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Peyman Setoodeh

McMaster University

Ontario, Canada

Saeid Habibi

McMaster University

Ontario, Canada

Simon Haykin

McMaster University

Ontario, Canada

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To the memory of RudolfEmil Kalman

Taking an algorithmic approach, this book provides a step towards bridging the gap between control theory, statistical signal processing, and machine learning regarding the state/parameter estimation problem. State estimation is an important concept that has profoundly influenced different branches of science and engineering. State of a system refers to a minimal record of the past history, which is required for predicting the future behavior. In this regard, a dynamic system can be described from the state perspective by selecting a set of independent variables as state variables. It is often desirable to know the state variables, and in control applications, to force them to follow desired trajectories in the state space. State estimation refers to the process of reconstructing the hidden or latent state variables, which cannot be directly measured, from system inputs and outputs in the minimum possible length of time. Filtering algorithms, which are deployed for state estimation, aim at minimizing the error between the estimated and the true values of the state variables.

The first part of the book is dedicated to classic estimation algorithms. A thorough presentation of the notion of observability, which refers to the ability to reconstruct the state variables from measurements, is followed by covering a number of observers as state estimators for deterministic systems. Regarding stochastic systems, optimal Bayesian filtering is presented that provides a conceptual solution for the general state estimation problem. Different Bayesian filtering algorithms have been developed based on computationally tractable approximations of the conceptual Bayesian solution. For the special case of linear systems with Gaussian noise, Kalman filter provides the optimal Bayesian solution. To extend the application of Kalman filter to nonlinear systems, two main approaches have been proposed to provide suboptimal solutions: using power series to approximate the nonlinear functions and approximating the probability distributions. While extended Kalman filter, extended information filter, and divided‐difference filter approximate the nonlinear functions, unscented Kalman filter, cubature Kalman filter, and particle filter approximate the probability distributions. Other Kalman filter variants include Gaussian‐sum filter, which handles non‐Gaussianity, and generalized PID filter. Among the mentioned filters, particle filter is capable of handling nonlinear and non‐Gaussian systems. Smooth variable‐structure filter, which has been derived based on a stability theorem, is able to handle model uncertainties. Moreover, it benefits from using a secondary set of performance indicators in addition to the innovation vector.