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Sliding modes in control and optimization. Transl. from the Russian. (English) Zbl 0748.93044
Communications and Control Engineering Series. Berlin etc.: Springer- Verlag. xvi, 286 p. with 24 fig. (1992).
The book addresses the behavior of discontinuous dynamic systems described by the equation \(\dot x=f(x,t)\), where \(x\) is a state vector in \({\mathfrak R}^ n\), \(t\) is time, and \(f(x,t)\) has discontinuities at a certain set within the \((n+1)\)-dimensional space \((x,t)\). The motion of such systems on discontinuity surfaces, called sliding mode, has properties useful for system linearization, reduction of the system differential equation order, and designing high-accuracy follow-up and stabilization systems. The book considers, from a control-theoretic viewpoint, the mathematical and application aspects of the theory of discontinuous dynamic systems and determine their place within the scope of the present-day control theory. The book follows a regularization approach to the sliding modes analysis through the introduction of a boundary layer. The book consists of 3 parts. Part 1 of 5 chapters is on mathematical tools and covers a wide range of topics on the theory of sliding modes. The topics include the regularization and the uniqueness problems, stability and robustness of discontinuous systems. Part 2 of 10 chapters, the major focus in the book, addresses control systems design methods. The topics covered in that part include decoupling in systems with discontinuous control, control of distributed-parameter plants, eigenvalue allocation, system optimization, and observation and filtering. Part 2 not only presents important results on design of discontinuous control systems, but also relates these results to the present-day control theory. Part 3 of 3 chapters is devoted to applications and provides numerous practical examples, such as the control of a robot arm and the control of electric motors. The book is theoretical and formal, and can be invaluable to researchers in control theory, physics and applied mathematics.

93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C41 Control/observation systems with incomplete information
93D09 Robust stability