Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

Abstract

Please cite this article as: Pérez-Molina, M., F.Pérez-Polo, M., Steady-state, self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/ 10. 1016/j.cnsns.2014.03.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThis paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov-Takens and Andronov-Poincaré-Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The BogdanovTakens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the AndronovPoincaré-Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.

Takens–Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system

“…This codimension-two bifurcation has been found and studied in a large number of systems considered in different fields (see, for instance, [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and references therein).…”

Precision concurrent speed and position tracking of brushed dc motors using nonlinear time-frequency control

“…Although parameters of the PID controller can be adjusted manually when the motor is running, it cannot guarantee robustness if the external loading changes dramatically. Pérez-Molina and Pérez-Polo (2014) studied the nonlinear dynamics of a PID controlled DC servomechanism under constant torque and random noise. Even though it was demonstrated that the chaotic behavior of the system could be used corporately with the PID controller to reach the desired set point, accuracy and practicability of the proposed methodology under complex working conditions remain an issue.…”