Necessary and sufficient conditions for crisp state-feedback design for fuzzy linear systems

In this paper, we examine the uncertain fuzzy continuous time linear systems. There is fuzzy uncertainty in the dynamics of the investigated system in the form of uncertain parameters represented by fuzzy granular numbers. A state feedback is designed to guarantee the stability of the closed-loop system with crisp gains and a necessary and sufficient condition is proposed to guarantee the stability. Furthermore, by using the proposed control input, the fuzzy eigenvalues of the closed-loop system are placed in a way that they have the greatest possibility of occurring in the desired poles. The state feedback gain which is presented in this paper is a crisp number and can be applied to real systems in practical applications. The proposed method ensures closed -loop stability whereas previous methods have presented fuzzy gains which are defuzzified to be applied to a real system without guaranteeing the stability. At the end, two practical simulation examples are given to verify the validity of the theoretical results.