Impact of fractional order integral performance indices in LQR based PID controller design via optimum selection of weighting matrices

A continuous and discrete time Linear Quadratic Regulator (LQR) based technique has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are selected as the optimal state-feedback gains corresponding to the standard quadratic cost function. Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator designs while minimizing another integral performance index which comprises of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. Next, the proposed methodology is applied with fractional order (FO) integral performance indices. The impact of these FO objective functions on the LQR tuned PID control loops is also highlighted, along with the achievable cost of control.