**Stability Study of Mamdani’s Fuzzy Controllers Applied to Linear Plants**

**Anis SAKLY**, **Basma ZAHRA**, **Mohamed BENREJEB**

Unité de recherche LARA Automatique

Ecole Nationale d’Ingénieurs de Tunis, BP 37, Belvédère 1002 Tunis, Tunisie

**Abstract**: A stability study approach of fuzzy control systems, in the continuous case and the discrete one, is presented in this paper. This approach is based on vector norms corresponding to a Lyapunov’s vector function and allowing to establish sufficient conditions for global asymptotic stability of controlled linear systems by Mamdani’s fuzzy controllers. The considered fuzzy regulators are of type PI and have particular fuzzy partition for input variables corresponding to Lur’e type systems.

**Keywords**: Mamdani’s fuzzy controllers, global asymptotic stability, vector norms, arrow’s form matrix.

**Anis Sakly** was born in 1970. He received his Electrical Engineering diploma in 1994 from the National Engineering School of Monastir (ENIM), Tunisia. Then, he received the doctorate in Electrical Engineering in 2005 from the National Engineering School of Tunis (ENIT). He is currently teaching in the ENIM. His research interests are in fuzzy control, particularly stability analysis and establishment of fuzzy control systems.

**Basma Zahra** was born in 1978. She received her Electrical Engineering diploma in 2003 from the National Engineering School of Sfax (ENIS), Tunisia. She is currently preparing her doctorate in Electrical Engineering in the National Engineering school of Tunis (ENIT) about analysis and synthesis of PI fuzzy control systems.

**Mohamed Benrejeb** was born in 1950. He received the Engineering diploma in 1973 from the North Industrial Institute (IDN currently central school of Lille), France. In 1976 he received the engineering doctor diploma in Automatic from Technology and Science university of Lille and the doctorate es physics sciences from the same university in 1980. He is currently professor in higher education in the National Engineering School of Tunis (ENIT) and the Central School of Lille. His fields of research include system control, modelisation, analysis and synthesis of certain and uncertain continuous systems by conventional or nonconventional approaches and concern recently discrete systems.

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**CITE THIS PAPER AS**:

Anis SAKLY, Basma ZAHRA, Mohamed BENREJEB, **Stability Study of Mamdani’s Fuzzy Controllers Applied to Linear Plants**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 17 (4), pp. 441-452, 2008.

**1. Introduction**

Although stability analyzes of a fuzzy model is particularly difficult, since it is naturally non linear, it is important when realizing the fuzzy controller to analyze the dynamic behavior of closed-loop system. Stability concept for fuzzy systems has been developed by many authors [1] [3] [5] [7] [10] [12] [17] [18] [19].

Since their appearance in the middle of 1970, many researches have been developed to analyze closed loop system’s stability with fuzzy regulators. In particular, Kichert and Mamdani [8] have developed an input/output model of type multi-relay for a fuzzy regulator to use first harmonic method to highlight existence conditions to periodic responses.

One of the approaches often used in the specialized literature consisting of a fuzzy controller which is the nonlinear element used with linear model to be controlled. In this approach the fuzzy model is considered as a particular class of nonlinear models.

In the case of single input controllers, the error, Ray and Majumder [14], after showing that the static input-output controller characteristic belongs to a bounded sector, used a particular case of the circle criterion to obtain sufficient conditions of asymptotic stability in the case of continuous linear systems. In the discrete case, Langari and Tomizuka [9] used the Lyapunov method to propose sufficient stability conditions for controllers sampling only the error. After this, Melin and Vidolov [11] extended the approach proposed in [14] to the case of fuzzy controllers realizing PD and PI type strategies to control SISO linear system.

They proposed sufficient stability conditions by using the Kalman-Yacubovitch theorem. In the same way, Rambault [13] presented sufficient stability conditions by using the Popov theorem in the case of PI fuzzy controllers.

Another approach was developed in [1] for the stability study of TSK fuzzy systems by using vector norms and exploiting the Borne and Gentina criterion.

In this work, we will present the studies proposed in [1] with another view and we will extrapolate them for the discrete case.

Thus, we are interested in PI fuzzy controllers whose inputs are the error and its variation and the output is the control variation . For a particular partition of inputs, we will show that this controller is equivalent to a linear PI with a nonlinear gain. These regulators correspond to systems of type Lur’e.

This paper is setup as follows: in the next section a description of the fuzzy system is presented, the third section is a setting of the system under the Lur’e problem form, the fourth section propose the stability conditions both in the continuous case and in the discrete one, an application example is presented in section five, and finally in the sixth section a conclusion is given.

**6. Conclusion**

In this paper we have studied the global asymptotic stability of a fuzzy system, where the fuzzy controller is a Mamdani’s one and for a particular partition of the input subsets. After presenting the particular class of fuzzy PI controller, we do release some properties allowing the verification of passivity relation of these controllers between the tuning grandeur and the observation , linear combination of the error and its variation.

The global asymptotic stability conditions obtained in both the continuous case and the discrete one are deduced from the application of the Borne and Gentina criterion and the use of vector norms as Lyapunov function.

Finally it suits to remark that this study can be extended to the stability study of non linear systems controlled by the same way, the Mamdani’s fuzzy controllers, with a few modifications on the fuzzy controller, and it remains with the report of the system to be controlled and the nature of its nonlinearity.

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