applications.
  • The target of estimation is the conditional pdf p(xk|Yk), which captures information about the state conveyed by measurements through time.
  • Prediction uses p(xk−1|Yk−1) to obtain p(xk|Yk−1), and the new measurement yk updates it to p(xk|Yk).
  • The Bayesian filtering equations rely on Bayes’ rule and Markovian state propagation, with prediction and update executed alternately through time.

III. FROM BAYESIAN FILTERING TO KALMAN FILTERING

Exact Bayesian filtering becomes difficult for nonlinear transformations, so Kalman filtering uses Gaussian approximations and tracks state means and covariances. Different approximations yield a family of nonlinear Kalman filters.

  • Nonlinear transformations can make exact or closed-form tracking of the conditional state pdf difficult or impossible.
  • Gaussianity approximations make nonlinear Bayesian filtering mathematically tractable and lead to a Kalman-filter formulation.
  • Prediction estimates xk using the conditional expectation E(xk|Yk−1), with an associated prediction-error covariance.
  • The conceptual KF estimates states by recursively tracking the mean and covariance of a random state vector.
  • For nonlinear systems, explicit mean and covariance calculations require approximations, producing EKF, UKF, and EnKF variants.

IV. STANDARD LINEAR KALMAN FILTER

The standard linear Kalman filter recursively estimates system states through prediction and measurement update, tracking both the state estimate and its uncertainty. Under suitable conditions, its covariance converges to a steady-state solution that enables a more computationally efficient implementation.

  • IV. STANDARD LINEAR KALMAN FILTER: The standard KF applies recursively to linear systems under Gaussian noise and initial-state assumptions.These assumptions ensure the relevant conditional distributions remain Gaussian.
  • IV. STANDARD LINEAR KALMAN FILTER: The Kalman gain combines predicted state uncertainty with measurement information during the update.The update is performed when the output measurement arrives.
  • IV. STANDARD LINEAR KALMAN FILTER: Prediction uses prior data to produce x̂_k|k−1 and P_k|k−1, while measurement update produces x̂_k|k and P_k|k.The covariance quantifies uncertainty in the updated estimate.
  • IV. STANDARD LINEAR KALMAN FILTER: Under detectability and stabilizability conditions, the prediction covariance converges to the solution of a discrete-time algebraic Riccati equation.This convergence provides the basis for steady-state filtering.
  • IV. STANDARD LINEAR KALMAN FILTER: A steady-state KF computes the gain and covariance offline, reducing online computation at the cost of being suboptimal.It can avoid recomputing the gain and covariance at every time step.

V. REVIEW OF NONLINEAR KALMAN FILTERS

The review introduces major nonlinear Kalman-filter techniques, focusing on EKF, UKF, and EnKF. These methods extend the Kalman-filter framework to nonlinear state-estimation problems.

  • V. REVIEW OF NONLINEAR KALMAN FILTERS: The section surveys EKF, UKF, and EnKF as major nonlinear Kalman-filter techniques.EKF and UKF are emphasized in control systems, while EnKF is prominent in data assimilation.

A. Extended Kalman Filter

The extended Kalman filter accommodates nonlinear dynamics and observations by locally linearizing them around current state estimates. Its simplicity and convergence guarantees support broad use, but first-order accuracy, derivative requirements, and computational cost constrain applicability.

  • A. Extended Kalman Filter: EKF linearizes nonlinear functions around the most recent state estimate using a first-order Taylor expansion.The resulting prediction and update retain the standard KF structure while incorporating local linearization.
  • A. Extended Kalman Filter: The EKF uses linearized approximations to estimate the means and covariances needed for prediction and measurement update.The state-transition function is linearized for prediction, and the measurement function is linearized around the predicted estimate for update.
  • A. Extended Kalman Filter: EKF retains the prediction-update structure of the standard KF while adding a linearization procedure for nonlinearities.This structural resemblance allows the nonlinear filter to follow the standard KF recursion.
  • A. Extended Kalman Filter: EKF offers relative design and execution ease, broad engineering use, and provable asymptotic convergence under some conditions.The cited applications include aerospace, robotics, biomedical, mechanical, chemical, electrical, and civil engineering.
  • A. Extended Kalman Filter: First-order accuracy becomes inadequate for highly nonlinear systems, while explicit derivative matrices exclude discontinuous or nondifferentiable functions.The computational complexity is O(n^3_x), limiting EKF application to low-dimensional systems.
  • A. Extended Kalman Filter: Second-order and iterated EKF variants trade increased computational cost for improved accuracy under stronger nonlinearities.The iterated EKF repeatedly refines the state estimate around the current point.

