Systems & Control: Foundations & Applications – serie

Congestion control algorithms were implemented for the Internet nearly two decades ago, but mathematical models of congestion control in such a large-scale network are relatively new. This text presents models for the development of new protocols that can help make Internet data transfers virtually loss- and delay-free. Introduced are tools from optimization, control theory, and stochastic processes integral to the study of congestion control algorithms.Intended for graduate students and researchers in systems theory and computer science, the text assumes basic knowledge of first-year, graduate-level control theory, optimization, and stochastic processes, but the key prerequisites are summarized in an appendix for quick reference. The work's wide range of applications to the study of both new and existing protocols and control algorithms make the book of interest to researchers and students concerned with many aspects of large-scale information flow on the Internet.

This self-contained monograph describes basic set-theoretic methods for control and provides a discussion of their links to fundamental problems in Lyapunov stability analysis and stabilization, optimal control, control under constraints, persistent disturbance rejection, and uncertain systems analysis and synthesis. New computer technology has catalyzed a resurgence of research in this area, particularly in the development of set-theoretic techniques, many of which are computationally demanding. The work presents several established and potentially new applications, along with numerical examples and case studies. A key theme is the trade-off between exact (but computationally intensive) and approximate (but conservative) solutions to problems. Mathematical language is kept to the minimum necessary for the adequate formulation and statement of main concepts. Efficient numerical algorithms, both in MATLAB and C++, will be available via the Internet to supplement the presentation and solution of problems in the book. "Set-Theoretic Methods in Control" is accessible to readers familiar with the basics of systems and control theory; prerequisites such as convexity theory are included.The text provides a solid foundation of mathematical techniques and applications and also features avenues for further theoretical study. While it is aimed primarily at graduate students and researchers in applied mathematics and engineering, the book will also appeal to practitioners, since it contains extensive references to the literature and supplies many recipes for solving real-world control problems.

The book deals with several closely related topics concerning approxima­ tions and perturbations of random processes and their applications to some important and fascinating classes of problems in the analysis and design of stochastic control systems and nonlinear filters. The basic mathematical methods which are used and developed are those of the theory of weak con­ vergence. The techniques are quite powerful for getting weak convergence or functional limit theorems for broad classes of problems and many of the techniques are new. The original need for some of the techniques which are developed here arose in connection with our study of the particular applica­ tions in this book, and related problems of approximation in control theory, but it will be clear that they have numerous applications elsewhere in weak convergence and process approximation theory. The book is a continuation of the author's long term interest in problems of the approximation of stochastic processes and its applications to problems arising in control and communication theory and related areas. In fact, the techniques used here can be fruitfully applied to many other areas. The basic random processes of interest can be described by solutions to either (multiple time scale) Ito differential equations driven by wide band or state dependent wide band noise or which are singularly perturbed. They might be controlled or not, and their state values might be fully observable or not (e. g. , as in the nonlinear filtering problem).

Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations.

The central focus of this work is the control of nonlinear systems. Using new techniques that employ the max-plus algebra, the author addresses several classes of nonlinear control problems, including nonlinear optimal control problems, nonlinear robust H-infinity control problems, and nonlinear risk-sensitive stochastic control problems.Well-known dynamic programming arguments show there is a direct relationship between the solution of a control problem, and the solution of a corresponding Hamilton--Jacobi--Bellman (HJB) partial differential equation (PDE). The max-plus-based methods examined in this monograph belong to an entirely new class of methods for the solution of nonlinear control problems and their associated HJB PDEs; they are not equivalent to either of the more commonly used finite element or characteristic approaches. The potential advantages of max-plus-based approaches lie in the fact that solution operators for nonlinear HJB problems are linear over the max-plus algebra, and this linearity is exploited in the construction of algorithms.The book will be of interest to applied mathematicians, engineers, and graduate students interested in the control of nonlinear systems through the implementation of recently developed numerical methods.Researchers and practitioners tangentially interested in this area will also find a readable, concise discussion of the subject through a careful selection of specific chapters and sections. Basic knowledge of control theory for systems with dynamics governed by differential equations is required; the material on risk-sensitive stochastic control requires knowledge of systems governed by stochastic differential equations.

