Possibility of Quenching of Limit Cycles in Multi Variable Nonlinear Systems with Special Attention to 3X3 Systems

Abstract

The present work proposes novel methods of Quenching self-sustained oscillations in the event of the existence of limit cycles (LC) in 3x3 non-linear systems. It explores the possibility of Stabilising/Quenching the LC by way of signal stabilization using high frequency dither signals both deterministic and random when 3X3 systems exhibit such self-sustained nonlinear oscillations under autonomous state. The present work also explores the suppression limit cycles of 3X3 systems using state feedback by either arbitrary pole placement or optimal selection of pole placement. The complexity involved, in implicit non-memory type nonlinearity for memory type nonlinearities, it is extremely difficult to formulate the problem. Under this circumstance, the harmonic linearization/harmonic balance reduces the complexity considerably. Furthermore, the method is made simpler assuming the whole 3X3 system exhibits the LC predominantly at a single frequency. It is equally a formidable task to make an attempt to suppress the limit cycles for 3X3 systems with memory type nonlinearity in particular. Backlash is one of the nonlinearities commonly occurring in physical systems that limit the performance of speed and position control in robotics, the automation industry, and other occasions of modern applications. The proposed methods are well illustrated through examples and substantiated by digital simulation (a program developed using MATLAB CODES) and the use of the SIMULINK Toolbox of MATLAB software.