Asymptotic stability and fuzzy control (Q2774241)

The author considers the controllability of a system with fuzzy input. Let \(S=(P,F)\) be a control system, where \(F\) is a controller and the process \(P\) has a transition function of the form \(Y(k+1)=P(Y(k), U(k))\), \(k=1,2, \dots\); with \(Y(0)=y_0\), \(U(0)=u_0\), where \(Y(k)\in \mathbb{R}^m\), \(U(k)\in \mathbb{R}^l\) denote the state and process input at time \(k\), respectively.NEWLINENEWLINENEWLINE\textit{J. J. Buckley} proved that there exists a fuzzy controller \(FC\) such that \(FS=(P,FC)\) is controllable over any finite time interval \([\tau,\tau +N]\) [Fuzzy Sets Syst. 77, 167-173 (1996); corrigendum ibid. 100, 377-379 (1998; Zbl 0874.93063)]. An open problem is whether the condition ``over any finite time interval'' can be relaxed.NEWLINENEWLINENEWLINEThe present author shows that any asymptotically controllable process can be attained by fuzzy control with the same control performance, and that fuzzy controllers are universal controllers. So, the results obtained yield affirmative answers to some open problems posed by J. J. Buckley in the above-mentioned paper.