# zbMATH — the first resource for mathematics

Nonlinear control systems. 3rd ed. (English) Zbl 0878.93001
Communications and Control Engineering Series. Berlin: Springer. xvi, 549 p. (1995).
The purpose of this book is to give a self-contained description of the fundamentals of the theory of nonlinear multivariable control systems on manifolds governed by an ODE $$\dot{x}=f(x)+g(x)u$$ with output $$y=h(x)$$. An aspect of major interest is the interaction between input-state and state-output. The focus is placed on feedback design methods for single-input single-output (SISO) as well as multi-input multi-output (MIMO) systems. The book is intended as a graduate text. It contains many illustrating remarks and examples. As a prerequisite the reader should be familiar with the basics of the linear theory. As a wide range of feedback design problems is covered as well as recent developments – in particular in the last three chapters, – it also serves as a reference handbook for scientists and engineers.
The first three chapters deal with the fundamental theory of nonlinear systems. In Chapter 1 the problem of decomposing a nonlinear system locally into reachable/non-reachable and observable/non-observable parts is analyzed. The analogy between the foliations locally obtained in the nonlinear setting and the linear results is stressed in order to keep the general ideas also accessible to a reader who is less familiar with differential geometry. Chapter 2 outlines to what extent global decompositions can be obtained and Chapter 3 deals with the input-output behavior and realization theory. As the author himself suggests, Chapter 2 and 3 might be skipped at first reading. In fact they do not differ very much from the first edition [Zbl 0569.93034].
The remaining chapters 4-9 which make up about two thirds of the book are an introduction and reference to nonlinear feedback design problems. Therefore a summary strongly resembles a table of contents.
Chapter 4 introduces and deals with standard feedback design problems for SISO systems: the problem of transforming a nonlinear system into a linear one by means of static state feedback and coordinate transformations (Static State Exact Linearization) is treated, the Output Zeroing Problem and the important role of the zero dynamics is explained. Further subjects are: Output Tracking, Asymptotic Output Tracking, Local Asymptotic Stabilization of an equilibrium, Disturbance Decoupling, and Observers with Linear Error Dynamics. In Chapter 5 MIMO feedback design problems are treated. A new aspect is the problem of Non-Interacting Control (NIC): each input affects precisely one output. This problem is solvable for a special class of systems. It is shown that general systems can be transformed into a system belonging to that class through the method of Dynamic Extension. Further topics are Static State Exact Linearization, Dynamic Linearization and Exact Linearization of Input Output Response.
Chapter 6 collects the tools necessary to treat the (NIC) problem with additional stability requirements via static state feedback as well as dynamic extension which is solved in Chapter 7.
The last two chapters are new (in comparison to the second edition). Chapter 8 deals with output tracking and regulation problems by means of full information feedback and by error feedback. Finally, in Chapter 9, which covers recent developments, several global feedback design problems for SISO systems are discussed, namely global and semi-global stabilization and global disturbance attenuation.
The book is completed by two appendices, Appendix A summarizing results from differential geometry and Appendix B containing results on center manifolds needed in Chapter 8.

##### MSC:
 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 93B52 Feedback control 93C10 Nonlinear systems in control theory 93B29 Differential-geometric methods in systems theory (MSC2000)