# zbMATH — the first resource for mathematics

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 exposition (monographs, survey articles) pertaining to systems and control theory 93B51 Design techniques (robust design, computer-aided design, etc.) 93C73 Perturbations in control/observation systems
Full Text:
##### References:
 [1] Minorsky, Journal of the American Society of Naval Engineering 34 pp 280– (1922) [2] Davison, IEEE Transactions on Automatic Control AC-21 pp 25– (1976) [3] Francis, Automatica 12 pp 457– (1976) [4] Francis, SIAM Journal of Control and Optimization 14 pp 486– (1977) [5] Linear Multivariable Control: A Geometric Approach (2nd edn). Springer: Berlin, 1979. [6] Devasia, IEEE Transactions on Automatic Control AC-41 pp 930– (1996) [7] Isidori, IEEE Transactions on Automatic Control AC-35 pp 131– (1990) [8] Huang, IEEE Transactions on Automatic Control AC-37 pp 1009– (1992) [9] Huang, IEEE Transactions on Automatic Control AC-37 pp 1395– (1992) [10] The construction of optimal linear and nonlinear regulators. In Systems, Models and Feedback, (eds). Birkhauser: Basel, 1992; 301-322. [11] Hepburn, Mathematical Systems Theory 17 pp 319– (1984) [12] On a robust nonlinear servomechanism problem. In Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, England, 1991; 2529-2530, (also on IEEE Transactions on Automatic Control 1994; AC-39:1510-1513). [13] Internal model principle and robust control of nonlinear systems. In Proceedings of the 32th IEEE Conference on Decision and Control, San Antonio, Texas, 1993; 1501-1506. [14] Huang, IEEE Transactions on Automatic Control AC-40 pp 1118– (1995) [15] Robust servomechanism output feedback controllers for a class of feedback linearizable systems. Workshop on Nonlinear Control Systems, St. Louis, Missouri, May 1992. [16] Khalil, Automatica 30 pp 1587– (1994) [17] Robust tracking for polynomial plants. In Proceedings of the Second European Control Conferences, Groeningen, The Netherlands, 1993; 369-373. [18] Sufficient conditions for robust tracking in nonlinear systems. Preprint, June 1993. [19] Sontag, European Journal of Control 1 pp 24– (1995) · Zbl 1177.93003 [20] Mahmoud, IEEE Transactions on Automatic Control AC-41 pp 1402– (1996) [21] Isidori, IEEE Transactions on Automatic Control AC-42 pp 1734– (1997) [22] Asymptotic tracking of a non-minumum phase nonlinear system with non-hyperbolic zero dynamics. IEEE Transactions on Automatic Control 2000, to appear. Also in Proceedings of 37th IEEE Conference on Decision and Control, Tampa, FL, 1998; 3064-3068. [23] Serrani, International Journal of Robust and Nonlinear Control 10 pp 379– (2000) [24] Serrani, Systems and Control Letters [25] The local solvability of a Hamilton-Jacobi-Bellman PDE around a nonhyperbolic critical point. Preprint, 1999. [26] Byrnes, IEEE Transactions on Automatic Control [27] Byrnes, International Journal of Robust and Nonlinear Control 9 pp 737– (1999) [28] Byrnes, Automatica 33 pp 369– (1997) [29] Hepburn, IEEE Transactions on Automatic Control AC-29 pp 397– (1984) [30] Huang, Systems and Control Letters 31 pp 215– (1997) [31] Huang, Automatica 26 pp 963– (1990) [32] Nonlinear Control Systems (3rd edn), Chapter 8. Springer: Berlin, 1995. [33] Semiglobal robust regulation of nonlinear systems. In Colloquium on Automatic Control, (eds). Springer: Berlin, 1996; 27-53. · Zbl 0886.93024 [34] Asymptotically robust perfect tracking for nonlinear systems. In Proceedings of the Fourth IFAC Symposium on Nonlinear Control Systems Design, Enschede, The Netherlands, 1998.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.