Comments on ‘Admissibility and static output‐feedback stabilization of singular Markovian jump systems with defective statistics of modes transitions’

Abstract

This document shows that the proof of Theorem 3.3 in the work of Liu et al (Int J Robust Nonlinear Control. 2015. 10.1002) is incorrect and, thus, the theorem is not proved. Since the results of the paper are strongly based on this theorem, the problem has to be addressed. If a natural correction is introduced, the problem still holds. Therefore, a new linear matrix inequality is proposed here to show its equivalence with the stochastic stability of a Markov jump linear singular system under a similar assumption as that in the work of Liu et al. KEYWORDSliner matrix inequality, Markov jump linear singular systems, stochastic stability COMMENTSIn the proof of the sufficiency of Theorem 3.3 in the work of Liu et al, 1 by using the transformation (k) = N i x (k), the authors claim that the stochastic stability of system (3), ie,is equivalent to that of system (15). However, the authors have confused E i and E r (k) . In this case, for a fixed mode i ∈  ≜ {1, … , s}, we would havewhereà r (k) = M −1 A r (k) N −1 i . Therefore, Equation (16) in Theorem 3.3 cannot be asserted and the proof is blocked. In regard with the necessity part of the proof, due to the same inaccuracy, it is not true that the stochastic stability of system (3) implies that of the system given in Equation (21).Even by applying a natural fix, that is, (k) = N r (k) x (k), so thatà i = M −1 A i N −1 i , the problem still holds [ I n r 0 0 0 ] (k + 1) = [ I n r 0 0 0 ] N r (k+1) x (k + 1) ≠ M −1 E r (k) x(k + 1) =à r (k) (k).