Multidirectional cascade of square MIMO LTI subsystems: Transfer function representation

Abstract

This paper considers continuous transfer function representation for a cascade composed of N mutually interconnected square multiple-input multiple-output (MIMO) linear time-invariant (LTI) subsystems with general multidirectional input–output configurations. Such cascaded structure can arise, e.g., from the spatial discretization of one-dimensional distributed parameter systems (DPSs) described by partial differential equations (PDEs) with boundary conditions representing different configurations of boundary input signals. As shown in the paper, the transfer function matrix of the resulting cascade can be calculated using the Redheffer star product of the subsystems’ transfer function matrices, which simplifies into the usual matrix product for the unidirectional cascade. For the particular case of 2 × 2 cascade composed of rational transfer function subsystems, the recursive formulas for its boundary and distributed transfer functions have been derived for both uni- and multidirectional configurations. The well-posedness and the stability criteria are also analyzed, proving to be more restrictive for the multidirectional cascade than for the unidirectional one due to the internal positive feedback in the first case. As shown in the attached example, the presented results can be used to approximate one-dimensional distributed parameter systems using rational transfer function subsystems representing the spatial sections of the original DPS.