Novel Adaptive Charged System Search algorithm for optimal tuning of fuzzy controllers Radu-Emil Precup a,⇑ , Radu-Codrut ß David a , Emil M. Petriu b , Stefan Preitl a , Mircea-Bogdan Ra ˘dac a a ‘‘Politehnica’’ University of Timisoara, Department of Automation and Applied Informatics, Bd. V. Parvan 2, RO-300223 Timisoara, Romania b University of Ottawa, School of Electrical Engineering and Computer Science, 800 King Edward, Ottawa, ON K1N 6N5, Canada article info Keywords: Fuzzy logic-based Adaptive Charged System Search algorithms Optimization problems Process gain sensitivity Sensitivity models Takagi–Sugeno PI-fuzzy controllers abstract This paper proposes a novel Adaptive Charged System Search (ACSS) algorithm for the optimal tuning of Takagi–Sugeno proportional–integral fuzzy controllers (T–S PI-FCs). The five stages of this algorithm, namely the engagement, exploration, explanation, elaboration and evaluation, involve the adaptation of the acceleration, velocity, and separation distance parameters to the iteration index, and the substitu- tion of the worst charged particles’ fitness function values and positions with the best performing particle data. The ACSS algorithm solves the optimization problems aiming to minimize the objective functions expressed as the sum of absolute control error plus squared output sensitivity function, resulting in opti- mal fuzzy control systems with reduced parametric sensitivity. The ACSS-based tuning of T–S PI-FCs is applied to second-order servo systems with an integral component. The ACSS algorithm is validated by an experimental case study dealing with the optimal tuning of a T–S PI-FC for the position control of a nonlinear servo system. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The design and tuning of optimal control systems is often based on linear or linearized models of the controlled processes. How- ever, in real life situations, many processes are subjected to para- metric variations which result in models that are either nonlinear or only locally linearized around several nominal operat- ing points or trajectories. In order to solve the problems occurring in these situations, a sensitivity analysis with respect to the para- metric variations of the controlled processes is required. As shown in Rosenwasser and Yusupov (2000), the uncontrolla- ble process parametric variations lead to undesirable behaviors of the control systems. While the fundamental deviation of a control system relative to its nominal trajectory is described by the control error, the additional deviation can be described by the output sen- sitivity function in the sensitivity model. Solving the optimization problem for the usually non-convex objective functions used in many control systems is not a trivial task as it can lead to several local minima (Angelov & Yager, 2013; Bayam, Liebowitz, & Agresti, 2005; Blaz ˇic ˇ et al., 2013; Hor- váth & Rudas, 2004; Iliadis, Kitikidou, & Skoularik, 2012; Linda & Manic, 2011; Srivastava, Chis, Deb, & Yang, 2012). Fuzzy control, as a relatively easily understandable nonlinear control strategy, is successfully embedded in these optimization problems as a conve- nient way to solve the systematic design and tuning of these con- trol systems (Castillo & Melin, 2012;Chiang & Liu, 2012; Feng, 2006; Precup & Hellendoorn, 2011;Precup et al., 2012c; Škrjanc, Blaz ˇic ˇ, & Agamennoni, 2005). The optimal tuning of fuzzy control- lers allows them to cope with non-convex or non-differentiable objective functions due to controllers’ structures and nonlineari- ties, and to process complexity, which can lead to multi-objective optimization problems. Some current evolutionary-based optimi- zation approaches to the parameter tuning of fuzzy control sys- tems include genetic algorithms (Onieva, Milanés, Villagrá, Pérez, & Godoy, 2012; Pérez, Milanés, Godoy, Villagrá, & Onieva, 2013), Simulated Annealing (Precup, David, Petriu, Preitl, & Ra ˘dac, 2012a; Jain, Sivakumaran, & Radhakrishnan, 2011), Particle Swarm Optimization (PSO) (Bingül & Karahan, 2011; Oh, Jang, & Pedrycz, 2011; Precup et al., 2013a), Gravitational Search Algorithms (GSAs) (Precup, David, Petriu, Preitl, & Ra ˘dac, 2012b; Precup et al., 2013a), Ant Colony Optimization (Chang, Chang, Tao, Lin, & Taur, 2012), cross-entropy (Haber, del Toro, & Gajate, 2010), migration algo- rithms (Vašc ˇák, 2012), chemical optimization (Melin, Astudillo, Castillo, Valdez, & Garcia, 2013), Charged System Search (CSS) algo- rithms (Precup, David, Petriu, & Preitl, 2011), etc., in several fuzzy control system structures. The appropriate tools specific to fuzzy systems (Ahmed, Shakev, Topalov, Shiev, & Kaynak, 2012; Akın, Khaniyev, Oruç, & Türks ßen, 2013; Baranyi et al., 2002; Göleç, Mur- at, Tokat, & Türks ßen, 2012; Johanyák, 2010; Precup et al., 2013a; 0957-4174/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.07.110 ⇑ Corresponding author. Tel.: +40 256 40 32 29/30/26; fax: +40 256 40 3214. E-mail addresses: radu.precup@aut.upt.ro (R.-E. Precup), davidradu@gmail.com (R.-C. David), petriu@eecs.uottawa.ca (E.M. Petriu), stefan.preitl@aut.upt.ro (S. Preitl), mircea.radac@aut.upt.ro (M.-B. Ra ˘dac). Expert Systems with Applications 41 (2014) 1168–1175 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa