### Efficient Sampling of PI Controllers in Evolutionary Multiobjective Optimization

Abstract

Proportional-Integral (PI) controllers remain as a practical and reliable solution for multivariable control for several industrial applications. Efforts to develop new tuning techniques fulfilling several performance indicators and guaranteeing robustness are worthwhile. Evolutionary multiobjective optimization (EMO) has been used for multivariable PI controller tuning, due to their flexibility and its advantages to depict the trade off among conflicting objectives. It is a regular practice bounding the search space as a hyperbox; nevertheless, the shape of the feasible space of PI parameters which are internally stable for a given control loop is irregular. Therefore, such hyperbox could enclose feasible and unfeasible solutions or contain a subset of the feasible set. In the former case, convergence capabilities of an algorithm could be compromised; in the latter case, search space is not fully explored. In this work, a coding mechanism is proposed in order to explore more efficiently the PI parameters feasible set (that is, all feasible solutions and only feasible solutions) in EMO. With the example provided, the advantages to approximate a Pareto front for 2, 3 and 5 objectives are shown, validating the mechanism as useful for EMO in multivariable PI controller tuning.