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Dynamics and control of trajectory tubes. Theory and computation. (English) Zbl 1336.93004

Systems & Control: Foundations & Applications. Cham: Birkhäuser/Springer (ISBN 978-3-319-10276-4/hbk; 978-3-319-10277-1/ebook). xviii, 445 p. (2014).
This book provides a self-contained presentation of different mathematical methods and computational schemes for solving a variety of control problems. The proposed deterministic approach allows efficient treatment of problems under state constraints and/or hard or double bounds on the controls. The concept of trajectory tubes is the key notion for describing control systems operating under uncertainty, in presence of unknown but bounded parameters and/or measurement errors. Using the Hamilton-Jacobi formalism and the dynamic programming approach, the authors investigate the control system dynamics taking into account its set-valued properties. The book is divided into 11 chapters. Chapter 1 describes the linear control theory in a form appropriate for using in the next chapters. Chapter 2 presents the dynamic programming approach. The emphasis is on the forward and backward reachability problems for “linear-convex” systems and on the design of closed-loop control strategies for optimal target and time-optimal feedback problems. Chapter 3 describes the basic properties of the ellipsoidal techniques for obtaining both external and internal ellipsoidal approximations of forward and backward reachability sets with any desired level of accuracy. Chapter 4 offers examples of problems related to the first three chapters. They are solved by ellipsoidal techniques and illustrated graphically. Chapter 5 considers nonconvex reachability sets and systems subjected to non-ellipsoidal constraints. The key idea is to approximate the corresponding Hamilton-Jacobi equation from above or below in order to obtain guaranteed upper or lower solution estimates. Chapter 6 deals with the problem of feedback impulse control. The importance of both a hard bound and an integral bound on the controls (used for approximating impulse controls) is emphasized. Chapter 7 studies problems of reachability and system dynamics under state constraints. A special attention is paid on linear systems under hard bounds on the controls. Chapter 8 presents the theory of trajectory tubes and their evolutionary dynamics. Also, the evolution of the so called “viability tubes” is considered. Chapter 9 describes briefly the problem of “guaranteed” state estimation. This problem is treated in both continuous and discrete time. Comparison with stochastic filtering is also discussed. Chapter 10 deals with problems of output feedback control based on available measurements under set-membership uncertainty. Chapter 11 considers a specific class of hybrid systems. Solutions to the reachability problem and their verification are indicated, followed by computational schemes.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93C41 Control/observation systems with incomplete information
93C15 Control/observation systems governed by ordinary differential equations
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