Coefficient-based conditions of matrix-valued real-rational proper negative imaginary functions

This paper mainly gives sufficient coefficient-based conditions of matrix-valued real-rational proper negative imaginary (NI) functions. The proper NI functions can be divided into three cases by relative order, namely, NI functions with relative order 0, 1, and 2. Based on the relationship between NI functions with poles at 0 and without poles at 0, as well as coefficient-based conditions of positive real functions, sufficient coefficient-based conditions are shown of NI functions with relative order 1 and 2. These main results are related to the numerical radius and Crawford number of matrices. Moreover, they are supplements to some existing results. NI functions with relative order 0 can be changed into the case of relative order 1, and the sufficient coefficient-based conditions are also obtained of NI functions with relative order 0. In addition, the coefficient-based conditions can be used to construct interval NI functions, and some examples are given to illustrate the main results.