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

VERÓNICA ESTELA PASTOR; GRACIELA ADRIANA GONZÁLEZ

CABA

Congreso; Mathematical Congress of the Americas 2021; 2021

FCEyN (UBA) y Mathematical Council of the Americas

The problem of Pyragas stabilizability was stated in [1] as follows. It is proposed to stabilize the nonlinear system given by:x_ = f(x) (1)in one of its (unknown) unstable equilibrium points by adding the feedback control:u(t) = K(x(t 􀀀 ) 􀀀 x(t)) (2)where the real constant matrix K and the real number are the control parameters.This method presents an essential constraint known as the "odd number limitation". Namely, if thejacobian 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, dierent methods have been designed by introducing a non-stationary feedbackcontrol. In particular, in [3], the constant gain K of (2) is replaced by a periodic K(t), dened bysome 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 ) 􀀀 x(t 􀀀 )) (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 eciency 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, Dierence feedback can stabilize uncer-tain steady states, EEE Trans. Autom. Control 46 (2001).[3] G.A. Leonov, M.M. Shumafov, Pyragas stabilizability of unstable equilibria by non-stationary timedelayed feedback, Autom. Remote Control 6 (2018).[4] V. E. Pastor, G. A. Gonzalez, Oscillating delayed feedback control schemes for stabilizing equilibrium points, Heliyon 5 (2019).