Persistency of excitation for uniform convergence in nonlinear control systems

In previous papers we have introduced a sufficient condition f or uniform attractivity of the origin for a class of nonlinear time-varying systems which is stated in terms of persistency of excitation (PE), a concept well known in the adaptive control and systems identification literature. The novelty of our condition, called uniform �-PE, is that it is tailored for nonlinear functions of time and state and it allows us to prove uniform asymptotic stability. In this paper we present a new definition of u�-PE which is conceptually similar to but technically different from its predecessors and give several useful characterizations. We make connections between this property and similar properties previously used in the literature. We also show when this condition is necessary and sufficient for uniform (global) asymptotic stability for a large class of nonlinear time-varying systems. Finally, we show the utility of our main results on some control applications regarding feedforward systems and systems with matching nonlinearities.

[1]  Andrew R. Teel,et al.  Uniform parametric convergence in the adaptive control of manipulators: a case restudied , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[2]  Bor-Sen Chen,et al.  A general stability criterion for time-varying systems using a modified detectability condition , 2002, IEEE Trans. Autom. Control..

[3]  Antonio Loría,et al.  Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications , 2002, Math. Control. Signals Syst..

[4]  Elena Panteley,et al.  UGAS of Skew-symmetric Time-varying Systems: Application to Stabilization of Chained Form Systems , 2002, Eur. J. Control.

[5]  Vladimir I. Vorotnikov,et al.  PARTIAL STABILITY, STABILIZATION AND CONTROL: A SOME RECENT RESULTS , 2002 .

[6]  Antonio Loría,et al.  Relaxed persistency of excitation for uniform asymptotic stability , 2001, IEEE Trans. Autom. Control..

[7]  Thor I. Fossen,et al.  A theorem for UGAS and ULES of (passive) nonautonomous systems: robust control of mechanical systems and ships , 2001 .

[8]  C. Samson,et al.  A characterization of the Lie algebra rank condition by transverse periodic functions , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[9]  Frédéric Mazenc,et al.  Asymptotic tracking of a reference state for systems with a feedforward structure , 2000, Autom..

[10]  Anuradha M. Annaswamy,et al.  Adaptation in the presence of a general nonlinear parameterization: an error model approach , 1999, IEEE Trans. Autom. Control..

[11]  Thor I. Fossen,et al.  Passive nonlinear observer design for ships using Lyapunov methods: full-scale experiments with a supply vessel , 1999, Autom..

[12]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[13]  Dirk Aeyels,et al.  Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria , 1998, Math. Control. Signals Syst..

[14]  Pascal Morin,et al.  Application of Backstepping Techniques to the Time-Varying Exponential Stabilisation of Chained Form Systems , 1997, Eur. J. Control.

[15]  L. Praly,et al.  Adding integrations, saturated controls, and stabilization for feedforward systems , 1996, IEEE Trans. Autom. Control..

[16]  Zhong-Ping Jiang,et al.  Iterative design of time-varying stabilizers for multi-input systems in chained form , 1996 .

[17]  Marko V. Jankovic,et al.  Adaptive output feedback control of nonlinear feedback linearizable systems , 1996 .

[18]  Bruno Siciliano,et al.  Modeling and Control of Robot Manipulators , 1995 .

[19]  C. Samson.  Control of chained systems application to path following and time-varying point-stabilization of mobile robots , 1995, IEEE Trans. Autom. Control..

[20]  Romeo Ortega,et al.  Asymptotic stability of a class of adaptive systems , 1993 .

[21]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[22]  R. Marino,et al.  Global adaptive output-feedback control of nonlinear systems. I. Linear parameterization , 1993, IEEE Trans. Autom. Control..

[23]  Jean-Baptiste Pomet.  Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift , 1992 .

[24]  R. Marino,et al.  Global adaptive output-feedback control of nonlinear systems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[25]  Mark W. Spong,et al.  Comments on "Adaptive manipulator control: a case study" by J. Slotine and W. Li , 1990 .

[26]  S. Sastry,et al.  Adaptive Control: Stability, Convergence and Robustness , 1989 .

[27]  Mark W. Spong,et al.  Robot dynamics and control , 1989 .

[28]  Mark W. Spong,et al.  Adaptive motion control of rigid robots: a tutorial , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[29]  Weiping Li,et al.  Adaptive manipulator control a case study , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[30]  Zvi Artstein,et al.  Stability, observability and invariance , 1982 .

[31]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[32]  Zvi Artstein,et al.  Uniform asymptotic stability via the limiting equations , 1978 .

[33]  B. Anderson.  Exponential stability of linear equations arising in adaptive identification , 1977 .

[34]  K. Narendra,et al.  On the Stability of Nonautonomous Differential Equations $\dot x = [A + B(t)]x$, with Skew Symmetric Matrix $B(t)$ , 1977 .

[35]  K. Narendra,et al.  On the uniform asymptotic stability of certain linear nonautonomous differential equations , 1976 .

[36]  James R. Munkres,et al.  Topology; a first course , 1974 .

[37]  R. A. Silverman,et al.  Introductory Real Analysis , 1972 .

[38]  Karl Johan Åström,et al.  Numerical Identification of Linear Dynamic Systems from Normal Operating Records , 1965 .

[39]  P. Hartman.  Ordinary Differential Equations , 1965 .

[40]  R. E. Kalman,et al.  Contributions to the Theory of Optimal Control , 1960 .

[41]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .