The accurate estimation of the noise covariance matrix (NCM) in a dynamic system is critical for state estimation and control, as it has a major influence in their optimality. Although a large number of NCM estimation methods have been developed, most of them assume the noises to be white. However, in many real-world applications, the noises are colored (e.g., they exhibit temporal autocorrelations), resulting in suboptimal solutions. Here, we introduce a novel brain-inspired algorithm that accurately and adaptively estimates the NCM for dynamic systems subjected to colored noise. Particularly, we extend the Dynamic Expectation Maximization algorithm to perform both online noise covariance and state estimation by optimizing the free energy objective. We mathematically prove that our NCM estimator converges to the global optimum of this free energy objective. Using randomized numerical simulations, we show that our estimator outperforms nine baseline methods with minimal noise covariance estimation error under colored noise conditions. Notably, we show that our method outperforms the best baseline (Variational Bayes) in joint noise and state estimation for high colored noise. We foresee that the accuracy and the adaptive nature of our estimator make it suitable for online estimation in real-world applications.

翻译：动态系统中噪声协方差矩阵的精确估计对状态估计与控制至关重要，因其直接影响系统的最优性。尽管已有大量噪声协方差矩阵估计方法被提出，但多数方法假设噪声为白噪声。然而，在实际应用中，噪声往往具有颜色特性（例如存在时间自相关），这会导致次优解。本文提出一种受脑科学启发的新型算法，能够针对有色噪声动态系统实现精确自适应的噪声协方差矩阵估计。具体而言，我们通过优化自由能目标函数，将动态期望最大化算法扩展为同时进行在线噪声协方差与状态估计。数学证明表明，所提噪声协方差矩阵估计器可收敛至该自由能目标函数的全局最优解。通过随机数值仿真，我们证明在有色噪声条件下，该方法能以最小噪声协方差估计误差优于九种基线方法。值得注意的是，在强色噪声联合噪声与状态估计场景中，本方法性能优于最优基线方法（变分贝叶斯）。我们预期该估计器的高精度与自适应特性使其适用于实际应用中的在线估计。