Abstract
In this paper, we derive conditions under which a nonlinear system can be rendered passive via smooth state feedback and we show that, as in the case of linear systems, this is possible if and only if the system in question has relative degree 1 and is weakly minimum phase. As an application of this analysis, we derive a stabilization result which incorporates and extends a number of stabilization schemes recently proposed in the literature for global asymptotic stabilization of certain classes of nonlinear systems.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Research supported in part by AFOSR, NSF, and MURST.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
C.I. Byrnes, A. Isidori and J.C. Willems, Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, submitted for publication.
C.I. Byrnes and A. Isidori, Local stabilization of minimum phase nonlinear systems, Syst Contr. Lett 11(1988), pp. 9–17.
C.I. Byrnes and A. Isidori, New results and examples in nonlinear feedback stabilization, Syst Contr. Lett 12(1989), pp. 437–442.
C.I. Byrnes and A. Isidori, Asymptotic stabilization of minimum phase nonlinear systems, to appear in IEEE Trans Ant Contr.
C.A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, 1975.
J.P. Gauthier and G. Bornard, Stabilisation des systemes nonlineaires, Outils et methodes mathematiques pour l’automatique… (I.D. Landau, ed.), C.N.R.S. (1981), pp. 307–324.
W. Hahn, Stability of Motion, Springer Verlag (1967).
D. Hill, Dissipativeness, stability theory and some remaining problems, Analysis and Control of Nonlinear Systems (C.I. Byrnes, C.F. Martin and R.E. Saeks eds.), North-Holland (1988), pp.443–452.
D. Hill and P. Moylan, The stability of nonlinear dissipative systems, IEEE Trans Aut Contr. 21(1976), pp. 708–711.
D. Hill and P. Moylan, Stability results for nonlinear feedback systems, Automatica 13 (1977), pp. 377–382.
D. Hill and P. Moylan, Dissipative dynamical systems: basic input-output and state properties, J. Franklin Inst 309 (1980), pp. 327–357.
D. Hill and P. Moylan, Connections between finite gain and asymptotic stability, IEEE Trans Aut Contr. 25(1980), pp. 931–936.
A. Isidori, Nonlinear Control Systems, 2nd ed., Springer Verlag (1989).
V. Jurdjevic and J.P. Quinn, Controllability and stability, J. Diff. Equations 28 (1978), pp. 381–389.
N. Kaloupsidis and J. Tsinias, Stability improvement of nonlinear systems by feedback, IEEE Trans Aut Contr. 29(1984), pp. 364–367.
P.V. Kokotovic and H.J. Sussmann, A positive real condition for global stabilization of nonlinear systems, Syst. Contr. Lett. 13(1989), pp. 125–134.
K.K. Lee and A. Araposthatis, Remarks on smooth feedback stabilization of nonlinear systems, Syst. Contr. Lett. 10(1988), pp. 41–44.
I.W. Sandberg, On the L2 boundedness of solutions of nonlinear functional equations, Bell System Tech. J. 43 (1964), pp. 99–104.
I.W. Sandberg, On the stability of interconnected systems, Bell System Tech. J. 57 (1978), pp. 3031–3046.
M. Vidyasagar, L 2 stability of interconnected systems using a reformulation of the passivity theorem, IEEE Trans. Circuits and Systems 24(1977), pp. 637–645.
M. Vidyasagar, New passivity-type criteria for large-scale interconnected systems, IEEE Trans Aut Contr. 24(1979), pp. 575–579.
A. Saberi, P.V. Kokotovic, H.J. Sussmann, Global stabilization of partially linear composite systems, 28th IEEE Conf. Dec. Contr. (1989), pp. 1385–1391.
E.D. Sontag, Feedback stabilization implies coprima factorization, IEEE Trans Aut Contr. 34(1989), pp. 435–443.
J.C. Willems, Dissipative dynamical systems Part I: general theory, Arch. Rational Mechanics and Analysis 45(1972), pp. 321–351.
J.C. Willems, Dissipative dynamical systems Part II: linear systems with quadratic supply rates, Arch. Rational Mechanics and Analysis 45(1972), pp. 352–393.
G. Zames, On the input-output stability of time-varying nonlinear feedback systems. Part I: conditions derived usning concepts of loop gain, conicity and positivity, IEEE Trans Aut Contr. 11(1966), pp. 228–238.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Birkhäuser Boston
About this chapter
Cite this chapter
Byrnes, C.I., Isidori, A., Willems, J.C. (1991). Feedback Equivalence to Passive Nonlinear Systems. In: Bonnard, B., Bride, B., Gauthier, JP., Kupka, I. (eds) Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory, vol 8. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-3214-8_10
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3214-8_10
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7835-1
Online ISBN: 978-1-4612-3214-8
eBook Packages: Springer Book Archive