Shift-invariant equivalents for a new class of shift-varying operators with applications to multi-rate digital control

A class of linear, shift-varying operators that generalizes the notion of N-periodicity is defined. It is shown that shift-invariant equivalents for these operators exist, and that the equivalence is in a strong sense, preserving both algebraic and analytic systems properties. It is shown that multi-rate sampled-data systems, though not generally periodic, fall into this class. Kranc vector switch decomposition and block filter implementations for single-input, single-output multi-rate systems are connected under the unifying framework of shift-invariant equivalents, and this framework is the way to extend them both to multi-input, multi-output systems. Possible applications include a parameterization of all stabilizing multi-rate controllers