Abstract
In this chapter, the problem of asymptotic tracking/rejection of exogenous commands/disturbances for nonlinear systems is discussed. Results that extend those developed earlier in Chap. 4 for linear systems are presented. The discussion follows very closely the analysis of necessary conditions presented in Sect. 4.3 and the construction of a regulator presented in the second part of Sect. 4.6. The construction of an internal model, though, requires a different and more elaborate analysis, for which two alternatives are offered. The chapter is complemented with a discussion of a simple problem of inducing consensus in a network of nonlinear agents.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Lemma B.7 in Appendix B.
- 2.
Even if the initial conditions are taken in a compact set, see in this respect Example B.7 in Appendix B.
- 3.
See section B.6 in Appendix B.
- 4.
See again section B.6 in Appendix B.
- 5.
See [6] for a proof.
- 6.
Strictly speaking, it is not necessary to consider here a linear stabilizer. In fact, as it will be seen in the sequel, it suffices that the stabilizer has an equilibrium state yielding \(\bar{u}=0\). However, since essentially all methods illustrated earlier in the book for nonlinear (robust) stabilization use linear stabilizers, in what follow we will consider a stabilizer of this form.
- 7.
Compare with a similar conclusion obtained in the proof of Proposition 4.6.
- 8.
It is instructive to compare these equations with (4.14).
- 9.
The function \(\gamma (\cdot )\) is only guaranteed to be continuous and may fail to be continuously differentiable. Closed-forms expressions for \(\gamma (\cdot )\) and other relevant constructive aspects are discussed in [13].
- 10.
Since W is a compact set, in the condition above only the values of \(\phi (\cdot )\) on a compact set matter. Thus, the assumption that the function is globally Lipschitz can be taken without loss of generality.
- 11.
Note that G is present in the function \(v(w,z,\xi )\) and hence in \(\bar{v}(w,z,\xi )\). Thus, this function depends on \(\kappa \) and, actually, its magnitude grows with \(\kappa \). However, this does not affect the conclusion. Once \(\kappa \) is fixed, the bound on \(\bar{v}(w,z,\xi )\) is fixed as well. The “gain function” \(\vartheta _1(\cdot )\) in (12.30) is influenced by the value of \(\kappa \), but an estimate of this form holds anyway.
- 12.
- 13.
See section B.4 in Appendix B for the definition of distance of a point from a set.
References
J. Huang, W.J. Rugh, On a nonlinear multivariable servomechanism problem. Automatica 26, 963–972 (1990)
A. Isidori, C.I. Byrnes, Output regulation of nonlinear systems. IEEE Trans. Autom. Control. AC-25, 131–140 (1990)
H. Khalil, Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica 30, 587–1599 (1994)
J. Huang, C.F. Lin, On a robust nonlinear multivariable servomechanism problem. IEEE Trans. Autom. Control AC-39, 1510–1513 (1994)
C.I. Byrnes, F. Delli, Priscoli, A. Isidori, W. Kang, Structurally stable output regulation of nonlinear systems. Automatica 33, 369–385 (1997)
C.I. Byrnes, A. Isidori, Limit sets zero dynamics and internal models in the problem of nonlinear output regulation. IEEE Trans. Autom. Control 48, 1712–1723 (2003)
A. Isidori, L. Marconi, A. Serrani, Robust Autonomous Guidance: And Internal-Model Approach (Springer, London, 2003)
J. Huang, Nonlinear Output Regulation: Theory and Applications (SIAM, Philadephia, 2004)
C.I. Byrnes, A. Isidori, Nonlinear internal models for output regulation. IEEE Trans. Autom. Control AC-49, 2244–2247 (2004)
Z. Chen, J. Huang, A general formulation and solvability of the global robust output regulation problem. IEEE Trans. Autom. Control AC-50, 448–462 (2005)
A. Pavlov, N. van de Wouw, H. Nijmeijer, Uniform Output Regulation of Nonlinear Systems: A Convergent Dynamics Approach (Birkhauser, Boston, 2006)
L. Marconi, L. Praly, A. Isidori A, Output stabilization via nonlinear luenberger observers. SIAM J. Control Optim. 45, 2277–2298 (2006)
L. Marconi, L. Praly, Uniform practical nonlinear output regulation. IEEE Trans. Autom. Control 53, 1184–1202 (2008)
A. Isidori, C.I. Byrnes, Steady-state behaviors in nonlinear systems, with an application to robust disturbance rejection. Annu. Rev Control 32, 1–16 (2008)
A. Isidori, L. Marconi, G. Casadei, Robust output synchronization of a network of heterogeneous nonlinear agents via nonlinear regulation theory. IEEE Trans. Autom. Control 59, 2680–2691 (2014)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Isidori, A. (2017). Regulation and Tracking in Nonlinear Systems. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-42031-8_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42030-1
Online ISBN: 978-3-319-42031-8
eBook Packages: EngineeringEngineering (R0)