Interlacing Properties of System-Poles, System-Zeros, and Spectral-Zeros in MIMO Systems

Single-input single-output (SISO), systems with zeros interlacing poles (ZIP) have been extensively studied in the literature and have received interest widely. However, the ZIP property for multiple-input and multiple-output (MIMO), systems has not been pursued sufficiently. In this article, we pursue the ZIP property for MIMO systems and link this to strict passivity. For the class of systems that admit a symmetric state-space realization, which for the SISO case is equivalent to ZIP, we obtain sufficient conditions under which MIMO systems too have ZIP. We also present new results in the context of “spectral-zero” of a system, a notion that plays a key role in many optimal control and estimation problems. We formulate conditions under which the spectral-zeros of a MIMO system are real, and further conditions that guarantee that the system-zeros, spectral-zeros, and the system-poles are all interlaced for MIMO systems.