Accounting for Nonlinearities in Open-Loop Protocols for Symmetry Fault Compensation

Abstract

In this paper, we consider model examples of dynamical systems with only a few degrees of freedom, and with desirable symmetry properties, and explore compensating control strategies for retaining robust symmetric system response even under symmetry-breaking defects. The analysis demonstrates the distinct differences between linear versions of these models, in which fault-compensating strategies are always found, and weakly nonlinear counterparts with varying degrees of asymmetry, for which a multitude of locally optimal solutions may coexist. We further formulate a candidate optimization protocol for fault compensation applied to self-healing systems, which respond to symmetry-breaking defects by a continuous process of fault correction. The analysis shows that such a protocol may exhibit discontinuous changes in the control strategy as the self-healing system successively regains its original symmetry properties. In addition, it is argued that upon return to a symmetric configuration, such a protocol may result in a different control strategy from that applied prior to the occurrence of a fault.

A multi-parametric recursive continuation method for nonlinear dynamical systems

“…Since then, several scenarios for the creation of ISs have been revealed. DiBerardino and Dankowicz [60] showed that ISs can be created by introducing asymmetry into a nonlinear system. In [61], the presence of IS is explained analytically by analyzing the 1:3 internal resonance configuration between two Duffing oscillators for different couplings.…”

Conditions for the existence of isolated backbone curves

“…An early study of isolas in engineering systems was carried out by Abramson [24] in 1955. Extensive work since this has focused on the mechanism for their creation, such as discontinuity [25,26], internal resonances [27] and symmetry breaking [28,29].…”

Isolated resonances and nonlinear damping

“…Lenci and Ruzziconi [23] showed the appearance of an IRC far from any resonance, while studying a single-DoF model of a suspended bridge, which includes quadratic and cubic hardening nonlinearity. DiBerardino and Dankowicz [24] identified IRCs related to symmetry breaking in a two-DoF system; adopting isola singularity identification, in combination with a multiple scale approach, they managed to predict the outbreak of an IRC. In [25], an IRC related to an internal resonance was encountered, while in [26] the phenomenon was explained through a frequency gap due to phase locking.…”

“…In fact, the onset of an IRC corresponds to an isola singularity in the frequency response function, while its merging with another branch corresponds to a simple bifurcation (see for example [59][60][61][62][63][64]). In spite of this, only few researchers [24,45] implemented singularity analysis for IRC identification.…”

“…In fact, the onset of an IRC corresponds to an isola singularity in the frequency response function, while its merging with another branch corresponds to a simple bifurcation (see for example [59][60][61][62][63][64]). In spite of this, only few researchers [24,45] implemented singularity analysis for IRC identification.In this study we attempt to fully exploit the potential of singularity theory for the prediction and identification of IRCs. In particular, we analyze their relation with the topology of the nonlinear damping force, which suggests the presence of general rules for their appearance.…”