Abstract: In order to overcome the limitation that the information loss performance index of the method for model reduction by minimizing information loss based on cross-Gramian matrix does not meet nonnegativity and any norm so that its physical meaning is not intuitive,the relationships between the Hilbert-Schmidt norm and the cross-Gramian are derived from the Hankel singular values of systems and the information property of the cross-Gramian.Via further theoretical reasoning,the information loss performance index based on Hilbert-Schmidt norm is obtained.Moreover,an approach to model reduction based on Hilbert-Schmidt norm and cross-Gramian is proposed.Finally,a numerical example is illustrated to verify that the performance of model reduction is greatly improved and the truncation error is smaller.Therefore,the rationality and the validity of the new performance index are proved.