Asian Journal of Control, Vol. 20, No. 2, pp. 1–14, March 2018 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1549 CONSTRAINED NONLINEAR-BASED OPTIMISATION APPLIED TO FUZZY PID CONTROLLERS TUNING Paulo Gil, Ana Sebastião, and Catarina Lucena ABSTRACT This paper aims at studying the optimal Fuzzy Proportional–Integral– Derivative controllers’ tuning problem by considering two different nonlinear constrained optimisation techniques. One relying on a Hessian-based analytical approach, and the other based on a differential evolutionary method. In the case of offine implementation, two basic frameworks are under assessment, depending on the controller parameters to be adjusted. For online scaling factors and membership functions’ width tuning, its implementation is based on the parallel computation paradigm. The perfor- mance index is described by a quadratic cost function, taking as arguments control errors and the increment of control actions. Constraints on the scaling factors, membership functions’ width, as well as on the system inputs and outputs are also included in the optimisation problem. Experiments carried out on a benchmark system favour the offine joint optimisation based on the differential evolutionary approach of scaling factors and membership functions’ width. Key Words: Fuzzy PID controllers tuning, constrained nonlinear optimisation, differential evolutionary computation, analytical optimisation, closed loop performance index. I. INTRODUCTION Proportional–Integral–Derivative (PID) controllers are still widely used in most automatic process con- trol applications. Several intrinsic factors come into play to foster this kind of technology, namely their func- tional/structural simplicity, easy tuning and low imple- mentation costs. It is well known that for linear invariant systems, or even when the system dynamics are slightly nonlinear, a PID controller with fxed gains can provide acceptable closed loop performance for a wide range of operating conditions. That, however, is not the case for moderate or signifcant nonlinearities, or even in the case of time varying systems. In such conditions PID con- trol invariably leads to closed loop under-performance or, ultimately, to the control system’s instability. In such cases, fuzzy logic theory [1] provides a straightforward Manuscript received February 15, 2016; revised October 7, 2016; accepted March 11, 2017. Paulo Gil is with the Department of Electrical Engineering, Faculty of Science and Technology, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal (corresponding author, e-mail: psg@fct.unl.pt). Paulo Gil and Ana Sebastião are with CTS-UNINOVA, Universidade NOVA de Lisboa, 2829-516 Caparica, Portugal. Paulo Gil and Catarina Lucena are with the CISUC - Centre for Informat- ics and Systems of the University of Coimbra, University of Coimbra, 3030-290 Coimbra, Portugal. Funding: This study was partially funded by Project CENTRO-07-ST24-FEDER-002003. Ethical approval: This article does not contain any studies with human partic- ipants or animals performed by any of the authors. framework for designing PID-like control structures, in particular Mamdani-type fuzzy controllers [2]. Fuzzy Logic Control (FLC) concerns a set of lin- guistic rules related by the dual concepts of fuzzy impli- cation and the compositional rule of inference. Fuzzy Logic was frst suggested by Lofti A. Zadeh [3], and it is recognised as having the built-in ability to deal with nonlinearities and uncertainties. This framework uses a methodology for reasoning similar to human decision- making, which makes FLC quit appealing in a number of ways, namely the fact that its design has lower costs than other nonlinear control techniques. Furthermore, it cov- ers a wide range of operating conditions, and allows the explicit incorporation of experts’ knowledge [4]. Among recent applications of fuzzy logic-based control systems are the clinker calcination process con- trol [5], smart grid voltage control [6], active and reactive power control [7], permanent magnet synchronous motor speed-regulation systems [8], dynamic balance control for biped robot walking [9], solar power plant control [10], load-frequency control in wind turbines [11], HVAC con- trol systems [12] and manipulators control [13], just to name a few. The reader is referred to [14] for a survey on industrial applications of FLC based control systems. Regardless the widespread of FLC systems, their implementation is still rather challenging, and in many instances it is particularly cumbersome to come up with a consistent set of linguistic variables, membership func- tions, rules and scaling factors. In addition, providing © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd