Further results on the boundedness of multidimensional systems

A sufficient condition is derived in this paper for the boundedness of a linear time invariant (LTI) multidimensional (MD) dynamic system over a special class of frequency domains. These frequency domains are able to include many practically adopted frequency domains, such as fans, rectangles, etc., as a special situation, as well as to give an approximation with a tolerable accuracy to many other physically significant frequency domains like ellipsoids and diamonds. This condition becomes also necessary when the prescribed frequency domain is path-connected. Moreover, it is expressed in a linear matrix inequality (LMI) form and can be directly applied to the optimization of system output matrix and direct transmission matrix, as well as the minimization of the frequency response bound. In addition, the dimension of the LMI is in the same order as that of the system matrices.