On the existence of frequency-interval gramians for bilinear systems

It is well-known that gramians are specific matrices which show the degree of controllability and observability. Therefore gramians are very popular in applications such as model reduction and control configuration selection. The frequency-interval controllability and observability gramians have been recently introduced for bilinear systems as the solutions to the generalized frequency-interval Lyapunov equations. Analogous to ordinary gramians for bilinear systems, it might happen that the frequency-interval Lyapunov equations have unique solutions which are not controllability and observability gramians of the bilinear systems. In other words, solvability of the frequency-interval Lyapunov equations does not guarantee the existence of the frequency-interval gramians. In this paper, the conditions which are required for the existence of frequency-interval gramians are obtained. Further, to cope with the problem of the existence of gramians, a scaling-based method is proposed. A proof for the theorem which suggests an iterative scheme for computing the frequency-interval generalized gramians is also presented in this paper.