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The equation of the IPID controller is

(2.8)

where K_pi, K_jj and K_di and are the proportional, integral, and differential coefficient, respectively, where e(t) = y_d (t) − y(t) is the system error, y_d (t) is the reference input, y(t) is system response, and u(t) is controlled output [6, 9]. The FOPID controller is an extension of the conventional IPID controller with the integral and the differential orders as fractional one [7].

FOPID controller is represented as

(2.9)

λe indicates integral order.

µ indicates the differential order.

K_pf, K_if, and K_df are fractional-order controller gains.

Laplace transfer function of the controller is given as

(2.10)

The FOPID has additionally more adjustable parameters, λ and µ, than IPID controller and have five control parameters (Kpf, Kif, Kdf, λe, and μ) to find a better control performance [9]. For optimization, the GA has a possibility to come with five optimum parameter space to achieve best control performance.