Balanced realisations and model reduction of periodically time-varying state-space digital filters

The authors discuss balanced realisations and model reduction of periodically time-varying (PTV) state-space digital filters. Controllability and observability Grammians of PTV state-space digital filters are discussed. It is extremely interesting to notice that although PTV state-space digital filters can be implemented by using a group of time-invariant coefficient sets, controllability and observability Grammians cannot be evaluated independently by using any one set of these time-invariant coefficients. Also, important physical interpretations of controllability and observability Grammians are considered. Based on these analysis results, balanced realisations of PTV state-space digital filters are defined and a synthesis method for balanced realisations is proposed. As one application of balanced realisations, a reduced-order model of a PTV state-space digital filter can be obtained by taking a subsystem of balanced realisation of the PTV state-space digital filter. Finally, a numerical example is given to illustrate the balanced model reduction procedure