Nonlinear Control of Chaotic Forced Duffing and Van der Pol Oscillators

Abstract

This paper discusses a novel technique and implementation to perform nonlinear control for two different forced model state oscillators and actuators. The paper starts by discussing the Duffing oscillator which features a second order non-linear differential equation describing complex motion whereas the second model is the Van der Pol oscillator with non-linear damping. A first order actuator is added to both models to expand on the chaotic behavior of the oscillators. In order to control the system without comprising linearization, Lyapunov non-linear control was used. A control Lyapunov function was tailored to the system. This led to improved maneuverability of the controller and the performance of the overall system. The controller was found to be highly efficient in system tracking and had swift response time. Simulations were performed on both the uncontrolled and controlled cases. Both simulation results ultimately confirmed the effectiveness of the proposed controller.

Dynamic Lyapunov Machine Learning Control of Nonlinear Magnetic Levitation System

“…Lyapunov control has proven successful in managing complex nonlinear oscillators to a certain extent of oscillator frequency ω = 2.5 Hz [7]. In order to improve the system stability at a higher ω, deep learning was introduced in [4].…”

“…The research completed in [7] revealed that the controller parameters had substantial impact on the desired system outcome and error; therefore, the research in [4] successfully presented a deep learning approach that would allow 1 of the controller parameters to change while the system is operational thus tackling sudden changes in stability. One of the disadvantages found through the survey in [10] is that the system learning/relearning process robustness is significantly reduced if more parameters require updates simultaneously.…”

“…Despite being efficient in managing midcomplexity system behaviors, they become susceptible to high complexity cyber-physical systems [6]. The control Lyapunov function (CLF) has been employed successfully to control complex nonlinear duffing and Van der Pol systems [7][8][9]. The idea behind using nonlinear controllers allows for a lossless control strategy to avoid system linearization while the integration of deep learning allows for the continuous adjustment and selection of system and control parameters.…”

A New Chaotic System with Stable Equilibrium: Entropy Analysis, Parameter Estimation, and Circuit Design

“…Near a bifurcation point, the system’s state becomes slower. In other words, the transient time increases near a bifurcation point [ 46 , 47 ]. In order to use Kolmogorov–Sinai entropy to anticipate a bifurcation point, we calculated it without removing the transient time of the trajectory.…”

Numerical Criterion for the Duration of Non-Chaotic Transients in ODEs

“…The purpose of these transformations is to simplify the equations so as to be able to use numerical methods to solve them [3,4]. A great number of physical systems are modelled by second-order ODEs, for example, the Duffing or Van der Pol equations [3,5,6,7,8,9]. These equations, after their transformation into a system of equations, are analysed on the phase plane.…”