In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Time-delay systems: an overview of some recent advances and open problems

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

This book provides an introduction to the mathematics needed to model, analyze, and design feedback systems. It is an ideal textbook for undergraduate and graduate students, and is indispensable for researchers seeking a self-contained reference on control theory. Unlike most books on the subject, Feedback Systems develops transfer functions through the exponential response of a system, and is accessible across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl strm and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. strm and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. They provide exercises at the end of every chapter, and an accompanying electronic solutions manual is available. Feedback Systems is a complete one-volume resource for students and researchers in mathematics, engineering, and the sciences.Covers the mathematics needed to model, analyze, and design feedback systems Serves as an introductory textbook for students and a self-contained resource for researchers Includes exercises at the end of every chapter Features an electronic solutions manual Offers techniques applicable across a range of disciplines

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Feedback Systems: An Introduction for Scientists and Engineers

A state observer is proposed for nonlinear continuous time systems which extends the well known Luenberger observer. In particular, on the basis of simple assumptions on the regularity of the system equations (observability and the global Holder condition for suitable functions), which are generally satisfied for physically meaningful dynamic systems, the global asymptotic convergence of the estimated state towards the true state is shown. Finally, some examples of applications are also reported showing the effectiveness of the proposed observer.

A Luenberger-like observer for nonlinear systems

The article presents a new approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a nonnegligible time delay. The proposed observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants (chain observer). Conditions are given for ensuring global exponential convergence to zero of the observation error for any given delay in the measurements. The implementation of the observer is simple and computer simulations demonstrate its effectiveness.

A new approach to state observation of nonlinear systems with delayed output

In this work we propose a new filtering approach for linear discrete time non-Gaussian systems that generalizes a previous result concerning quadratic filtering [A. De Santis, A. Germani, and M. Raimondi, IEEE Trans. Automat. Control, 40 (1995) pp. 1274--1278]. A recursive $\nu$th-order polynomial estimate of finite memory $\Delta$ is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical Kalman linear scheme. The resulting estimate will be the mean square optimal one among those estimators that take into account $\nu$-polynomials of the last $\Delta$ observations. Numerical simulations show the effectiveness of the proposed method.

Polynomial Filtering for Linear Discrete Time Non-Gaussian Systems

The problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied. For such models, a technique of state augmentation is used, leading to the definition of a general polynomial filter. The theory is developed for time-varying systems with nonstationary and non-Gaussian noises. Moreover, the steady-state polynomial filter for stationary systems is also studied. Numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques.

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Alfredo Germani

Papers

A Luenberger-like observer for nonlinear systems

A new approach to state observation of nonlinear systems with delayed output

Polynomial Filtering for Linear Discrete Time Non-Gaussian Systems

Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise

An observer for a class of nonlinear systems with time varying observation delay