Stability criteria based on generalized positive definite

In this paper we consider necessary and sufficient conditions for the stability of linear time invariant(LTI) systems and linear time varying(LTV) systems. Stability results based on generalized positive definite are derived. Two necessary and sufficient conditions of stability, in terms of generalized positive definite of matrices, for the class of linear time invariant systems which possess real eigenvalues is presented. The incorrect criteria based on generalized positive definite are pointed and corrected. The formulated reason of incorrect criteria is analyzed. Utilizing this result, the problem of stability of linear time varying systems via generalized positive definite is converted into linear matrix inequalities (LMI) techniques. Based on the same idea, the special issues of stability criteria of LTV systems has been discussed and the differences between LTI systems and LTV systems also been dealt with by several examples.