B. Unscented Kalman Filter

The unscented Kalman filter handles nonlinear transformations by deterministically propagating sigma points rather than explicitly calculating derivatives. It offers second-order accuracy and derivative-free operation, but its computational burden grows substantially with state dimension.

  • B. Unscented Kalman Filter: The unscented transform deterministically selects sigma points from an input mean and covariance, propagates them through a nonlinear function, and estimates the transformed mean and covariance.The sigma-point spread depends on the covariance and scaling parameters.
  • B. Unscented Kalman Filter: The parameter β incorporates prior distribution information, with β = 2 optimal for Gaussian inputs.The UT procedure uses weighted sigma points to recover transformed statistics.
  • B. Unscented Kalman Filter: UKF applies the unscented transform at both prediction and update steps to handle nonlinear state and measurement transformations.Sigma points are propagated through the state function f and measurement function h before Gaussian updates are performed.
  • B. Unscented Kalman Filter: UKF computes predicted measurement statistics and state-measurement cross-covariance before applying the Gaussian update.These quantities connect the sigma-point transformation to the updated state estimate.
  • B. Unscented Kalman Filter: UKF has computational complexity O(n^3_x), offers second-order accuracy, and avoids Jacobian calculations through derivative-free operations.These properties support use with discontinuous or nondifferentiable nonlinear transformations.
  • B. Unscented Kalman Filter: UKF variants modify numerical propagation, state refinement, or sigma-point selection to improve stability, accuracy, or efficiency.Examples include square-root UKF, iterated UKF, and broader sigma-point Kalman-filter methods.

C. Ensemble Kalman Filter

EnKF approximates the conditional state distribution with an ensemble, recursively propagating and updating its members and estimating the state from their sample mean and covariance. Its derivative-free, ensemble-based structure is attractive for large-scale nonlinear systems, with computational advantages and convergence toward KF under stated conditions.

  • Ensemble representation: EnKF combines Kalman filtering with Monte Carlo sampling to represent a conditional state distribution using an ensemble.The ensemble is sampled from the prior conditional distribution and propagated through process and measurement updates.
  • Prediction and update: The algorithm recursively predicts and updates ensemble members, then computes the state mean and covariance from the updated ensemble.The posterior ensemble approximates p(xk|Yk), while its sample statistics provide the updated estimate.
  • Computational structure: EnKF operates directly on ensembles and avoids derivative-based computation and explicit covariance-matrix propagation.These design choices distinguish it from EKF and UKF implementations based on covariance propagation and make parallel implementation advantageous.
  • Computational structure: For high-dimensional systems, EnKF has complexity O(n_s^3) under the stated dimensional conditions, contrasting with O(n_x^3) for EKF and UKF.The cited setting assumes n_x ≫ n_y and n_x ≫ n_s.
  • Convergence: EnKF convergence can be fast with a reasonably small ensemble and converges to KF for large ensemble size with Gaussian state distributions.The latter convergence result is stated for the limit of large ensemble size and Gaussian state probability distributions.

VI. APPLICATION TO SPEED SENSORLESS INDUCTION MOTORS

The paper evaluates EKF, UKF, and EnKF for state estimation in speed sensorless induction motors, including the effect of EnKF ensemble size. In this simulation, UKF gives the most accurate overall estimates, while EnKF accuracy improves with larger ensembles; filter behavior also depends on initialization, tuning, and the specific problem.

  • Application setup: The case study applies EKF, UKF, and EnKF to estimate induction-motor states, including rotor speed, from stator-current outputs.The model includes stator currents, rotor fluxes, and rotor speed among its state variables.
  • Experimental design: EnKF is evaluated with ensemble sizes Ns = 40, 100, 200 and 400, with each size run 100 times to reduce random-noise effects.The final comparison averages the estimation errors across the 100 runs.
  • Results: UKF provides the best rotor-speed estimation overall and the most accurate estimation when all state variables are considered.The rotor-speed comparison uses EKF, UKF, and EnKF with ensemble size 100 and includes ±3σ confidence bounds.
  • Results: EnKF estimation accuracy improves as the ensemble size increases.This trend is reported for the average estimation errors summarized in Table I.
  • Practical considerations: Filter performance depends on initialization, tuning, computational efficiency, structural complexity, problem size, and the specific estimation problem.The authors report that EKF and EnKF are more sensitive to the initial guess, while covariance and filter-parameter tuning can strongly influence estimation.

