On Bifurcation Control in Time Delay Feedback Systems

Abstract

The paper addresses bifurcations of limit cycles for a class of feedback control systems depending on parameters. A set of simple approximate analytical conditions characterizing all the generic limit cycle bifurcations is determined via a first-order harmonic balance analysis in a suitable frequency band. Based on the results of this analysis, an approach to limit cycle bifurcation control is proposed. In particular, an example concerning a biological delay model is developed, where a flip bifurcation control is designed via a modified Pyragas technique.

Chaos stabilization with TDAS and FPIC in a buck converter controlled by lateral PWM and ZAD

“…The goal is to feedback a time-delayed variable. Several papers have used this strategy [17], Batlle [18] and Illing [19].…”

Bifurcation Control: Theories, Methods, and Applications

“…For this purpose, the harmonic balance approximation technique is very efficient. This technique is useful in controlling bifurcations [Basso et al, 1998], such as for delaying and stabilizing the onset of period-doubling bifurcations [Genesio et al, 1993;Tesi et al, 1996]. Consider the Lur'e system described by…”

“…Bifurcation control refers to the task of designing a controller to modify the bifurcation properties of a given nonlinear system, thereby achieving some desirable dynamical behaviors. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation [Tesi et al, 1996;, introducing a new bifurcation at a preferable parameter value [Abed, 1995;Chen et al, 1998b], changing the parameter value of an existing bifurcation point [Chen & Dong, 1998;Moiola & Chen, 1996], modifying the shape or type of a bifurcation chain , stabilizing a bifurcated solution or branch [Abed & Fu, 1986, 1987Abed et al, 1994;Kang, 1998aKang, , 1998bLaufenberg et al, 1997;Littleboy & Smith, 1998;Nayfeh et al, 1996;Senjyu & Uezato, 1995], monitoring the multiplicity [Calandrini et al, 1999;Moiola & Chen, 1998], amplitude [Berns et al, 1998a;Moiola et al, 1997a], and/or frequency of some limit cycles emerging from bifurcation [Cam & Kuntman, 1998;Chen & Moiola, 1994;Chen & Dong, 1998b], optimizing the system performance near a bifurcation point [Basso et al, 1998], or a combination of some of these objectives Chen 1998Chen , 1999aChen , 1999b. Bifurcation control with various of objectives have been implemented in experimental systems or tested by using numerical simulations in a great number of engineering, biological, and physicochemical systems; examples can be named in chemical engineering [Alhumaizi & Elnashaie, 1997;Moiola et al, 1991], mechanical engineering [Liaw & Abed, 1996;Wang et al, 1994b;Cheng, 1990;Gu et al, 1997, Hackl et al, 1993…”

Application of Order Parameter Equations for the Analysis and the Control of Nonlinear Time Discrete Dynamical Systems

“…The calculations in this work show, that it is possible to derive the order parameter equations for nonlinear maps with dimensionalities 2, 3 and 4 analytically. From the point of view of control theory, we have shown the efficiency of the stabilization by shifting the bifurcation point using the method of delayed feedback mentioned above and in addition derived the modified control scheme (see also [Basso et al, 1998]). The advantage of this method is that we are able to calculate the appropriate value of the parameter k crit for the specific bifurcation point a n crit .…”