Fuzzy Gain Scheduling Design Based on Multiobjective Particle Swarm Optimization Edson B.M. Costa Federal University of Maranh˜ ao Av. dos Portugueses, s/n, Bacanga CEP: 65001-970 S˜ ao Lu´ ıs-MA, Brazil Email: edsonbmarquesc@hotmail.com Ginalber L.O. Serra Federal Institute of Education, Science and Technology Av. Get´ ulio Vargas, 04, Monte Castelo, CEP 65030-005, S˜ ao Lu´ ıs - MA, Brazil Email: ginalber@ifma.edu.br Abstract—In this paper, a fuzzy gain scheduling control approach based on gain and phase margins specifications for nonlinear systems with time varying delay, is proposed. A multi- objective particle swarm optimization (MPSO) strategy is defined to tune the fuzzy gain scheduling controller parameters for each operating condition, so the gain and phase margins of the fuzzy control system are close to specified ones. Experimental results show the efficiency of the proposed methodology for control of a thermal plant with time varying delay. I. I NTRODUCTION Fuzzy control was developed based on the fuzzy set theory proposed by Lotfi A. Zadeh [1] and has been widely applied to various areas including power systems, telecommunications, mechanical/robotic systems, automobile, industrial/chemical processes, aircrafts, motors, medical services, consumer elec- tronics, chaos control, nuclear reactors, and others areas [2]. Modern processes in industry presents complexities such as little knowledge about the system, strong nonlinear dynamics, the high degree of uncertainty and time varying characteristics. Several studies have shown that fuzzy gain scheduling systems have become an important tool to control these types of processes [3], [4], [5], [6], [7]. In general, gain-scheduling encompasses the attenuation of the nonlinear dynamics over a range of operations, the attenuation of the environmental time- variations or the attenuation of the parameter variations and uncertainties. Fuzzy gain scheduling involves offline lineariza- tion of the nonlinear dynamic system at multiple operating conditions and the design of corresponding linear controllers at each operating condition [8]. In this paper, a fuzzy gain-scheduling controller design based on robust stability criterion via MPSO for Takagi- Sugeno (TS) fuzzy model [9], is proposed. The plant to be controlled is identified from input-output experimental data, by using the fuzzy C-Means clustering algorithm and least- squares estimator for antecedent and consequent parameters estimation, respectively. The MPSO algorithm is used to tune the fuzzy gain scheduling controller parameters, via Parallel Distributed Compensation (PDC) strategy [10], based on gain and phase margins specifications, according to identified fuzzy model parameters of the plant to be controlled for each operat- ing condition. Experimental results for fuzzy gain scheduling control of a thermal plant with time varying delay is presented to illustrate the efficiency and applicability of the proposed methodology. This paper is organized as follows: In section II the Takagi- Sugeno fuzzy modelling approach, is presented. In section III the fuzzy gain scheduling control design approach, is presented. In section IV experimental results for fuzzy gain scheduling control of a thermal plant with time varying delay are shown. Finally, in section V the conclusions of this work are discussed. II. TAKAGI -SUGENO FUZZY MODELLING APPROACH: OPERATING CONDITIONS ESTIMATION In this paper, the fuzzy gain scheduling controller design procedure is based on the representation of a given nonlinear plant in terms of the TS fuzzy model given by:  ∣=1,2,..., : IF  1    1 AND ⋅⋅⋅ AND       THEN    ()=   0 +  1  −1 +⋅⋅⋅+    − 1+  1  −1 +  2  −2 +⋅⋅⋅+    −  −   / (1) where    is the time delay for -th inference rule   ,  is the number of rules,   and   are the orders of the numerator and denominator of the transfer function with   1,2,⋅⋅⋅ , and   1,2,⋅⋅⋅ , its parameters, respectively. The variable   belongs to a fuzzy set    with a truth value given by a membership function    : ℝ → [0, 1]. The methodology for estimation of the parameters of the antecedent and the consequent of the fuzzy rules, given in equation (1), is discussed below. A. Antecedent parameters estimation The antecedent parameters of the TS fuzzy model are estimated by Fuzzy clustering. The fuzzy clustering algorithms are used to construct fuzzy models from experimental data. Among the most popular algorithms are the following: Fuzzy C - Means (FCM), Gustafson-Kessel (GK) and Fuzzy Maxi- mum Likelihood Estimates (FLME) algorithms [11], [12]. The Fuzzy C-Means (FCM) clustering algorithm is used in this paper, and for more detail it can be seen in [13]. B. Consequent parameters estimation The TS fuzzy model output ˆ  is computed by taking the weighted average of the individual rules contributions. Given the normalized degree of fulfillment,   , then the inference formula of the TS fuzzy model in (1) can be written as the difference equation