VII. OTHER FILTERING APPROACHES AND ESTIMATION PROBLEMS

The section surveys Gaussian, Gaussian-sum, particle, and moving-horizon approaches as extensions of Bayesian nonlinear state estimation, alongside simultaneous state and parameter/input estimation. These methods trade approximation structure, sampling, constraints, and computational cost to address nonlinear, non-Gaussian, or incompletely known systems.

  • Gaussian filters: Gaussian filters approximate Bayesian distributions with Gaussians and use numerical quadrature to preserve a Kalman-filter prediction-update structure.Gauss-Hermite and cubature rules produce KF realizations, with the cubature filter a special UKF case under specified parameters.
  • Gaussian-sum filters: Gaussian-sum filters represent non-Gaussian probability densities as weighted sums of Gaussian basis functions.They can be combined with UKF, EnKF, Gaussian, particle, or optimization methods, while EKF-based updates assume nonlinear dynamics are adequately represented by local linearizations.
  • Particle filters: Particle filters use weighted random samples to approximate the Bayesian state distribution through propagation, likelihood-based weight updates, normalization, and resampling.Resampling addresses degeneracy, where many samples acquire nearly zero weights.
  • Particle filters: With sufficiently large Ns, particle filters can achieve superior estimation accuracy, with an error bound independent of system dimension, but their complexity is O(Nsnx^2) when Ns ≫ nx.Rao-Blackwellization, KF-PF combinations, and implicit sampling are presented as ways to reduce computational or sampling difficulties.
  • Moving horizon estimation: Moving horizon estimation performs constrained optimization over a receding horizon, reduces to the standard KF for unconstrained linear systems, and is computationally expensive because of constrained optimization and arrival-cost requirements.It is therefore usually better suited to slow dynamic processes, although real-time optimization may accelerate implementation.
  • Other estimation problems: Simultaneous state and parameter or input estimation extends state estimation to systems whose model parameters or inputs are unknown or time-varying.The motivation includes industrial monitoring under unknown disturbances and adaptation to incomplete model knowledge.

VIII. CONCLUSION

The conclusion presents the article as a state-of-the-art review that develops nonlinear state estimation from Bayesian principles and connects major filtering approaches through that perspective. It also identifies unresolved challenges involving nonlinear probability transformations, uncertainty, sensing design, and cyber-physical settings.

  • Conclusion: The article reviews Bayesian nonlinear state estimation from Bayesian updating and the general Kalman filter through KF variants and broader filtering approaches.Its coverage includes KF, EKF, UKF, EnKF, GF, GSF, PF, and MHE, with a speed-sensorless induction-motor case study comparing EKF, UKF, and EnKF.
  • Future directions: No generally acknowledged satisfactory solution yet exists for accurately characterizing nonlinear transformations of probability distributions.The conclusion identifies this issue as a continuing research direction for methods such as EKF, UKF, and EnKF.
  • Future directions: Future work must also address uncertainty, sensing-structure and estimator co-design, and Bayesian estimation in cyber-physical settings.The discussion links uncertainty to unmodeled dynamics, external disturbances, process variability, and sensor noise, while noting that sensor deployment affects data informativeness and estimation effectiveness.

APPENDIX

The appendix summarizes basic properties of Gaussian random vectors. It covers affine transformations, sums of independent Gaussian vectors, and conditional distributions of jointly Gaussian vectors.

  • Gaussian distribution: A Gaussian random vector is characterized by its mean and covariance and has an explicitly defined probability density function.The appendix considers z ∈ R^n with z ∼ N(z̄, Pz).
  • Gaussian properties: Affine transformations of Gaussian vectors remain Gaussian.The stated transformation is Az + b.
  • Gaussian properties: The sum of two independent Gaussian random vectors is Gaussian.The property applies when the two vectors are independent and Gaussian.
  • Gaussian properties: For jointly Gaussian random vectors, the conditional distribution of one vector given the other is Gaussian.This property supports conditional Gaussian modeling in estimation.
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