Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

TLDR

The Kalman filter is the minimum‑variance state estimator for linear dynamic systems with Gaussian noise, remains the best linear estimator even with non‑Gaussian noise, and for nonlinear systems optimal closed‑form estimators are generally unavailable but various Kalman‑filter modifications can be used, yet it does not incorporate known state constraints such as equality or inequality constraints. This paper aims to modify the Kalman filter to exploit known state constraints and thereby achieve better filtering performance than the unconstrained filter. The authors review several approaches that incorporate state constraints into the Kalman filter and its nonlinear extensions—including the extended, unscented, and particle filters—by modifying the estimation process to enforce the constraints. When both the system dynamics and the state constraints are linear, all constrained filtering approaches yield the same optimal constrained linear state estimate, whereas with nonlinear systems or constraints the approaches are generally suboptimal and produce differing results.

Abstract

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.