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The optimum design of a controller plays a significant role in obtaining an adequate response from the process plant. However, in several areas of fractional order control system design, the coefficient diagram method (CDM) has not been thoroughly explored. The article proposes an optimal non-integer PIλDμ controller integrated with the CDM to accurately regulate the location of an unstable Magnetic Ball Levitation System (MLS). The controller maintains a predetermined force on the electromagnetic coil, allowing the ball to levitate. The controller's design entities are derived through the use of the CDM and a recently proposed Mayfly optimizer algorithm(MOA) that incorporates IAE, ITAE, ISE, and ITSE performance criteria, offering efficient, intelligent, and speedy solutions to complex design problems. The paper aims to leverage the powerful computational capabilities of fractional calculus (FC) in combination with the benefits of the CDM control strategy across various fields of control theory. The article also proposes a design methodology for the CDM-PIλDµ controller, which offers improved performance for systems containing non-linear and time-varying parameters. Simulation results demonstrate a comparativeanalysis of the transient response, including optimum overshoot, rise time, and settling time. [ABSTRACT FROM AUTHOR]

In this paper, the homotopy perturbation method will be applied to find the approximate solution of fractional order random ordinary differential equations, in which the fractional order derivatives and integrals are defined using Caputo and Riemann-Liouville definitions of fractional derivatives and integrals, respectively. Also, the convergence of the approximated solution is stated and proved in this work. The work is verified for three different examples, which are simulated for different generations of Brownian motion. [ABSTRACT FROM AUTHOR]