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AUTHOR(S): Kartik Chandra Patra, Asutosh Patnaik
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ABSTRACT The paper describes a systematic approach for analysing and controlling limit cycles (LC) in three-dimensional multivariable systems with memory nonlinearities. The main contributions of the paper can be summarized as follows: A novel graphical method is introduced, utilizing computer graphics and geometric tools to predict the occurrence of LC in systems with memory nonlinearities. This technique provides a systematic way of analysing the behaviour of these systems, particularly in the context of their dynamic response and limit cycle behaviour. Once limit cycles are detected in an autonomous system, the paper explores techniques for quenching/suppression these limit cycles. The primary technique proposed for quenching involves the application of high-frequency dither signals, which can be either deterministic or random. These signals help to stabilize the system and prevent undesirable oscillations. Another approach explored in the paper for suppressing limit cycles is the pole placement technique. This technique involves the arbitrary/optimal selection of a state feedback gain matrix K, satisfying state controllability conditions and using the Ricatti equation respectively. This modifies the system's dynamics to suppress the limit cycle behaviour. By adjusting the poles of the system's transfer function, the authors aim to shift the system's behaviour away from oscillatory states. To deal with the complexity of the nonlinearities (especially memory-type nonlinearities), the paper introduces a reduction in complexity through harmonic linearization or harmonic balance. This technique allows for a simplified model of the system by approximating nonlinear effects with harmonic terms, making the problem more tractable. The paper further simplifies the analysis by assuming that the three-dimensional system predominantly exhibits limit cycles at a single frequency. This assumption reduces the complexity of the system and makes the prediction of limit cycles more manageable. The proposed methods are validated through digital simulations, which were implemented using MATLAB codes and the SIMULINK Toolbox. These simulations serve as proof-of-concept for the proposed techniques, demonstrating their effectiveness in practical scenarios. The paper offers innovative methods for predicting, quenching, and suppressing LC in complex systems with memory nonlinearities. The use of graphical tools, high-frequency dithering, pole placement techniques, and harmonic linearization all contribute to a more efficient and understandable approach to controlling limit cycle behaviour in multivariable systems. The approach is validated through simulations, making it a promising framework for future applications in control systems and nonlinear dynamics. |
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KEYWORDS Signal stabilization, Quenching, Suppression, Limit Cycles, 3×3 Nonlinear Systems, Backlash Nonlinearities |
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Cite this paper Kartik Chandra Patra, Asutosh Patnaik. (2025) Analysis and Control of 3×3 Systems with Memory Nonlinearities: Estimation, Stabilization, and Suppression Limit Cycle. International Journal of Mathematical and Computational Methods, 10, 93-124 |
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