Session S25 - New Methods and Emerging Applications in Dynamics, Networks, and Control

Tuesday, July 13, 19:30 ~ 20:00 UTC-3

An alternative approach to overcome the ''odd number limitation'' of Pyragas stabilizability problem.

Verónica Estela Pastor

Universidad de Buenos Aires, Facultad de Ingeniería, Departamento de Matemática, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The problem of Pyragas stabilizability was stated in [1] as follows. It is proposed to stabilize the nonlinear system given by: \[ \dot x = f(x) \ \ \ \ \ (1) \] in one of its (unknown) unstable equilibrium points by adding the feedback control: \[ u(t)= K (x(t- \tau)-x(t)) \ \ \ \ \ (2) \] where the real constant matrix $K$ and the real number $\tau$ are the control parameters.

This method presents an essential constraint known as the ''odd number limitation''. Namely, if the jacobian matrix of the system evaluated on the equilibrium point has an odd number of positive eigenvalues or a zero eigenvalue, stabilization is not achieved for any value of the control parameters ([2]).

To overcome this drawback, different methods have been designed by introducing a non-stationary feedback control. In particular, in [3], the constant gain $K$ of (2) is replaced by a periodic $K(t)$, defined by some adequate constants. The choice of these constants is based on analytical arguments but a complete characterization of the available set of stability parameters is not determined.

As an alternative, we propose the following scheme: \[ u(t)= K(t) (x(t- 2\tau)-x(t- \tau)) \ \ \ \ \ (3) \] where the periodic gain yields to an oscillatory type control.

This proposal keeps the non-invasive feature of its antecedents and it is based on the methodology developed in [4] for the one dimensional case. Its efficiency for any hyperbolic equilibrium point is proved and a full description of the set of stability parameters is deduced.

References:

$[1]$ K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A 206 (1995).

$[2]$ H. Kokame, K. Hirata, K. Konishi, T. Mori, Difference feedback can stabilize uncer-tain steady states, IEEE Trans. Autom. Control 46 (2001).

$[3]$ G.A. Leonov, M.M. Shumafov, Pyragas stabilizability of unstable equilibria by non-stationary time-delayed feedback, Autom. Remote Control 6 (2018).

$[4]$ V. E. Pastor, G. A. González, Oscillating delayed feedback control schemes for stabilizing equilibrium points, Heliyon 5 (2019).

Joint work with Graciela Adriana González (Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Matemática y CONICET, Argentina).

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