Tuning of complex coefficient PI/PD/PID controllers for a universal plant structure

Controllers are tuned either analytically or through optimisation techniques for integer, fractional and fractional complex order plants with dead time by considering only the positive frequency information of the plant. This is applicable only for integer / fractional order plants containing real coefficients which have an even symmetrical magnitude and odd symmetrical phase behaviour in frequency responses. The frequency response for (i) integer / fractional order plants containing complex coefficients and (ii) fractional complex order plants containing real / complex coefficients have unsymmetrical magnitude and phase behaviour. Tuning of controllers for such plants by considering only its positive frequency response produces reduced stability margins and in turn deteriorates its time response. Hence, tuning of controllers for such plants requires both positive and negative frequency information which in turn demands complex coefficient controllers. To address the problem, complex coefficient PI/PD/PID controllers are proposed in this paper. Unified expressions are derived for complex coefficient PI/PD/PID controllers by considering both positive and negative frequency information of plant with complex coefficients and complex order derivatives plus dead time. Numerical simulations are performed for three different case studies with the proposed complex coefficient integer order controllers and results are compared with real coefficient PI/PD/PID controllers under various operating conditions.