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2 International Journal of Applied Mathematics and Computer Science

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A new fuzzy Lyapunov approach to non-quadratic stabilization of Takagi-Sugeno fuzzy models

Abdelmalek I., Goléa N., Hadjili M. L.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no. 1

39-51

In this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.

Neural network-based MRAC control of dynamic nonlinear systems

Debbache G., Bennia A., Goléa N.

International Journal of Applied Mathematics and Computer Science

2006

Vol. 16, no. 2

219-232

This paper presents direct model reference adaptive control for a class of nonlinear systems with unknown nonlinearities. The model following conditions are assured by using adaptive neural networks as the nonlinear state feedback controller. Both full state information and observer-based schemes are investigated. All the signals in the closed loop are guaranteed to be bounded and the system state is proven to converge to a small neighborhood of the reference model state. It is also shown that stability conditions can be formulated as linear matrix inequalities (LMI) that can be solved using efficient software algorithms. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. Simulation results are presented to show the effectiveness of the approach.