Over the last two decades, moving horizon estimation (MHE) has increasingly been used by researchers and industrial practitioners of nonlinear model predictive control as an alternative to recursive Bayesian estimation schemes. The MHE formulations available in the literature assume that uncertainties in the state dynamics can be modelled as additive and Gaussian stochastic processes. In practice, the unmeasured disturbances affect the system dynamics in a much more complex way and Gaussian assumption may prove inadequate to represent their behaviour. However, unlike the recursive Bayesian estimation area, handling of non-additive and non-Gaussian state disturbances has not received much attention in the MHE literature. In this work, a novel formulation of MHE is proposed in the Bayesian framework to solve state estimation problems associated with systems subjected to nonlinear non-Gaussian stochastic disturbances. To begin with, a Bayesian formulation that maximizes the joint posterior density function of the initial state and stochastic inputs is developed for a batch of data, which is further extended to the moving horizon framework. The proposed formulation associates the random fluctuations affecting the system dynamics with the unmeasured inputs entering the systems. The probabilistic formulation of MHE together with modelling of the state disturbances arising from physical sources enables us to work with any (non-Gaussian or Gaussian) probability density function (PDF) and systematically handle the stochastic inputs affecting the dynamics in a nonlinear manner without making any simplifying assumptions. We proceed to show that the disturbances/states with constraints can also be incorporated in our proposed formulation and can be interpreted as following truncated probability density functions. Efficacy of the proposed MHE approach is demonstrated using three benchmark simulation case studies and an experimental case study. Simulation studies show that the proposed MHE framework is able to perform similar to EnKF, a sampling based sequential Bayesian estimation approach which is capable of handling nonlinear non-Gaussian disturbances. It is further shown that the proposed MHE performs much better than the conventional MHE approach which assumes additive Gaussian noise in the state dynamics. Thus, the proposed MHE scheme provides an optimization based alternative to sampling based estimation schemes, such as Ensemble Kalman filtering, which can handle state estimation problems when state and measurement densities are non-Gaussian.

在过去的二十年里，移动水平估计 (MHE) 越来越多地被非线性模型预测控制的研究人员和工业从业者用作递归贝叶斯估计方案的替代方案。文献中可用的 MHE 公式假设状态动力学中的不确定性可以建模为加性和高斯随机过程。在实践中，未测量的干扰以更复杂的方式影响系统动力学，高斯假设可能不足以代表它们的行为。然而，与递归贝叶斯估计领域不同，非加性和非高斯状态扰动的处理在 MHE 文献中并未受到太多关注。在这项工作中，在贝叶斯框架中提出了一种新的 MHE 公式，以解决与受到非线性非高斯随机扰动的系统相关的状态估计问题。首先，为一批数据开发了一个最大化初始状态和随机输入的联合后验密度函数的贝叶斯公式，该公式进一步扩展到移动水平框架。所提出的公式将影响系统动力学的随机波动与进入系统的未测量输入联系起来。MHE 的概率公式以及对物理源产生的状态扰动的建模使我们能够使用任何（非高斯或高斯）概率密度函数 (PDF)，并系统地处理以非线性方式影响动力学的随机输入，而无需任何简化的假设。我们继续表明，具有约束的干扰/状态也可以纳入我们提出的公式中，并且可以解释为以下截断的概率密度函数。使用三个基准模拟案例研究和一个实验案例研究证明了所提出的 MHE 方法的有效性。仿真研究表明，所提出的 MHE 框架能够执行类似于 EnKF，一种基于采样的顺序贝叶斯估计方法，能够处理非线性非高斯扰动。进一步表明，所提出的 MHE 比在状态动态中假设加性高斯噪声的传统 MHE 方法表现得更好。因此，所提出的 MHE 方案为基于采样的估计方案提供了一种基于优化的替代方案，例如集成卡尔曼滤波，当状态和测量密度为非高斯时，该方案可以处理状态估计问题。