Synthesis of optimal fuzzy estimation algorithms (Q1288673)

The study deals with a discrete-time stationary system governed by the relationship \[ {\mathbf x}(k+ 1)= {\mathbf x}(k),\;{\mathbf x}(k= 0)= {\mathbf x}^*\in\mathbb{R}^n,\quad k=0,1,2,\dots\;. \] The state \({\mathbf x}(k)\) is observed by some observer \({\mathbf y}(k)\) whose error properties are unknown and lead to the relationship \[ ({\mathbf y}(k),{\mathbf x}(k))\in S(k),\quad k=1,2,\dots \] with \(S(k)\) being a closed subset in the Cartesian product of the vector spaces of \(\mathbb{R}^n\) and \(\mathbb{R}^m\) \((m\leq n)\). This subject could be also regarded as the set of all possible states compatible with \({\mathbf y}(k)\). A fuzzy set-based estimation approach producing estimates in the form of fuzzy ellipsoidal sets is proposed. It is shown how this helps overcome inconsistencies between current and accumulated estimation results. Subsequently, the author relates the set-theoretic estimation procedure and dynamic programming.