Abstract
Abstract In this paper, we have studied numerically free vibration of modified Mathieu Equation with quadratic damping using time-delayed FOPID (fractional-order Proportional-Integral-Derivative) controller focussed on displacement feedback. A special filter known as Oustaloup filter is employed to carry out the fractional order differetiation and integration numerically with the help of Matlab Simulink. The effect of time delayed FOPID with displacement feedback on control performance of the parametrically excited system based on modified Mathieu equation are studied by time response. The results show that the system becomes unstable when time delay increases. It also shows that the use of FOPID becomes quite advantageous over traditional integer order PID (IOPID) control only when the natural frequency is kept low. Equivalent damping of the system could be enhanced to increase the differential gain in order to stabilize the system. On increasing the proportional and integral gain, the system gets destabilized which is in good agreement with the classical integer-order PID control
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