Identifying the input-output relationship of a system or discovering the evolutionary law of a signal on the basis of observation data, and applying the constructed mathematical model to predicting, controlling or extracting other useful information constitute a problem that has been drawing a lot of attention from engineering and gaining more and more importance in econo­ metrics, biology, environmental science and other related areas. Over the last 30-odd years, research on this problem has rapidly developed in various areas under different terms, such as time series analysis, signal processing and system identification. Since the randomness almost always exists in real systems and in observation data, and since the random process is sometimes used to model the uncertainty in systems, it is reasonable to consider the object as a stochastic system. In some applications identification can be carried out off line, but in other cases this is impossible, for example, when the structure or the parameter of the system depends on the sample, or when the system is time-varying. In these cases we have to identify the system on line and to adjust the control in accordance with the model which is supposed to be approaching the true system during the process of identification. This is why there has been an increasing interest in identification and adaptive control for stochastic systems from both theorists and practitioners.

This volume provides an integrated development of the mathematical connections between nonlinear control, dynamical systems and time-varying perturbed systems for scientists and engineers. It includes examples and definitions and is designed for use as a self-study or reference guide for all scientists and engineers.

This text discusses mathematical modelling, analysis and control of the immune system and disease dynamics. The purpose of the book is the practical application of mathematics to immunology and medicine in order to establish a basis for more effective treatment, to provide a tutorial systematic description of how the immune system controls diseases and to present several significant examples such as malignant tumour dynamics and control, and viral hepatitis. The book is multidisciplinary in content, with the intended readers including biomathematicians, biologists and physicists. It combines immunological principles, mathematical models, computer simulations and methods of analysis.

This text gives an account of an ellipsoidal calculus and ellipsoidal techniques that allows presentation of the set-valued solutions to these problems in terms of approximating ellipsoidal-valued functions. Such an approach leads to effective computation schemes, an dopens the way to applications and implementations with computer animation, particularly in decision support systems. The problems treated here are those that involve calculation of attainability domains, of control synthesis under bounded controls, state constraints and unknown input disturbances, as well as those of "viability" and of the "bounding approach" to state estimation. The text ranges from a specially developed theory of exact set-valued solutions to the description of ellipsoidal calculus, related ellipsoidal-based methods and examples worked out with computer graphics. the calculus given here may also be interpreted as a generalized technique of the "interval analysis" type with an impact on scientific computation.

A typical subsystem found in almost all aircraft and space vehicles consists of beam, plate and/or shell elements attached to each other in a rigid or flexible manner. Due to limitations on their weights, the elements themselves must be highly flexible, and due to limitations on their initial configuration (i.e., before deployment), those aggregates often have to contain several links so that the substructure may be unfolded or telescoped once it is deployed. The defining philosophy of this monograph is that in order to understand completely the dynamic response of such a complex elastic structure, it is not sufficient to consider only its global motion but also necessary to take into account the flexibility of individual elements and the interaction and transmission of elastic effects such as bending, torsion, and axial deformations at junctions where members are connected to each other.Therefore, the purposes of this monograph are: to derive distributed parameter models of the transient behavior of some or all of the state and interval variables which describe the dynamic response of multiple-link flexible structures such as trusses, frames, robot arms, solar panels, antennae and deformable mirrors, based on the principles of continuum mechanics and under reasonable constraints on the geometry of the admissible deformations; to provide rigorous mathematical analyses of the resulting models; and to develop control theoretic properties of multiple-link flexible structures based on the control theoretic properties of the models. The modelling and analysis of these complicated and realistic structural configurations should be of interest to a diverse group of applied mathematicians, structural, aeronautical, aerospace, and mechanical engineers and to advanced graduate students working on such problems.

Infinite dimensional systems can be used to describe many physical phenomena in the real world. Well-known examples are heat conduction, vibration of elastic material, diffusion-reaction processes, population systems and others. Thus, the optimal control theory for infinite dimensional systems has a wide range of applications in engineering, economics and some other fields. On the other hand, this theory has its own mathematical interests since it is regarded as a generalization for the classical calculus of variations and it generates many interesting mathematical questions. The Pontryagin maximum principle, the Bellman dynamic programming method and the Kalman optimal linear quadratic regulator theory are regarded as the three milestones of modern (finite dimensional) control theory. Since the 1960s, the corresponding theory for infinite dimensional systems has also been developed. The essential difficulties for the infinite dimensional theory come from two aspects: the unboundedness of the differential operator or the generator of the strongly continuous semigroup and the lack of the local compactness of the underlying spaces.The purpose of this book is to introduce optimal control theory for infinite dimensional systems. The authors present the existence theory for optimal control problems. Some applications are also included in this volume.

