Kalman smoothing and block tridiagonal systems: new connections and numerical stability results

Kalman smoothing and block tridiagonal systems: new connections and numerical stability results
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卡尔曼平滑和分块三对角系统：新连接和数值稳定性结果

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发表时间：
2013
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通讯作者：
G. Pillonetto
G. Pillonetto
中科院分区：
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作者：
A. Aravkin;Bradley B. Bell;J. Burke;G. Pillonetto

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Rauch-Tung-Striebel（RTS）算法和Mayne-Fraser（MF）算法是两种最流行的平滑算法，用于从在固定时间间隔上收集的测量数据重建动态线性系统的状态。另一种（不太流行的）方法是Mayne（M）算法，在他的原始论文中以算法A的名义介绍。在本文中，我们分析了这三个平滑从优化和代数的角度来看，揭示了新的见解，其数值稳定性。在这样做时，我们重新解释经典的递归作为块三对角矩阵的矩阵分解方法。 首先，我们证明了经典的RTS平滑器是特定块三对角系统的前向块三对角（FBT）算法（也称为托马斯算法）的实现。我们研究了该方案的数值稳定性，连接的条件数的完整系统的性能遇到的各个块在标准递归。其次，我们研究了M平滑器，证明了它等价于一个后向块三对角（BBT）算法，并具有比RTS更强的稳定性保证。第三，我们说明了如何MF平滑解决块三对角系统，并证明它具有相同的RTS（但不是那些M）的数值稳定性。最后，我们提出了一种新的混合RTS/M（FBT/BBT）光滑格式，它比MF更快，并且具有与RTS和MF相同的数值稳定性保证。
The Rauch-Tung-Striebel (RTS) and the Mayne-Fraser (MF) algorithms are two of the most popular smoothing schemes to reconstruct the state of a dynamic linear system from measurements collected on a fixed interval. Another (less popular) approach is the Mayne (M) algorithm introduced in his original paper under the name of Algorithm A. In this paper, we analyze these three smoothers from an optimization and algebraic perspective, revealing new insights on their numerical stability properties. In doing this, we re-interpret classic recursions as matrix decomposition methods for block tridiagonal matrices. First, we show that the classic RTS smoother is an implementation of the forward block tridiagonal (FBT) algorithm (also known as Thomas algorithm) for particular block tridiagonal systems. We study the numerical stability properties of this scheme, connecting the condition number of the full system to properties of the individual blocks encountered during standard recursion. Second, we study the M smoother, and prove it is equivalent to a backward block tridiagonal (BBT) algorithm with a stronger stability guarantee than RTS. Third, we illustrate how the MF smoother solves a block tridiagonal system, and prove that it has the same numerical stability properties of RTS (but not those of M). Finally, we present a new hybrid RTS/M (FBT/BBT) smoothing scheme, which is faster than MF, and has the same numerical stability guarantees of RTS and MF.