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Top1. Introduction
There are two types of controller operation. One is a regulator (Alfaro, 2002) when the reference value R(s) remains constant (a system behavior insensitive to disturbances is preferred). The other one is a servomechanism (Alfaro, 2003) when the reference R(s) may change over time (therefore good tracking is expected). For a process matching some predetermined design specifications and robustness according with a performance criteria, the controller tuning based on PID structure consists on setting their parameters 
, 
and 
, representing the proportional, integral and derivative criteria, respectively.
Before the controller tuning it is necessary to identify the dynamics of the process to be controlled and representing that by some model 
. It is common that the model is a low order transfer function (first or second order, plus time delay). After having characterized the dynamics of the process and the type of operation, the choosing of the method for controller tuning proceeds. The process model has great influence on the selected tuning rule. The selection of the tuning rule is based on the required performance and the desired robustness. O'Dwyer (2009) reports that 90% of the tuning rules developed to date are based on a model of first and second order plus time delay. The controller tuning rules most frequently used are not based on an integral performance criterion. The optimal tuning rules based on second-order models are just 14 of the 84 reported until 2009.
A comparative study of performance of different tuning classical methods for PI and PID controllers is achieved in Desanti (2004). This study concludes that tuning methods that require a Second Order System Plus Time Delay model (SOSPD) perform better than those that require a First Order Lag Plus time Delay model (FOLPD). This is the reason why the second order models plus time delay are dealt in this paper. The second order system plus time delay model was selected as representing the plants in order to compare the performance of a heuristic algorithm with the “best” techniques developed for PID controllers optimal tuning. For SOSPD models represented by Equations (1) and (2), 147 tuning rules have been defined based on the ideal structure of a controller PI/PID (O’Dwyer, 2009). In general, those rules are based on several relationships and/or conditions of the parameters defining the process model.
(1) (2)Where, Kp: Plant Gain; τm : Time Delay; Tm1, Tm2: Time constants of the plant; ξ: Damping factor of the plant.
In Mora (2004) and Solera (2006) the performance and robustness of some tuning rules are evaluated, and a complete analysis of the methods of tuning controllers based on SOSPD is made.
Each of the developed tuning rules for PI and PID controllers has only been applied to a certain group of processes. Alternative methodologies, such as design based on the root locus, tuning by pole-zero cancellation, tuning by the location of the closed-loop poles, among others, require cumbersome procedures and specialized knowledge in control theory. Additionally, most methods for optimal tuning of SOSPD systems require additional system information from experiments carried out directly on the plant; activities that are not always possible to perform because the presence of extreme stresses and oscillations which may create instability and damage to the system.