IAE optimization of delayed PID control loops using dimensional analysis approach

A specific issue of the PID control loop with time delay is the contradiction of its infinite-order dynamics with the only three controller parameters used to adjusting its behaviour. For selecting the optimum PID parameters the IAE criterion has been used as performance measure of the disturbance rejection in the investigated control loop. In order to obtain the results in a generic form the dimensionless description of the control loop, originally introduced in [14], was applied. The plant model is based on the dimensional analysis, reducing the relevant parameters of the control loop to a pair of similarity numbers, namely the so-called laggardness (ϑ) and swingability (λ) numbers. The IAE optimum search is performed by means of the gradient-based method over a representative set of options of λ, ϑ, and the optimum PID controller parameters are assessed for the whole considered area of λ, ϑ. To each of the optimum rejection responses a characteristic quasi-polynomial corresponds and then the rightmost part of its spectrum can be evaluated. The large scale spectral analysis has shown that for all of the investigated options a double pair group of poles results as dominant in the control loop dynamics. The final result of the paper consists in summarizing the IAE optimum settings of PID and comparing the obtained natural frequency angles of the control responses and their damping.