The problem of a posteriori estimation for linear discrete systems with noise described by fuzzy sets (Q1900777)

The paper is concerned with linear nonstationary systems governed by the relationship \[ x(k+1)= A(k) x(k)+ v(k), \qquad y(k)= C(k) x(k)+ z(k), \] \(k=0, 1, 2, \dots, N-1\). \(x(k)\) is an \(n\)-dimensional vector in \(\mathbb{R}^n\), \(A\) is a given \((n\times n)\) matrix, \(v(k)\) describes an \(n\)- dimensional noise characterized by fuzzy sets in \(\mathbb{R}^n\) with continuous membership functions. Similarly, \(y(k)\) is an \(m\)-dimensional output vector, \(C\) stands for an \((m\times n)\) matrix and \(z(k)\) describes a measurement noise defined as a fuzzy set. The initial state \(x(0)\) belongs to a given compact set \(X_0\) in \(\mathbb{R}^n\). The paper summarizes some properties of a trajectory, it is regarded as a sequence of fuzzy sets. The main result deals with an estimation of fuzzy sets in \(\mathbb{R}^n\), specifying admissible initial states \(X_0\), given the observation results \(Y^*= \{y^*(t)\), \(t\in T\}\). A recurrent procedure for constructing a solution to this estimation problem is provided.