A reliability-based approach for the Pareto optimum design of proportional-integral-derivative (PID) controllers for systems with probabilistic uncertainty is presented. In this way, some non-dominated optimum PID controllers in the Pareto sense are found using four non-commensurable objective functions in frequency domain based on stochastic behaviour of a system with parametric uncertainties. Such conflicting objective functions are namely the probability of instability, the probability of failure to a desired frequency response, the variability of frequency response about deterministic behaviour, and the value of degree of stability from the Nyquist diagram's percentiles. The first three objective functions have to be minimized whilst the last has to be maximized simultaneously. It is shown that multi-objective Pareto optimization of such robust PID controllers using a recently developed diversity-preserving mechanism genetic algorithm unveils some very important and informative trade-offs among those objective functions. Consequently, some optimum PID controllers can be compromised and chosen from the Pareto frontiers.
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