Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution
Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution
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DOI：
10.1109/tac.2011.2161834
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发表时间：
2009-03
影响因子：
6.8
通讯作者：
S. Kar;B. Sinopoli;José M. F. Moura
中科院分区：
文献类型：
--
作者：
S. Kar;B. Sinopoli;José M. F. Moura
本文研究了卡尔曼滤波中离散时间随机Riccati方程(RRE)在观测值的到达用Bernoulli独立同分布(I.I.D.)描述时的渐近行为。进程。我们将RRE建模为保序的强次线性随机动力系统(RDS)。在随机有界性的充分条件下，利用保序、强次线性RDS的极限集二分结果，我们建立了RRE的渐近性质：随机预测误差协方差矩阵序列弱收敛于唯一的不变分布，其支持度具有分形性。对于可稳定和可检测的系统，对于任何非零观测分组到达概率，随机有界性(因此弱收敛)都成立，特别地，我们可以在远低于平均稳定的临界概率的操作到达率下建立弱收敛(在这种情况下得到的不变度量不具有一阶矩)。我们利用控制RRE的马尔可夫过程的弱Feller性质来刻画极限不变分布的支撑性，作为可数点集的拓扑闭包，通常在正半正定矩阵集中是不稠密的。我们使用不变分布的支撑性的显式刻画和几乎必然的(A.S.)样本路径的遍历性，以便于计算不变分布的统计量。一个一维的例子表明，支撑体是具有自相似性质的非负实数的一个分裂子集。
The paper studies the asymptotic behavior of discrete time Random Riccati Equations (RRE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli independent and identically distributed (i.i.d.) process. We model the RRE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RRE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. For stabilizable and detectable systems, stochastic boundedness (and hence weak convergence) holds for any nonzero observation packet arrival probability and, in particular, we can establish weak convergence at operating arrival rates well below the critical probability for mean stability (the resulting invariant measure in that situation does not possess a first moment). We apply the weak-Feller property of the Markov process governing the RRE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure (a.s.) ergodicity of the sample paths to easily compute statistics of the invariant distribution. A one-dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.