Most manufacturing systems are large, complex, and subject to uncertainty. Obtaining exact feedback policies to run these systems is nearly impossible, both theoretically and computationally. It is a common practice, therefore, to manage such systems in a hierarchical fashion. This book articulates a theory that shows that hierarchical decision making in the context of a goal-seeking manufacturing system can lead to near optimization of its objective. The approach in this book considers manufacturing systems in which events occur at different time scales. For example, changes in demand may occur far more slowly than breakdowns and repairs of production machines. This suggests that capital expansion decisions that respond to demand are relatively longer term than those decisions regarding production. Thus, long-term decisions such as those dealing with capital expansion can be based on the average existing production rapacity, and can be expected to be nearly optimal even though the short-term capacity fluctuations are ignored. Having the long-term decisions in hand, one can then solve the simpler problem of obtaining production rates.Increasingly complex and realistic models of manufacturing systems with failure-prone machines facing uncertain demands are formulated as stochastic optimal control problems. Partial characterization of their solutions is provided when possible along with their hierarchical decomposition based on event frequencies. In the latter case, multilevel decisions are constructed in the manner described above and these decisions are shown to be asymptotically optimal as the average time between successive short-term events becomes much smaller than that between successive long-term events. Much attention is given to establish that the order of deviation of the cost of the hierarchical solution from the optimal cost is small. The striking novelty of the approach is that this is done without solving for the optimal solution, which as stated earlier is an insurmountable task. This approach presents a paradigm in convex production planning, whose roots go back to the classical work of Arrow, Karlin and Scarf (1958). It also represents a new research direction in control theory.Finally, the material covered in the book cuts across the disciplines of Operations Management, Operations Research, System and Control theory, Industrial Engineering, Probability and Statistics, and Applied Mathematics. It is anticipated that the book would encourage development of new models and techniques in these disciplines.

Hamilton-Jacobi equations and other types of partial differential equa- tions of the first order are dealt with in many branches of mathematics, mechanics, and physics. These equations are usually nonlinear, and func- tions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. An example of such a situation can be provided by the value function of a differential game or an optimal control problem. It is known that at the points of differentiability this function satisfies the corresponding Hamilton-Jacobi-Isaacs-Bellman equation. On the other hand, it is well known that the value function is as a rule not everywhere differentiable and therefore is not a classical global solution. Thus in this case, as in many others where first-order PDE's are used, there arises necessity to introduce a notion of generalized solution and to develop theory and methods for constructing these solutions.In the 50s-70s, problems that involve nonsmooth solutions of first- order PDE's were considered by Bakhvalov, Evans, Fleming, Gel'fand, Godunov, Hopf, Kuznetzov, Ladyzhenskaya, Lax, Oleinik, Rozhdestven- ski1, Samarskii, Tikhonov, and other mathematicians. Among the inves- tigations of this period we should mention the results of S.N. Kruzhkov, which were obtained for Hamilton-Jacobi equation with convex Hamilto- nian. A review of the investigations of this period is beyond the limits of the present book. A sufficiently complete bibliography can be found in [58, 126, 128, 141].

Over the past decade a new approach has been introduced to the theory of systems representations. Introduced by Jan C. Willems, it is called the "behavioural" approach. One of its main features is that it is well suited for modelling interconnections of systems. In this book, the author develops representation theory from a behavioural point of view and focuses on various types of ("generalized state space") first-order representations that commonly arise in the process of modelling. It covers minimality, derives transformation groups and offers realization methods that lead directly to minimal realizations. The book further presents generalized notions of controllability indices and observability indices and gives methods to calculate these indices. The book includes a procedure to derive a standard state space description from a general first-order representation. It fortifies the reader's understanding with basic examples from electrical networks and mechanics.Mathematicians and control engineers doing research on systems that are linear, time-invariant, deterministic, and finite dimensional should find this book a firm basis for understanding both the theory and applications of this behavioural approach.

