What is the Lagrangian for Nonlinear Filtering?
What is the Lagrangian for Nonlinear Filtering?
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DOI：
10.1109/cdc40024.2019.9030206
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发表时间：
2019-03
期刊：
影响因子：
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通讯作者：
J. W. Kim;P. Mehta;Sean P. Meyn
中科院分区：
文献类型：
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作者：
J. W. Kim;P. Mehta;Sean P. Meyn
估计与最优控制的对偶性是一个具有丰富历史意义的问题。第一个对偶原理出现在Kalman-Bucy的开创性论文中，其中最小方差估计问题被证明是线性二次（LQ）最优控制问题的对偶。对偶性提供了一种从最优控制解推导卡尔曼滤波方程的构造性证明技术，本文将卡尔曼-布西的经典对偶结果推广到非线性滤波器：状态作为连续时间马尔可夫过程演化，观测值是受加性高斯噪声污染的状态的非线性函数。引入一个对偶过程作为倒向随机微分方程。该过程用于将最小方差估计问题转化为最优控制问题。它的解决方案是从最大值原理的应用，并随后用于推导方程的非线性滤波器。Kalman-Bucy的经典对偶结果是其特例。
Duality between estimation and optimal control is a problem of rich historical significance. The first duality principle appears in the seminal paper of Kalman-Bucy, where the problem of minimum variance estimation is shown to be dual to a linear quadratic (LQ) optimal control problem. Duality offers a constructive proof technique to derive the Kalman filter equation from the optimal control solution.This paper generalizes the classical duality result of Kalman-Bucy to the nonlinear filter: The state evolves as a continuous-time Markov process and the observation is a nonlinear function of state corrupted by an additive Gaussian noise. A dual process is introduced as a backward stochastic differential equation (BSDE). The process is used to transform the problem of minimum variance estimation into an optimal control problem. Its solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The classical duality result of Kalman-Bucy is shown to be a special case.