Applications of Homogeneity to Nonlinear Adaptive Control

  • Matthias Kawski
Part of the Progress in Systems and Control Theory book series (PSCT, volume 11)


This article is intended to bring together ideas from the geometric theory of nonlinear control systems, in particular the employment of homogeneity properties for asymptotic feedback stabilization, with a classical approach to adaptive control, in particular the regulation problem in the presence of unknown parameters. The control systems under consideration are of the form
$$\dot x\left( t \right) = f\left( {x\left( t \right),p} \right) + u\left( t \right)g\left( {x\left( t \right),p} \right)$$
The control u takes vaLues in R, x E R“ and the vectorfields f and g are explicitly linearly parametrized by the supposedly unknown parameter p E R, i.e.
$$f\left( {x,p} \right) = {f_0}\left( x \right) + p{f_1}\left( x \right)andg\left( {x,p} \right) = {g_0}\left( x \right) + p{g_1}\left( x \right)$$
where k and gi are smooth vectorfields on R“, and we suppose that f°(0) _ f1(0) = O. (Note, that this notion of linear parametrization is very much coordinate dependent.)


Filter Design Nonlinear Control System Feedback Stabilization Lyapunov Function Versus Open Neighbourhood Versus 
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  1. 1.
    Z. Artstein, “Stabilization with relaxed controls,” Nonlinear Analysis: Theory, Methods and Applications, v. 7 no.11, 1983, pp.1163–1173.CrossRefGoogle Scholar
  2. 2.
    W. Boothby and R. Marino “The center manifold theorem in feedback stabilization of planar SISO systems,” Control Theory and Advanced Technology (CTAT),v. 6 no.4, 1990.Google Scholar
  3. 3.
    C. I. Byrnes and A. Isidori, “The analysis and design of nonlinear feedback systems,” to appear in: IEEE Transactions Ant. Control. Google Scholar
  4. 4.
    B. D’andrea-Novel, J. M. Coron and L. Praly, “Lyapunov design of stabilizing controllers for cascaded systems”, to appear in IEEE Transactions Ant. Control. Google Scholar
  5. 5.
    W. Dayawansa, C. F. Martin and G. Knowles, “Asymptotic stabilization of a class of two-dimensional systems”, SIAM J. Control & Opt., v. 26 no.6, 1990, pp.1321–1349.CrossRefGoogle Scholar
  6. 6.
    M. Kawski, “High-order small-time local controllability,” Nonlinear Controllability and Optimal Control, H. J. Sussmann, ed., 1990, pp. 441–477 (Dekker).Google Scholar
  7. 7.
    M. Kawski, “Stabilization of nonlinear systems in the plane,” Systems and Control Letters, v. 12, 1989, pp.169–175.CrossRefGoogle Scholar
  8. 8.
    M. Kawski, “Homogeneous stabilizing feedback laws,” Control Theory and Advanced Technology (CTAT), v. 6 no.4, 1990.Google Scholar
  9. 9.
    M. Kawski, “Families of dilations and asymptotic stability,” Proc. of Conf. Analysis of Controlled Dynamical Systems, Lyon, France, 1990.Google Scholar
  10. 10.
    P. J. Olver, “Applications of Lie Groups to Differential Equations,” Graduate Texts Mathematics, v. 107, 1986. (Springer).Google Scholar
  11. 11.
    P. C. Parks, “Lyapunov Redesign of Model Reference Adaptive Control Systems,” IEEE Transact. Ant. Control, v. 11 no.3, 1966.Google Scholar
  12. 12.
    L. Praly, G. Bastin, J.-B. Pomet and Z.P. Jiang; “ Adaptive Stabilization of Nonlinear Systems,” CAI Report No. A236 (Ecole Nationale Superieure des Mines de Paris, Centre d’Automatique et Informatique), 1990.Google Scholar
  13. 13.
    L. P. Rothschild and E. M. Stein, “Hypoelliptic differential operators and nilpotent groups,” Acta Math., v. 137, 1976, pp.247–320.CrossRefGoogle Scholar
  14. 14.
    E. D. Sontag “A `universal’ construction of Artstein’s theorem on nonlinear stabilization,” Report 89–03, SYCON - Rutgers Center for Systems and Control, 1989.Google Scholar
  15. 15.
    H. J. Sussmann, “A general theorem on local controllability,” SIAM J. Control & Opt., v. 25 no.1, 1987, pp. 158–194.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Matthias Kawski
    • 1
  1. 1.Department of MathematicsArizona State UniversityTempeUSA

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