On factorization of the nonlinear drift term for SDRE approach

This study presents necessary and sufficient conditions for the existence of a constant matrix A that satisfies Ax = f, (A,B) is stabilizable and (A,C) is observable (resp., detectable), where x and f are two constant vectors with x ≠ 0, while B and C are two constant matrices with approximate dimension. The conditions presented are very easy to check and a set of such feasible matrices A are also provided explicitly when the existence conditions are satisfied. In practical applications, this study may cooperate with the state-dependent Riccati equation (SDRE) implementation at those states where the SDRE scheme fails to operate, yet the presented existence conditions hold there.