New Foundation of Automatic Control Modeling and Analysis Using Arithmetic Fuzzy Logic-based Representation in Fully Fuzzy Environment Hassen Taher Dorrah, M. IEEE, Ph.D. Department of Electrical Engineering, Faculty of Engineering, Cairo University, Giza, Egypt dorrahht@aol.com Walaa Ibrahim Mahmoud Gabr, Ph. D. Egyptian Holding Company of Electricity, Ministry of Electricity and Energy, Abbassia, Cairo, Egypt, On temporary leave to SDA Engineering Canada Inc. walaa_gabr@yahoo.com Abstract - This paper presents the new foundation of the fuzzy modeling and analysis of automatic control systems in a fully fuzzy environment. The approach is based on the normalized fuzzy matrices and an extension of the Arithmetic and Visual Fuzzy Logic-based Representations developed recently by Gabr and Dorrah. The approach is also suitable for fuzzy dynamical systems where all system coefficients are expressed as fuzzy parameters. It is simply based on the assignment of corresponding fuzzy levels for parameters uncertainty that is made in a heuristic way circumventing the previous difficulties in assuming probabilistic or membership functions. Implementations of the approach are carried out for solving selected automatic control problem with their parameters expressed in fully fuzzy environments. These problems involve fuzzy impulse response of systems, fuzzy Routh-Hurwitz stability criteria, fuzzy controllability and observability, and the stabilization of inverted pendulum through pole placement technique. The results demonstrated the robustness of the proposed formulation and illustrated in a systematic way how the system parameters fuzziness effect on output results can be effectively tracked for monitoring and control. Finally, it is pointed out that the suggested new arithmetic and visual fuzzy logic-based representations will open the door for a unified theory for fuzzy modeling, analysis and control for continuous and discrete automatic control systems operating in fully fuzzy environments. Index Terms:- Normalized Fuzzy Matrices, Arithmetic Fuzzy Logic-based Representations, Automatic Control Systems, and Inverted Pendulum System. I. INTRODUCTION he modeling and analysis of automatic control systems operating in a fully fuzzy environment is not effectively solved in the literature [1]-[3]. There are many approaches that are carried out to handle this problem using the Conventional Fuzzy Theory. These approaches suffer many drawbacks such as with the increase of system dimensionality, the solution processing becomes very cumbersome. Moreover, the results gained by such approach are not linear and thus not reversible, leading that the results obtained in the forward path will be different than the backward path [4]-[5]. Other approaches, using the direct implementation of fuzzy matrices also has many shortcomings [6]-[8]. The main hindrance of their spread is heavily related to their implacability of its present operations (Max, Min, Max.Min, and Min.Max) as they do not reflect a corresponding real life physical meaning and causing irreversibility and nonlinearity in their processing. Recently, Gabr and Dorrah presented their new notion of Arithmetic and Visual Fuzzy Logic-based Representations [9]-[14]. The approach is based on the normalized fuzzy matrices, where every parameter is expressed by it value and corresponding fuzzy level. It is shown by Gabr [14] that the proposed normalized fuzzy logic-based arithmetic representation has the same mathematical function of the conventional fuzzy theory. However, the new approach provides a much easier arithmetic rather than logic calculations forum that makes its application much practical and effective. Moreover, the Fuzzy Logic-based Arithmetic Representation approach possesses the key features over the Conventional Fuzzy Theory, namely; linearity, reversibility, simplicity, and applicability. In the present work, the Fuzzy Logic-based Arithmetic Representations based on the Normalized Fuzzy Matrices as originally developed by Gabr and Dorrah [9]-[14] is extended for establishing the new foundation of fuzzy automatic control theory modeling and analysis. This new foundation will be based on handling systems with all their coefficients expressed in fully fuzzy environments. In the following section, a short introduction of the Arithmetic Fuzzy Logic- based Representation is firstly presented. II. BRIEF DESCRIPTION OF ARITHMETIC FUZZY LOGIC-BASED REPRESENTATION The normalized fuzzy logic-based is based on representing each parameter X by two components: o X the deterministic equivalence, and f X the fuzzy equivalence representing a small uncertainty or value tolerance in the parameter X [9]-[12]. The term f X is modeled by the formula: o x r f X f X l = where r f is the relative unit fuzziness (usually a certain small percentage), and x l is the T