Suppression Limit Cycles in 2x2 Nonlinear Systems with Memory Type Nonlinearities
Authors: Patra Kartik Chandra , Kar Namrata , Patnaik Asutosh 
Published: November 2024

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) systems. 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’ people have started discussing suppression of LC which limits the performance of most of the physical systems in the world. It is a formidable task to suppress the limit cycles for 2x2 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, automation industry and other occasions like Load Frequency Control (LFC) in multi area power systems. The feasibility of suppression of such nonlinear self-oscillations has been explored in case of the memory type nonlinearities. Backlash is a common memory type nonlinearity which is an inherent Characteristic of a Governor, used for usual load frequency control of an inter-connected power system and elsewhere. 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 and is done through state feedback. The Governing equation is d/dt [X(t)] =(A-BK) X: which facilitates the determination of feedback gain matrix K for close loop Poles / Eigen values placement where the limit cycles are suppressed/eliminated in the general multi variable systems. The complexity involved in implicit non-memory type or memory type nonlinearities, it is extremely difficult to formulate the problem for 2x2 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 2x2 system and the method is made further simpler assuming a 2x2 system exhibits the LC predominately at a single frequency. Hence in the proposed work an alternative attempt has been made to develop a graphical method for the prediction of Limit Cycling Oscillations in 2x2 memory type Nonlinear systems which not only reduces the complexity of formulations but also facilitates clear insight into the problem and its solution. The present techniques are well illustrated with an example and validated / substantiated by digital simulation (developed program using MATLAB codes) and use of SIMULINK Tool Box of MATLAB software. The present work has the brighter future scope of: Adapting the Techniques like Signal Stabilization and Suppression LC for 3x3 or higher dimensional nonlinear systems through an exhaustive analysis. Analytical/Mathematical methods 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. 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.