Bayesian Real-Time System Identification

This chapter introduces the basis and fundamental concepts about system identification. First, dynamical systems and their basic properties are introduced. System identification is the problem of building mathematical models describing the behavior of a dynamical system based on the observations from the system. Traditionally, there are two levels of system identification problems, i.e., estimation of the uncertain parameters governed in the mathematical model and selection of an appropriate model class representing the underlying dynamical system. In addition, one of the special features of this book is that it will introduce the new third level of system identification using self-calibratable model classes. Real-time system identification is desirable for promptly acquiring information from the system for monitoring and control purposes. The Bayes’ theorem and system identification using Bayesian methods are briefly introduced. Finally, an overview of this book is given with outline of each chapter for the convenience of readers. This book introduces some recent developments in Bayesian real-time system identification. It contains two different perspectives on data processing for system identification, namely centralized and distributed.

This chapter presents the standard Kalman filter (KF) and the extended Kalman filter (EKF). The detailed derivations of the discrete-time KF and EKF are presented from a Bayesian perspective. In order to formulate the KF algorithm, the state space model of a linear dynamical system is introduced. By using the Bayes’ theorem, the conditional probability density function for prediction can be obtained in a recursive manner and the analytical solutions can be obtained. Applications for a vehicle tracking system and a structural state estimation problem are provided to demonstrate the state tracking capability of the KF algorithm. Afterwards, the nonlinear state space model for general nonlinear dynamical systems is introduced and it will be utilized in the EKF. By expanding the nonlinear equations using the Taylor series expansion, the locally linearized state space model can be obtained and the procedures of the EKF algorithm are formulated in the same manner as the standard KF algorithm. The EKF with fading memory is introduced to enhance the tracking capability for time-varying systems. Applications to simultaneous states and model parameters estimation are presented. The KF algorithm and its variants provide an effective and efficient way for recursive estimation. More importantly, the EKF is the fundamental algorithm in this book.

This chapter presents the algorithm for real-time updating of the noise covariance matrices in the extended Kalman filter. This content is motivated from practical applications, in which the noise statistics of the Kalman filter or its variants are usually not known a priori. To address this issue, a Bayesian probabilistic algorithm is developed to estimate the noise parameters which are utilized to parameterize the noise covariance matrices in the extended Kalman filter. A computationally efficient algorithm is then introduced to resolve the optimization problem formulated by using the Bayesian inference. The proposed method not only estimates the optimal noise parameters but also quantifies the associated estimation uncertainty in a real-time manner. This method does not impose any stationarity condition of the process noise and measurement noise. By removing the stationarity constraint in the extended Kalman filter, the proposed method enhances the applicability of the real-time system identification algorithm for nonstationary circumstances generally encountered in practice. Examples using stationary/nonstationary response of linear/nonlinear time-varying dynamical systems are presented to illustrate the practical aspects in real-time system identification.

This chapter introduces an algorithm for detecting anomalous data in the measurements from time-varying systems. The probability of outlier of a data point is defined and derived and this algorithm utilizes it to evaluate the outlierness of each data point. The probability of outlier integrates the normalized residual, the measurement noise level and the size of the dataset, and provides a systematic and objective criterion to effectively screen the possibly abnormal data points in the observations. Instead of using other adhoc judgement on selecting outliers, the proposed method provides an intuitive threshold 0.5 for outlier detection. Computationally efficient techniques are introduced to alleviate the heavy burden encountered in the identification using long-term monitoring data. The proposed outlier detection algorithm is embedded into the extended Kalman filter. Therefore, it can remove the outliers in the measurements and identify the time-varying systems simultaneously. By excluding the outliers in the measurements, the proposed algorithm ensures the stability and reliability of the estimation. Examples are presented to illustrate the practical aspects of detecting outliers in the measurements and identifying time-varying systems in a real-time manner. The algorithm presented in this chapter is suitable for centralized identification while the algorithm presented in Chap. 7 is suitable for distributed identification.

This chapter introduces the Bayesian model class selection for real-time system identification and an adaptive reconfiguring strategy for the model classes. In addition to parametric estimation, another critical problem in system identification is to determine a suitable model class for describing the underlying dynamical system. By utilizing the Bayes’ theorem to obtain the plausibilities of a set of model classes, model class selection can be performed accordingly. The proposed method provides simultaneous model class selection and parametric identification in a real-time manner. On the other hand, although Bayesian model class selection allows for determination of the most suitable model class among a set of prescribed model class candidates, it does not guarantee a good model class to be selected. It is possible that all the prescribed model class candidates are inadequate. Thus, a new third level of system identification is presented to resolve this problem by using self-calibratable model classes. This self-calibrating strategy can correct the deficiencies of the model classes and achieve reliable real-time identification results for time-varying dynamical systems. On the other hand, the large number of prescribed model class candidates will hamper the performance of real-time system identification. In order to resolve this problem, a hierarchical strategy is proposed. It only requires a small number of model classes but a large solution space can be explored. Although the algorithms presented in this chapter are based on the EKF, the real-time model class selection component and the adaptive reconfiguring strategy for model class selection can be easily embedded into other filtering tools.

This chapter introduces an online dual-rate distributed identification framework for wireless sensor networks. Distributed identification is a concept that allows an individual unit to obtain local estimation using part of the data, and the obtained local estimation can then be used as a basis for global estimation. In this chapter, typical architectures of wireless sensor networks will first be introduced, including centralized, decentralized and distributed networks. Then, the online dual-rate distributed identification approach is introduced for wireless sensor networks. Filtering method using only raw observations collected at each sensor is introduced. The preliminary local identification results are then compressed before transmitting to the central station for fusion. At the central station, Bayesian fusion is developed to integrate the compressed local identification results transmitted from the sensor nodes in order to obtain reliable global estimation. As a result, the large identification uncertainty in the local identification results can be substantially reduced. In addition to data compression, a dual-rate strategy for sampling and transmission/fusion is used to alleviate the data transmission burden so that online model updating can be realized efficiently for wireless sensor networks. The computational framework in this chapter will be followed in the next chapter, where specific algorithms for handling asynchronous data and multiple outlier-corrupted data will be introduced.

In this chapter, two novel methods are presented to handle two typical challenging problems in system identification, including asynchronous measurements and multiple outlier-corrupted measurements. These two methods are built based on the online dual-rate distributed identification framework elaborated in Chap. 6. First, due to unavoidable imperfection of data acquisition systems, the measurements among different channels are generally asynchronous. Furthermore, since the sensing systems for civil engineering structures are usually exposed to severe service environment, the measurements are inevitable to be corrupted with outliers. Effective methods in an online manner are proposed to settle these challenging issues. Regarding the first issue of asynchronous data, since each sensor node uses only the data of its own in the proposed distributed identification method, the local model identification results are not affected by asynchronism of different sensor nodes. The proposed approach utilizes directly asynchronous data for online system identification. Regarding the second issue of outlier contamination, a hierarchical outlier detection approach is introduced. It detects the local outliers according to the outlier probability of the raw observations at the sensor nodes while it detects the global outliers according to the outlier probability of the local estimation results. The proposed methods can resolve the challenging problems of asynchronous measurements and multiple outlier-corrupted measurements effectively and achieve reliable identification results for time-varying dynamical systems in an online manner.