Design of an effective and efficient fractional order PID (FOPID) controller, as a generalization of a standard PID controller based on fractional order calculus, for an industrial control system to obtain high-quality performances is of great theoretical and practical significance. From the perspective of multi-objective optimization, this paper presents a novel FOPID controller design method based on an improved multi-objective extremal optimization (MOEO) algorithm for an automatic regulator voltage (AVR) system. The problem of designing FOPID controller for AVR is firstly formulated as a multi-objective optimization problem with three objective functions including minimization of integral of absolute error (IAE), absolute steady-state error, and settling time. Then, an improved MOEO algorithm is proposed to solve this problem by adopting individual-based iterated optimization mechanism and polynomial mutation (PLM). From the perspective of algorithm design, the proposed MOEO algorithm is relatively simpler than NSGA-II and single-objective evolutionary algorithms, such as genetic algorithm (GA), particle swarm optimization (PSO), chaotic anti swarm (CAS) due to its fewer adjustable parameters. Furthermore, the superiority of proposed MOEO-FOPID controller to NSGA-II-based FOPID, single-objective evolutionary algorithms-based FOPID controllers, MOEO-based and NSGA-II-based PID controllers is demonstrated by extensive experimental results on an AVR system in terms of accuracy and robustness.
In the past decades, a great many advancements have been gained in control theories and practices -, proportional-integral-derivative (PID) control is still widely recognized as one of the simplest yet most effective control strategies in the control industry -. As a generalization of a standard PID controller based on fractional order calculus, fractional order PID (FOPID) controller namely PIλD µ controller was firstly proposed by Podlubny , and it has been demonstrated to provide better control performance than standard integer order PID controller due to extra degrees of freedom introduced by an integrator of fractional order λ and a differentiator of fractional order µ. As a consequence, FOPID controller has attracted increasing attentions by the academic and industrial community -. On the other hands, the introduction of extra parameters in a FOPID controller also increases the difficulty of tuning satisfied values of parameters, so how to design and tune an optimal FOPID controller to obtain high-quality performances, such as high stability, satisfied transient response, excellent steady performance, and good robustness, is of great theoretical and practical significance, but is still far from well-understood. In the attempt to address this issue, some researchers have made a great deal of efforts from the following different respective of analytical methods - and evolutionary algorithms-based methods , -. More specifically, the evolutionary algorithms, such as genetic algorithm (GA) , chaotic ant swarm (CAS) , particle swarm optimization (PSO) , , differential evolution (DE) , artificial been colony algorithm , hybrid algorithm combing with electromagnetism-like algorithm and GA , have been utilized for the design of FOPID controllers. Nevertheless, most of the reported research works focus on single-objective optimization for the design of FOPID controllers. In practice, multi-objective optimization algorithms - are required to design FOPID and PID controllers because of contradictory objective functions and performance metrics, e.g., integral of the time multiplied squared error (ITSE) and the integral of the squared deviation of controller output (ISDCO) . However, the reported studies concerning design of FOPID controller based on multi-objective evolutionary algorithms (MOEAs) is considered as just a beginning because only NSGA-II  has been extended to design FOPID controllers so far. This paper presents an alternative effective MOEA method based on multi-objective extremal optimization called MOEO for the design of FOPID controller in an automatic regulator voltage (AVR)  system, which is used to maintain the terminal voltage of a synchronous generator at a desired level.