Abstract
Summary. Among Alberto Isidori’s many seminal contributions, his solution of the nonlinear tracking problem and the underlying concept of zero dynamics have had the widest and strongest impact. Here we use these results to investigate and quantify the limit to tracking performance posed by unstable zero dynamics. While some aspects of this limit are nonlinear analogs of Bode’s T-integral formula, the dependence on the exosystem dynamics is an added complexity of nonlinear tracking.
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
Preview
Unable to display preview. Download preview PDF.
References
A.P. Aguiar, D.B. Dačić, J.P. Hespanha, and P.V. Kokotović. Path-following or reference-tracking? An answer relaxing the limits to performance. In Proc. of IAV2004 - 5th IFAC/EURON Symp. on Intel. Auton. Vehicles, Lisbon, Portugal, 2004.
A.P. Aguiar, J.P. Hespanha, and P.V. Kokotović. Path-following for non-minimum phase systems removes performance limitations. IEEE Trans. on Automat. Contr., 50(2):234–239, 2005.
A.P. Aguiar, J.P. Hespanha, and P.V. Kokotović. Performance limitations in reference-tracking and path-following for nonlinear systems. Automatica, 2007. In press.
J. Chen, L. Qiu, and O. Toker. Limitations on maximal tracking accuracy. IEEE Trans. on Automat. Contr., 45(2):326–331, 2000.
D.B. Dačić and P.V. Kokotović. Path-following for linear systems with unstable zero dynamics. Automatica, 42(10):1673–1683, 2006.
D.B. Dačić, M.V. Subbotin, and P.V. Kokotović. Path-following for a class of nonlinear systems with unstable zero dynamics. In Proc. of the 43rd IEEE Conf. on Decision and Contr., Paradise Island, Bahamas, 2004.
E.J. Davison. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. on Automat. Contr., 21(1):25–34, 1976.
B.A. Francis. The linear multivariable regulator problem. SIAM J. Contr. Optimization, 15(3):486–505, 1977.
B.A. Francis. The optimal linear-quadratic time-invariant regulator with cheap control. IEEE Trans. on Automat. Contr., 24(4):616–621, 1979.
B.A. Francis and W.M. Wonham. The internal model principle of control theory. Automatica, 12(5):457–465, 1976.
A. Isidori. The matching of a prescribed linear input-output behavior in a nonlinear system. IEEE Trans. on Automat. Contr., 30(3):258–265, 1985.
A. Isidori. Nonlinear control systems: An introduction. Springer-Verlag, New York, NY, 1985.
A. Isidori. Nonlinear Control Systems. Communications and Control Engineering Series. Springer-Verlag, Berlin, 3rd edition, 1995.
A. Isidori and C.I. Byrnes. Output regulation of nonlinear systems. IEEE Trans. on Automat. Contr., 35(2):131–140, 1990.
A. Isidori and J.W. Grizzle. Fixed modes and nonlinear noninteracting control with stability. IEEE Trans. on Automat. Contr., 33(10):907–914, 1988.
A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Monaco. Nonlinear decoupling via feedback: A differential geometric approach. IEEE Trans. on Automat. Contr., 26(2):331–345, 1981.
A. Isidori and C. Moog. On the nonlinear equivalent of the notion of transmission zeros. In Byrnes C. and A. Kurzhanski, editors, Modelling and Adaptive Control, Lecture notes in information and control, pages 146–157. Springer-Verlag, Berlin, 1988.
A. Jameson and R.E. O’Malley. Cheap control of the time-invariant regulator. Appl. Math. Optim., 1(4):337–354, 1975.
A.J. Krener. The local solvability of a Hamilton-Jacobi-Bellman PDE around a nonhyperbolic critical point. SIAM J. Contr. Optimization, 39(5):1461–1484, 2001.
M. Krstić, I. Kanellakopoulos, and P.V. Kokotović. Nonlinear and Adaptive Control Design. John Wiley & Sons, Inc., New York, USA, 1995.
H. Kwakernaak and R. Sivan. The maximal achievable accuracy of linear optimal regulators and linear optimal filters. IEEE Trans. on Automat. Contr., 17(1):79–86, 1972.
R.H. Middleton. Trade-offs in linear control systems design. Automatica, 27(2):281–292, 1991.
L. Qiu and E.J. Davison. Performance limitations of nonminimum phase systems in the servomechanism problem. Automatica, 29:337–349, 1993.
M.M. Seron, J. H. Braslavsky, P.V. Kokotović, and D.Q. Mayne. Feedback limitations in nonlinear systems: From Bode integrals to cheap control. IEEE Trans. on Automat. Contr., 44(4):829–833, 1999.
W. Su, L. Qiu, and J. Chen. Fundamental performance limitations in tracking sinusoidal signals. IEEE Trans. on Automat. Contr., 48(8):1371–1380, 2003.
K. Young, P.V. Kokotović, and V. Utkin. A singular perturbation analysis of high-gain feedback systems. IEEE Trans. on Automat. Contr., 22:931–938, 1977.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Aguiar, A., Hespanha, J., Kokotovic, P. (2008). Zero Dynamics and Tracking Performance Limits in Nonlinear Feedback Systems. In: Astolfi, A., Marconi, L. (eds) Analysis and Design of Nonlinear Control Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74358-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-74358-3_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74357-6
Online ISBN: 978-3-540-74358-3
eBook Packages: EngineeringEngineering (R0)