l_∞-based stability criteria and its applications on FLC systems

In this paper, a new stability analysis and design method for the fuzzy logical systems described by the Takagi-Sugeno model is proposed. The proposed fuzzy controller is implemented by a parallel distributed compensation structure. The closed-loop system can be expressed as a linear time-variant system. A minimum transition matrix set can be determined by the fuzzy rule structure, so that it contains the transition matrices of the closed-loop system. Thus the stability of the system defined over the set is sufficient to ensure the stability of the closed-loop system. A common l_∞ Lyapunov function method is employed for the stability analysis of the system defined over the transition matrix set. It is proved that the existence of the l_∞ Lyapunov function is the sufficient and necessary stability condition of the system defined over the transition matrix set. Thus the l _∞ Lyapunov function can be applied to more cases than quadratic Lyapunov functions. A convenient stability criterion based on l _∞ Lyapunov function is also given in this paper. By using geometric properties of l_∞ Lyapunov functions, an iterative algorithm is introduced to construct the required l_∞ Lyapunov function. © 2002 Elsevier B.V. All rights reserved.