Some Remarks on Control Strategies for Continuous Gradient Play Dynamics

This work is motivated by the previous studies of approximating the derivative actions on continuous gradient play (GP) dynamics. The novelty lies on the implementation of derivative actions by small time delays. The stability/instability behavior of the GP dynamics with respect to the spectrum of its characteristic matrix is rendered and the corresponding phenomena are explained (well- and ill-posed approximation). The development is represented via a simple geometric approach that surfaces from the analysis. It is proven that sufficiently small delays do not necessarily guarantee the stability of GP dynamics, but by tuning some parameters one may eliminate such instabilities. Illustrative examples complete the paper

[1]  G. M. Schoen,et al.  Stability and stabilization of time-delay systems , 1995 .

[2]  Dirk Roose,et al.  Limitations of a class of stabilization methods for delay systems , 2001, IEEE Trans. Autom. Control..

[3]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[4]  R. Datko.  A procedure for determination of the exponential stability of certain differential-difference equations , 1978 .

[5]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[6]  Hartmut Logemann,et al.  Conditions for Robustness and Nonrobustness of theStability of Feedback Systems with Respect to Small Delays inthe Feedback Loop , 1996 .

[7]  Rifat Sipahi,et al.  Cluster treatment of characteristic roots, CTCR, a unique methodology for the complete robustness analysis of linear time invariant multiple time delay systems against delay uncertainties , 2005 .

[8]  Wim Michiels,et al.  Stabilizing a chain of integrators using multiple delays , 2004, IEEE Transactions on Automatic Control.

[9]  Wim Michiels,et al.  Finite spectrum assignment of unstable time-delay systems with a safe implementation , 2003, IEEE Trans. Autom. Control..

[10]  Jeff S. Shamma,et al.  Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria , 2005, IEEE Transactions on Automatic Control.

[11]  Denis Dochain,et al.  Sensitivity to Infinitesimal Delays in Neutral Equations , 2001, SIAM J. Control. Optim..

[12]  Gürdal Arslan,et al.  Distributed convergence to Nash equilibria with local utility measurements , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[13]  Takehiro Mori,et al.  STABILITY PRESERVING TRANSITION FROM DERIVATIVE FEEDBACK TO ITS DIFFERENCE COUNTERPARTS , 2002 .

[14]  Silviu-Iulian Niculescu,et al.  Additional dynamics in transformed time-delay systems , 2000, IEEE Trans. Autom. Control..

[15]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[16]  Wim Michiels.  Stability and stabilization of time-delay systems , 2002 .

[17]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[18]  Miguel V. S. Frasson,et al.  Large Time Behaviour of Linear Functional Differential Equations , 2003 .

[19]  Sabine Mondié,et al.  Approximation of control laws with distributed delays: a necessary condition for stability , 2001, Kybernetika.

[20]  Rifat Sipahi,et al.  An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems , 2002, IEEE Trans. Autom. Control..

[21]  Dimitri Breda,et al.  Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations , 2005, SIAM J. Sci. Comput..

[22]  Qing-Chang Zhong,et al.  On distributed delay in linear control Laws-part I: discrete-delay implementations , 2004, IEEE Transactions on Automatic Control.