Boundedness of Multi-dimensional Systems over a Prescribed Frequency Domain

On the basis of a parameterization of the span of a multivariate matrix polynomial, a sufficient condition is derived in this paper for a multi-dimensional multi-input multi-output (MIMO) IIR system being upper bounded over a cuboid frequency domain. This condition is expressed through a linear matrix inequality (LMI) and can be computationally verified. Moreover, by means of parameter dependent LMIs, two necessary and sufficient conditions are also obtained for this boundedness verification problem, which are again expressed by LMIs. Furthermore, LMI based conditions are derived for system output matrix and direct transmission matrix. Numerical examples are included to illustrate the efficiency and characteristics of the derived theoretical results.