PIλDµ Controllers for Suppression of Limit Cycle in a Plant with Time Delay and Backlash Nonlinearity

This paper evaluates the existence of a periodic limit cycle oscillation in a system with backlash nonlinearity in the presence of time delay. An armature voltage-controlled DC motor system is studied in this regard whose output signifies accuracy in position control. An analytical solution for the limit cycle based on the Describing Function (DF) method is obtained whose authenticity is verified with the Nyquist contour-based graphical method and the digital simulations. The effect of parametric changes on the magnitude and frequency of the limit cycle is examined in this article. Integer and non-integer order proportional-integral-derivative (PID) controllers are designed to eliminate these undesirable periodic oscillations present in the system. Multiple optimization techniques considering error-based, time domain specification-based objective functions are scrutinized through statistical tests towards the parameter estimation of the applied controllers. Observations reveal that while the Moth flame optimizer (MFO) with Integral time absolute error (ITAE) produces superior results for the PID controller, the MFO with the Integral time square error (ITSE) provides better results for the FOPID controller. Further, the gain and phase margin-based loop shaping method is also used for an analytical calculation of the controller parameters. Out of the five loop shaping constraints, superior results are obtained by considering robustness towards gain variation constraint as an objective function, and the rest as nonlinear constraints. Simulation studies suggest the efficiency of the utilized controllers in quenching the periodic limit cycle oscillations. The superiority of the FOPID controller over the PID controller is validated by considering suitable performance-based comparisons. The effectiveness of the designed controllers is also tested against the variations in system parameters. Further, the physical realizations of the integer and fractional order PID controllers are performed through Oustaloup recursive filter approximation.