Designing an Undamped Oscillator Using Fractional Order Delayed Systems

This paper studies undamped oscillation in fractional order delayed systems. Laplace transformation and Mittag-Leffler function is adopted to solve the fractional order delayed differential equations. We seek to find a relation between the system parameters and the oscillation frequency and then adjust the system's coefficients so that all the roots of the characteristic equation, except the roots on the imaginary axis, lie on the left-hand side of the imaginary axis. We also obtain amplitude of the oscillation using the asymptotic behavior of the Mittag-Leffler function and calculate the transfer function residue at the frequency of oscillation. Since the other roots of the characteristic equation lie in the left-hand side of the imaginary axis, their contribution decay to zero as time tends to infinity. Finally, numerical simulation results are provided to verify the analysis.