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ASOC-733; No. of Pages 9

ARTICLE IN PRESS
Applied Soft Computing xxx (2009) xxx–xxx

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Applied Soft Computing
journal homepage: www.elsevier.com/locate/asoc

A framework for the automatic synthesis of hybrid fuzzy/numerical controllers
Daniele Magazzeni
Department of Computer Science, University of L’Aquila, Via Vetoio, Coppito,L’Aquila, Italy

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In this paper, a framework for the automatic synthesis of hybrid fuzzy/numerical controllers is proposed. The methodology is based on model checking and on a very precise analysis of a system. This allows one to synthesize optimal numerical controllers and then use them to consistently improve fuzzy controllers. Moreover, we present a newapproach that integrates the numerical and the fuzzy components and automatically outputs a hybrid controller. Such a hybrid controller exploits the optimality of numerical controllers and the robustness of fuzzy ones, and it is very compact and fast to read thanks to the use of OBDDs. We apply our methodology to two benchmark problems, the dc motor and the inverted pendulum. The results show thatthe hybrid controller can handle linear as well as nonlinear systems outperforming both the numerical and the fuzzy controllers. © 2009 Elsevier B.V. All rights reserved.

Article history: Received 22 December 2008 Received in revised form 19 October 2009 Accepted 16 November 2009 Available online xxx Keywords: Hybrid control Fuzzy logic Numerical control Controller synthesis Model checking

1.Introduction Soft computing refers to the integration of artificial intelligence techniques in hybrid frameworks for solving real world problems [1]. In particular, much work is being done to combine different control paradigms in order to obtain hybrid controllers with better performance. For example, in [2], neural networks and dynamic programming are used to control a nonlinear process. In thiscontext, the use of fuzzy logic together with other approaches has been extensively worked out. For example, in [3] hybrid controllers making use of fuzzy logic and neural networks are proposed, while in [4] genetic algorithms are used to generate and/or calibrate fuzzy controllers. Indeed, fuzzy control represents a powerful technique to cope with continuous systems. In this paper, we focus onthe integration of fuzzy and numerical control, starting from previous works based on cell mapping. Cell mapping [5] is based on the discretization of state variables of the system, partitioned into cells, and represents a very efficient computational technique for the global analysis of continuous systems. Hsu used cell mapping to generate a numerical controller, that is a control table containingstate-action pairs [6]. However, the values in the table are discretized and this can introduce steady state errors when dealing with continuous systems. Furthermore, even using a very fine discretization, the table can require a lot of memory for a large number of cells. In contrast, a fuzzy logic

controller is continuous and only requires memory for the input fuzzy set definitions and theoutput function parameters. Much work is being done to provide methodologies for the automatic calibration of fuzzy logic controllers in order to improve their performance. In particular, cell mapping has been used to generate control tables to improve the quality of fuzzy controllers [7–9]. However, to the best of our knowledge, no previous works have addressed the problem of simultaneously using afuzzy controller together with an optimal control table. In this context, a very important issue is to provide a methodology for the automatic generation of such a hybrid controller. 1.1. Our contribution In this paper, we propose a new approach based on the following considerations: 1. Model checking techniques [10,11] allow one to explore huge state spaces (also up to billions of states) and...