zbMATH — the first resource for mathematics

On periodic solutions of adaptive systems in the presence of periodic forcing terms. (English) Zbl 0712.93031
This paper studies solutions of adaptive feedback loops in the presence of unmodeled dynamics if the discrete-time linear plant. It assumes that the reference output is periodic with period K. The aim is to find K- periodic solutions of the adaptive system and study their stability. Sufficient conditions to have K-periodic solutions are derived. Also, the stability of these solutions is studied assuming that adaptation is slow. A wide class of direct adaptive controllers is used for checking the sufficient conditions for the existence of a periodic solution. It is proved that “almost always” the periodic solution exists, but no tools to study its stability, except in the exact modeling case, are available. The present study would be a first step toward a complete description of the behavior of an adaptive controller in feedback with a high-order linear plant.
Reviewer: P.Stoica

93C40 Adaptive control/observation systems
93C05 Linear systems in control theory
93C55 Discrete-time control/observation systems
Full Text: DOI
[1] B. D. O. Anderson, R. R. Bitmead, C. R. Johnson, P. V. Kokotovic, R. L. Kosut, I. M. Y. Mareels, L. Praly, and B. D. Riedle,Stability of Adaptive Systems: Passivity and Averaging Analysis, MIT Press, Cambridge, MA, 1986.
[2] V. Arnold,Equations diffèrentielles ordinaires, Mir Editions, Moscow, 1974. · Zbl 0296.34002
[3] J.-P. Aubin and I. Ekeland,Non-Linear Applied Analysis, Wiley, New York, 1984. · Zbl 0641.47066
[4] E. W. Bai, L. C. Fu, and S. S. Sastry, Averaging analysis for discrete-time and sampled data adaptive systems,IEEE Trans. Circuits and Systems,34 (1988), 137–148. · Zbl 0649.93036 · doi:10.1109/31.1715
[5] E. W. Bai and S.S. Sastry, Persistency of excitation, sufficient richness and parameter convergence in discrete time adaptive control,Systems Control Lett.,6 (1985), 153–163. · Zbl 0568.93043 · doi:10.1016/0167-6911(85)90035-0
[6] A. Benveniste, M. Métivier, and P. Priouret,Algorithmes adaptatifs et approximations stochastiques; théorie et applications, Masson, Paris, 1987.
[7] G. C. Goodwin and K. S. Sin,Adaptive Filtering, Prediction, and Control, Prentice-Hall, Englewood Cliffs, NJ, 1984. · Zbl 0653.93001
[8] P. V. Kokotovic, A Riccati equation for block-diagonalization of ill conditioned systems,IEEE Trans. Automat. Control,20 (1975), 812–814. · Zbl 0317.93043 · doi:10.1109/TAC.1975.1101089
[9] P. V. Kokotovic, B. Riedle, and L. Praly, On a stability criterion for continuous slow adaptation,Systems Control Lett.,6, (1985), 7–14. · Zbl 0569.93050 · doi:10.1016/0167-6911(85)90047-7
[10] S. Lefschetz,Differential Equations: A Geometric Theory. Dover Publications, New York, 1977. · Zbl 0080.06401
[11] N. G. Lloyd,Degree Theory, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1978.
[12] R. K. Miller and A. N. Michel,Ordinary Differential Equations, Academic Press, New York, 1982. · Zbl 0552.34001
[13] J. Milnor,Topology from the Differentiable Point of View, University of Virginia Press, Charlottesville, VA, 1965. · Zbl 0136.20402
[14] L. Praly and J.-B. Pomet, Periodic solutions in adaptive systems: the regular case,Proceedings of the 10th Triennal IFAC Congress, Munich, 1987, pp. 40–45.
[15] L. Praly and D. Rhode, Local analysis of a one step ahead adaptive controller,Proceedings of the 24th IEEE Conference on Decision and Control, Ft. Lauderdate, FL, 1985, pp. 1862–1867.
[16] B. D. Riedle and P. V. Kokotovic, Stability of slow adaptation for non-SPR systems with disturbances,Proceedings of the 1986 American Control Conference, Seattle, WA, pp. 97–101. · Zbl 0621.93038
[17] B. D. Riedle, L. Praly, and P. V. Kokotovic, Examination of the SPR condition in output error parameter estimation,Automatica,22 (1986), 495–498. · Zbl 0626.93068 · doi:10.1016/0005-1098(86)90055-5
[18] M. Shub,Stabilité globale des systèmes dynamiques, Astérisque, Vol. 56, Soc. Math. Francaise, Paris, 1978; English translation,Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. · Zbl 0396.58014
[19] B. L. van der Waerden,Modern Algebra, Ungar, New York, 1949.
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.