Tuning of Complex Coefficient Fractional Complex Order Controllers for a Generalized System Structure—An Optimisation Approach

Abstract

In general, controllers for integer, fractional and fractional complex order systems with dead time are tuned either analytically or through optimisation approaches. This is performed by accounting only the positive frequency data of the system. However, this way of controller tuning is applicable only for integer/fractional order systems containing real coefficients. These real-valued systems have an even symmetrical magnitude and odd symmetrical phase behaviour in frequency responses. Whereas for integer/fractional order systems containing complex coefficients and fractional complex order containing real/complex coefficients, an unsymmetrical behaviour is seen in its magnitude and phase responses. The conventional way of tuning controllers for such systems by considering only its positive frequency response results with reduced stability margins and in turn degrades its time response behaviour. Hence, controllers with complex coefficients are required for such systems to improve its time response and have a better stability margins. Hence, this chapter proposes complex coefficient fractional complex order controllers for such systems. An optimization approach is used to tune these controller parameters by considering both positive and negative frequency information of system with complex coefficients and complex order derivatives plus dead time. Numerical simulations are performed for a case study with the proposed and real-valued fractional order controllers.