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On a possibilistic approach to the analysis of fuzzy feedback systems. (English) Zbl 0532.93048
An analysis of discrete-time closed loop fuzzy systems is performed. Relations between the control $$U_ t$$, the state $$X_ t$$, and the output $$Y_ t$$, all of them fuzzy variables, are characterized by conditional possibility distributions $$\mu_{Y_ t| X_ t}(y_ t| x_ t)$$, $$\mu_{U_ t| X_ tY_ t}(u_ t| x_ t,y_ t)$$, $$\mu_{X_{t+1}| X_ tY_ tU_ t}(x_{t+1}| x_ t,y_ t,u_ t)$$, $$\mu_{U_ t| Y_ t}(u_ t| y_ t)$$, $$\mu_{X_{t+1}| X_ tU_ t}(x_{t+1}| x_ t,u_ t)$$. Some properties of fuzzy systems described by the equation $$X_{t+1}=X_ t\circ Q_ t$$ are studied. $$Q_ t$$ stands for a nonstationary fuzzy relation equal to $\mu_{Q_ t}(x,z)=\sup_{u,y}[\min(\mu_{Y_ t| X_ t}(y| x),\quad \mu_{U_ t| X_ tY_ t}(u| x,y),\quad \mu_{X_{t+1}| X_ tY_ tU_ t}(z| x,y,u))],$ x, $$z\in {\mathcal X}$$, with $${\mathcal X}$$ being the state space. It is remarkable that the proposed system description assumes interactivity existing between appopriate system variables that is handled by their conditional possibility distributions. From a formal point of view this probabilistic approach to system analysis may be also treated by means of the theory of fuzzy relational equations.
Reviewer: W.Pedrycz

##### MSC:
 93E03 Stochastic systems in control theory (general) 03E72 Theory of fuzzy sets, etc. 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory) 60E99 Distribution theory 03C55 Set-theoretic model theory
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