LM²MSE State Estimation - Kalman Filtering under Stochastic and Unknown but Bounded Uncertainties

许多研究方向和应用都有一个共同的目标，那就是计算动态系统未知状态的估计。卡尔曼滤波已经发展成为一个基于噪声测量来估计系统状态的成熟概念。然而，应用卡尔曼滤波的一个主要限制不是假设表征相关不确定性的高斯分布，而是更普遍地使用纯粹的概率模型。当必须处理不同类型的不确定性时，这个模型可能过于具体和过于严格。重要的例子包括由未知的系统误差、量化和线性化误差或与不可靠的通信网络有关的干扰引起的不确定性。卡尔曼滤波和大多数众所周知的估计技术需要对不确定性进行纯粹的概率建模，因此会使许多估计问题的解决变得更加困难。在前期工作中，对卡尔曼滤波算法进行了扩展，以补充依赖于集员不确定性描述的非随机模型对常见随机不确定性的描述。这种不确定性模型特别适用于未知但有界的干扰。与以概率分布为特征的随机错误不同，集员模型不需要精确定位任何特定的错误行为--除了有界性。这种组合方法，在本方案中被称为LMMSE估计器(LMMSE：线性最小-最大均方误差)，能够同时处理随机和未知但有界的不确定性，并且仍然易于实现。本研究项目的主要目的是通过集员不确定性模型将常见的纯随机方法扩展到状态估计。对LMMSE概念的进一步分析和发展分为三个方面需要解决。与第一个方面一致的是，需要指出的是LMMSE滤波器的重要特性。这些特别包括收敛性质和对其估计质量的评估。这一方面还包括有关过滤器在控制问题中的应用的重要问题。与能够同时处理不同类型不确定性的其他方法的比较在于第二个方面的重点。第三个方面致力于重要的进一步发展，以解决连续时间和非线性状态估计问题，并允许分布式计算状态估计。这一研究项目的结果有望为处理测量不确定度提供一个新的视角，并为许多估计和控制问题提供更简单的解决方案。伴随而来的应用和案例研究突出了这一新概念的好处。

Many research directions and applications share the common objective of computing an estimate for the unknown state of a dynamic system. The Kalman filter has evolved to a well-established concept for estimating the system's state on the basis of noisy measurements. However, a major limitation in applying the Kalman filter is less the assumption of Gaussian distributions that characterize the involved uncertainties but more generally the use of a purely probabilistic model. This model can be too specific and too restrictive when different types of uncertainties have to be dealt with. Important examples include uncertainties originating from unknown systematic errors, quantization and linearization errors, or disturbances related to an unreliable communication network. The Kalman filter and the majority of well-known estimation techniques require a purely probabilistic modeling of uncertainties and thus can render the solving of many estimation problems more difficult. In the course of preliminary work, the Kalman filter algorithm has been extended in order to complement the common stochastic uncertainty characterization by a non-stochastic model that relies on set-membership uncertainty descriptions. This uncertainty model is particularly tailored to unknown but bounded disturbances. Unlike random errors, which are characterized by probability distributions, a set-membership model does not require any specific error behavior to be pinpointed - except for boundedness. This combined approach, which is referred to as LM²MSE estimator (LM²MSE: Linear Min-Max Mean Squared Error) in this proposal, enables a simultaneous treatment of stochastic as well as unknown but bounded uncertainties and remains simple to implement.This research project primarily aims at extending the common purely stochastic approach to state estimation by a set-membership uncertainty model. The further analysis and development of the LM²MSE concept is subdivided into three aspects to be addressed. In line with the first aspect, important properties of the LM²MSE filter are to be pointed out. These include, in particular, convergence properties and an assessment of its estimation quality. This aspect also covers important questions concerning the filter's application to control problems. A comparison with other approaches that allow for a simultaneous treatment of different types of uncertainties lies in the focus of the second aspect. The third aspect is dedicated to important further developments in order to tackle continuous-time as well as nonlinear state estimation problems and to allow for distributed computations of state estimates. The results of this research project are expected to spur a new perspective on the treatment of measurement uncertainties and to provide simpler solutions to many estimation and control problems. Accompanying application and case studies highlight the benefits of this new concept.