Stable Inversion of Discrete-Time Linear Periodically Time-Varying Systems via Cyclic Reformulation

Inverse systems for discrete-time linear periodically time-varying (LPTV) plants are fundamental to feedforward control and iterative learning control of multirate and periodic systems. Building on the classical cyclic reformulation, which converts an N-periodic system into an equivalent LTI system at the original sampling rate, this paper derives an explicit closed-form N-periodic state-space realization of the inverse for an arbitrary uniform periodic relative degree r >= 0 (defined through the periodic Markov parameters). The technical basis is a structure-preservation property: after absorbing a phase shift for r >= 1, the LTI inverse of the cycled plant provably retains the cyclic (block-circulant/block-diagonal) structure, so that the periodic inverse matrices can be read off block-by-block, in exact correspondence with the LTI inverse. The resulting inverse system is real-valued, causal for r = 0 and r-step-delayed for r >= 1, operates at the original sampling rate, and reconstructs the input exactly under matched initial conditions, with geometric error decay otherwise. Through this correspondence, its stability is shown to be equivalent to the invariant zeros of the cycled plant - the periodic invariant zeros of the plant - lying in the open unit disk, which generalizes the minimum phase condition of the LTI case and is verifiable on a single constant matrix pencil. Numerical examples illustrate the construction, the stability characterization, and the implementation as an online periodic filter.