H-infinity engineering continues to establish itself as a discipline of applied mathematics. As such, this extensively illustrated monograph makes a significant application of H-infinity theory to electronic amplifier design, demonstrating how recent developments in H-infinity engineering equip amplifier designers with new tools and avenues for research.The amplification of a weak, noisy, wideband signal is a canonical problem in electrical engineering. Given an amplifier, matching circuits must be designed to maximize gain, minimize noise, and guarantee stability. These competing design objectives constitute a multiobjective optimization problem. Because the matching circuits are H-infinity functions, amplifier design is really a problem in H-infinity multiobjective optimization.

This book presents a generalized state-space theory for the analysis and synthesis of finite horizon suboptimal H...controllers. Expressions for a suboptimal controller are derived in a general setting and an approximate solution to the H...performance robustness problem is proposed. Chapter 1 treats nonlinear optimal control problems in which the cost functional is of the form of a quotient or a product of powers of definite integrals. Chapter 2 gives a criterion for the evaluation of the infimal H...norm in the finite horizon case. A general suboptimal control problem is then posed, and an expression for a suboptimal H...state feedback controller is derived. Chapter 3 develops expressions for a suboptimal H...output feedback controller in a very general case via the solution of two dynamical Riccati equations. Assuming the adequacy of linear expressions, chapter 4 gives an iterative procedure for the synthesis of a suboptimal H...controller which yields the required performance even under parameter variations. Chapter 5 treats a general minimization problem in which the cost functional is a quotient of definite integrals.The results are useful in computing the actual performance of a given controller. In chapter 6, a design of the F/A-18A automatic carrier landing system is accomplished using finite horizon H...techniques. The book is based mainly on the author's research papers and presents a very practical point of view toward the applications. It can be used as a one semester or one quarter graduate level course in electrical, mechanical or chemical engineering.

Introducing readers to adaptive systems in a rigorous but elementary fashion, this text emphasizes the mainstream developments in adaptive control and signal processing of linear discrete time systems. A unified framework is developed whereby the reader can analyze and understand any adaptive system in the literature. The so-called equilibrium analysis facilitates an understanding of the limitations and potential of adaptive systems in a transparent fashion; while the behavioural approach to linear systems plays an essential role at some key points in the text. So-called universal controllers are presented in some detail. Each chapter is accompanied by exercises that aim to develop certain aspects of the theory, as well as to give the reader a better understanding of the actual behaviour of adaptive systems.

The analysis, processing, evolution, optimization and/or regulation, and control of shapes and images appear naturally in engineering (shape optimization, image processing, visual control), numerical analysis (interval analysis), physics (front propagation), biological morphogenesis, population dynamics (migrations), and dynamic economic theory.   These problems are currently studied with tools forged out of differential geometry and functional analysis, thus requiring shapes and images to be smooth.  However, shapes and images are basically sets, most often not smooth.  J.-P. Aubin thus constructs another vision, where shapes and images are just any compact set.  Hence their evolution -- which requires a kind of differential calculus -- must be studied in the metric space of compact subsets.  Despite the loss of linearity, one can transfer most of the basic results of differential calculus and differential equations in vector spaces to mutational calculus and mutational equations in any mutational space, including naturally the space of nonempty compact subsets.   "Mutational and Morphological Analysis" offers a structure that embraces and integrates the various approaches, including shape optimization and mathematical morphology.   Scientists and graduate students will find here other powerful mathematical tools for studying problems dealing with shapes and images arising in so many fields.

One of the main problems in control theory is the stabilization problem consisting of finding a feedback control law ensuring stability; when the linear approximation is considered, the nat­ ural problem is stabilization of a linear system by linear state feedback or by using a linear dynamic controller. This prob­ lem was intensively studied during the last decades and many important results have been obtained. The present monograph is based mainly on results obtained by the authors. It focuses on stabilization of systems with slow and fast motions, on stabilization procedures that use only poor information about the system (high-gain stabilization and adaptive stabilization), and also on discrete time implementa­ tion of the stabilizing procedures. These topics are important in many applications of stabilization theory. We hope that this monograph may illustrate the way in which mathematical theories do influence advanced technol­ ogy. This book is not intended to be a text book nor a guide for control-designers. In engineering practice, control-design is a very complex task in which stability is only one of the re­ quirements and many aspects and facets of the problem have to be taken into consideration. Even if we restrict ourselves to stabilization, the book does not provide just recipes, but it fo­ cuses more on the ideas lying behind the recipes. In short, this is not a book on control, but on some mathematics of control.

