Linear estimation in Krein spaces. Part II. Applications

We have shown that several interesting problems in H∞-filtering, quadratic game theory, and risk sensitive control and estimation follow as special cases of the Krein-space linear estimation theory developed in Part I. We show that all these problems can be cast into the problem of calculating the stationary point of certain second-order forms, and that by considering the appropriate state space models and error Gramians, we can use the Krein-space estimation theory to calculate the stationary points and study their properties. The approach discussed here allows for interesting generalizations, such as finite memory adaptive filtering with varying sliding patterns

Array algorithms for H-infinity estimation

In this paper we develop array algorithms for H-infinity filtering. These algorithms can be regarded as the Krein space generalizations of H-2 array algorithms, which are currently the preferred method for implementing H-2 biters, The array algorithms considered include typo main families: square-root array algorithms, which are typically numerically more stable than conventional ones, and fast array algorithms which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H-infinity filters, as these conditions are built into the algorithms themselves. However, since H-infinity square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H-2 case, further investigation is needed to determine the numerical behavior of such algorithms

Square-root arrays and Chandrasekhar recursions for H∞ problems

Using their previous observation that H∞ filtering coincides with Kalman filtering in Krein space the authors develop square-root arrays and Chandrasekhar recursions for H∞ filtering problems. The H∞ square-root algorithms involve propagating the indefinite square-root of the quantities of interest and have the property that the appropriate inertia of these quantities is preserved. For systems that are constant, or whose time-variation is structured in a certain way, the Chandrasekhar recursions allow a reduction in the computational effort per iteration from O(n^3) to O(n^2), where n is the number of states. The H∞ square-root and Chandrasekhar recursions both have the interesting feature that one does not need to explicitly check for the positivity conditions required of the H∞ filters. These conditions are built into the algorithms themselves so that an H∞ estimator of the desired level exists if, and only if, the algorithms can be executed

Receding horizon filtering for a class of discrete time-varying nonlinear systems with multiple missing measurements

This paper is concerned with the receding horizon filtering problem for a class of discrete time-varying nonlinear systems with multiple missing measurements. The phenomenon of missing measurements occurs in a random way and the missing probability is governed by a set of stochastic variables obeying the given Bernoulli distribution. By exploiting the projection theory combined with stochastic analysis techniques, a Kalman-type receding horizon filter is put forward to facilitate the online applications. Furthermore, by utilizing the conditional expectation, a novel estimation scheme of state covariance matrices is proposed to guarantee the implementation of the filtering algorithm. Finally, a simulation example is provided to illustrate the effectiveness of the established filtering scheme.This work was supported in part by the Deanship of Scientific Research (DSR) at King Abdulaziz University in Saudi Arabia [grant number 16-135-35-HiCi], the National Natural Science Foundation of China [grant number 61329301], [grant number 61203139], [grant number 61134009], and [grant number 61104125], Royal Society of the U.K., the Shanghai Rising-Star Program of China [grant number 13QA1400100], the Shu Guang project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation [grant number 13SG34], the Fundamental Research Funds for the Central Universities, DHU Distinguished Young Professor Program, and the Alexander von Humboldt Foundation of Germany

Array algorithms for H^2 and H^∞ estimation

Currently, the preferred method for implementing H^2 estimation algorithms is what is called the array form, and includes two main families: square-root array algorithms, that are typically more stable than conventional ones, and fast array algorithms, which, when the system is time-invariant, typically offer an order of magnitude reduction in the computational effort. Using our recent observation that H^∞ filtering coincides with Kalman filtering in Krein space, in this chapter we develop array algorithms for H^∞ filtering. These can be regarded as natural generalizations of their H^2 counterparts, and involve propagating the indefinite square roots of the quantities of interest. The H^∞ square-root and fast array algorithms both have the interesting feature that one does not need to explicitly check for the positivity conditions required for the existence of H^∞ filters. These conditions are built into the algorithms themselves so that an H^∞ estimator of the desired level exists if, and only if, the algorithms can be executed. However, since H^∞ square-root algorithms predominantly use J-unitary transformations, rather than the unitary transformations required in the H^2 case, further investigation is needed to determine the numerical behavior of such algorithms

Damage Localization of Mechanical Structures by Subspace Identification and Krein Space Based H-infinity Estimation

This dissertation is devoted to the theoretical development and experimental laboratory verification of a new damage localization method: The state projection estimation error (SP2E). This method is based on the subspace identification of mechanical structures, Krein space based H-infinity estimation and oblique projections. To explain method SP2E, several theories are discussed and laboratory experiments have been conducted and analysed. A fundamental approach of structural dynamics is outlined first by explaining mechanical systems based on first principles. Following that, a fundamentally different approach, subspace identification, is comprehensively explained. While both theories, first principle and subspace identification based mechanical systems, may be seen as widespread methods, barely known and new techniques follow up. Therefore, the indefinite quadratic estimation theory is explained. Based on a Popov function approach, this leads to the Krein space based H-infinity theory. Subsequently, a new method for damage identification, namely SP2E, is proposed. Here, the introduction of a difference process, the analysis by its average process power and the application of oblique projections is discussed in depth. Finally, the new method is verified in laboratory experiments. Therefore, the identification of a laboratory structure at Leipzig University of Applied Sciences is elaborated. Then structural alterations are experimentally applied, which were localized by SP2E afterwards. In the end four experimental sensitivity studies are shown and discussed. For each measurement series the structural alteration was increased, which was successfully tracked by SP2E. The experimental results are plausible and in accordance with the developed theories. By repeating these experiments, the applicability of SP2E for damage localization is experimentally proven

Design of H-infinity Extended Recursive Wiener Estimators in Discrete-Time Stochastic Systems

This paper designs (1) the H-infinity RLS Wiener fixed-point smoother and filter for the observation equation with the linear modulation and (2) the extended H-infinity recursive Wiener fixed-point smoother and filter in discrete-time wide-sense stationary stochastic systems. ln the extended estimatars, it is assumed that the signal is observed with the nonlinear modulation and with additional white observation noise. In the estimators, the system matrix Φ for the state vector x(k), the observation vector C for the state vector, the variance K(k,k) = K(0) of the state vector, the nonlinear observation function and the variance of the white observation noise are used. Φ, C and K(0) axr calculated from the auto covariance data of the signal. A simulation example, on the estimation of a speech signal in the phase demodulation problem, is demonstrated to show the estimation characteristics of the proposed extended H-infinity recursive Wiener estimatoxs