Stability analysis of discrete singular fuzzy systems (Q1770777)

The author discusses is a problem of stability of rule-based Takagi-Sugeno models assuming the following form: \[ \text{If }x_1(k)\text{ is }F_{1i}\text{ and }x_2(k)\text{ is }F_{2i}\text{ and }\dots\;x_n(k)\text{ is }F_{ni}\text{ then }Ex(k+ 1)= A_ix(k),\;i= 1,2,\dots, r, \] where \(x(k)\) is the state vector, \(F_{ij}\) is a corresponding fuzzy set in the condition part and \(E\) and \(A_i\) are matrices, \(\dim(A)= n\times n\), \(\dim(E)= n\times n\). The case when \(E\) is singular is considered too. For this type of singular fuzzy models, the author introduces conditions of stability stating that the above system is stable if there exists an invertible symmetric matrix \(P\) such that \(E^TPE\geq 0\), \((A_i-A_j)^T P(A_i-A_j)\geq 0\) for all \(i< j\), and \(A_i^TPA_i- E^TPE< 0\) for all \(i\). Owing to the occurrence of the non-strict inequalities, the stability analysis requires the use of the projection technique. Shown are numerical methods exploiting some projection operators.