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Output regulation for nonlinear systems: An overview. (English) Zbl 0962.93007

This survey on robust output regulation (asymptotic tracking, disturbance rejection) via the internal model approach presents first the linear case, which has been treated by Davison, Wonham and Francis, and then results of the authors in the nonlinear case, and mentions the further efforts of Huang, Lin, Khalil (approximation procedure, robustification, semiglobal aspects when the zero dynamics is stable). A section stresses via a linear example that perfect tracking via an inversion procedure is not robust to parametric uncertainties in the system but asymptotic tracking will do. The authors introduce the definitions of “structurally stable output regulation” and the stronger “robust output regulation”, but the latter concept is not developed explicitly here. A last section deals with current research (semiglobal aspects, cases when the dynamics of the exogenous input is uncertain, partial differential equations).

It should be mentioned that this publication has not been written with the greatest care. The title without the word “nonlinear” would be more accurate since the linear situation takes almost half of the paper. In Theorem 2, the statement is not valid for all $P$ and $Q$ as written but only locally as expressed earlier. The locations of the input and of the disturbance in (15) and (20) are interchanged. Unfortunately, the dimensions of introduced quantities are omitted sometimes. The transition from the linear to the nonlinear situation is not transparent. The examples are superfluous. Worse, the presentation is quite formal masking the essence of the design: The internal model (for a first compensator) allows to catch the uncertainties in the parameters of the dynamics and in the initial condition of the disturbance dynamics in its initial condition; this leads to a second stabilizing compensator to get rid of it.

##### MSC:
 93-02 Research monographs (systems and control) 93B51 Design techniques in systems theory 93C73 Perturbations in control systems