Quenching and Suppression of Limit Cycles in 3x3 Nonlinear Systems

Abstract

For several decades, the importance and weight-age of prediction of nonlinear self-sustained oscillations or Limit Cycles (LC) and their quenching by signal stabilization have been discussed which is confined to Single Input and Single Output (SISO) system. However, for the last five to six decades, the analysis of 2x2 Multi Input and Multi Output (MIMO) Nonlinear Systems gained importance in which a lot of literature available. In recent days few literatures are available which addresses the exhibition of LC and their quenching/suppression in 3x3 MIMO Nonlinear systems. Poor performances in many cases like Load Frequency Control (LFC) in multi area power system, speed and position control in robotics, automation industry and other occasions have been observed which draws attention of Researchers. The complexity involved, in implicit nonmemory type and memory type nonlinearities, it is extremely difficult to formulate the problem in particular for 3x3 systems. Under this circumstance, the harmonic linearization/ harmonic balance reduces the complexity considerably. Still the analytical expressions are so complex which loses the insight into the problem particularly for memory type nonlinearity in 3x3 system. Hence in the present work a novel graphical method has been developed for prediction of limit cycling oscillations in a 3x3 nonlinear system. The quenching of such LC using signal stabilization technique using deterministic (Sinusoidal) and random (Gaussian) signals has been explored. Suppression LC using pole placement technique through arbitrary selection and optimal selection of feedback Gain Matrix K with complete state controllability condition and Riccati Equation respectively. The method is made further simpler assuming a 3x3 system exhibits the LC predominantly at a single frequency, which facilitates clear insight into the problem and its solution. The proposed techniques are well illustrated with example and validated/substantiated by digital simulation (a developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The Signal stabilization with Random (Gaussian) Signals and Suppression LC with optimal selection of state feedback matrix K using Riccati Equation for 3x3 nonlinear systems have never been discussed elsewhere and hence it claims originality and novelty. The present work has the brighter future scope of: i. Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. ii. Analytical/Mathematical method may also be developed for signal stabilization using both deterministic and random signals based on Dual Input Describing function (DIDF) and Random Input Describing Function (RIDF) respectively. iii. The phenomena of Synchronization and De-synchronization can be observed/identified analytically using Incremental Input Describing Function (IDF), which can also be validated by digital simulations.