Abstract: The paper presents an innovative graphical technique that uses computer graphics in conjunction with geometric tools in the estimation of limit cycles (LC) in 3×3 nonlinear (NL) systems with rectangular hysteresis and backlash Nonlinearities. In the event the system exhibits LCs under autonomous conditions, the work explores the possibility of mitigating/quenching limit cycling oscillations through signal stabilization using high-frequency deterministic/Gaussian signals. In addition, another attempt is taken to suppress LCs by the pole placement (PP) technique through state feedback. The state feedback gain matrix K is determined either by arbitrary selection ensuring a complete state controllability condition or by optimal selection from the Riccati equation. To avoid the mathematical complexity of formulation for 3×3 memory-type Nonlinearities, in particular for estimation/prediction of LCs, a novel graphical method is developed using the most common and popular describing function (DF), assuming harmonic balance with proper justification that the linear components of systems possess low-pass characteristics. The complexity and criticality in the mathematical expression (which diminish problem insight) are mitigated when the system exhibits LC predominantly at a single frequency. The proposed methods are demonstrated through practical examples and validated using digital simulations, implemented via a program developed in MATLAB. These results are further substantiated by those obtained using the SIMULINK toolbox in the MATLAB environment.

Keywords: Describing function, Limit cycles, Signal Stabilization, Pole Placement

Pages: 38-62

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