# zbMATH — the first resource for mathematics

Nonlinear output regulation: Theory and applications. (English) Zbl 1087.93003
Advances in Design and Control 8. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-562-8/hbk; 978-0-89871-868-3/ebook). xvi, 318 p. (2004).
This book presents problems associated with the regulation of nonlinear control systems in finite dimension. Actually, it addresses many issues outlined by C. I. Byrnes and A. Isidori [Int. J. Robust Nonlinear Control 10, No. 5, 323–337 (2000; Zbl 0962.93007)]. Conditions for solvability are presented as well as some attempts to obtain solutions. The author follows the “internal model principle” approach, which was introduced in the 1970s for solving the linear case and which is outlined in the first chapter. Chapter two is a “standard” (to quote the author) introduction to nonlinear systems (stability, center manifold, normal forms, zero dynamics, the discrete case). Chapter three presents conditions for solvability for the output regulation of a nonlinear plant with a linear internal model (some sections examine the cases related to hyperbolicity of the zero dynamics). The next chapter presents an approximation method for the associated partial differential equation (lower order terms). This is extended in the subsequent chapter when the plant has parametric uncertainties. But a solution to the corresponding partial differential equation is obtained only in specific cases. Also global aspects are not handled. The author thus introduces a “new design framework” whose natural idea consists in handling a nonlinear internal model. The regulation problem is converted into a stabilization problem (see the last sentence of the above-mentioned review). Robust regulation is addressed again from this perspective and solved in specific cases only. Chapter eight addresses the singular case and chapter nine the discrete case.
Overall, robust regulation remains a difficult and quite open problem, even if a perspective for studying it in the nonlinear case is presented in this book. Even if robustness issues, so dear to engineers, are considered, still the development remains quite academic here because only parametric uncertainties are considered while in practice, uncertainties are dynamical.

##### MSC:
 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 93B51 Design techniques (robust design, computer-aided design, etc.) 93C10 Nonlinear systems in control theory 93C73 Perturbations in control/observation systems
Matlab
Full Text: