Abstract

Kalman filters are widely used for estimating the state of a system based on noisy or inaccurate sensor readings, for example in the control and navigation of vehicles or robots. However, numerical instability or modelling errors may lead to divergence of the filter, leading to erroneous estimations. Establishing robustness against such issues can be challenging. We propose novel formal verification techniques and software to perform a rigorous quantitative analysis of the effectiveness of Kalman filters. We present a general framework for modelling Kalman filter implementations operating on linear discrete-time stochastic systems, and techniques to systematically construct a Markov model of the filter's operation using truncation and discretisation of the stochastic noise model. Numerical stability and divergence properties are then verified using probabilistic model checking. We evaluate the scalability and accuracy of our approach on two distinct probabilistic kinematic models and four Kalman filter implementations.

中文翻译：

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卡尔曼滤波器的定量验证

摘要

卡尔曼滤波器广泛用于根据噪声或不准确的传感器读数估计系统状态，例如在车辆或机器人的控制和导航中。然而，数值不稳定性或建模错误可能会导致滤波器发散，从而导致错误估计。建立针对此类问题的稳健性可能具有挑战性。我们提出了新颖的形式验证技术和软件来对卡尔曼滤波器的有效性进行严格的定量分析。我们提出了一个通用框架，用于对在线性离散时间随机系统上操作的卡尔曼滤波器实现进行建模，以及使用随机噪声模型的截断和离散化系统地构建滤波器操作的马尔可夫模型的技术。然后使用概率模型检查验证数值稳定性和发散特性。我们在两个不同的概率运动模型和四个卡尔曼滤波器实现上评估我们的方法的可扩展性和准确性。