Multi-objective control design of the nonlinear systems using genetic algorithm

Abstract

Abstract—The problem of multi–objective feedback controllerdesign of nonlinear systems is solved in this paper. The T–S fuzzymodel is adopted to describe the nonlinear systems and geneticalgorithm is used to identify the T–S fuzzy model. The identifiedT–S fuzzy model is reduced by applying Higher Order SingularValue Decomposition (HOSVD) method. Based on the reduced T–S fuzzy model, an optimal state feedback controller is designed byachieving the trade-off among three conflicting object functionsusing the optimal Pareto frontier. The simulation results revealthe effectiveness of the proposed method. I. I NTRODUCTION Recently, multi–objective optimization design has attractedgreat interest in engineering problems that have conflictingobjectives [1], [2]. For example, there always exist morethan one performance measures in practical control problemwhich should be optimized simultaneously (e.g., rise time,overshoot, control efforts, ...). These performance measuresare posed as control deign objectives that are normally inconflict with each other. Since the conflicts exist among theobjective functions, controller design can be formulated as amulti–objective optimization problem (MOP) so that the trade–offs among the objectives can be found consequently [3].To design high performance controllers for highly nonlinearsystems is one of the top challenges faced by control engi-neers. Although nonlinear controllers have demonstrated verypromising performance and robustness, the stability analysisand controller synthesis are often difficult both in theoryand practice. Therefore, many efforts have been made tomodel the dynamics of complex systems which allows thedesigners use the linear stability analysis and control designtechniques [4], [5]. One of the powerful tools in approximatinga complex nonlinear system is the well–known Takagi–Sugeno(T–S) fuzzy modelling [6], [7]. By using T–S fuzzy model, anonlinear system can be modelled with N linear models, andthe system behaviour is described by a Convex Combinationof the N linear systems [6]. In fact, such models containa polytope consisting of N linear models (vertices). Thesemodels are also known as polytopic linear parameter–varying(LPV) systems [4]. However, one common drawback remainsin these models. When the system has a multidimensionalvector of the varying parameters, it results in using the largenumber of IF–THEN rules which increase the computationalcomplexity of the T-S fuzzy model. To reduce the large rulebase number, various attempts have been made over the years[8], [9], [10]. The rule base reduction is very important for thecontrol system design especially when a fuzzy system has to beimplemented in real time. Yam et al. presented a method basedon Singular Value Decomposition (SVD) to find a minimalnumber of fuzzy rules from sampled data [9].In this paper we use genetic algorithms to identify theoptimal T–S fuzzy model of the nonlinear system. A setof input–output is considered for modelling the nonlinearsystem. Then, genetic algorithm is used to minimize the errorbetween the measured output and the model output. Also, thereduced T–S fuzzy model is obtained from the identified oneby HOSVD method. Multi–objective genetic algorithm opti-mization approach is used to optimally design state–feedbackcontroller for nonlinear system based on the reduced T–S fuzzymodel. Both state response and control signal performances areconsidered in formulating the objective functions. The trade–offs among design objectives are found using the optimalPareto frontier for the optimal state–feedback controllers. Thedeveloped controller is tested on the benchmark–TranslationalOscillator with an eccentric Rotational proof of mass Actuator(TORA). The simulation results verify the performance of thedeveloped controller.This paper is organized as follows. Section 2 presentsa nonlinear system and GA–based method to identify theT–S fuzzy model. Multi–objective control design method isproposed in section 3. In section 4, the modelling results andthe reduced T–S fuzzy model obtained by HOSVD method arepresented. Also, simulation results of the optimal controllerdesigns are given in section 4. Finally, Section 5 gives someconcluding remarks of this work.II. O