One of the main goals of optimal control theory is to provide a theoretical basis for choosing an appropriate controller for whatever system is under consideration by the researcher or engineer. Two popular norms that have proved useful are known as H-2 and H - infinity control. The first has been particularly applicable to problems arising in the aerospace industry. However, most industrial problems are badly modeled and the second norm proved to be more appropriate when the actual conditions of the problem did not conform to the stipulated conditions of the theory. This book takes the topic of H-infinity control as a point of departure and pursues an improved controller design which has been suggested in the mainstream of robust control. Its main theme, minimum entropy control, provides a means of trading off some of the features of other control problems. The book is aimed at research workers in networking systems as well as those in operator theory and linear multivariable control. The use of stochastic methods makes the book also of importance to the circuits and systems community. CONTENTS: Preface • Introduction • Preliminaries • Induced Operator Norms • Discrete-Time Entropy • Connections With Related Optimal Control Problems • Minimum Entropy Control • Continuous-Time Entropy • A. Proof of Theorem • B. Proof of Theorem • Bibliography • Notation • Index

The problem of controlling the output of a system so as to achieve asymptotic tracking of prescribed trajectories and/or asymptotic re­ jection of undesired disturbances is a central problem in control the­ ory. A classical setup in which the problem was posed and success­ fully addressed - in the context of linear, time-invariant and finite dimensional systems - is the one in which the exogenous inputs, namely commands and disturbances, may range over the set of all possible trajectories ofa given autonomous linear system, commonly known as the exogeneous system or, more the exosystem. The case when the exogeneous system is a harmonic oscillator is, of course, classical. Even in this special case, the difference between state and error measurement feedback in the problem ofoutput reg­ ulation is profound. To know the initial condition of the exosystem is to know the amplitude and phase of the corresponding sinusoid. On the other hand, to solve the output regulation problem in this case with only error measurement feedback is to track, or attenu­ ate, a sinusoid ofknown frequency but with unknown amplitude and phase. This is in sharp contrast with alternative approaches, such as exact output tracking, where in lieu of the assumption that a signal is within a class of signals generated by an exogenous system, one instead assumes complete knowledge of the past, present and future time history of the trajectory to be tracked.

The engineering objective of high performance control using the tools of optimal control theory, robust control theory, and adaptive control theory is more achiev­ able now than ever before, and the need has never been greater. Of course, when we use the term high peiformance control we are thinking of achieving this in the real world with all its complexity, uncertainty and variability. Since we do not expect to always achieve our desires, a more complete title for this book could be "Towards High Performance Control". To illustrate our task, consider as an example a disk drive tracking system for a portable computer. The better the controller performance in the presence of eccen­ tricity uncertainties and external disturbances, such as vibrations when operated in a moving vehicle, the more tracks can be used on the disk and the more memory it has. Many systems today are control system limited and the quest is for high performance in the real world.

In view of Professor Wendell Fleming's many fundamental contributions, his profound influence on the mathematical and systems theory communi­ ties, his service to the profession, and his dedication to mathematics, we have invited a number of leading experts in the fields of control, optimiza­ tion, and stochastic systems to contribute to this volume in his honor on the occasion of his 70th birthday. These papers focus on various aspects of stochastic analysis, control theory and optimization, and applications. They include authoritative expositions and surveys as well as research papers on recent and important issues. The papers are grouped according to the following four major themes: (1) large deviations, risk sensitive and Hoc control, (2) partial differential equations and viscosity solutions, (3) stochastic control, filtering and parameter esti­ mation, and (4) mathematical finance and other applications. We express our deep gratitude to all of the authors for their invaluable contributions, and to the referees for their careful and timely reviews. We thank Harold Kushner for having graciously agreed to undertake the task of writing the foreword. Particular thanks go to H. Thomas Banks for his help, advice and suggestions during the entire preparation process, as well as for the generous support of the Center for Research in Scientific Computation. The assistance from the Birkhauser professional staff is also greatly appreciated.

"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is quite satisfactory and natural for scalar systems, the study of multi-input, multi-output linear time­ invariant control systems requires projective algebraic geometry. Thus, this second volume deals with multi-variable linear systems and pro­ jective algebraic geometry. The results are deeper and less transparent, but are also quite essential to an understanding of linear control theory. A review of * From the Preface to Part 1. viii Preface the scalar theory is included along with a brief summary of affine algebraic geometry (Appendix E).

