Kalman Filter Auto-tuning through Enforcing Chi-Squared Normalized Error Distributions with Bayesian Optimization

The nonlinear and stochastic relationship between noise covariance parameter values and state estimator performance makes optimal filter tuning a very challenging problem. Popular optimization-based tuning approaches can easily get trapped in local minima, leading to poor noise parameter identification and suboptimal state estimation. Recently, black box techniques based on Bayesian optimization with Gaussian processes (GPBO) have been shown to overcome many of these issues, using normalized estimation error squared (NEES) and normalized innovation error (NIS) statistics to derive cost functions for Kalman filter auto-tuning. While reliable noise parameter estimates are obtained in many cases, GPBO solutions obtained with these conventional cost functions do not always converge to optimal filter noise parameters and lack robustness to parameter ambiguities in time-discretized system models. This paper addresses these issues by making two main contributions. First, we show that NIS and NEES errors are only chi-squared distributed for tuned estimators. As a result, chi-square tests are not sufficient to ensure that an estimator has been correctly tuned. We use this to extend the familiar consistency tests for NIS and NEES to penalize if the distribution is not chi-squared distributed. Second, this cost measure is applied within a Student-t processes Bayesian Optimization (TPBO) to achieve robust estimator performance for time discretized state space models. The robustness, accuracy, and reliability of our approach are illustrated on classical state estimation problems.