不可镇定线性系统时变卡尔曼滤波渐近稳定性的优化视角研究
On asymptotic stability of the time-varying Kalman filter for unstabilizable linear systems: an optimization perspective
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- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
- Leibniz University Hannover（汉诺威莱布尼茨大学）
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中文总结 AI 辅助
本文从优化视角推导时变卡尔曼滤波渐近稳定性的充要条件，提出修正Q函数，避免Riccati迭代，便于推广至非线性系统。
中文摘要 AI 辅助
本文针对具有半正定初始状态协方差以及正定过程噪声和测量噪声的线性时不变系统，建立了时变卡尔曼滤波渐近稳定性的充分必要条件。与经典文献中分析离散Riccati方程的方法不同，本文提出了等价的状态平滑优化问题，并利用该优化问题的性质建立了所有结果。推导了一个类似Lyapunov的函数，称为修正的$Q$-函数，并用于此分析。这种优化方法无需经典但繁琐的Riccati迭代代数运算，并为非线性系统提供了更好的推广和应用。
英文摘要
This paper establishes the necessary and sufficient conditions for asymptotic stability of the time-varying Kalman filter applied to a linear time invariant system with semidefinite initial state covariance and positive definite process and measurement noise. Rather than analyze the discrete Riccati equation as in the classic literature, the equivalent state smoothing optimization problem is stated and all results are established using properties of this optimization problem. A Lyapunov-like function, termed a modified $Q$-function is derived and used for this analysis. This optimization approach removes the need for the classic but cumbersome Riccati iteration algebra and provides better generalization and application for nonlinear systems.