This volume presents state-of-the-art methods for the synthesis of nonlinear and robust feedback controllers for transport-reaction processes. It should be a useful resource for chemical engineers, control engineers and mechanical engineers. The interest in control of nonlinear partial differential equation (PDE) systems has been triggered by the need to achieve tight distributed control of transport-reaction processes which exhibit strong spatial variations. Drawing from advances in dynamics of PDE systems and nonlinear control theory, control of nonlinear PDEs has evolved into a very active research area of systems and control. This volume presents general methods for the synthesis of nonlinear and robust feedback controllers for broad classes of nonlinear PDE systems and illustrates their applications to transport-reaction processes of industrial interest. The controllers use a finite number of measurement sensors and control actuators to achieve stabilization of the closed-loop system, output tracking and attenuation of the effect of model uncertainty.Beginning with an introduction to control of PDE sy stems, the book discusses nonlinear and robust control of hyperbolic and parabolic PDEs with fixed spatial domains and parabolic PDEs with time-dependent spatial domains. The synthesis of the controllers is performed by using geometric and Lyapunov-based control techniques. The book includes comparisons of the methods followed for controller synthesis with other approaches and discussions of practical implementation issues that can help researchers and engineers understand the development and application of the methods in greater depth.

This work focuses on the contemporary developments and results in mathematical systems theory and control. The book is written in honour of Diederich Hinrichsen, for his fundamental contributions and achievements in the fields of linear systems theory and control theory, and for his long-term achievements in estabilshing mathematical systems theory in Germany. A key theme of the book is the stability and robustness of linear and nonlinear systems using the concepts of stability radii and spectral value sets. Chapters survey advances in linear and nonlinear systems theory, including parametrizations, problems and behaviours of linear systems, convolutional codes, and complementary and hybrid systems. In addition, the volume examines controllability and stabilization of infinite dimensional systems (allowing for hysteresis nonlinearities) with functional analytic and algebraic approaches.

"This volume is a textbook on linear control systems with an emphasis on stochastic optimal control with solution methods using spectral factorization in line with the original approach of N. Wiener. Continuous-time and discrete-time versions are presented in parallel!. Two appendices introduce functional analytic concepts and probability theory, and there are 77 references and an index. The chapters (except for the last two) end with problems!. [T]he book presents in a clear way important concepts of control theory and can be used for teaching." --Zentralblatt Math "This is a textbook intended for use in courses on linear control and filtering and estimation on (advanced) levels. Its major purpose is an introduction to both deterministic and stochastic control and estimation. Topics are treated in both continuous time and discrete time versions!. Each chapter involves problems and exercises, and the book is supplemented by appendices, where fundamentals on Hilbert and Banach spaces, operator theory, and measure theoretic probability may be found.The book will be very useful for students, but also for a variety of specialists interested in deterministic and stochastic control and filtering." --Applications of Mathematics "The strength of the book under review lies in the choice of specialized topics it contains, which may not be found in this form elsewhere. Also, the first half would make a good standard course in linear control." --Journal of the Indian Institute of Science

l\lany systems encountered in practice involve a coupling between contin­ uous dynamics and discrete events. Researchers with a background and interest in continuous-time systems and control theory are concerned primarily with properties of the contin­ uous dynamics, such as Lyapunov stability.

As soon as we (AS and PS) completed writing the book on H Optimal Con- 2 trol, another task of equal magnitude was laid to our charge. This task was to work on ?ltering and related topics. This book releases us from this charge. In this endeavor, we are fortunate to have found a capable person in our friend and colleague (AAS) who helped us release our burden. The subject of ?ltering is indeed vast and immense, much more so than the subject of H Optimal Control. In this work, we have tried to present what we 2 believe to be the fundamental issues of ?ltering. The book is not intended to give a chronological development of ?ltering from a historical point of view. A vast number of books already do so. Our intent here is to develop from our perspective the complete theory of ?ltering and various design methodologies associated with it along with their practical implementations. In this respect, we present here a state-of-the-art view of exact and almost input-decoupled ?ltering, H ,and H 2 1 ?ltering and inverse ?ltering issues, and include an application of ?ltering and inverse ?ltering to fault detection, isolation, and estimation. Most of the work reported here arose out of the research conducted by one or more of us and so- times in collaboration with our students and colleagues. Supposedly, young F.