Suppressing Chaos Using Fractional Controllers

Summary:

Chaos in dynamical systems refers to a type of undesirable behavior where the output of the system becomes highly sensitive to the initial conditions. Chaos results in a complicated and unpredictable wandering of the system over longer periods of time. This undesirable behavior of dynamical systems underscores the crucial need to design controllers to suppress chaotic behaviors, which constitutes the main objective of this project. In this project, we design fractional controllers to suppress chaos in a class of dynamical systems. More specifically, for a general three-dimensional nonlinear integer-order system, we take two approaches in [1, 2] toward suppressing chaos, i.e., making the closed loop system locally stable at its fixed point.

For the controller we designed in [1], the control input is a fractional integrator of a linear combination of the states in the system when linearized around its fixed point. In [2], on the other hand, the control input is a combination of an integer-order derivative of one state and a fractional-order derivative of another state of the system. Our controllers can be applied to various well-known three-dimensional system such as the Lorenz system, Rossler system, Chua system, Chen system, and Lü system. The performance of our first controller is examined for the Chen system and a chaotic oscillator of canonical structure in [1], while our second controller is shown to perform well for a chaotic circuit in [2].

Reference

  1. S. Tavazoei, M. Haeri, S. Bolouki, M. Siami. “Using Fractional Order Integrator to Control Chaos in Single-Input Chaotic Systems.” Nonlinear Dynamics, vol. 55, no. 1-2, pp. 179-190. January 2009.
  2. S. Tavazoei, M. Haeri, S. Jafari, S. Bolouki, M. Siami. “Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations.” IEEE Transactions on Industrial Electronics, vol. 55, no. 11, pp. 4094-4101. November 2008.