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

TL;DR

This paper introduces a systematic cyclic reformulation method to construct stable inverses of discrete-time linear periodically time-varying systems, avoiding complex Floquet factors and noncausal processing, with explicit formulas and stability analysis.

Contribution

It presents a novel approach transforming LPTV systems into LTI form for stable inverse construction, including explicit formulas and stability conditions based on transmission zeros.

Findings

Explicit closed-form inverse matrices for relative degree zero systems.

Stable inverse systems characterized by transmission zeros.

Numerical examples confirm effectiveness and stability conditions.

Abstract

Stable inverse systems for periodically time-varying plants are essential for feedforward control and iterative learning control of multirate and periodic systems, yet existing approaches either require complex-valued Floquet factors and noncausal processing or operate on a block time scale via lifting. This paper proposes a systematic method for constructing stable inverse systems for discrete-time linear periodically time-varying (LPTV) systems that avoids these limitations. The proposed approach proceeds in three steps: (i) cyclic reformulation transforms the LPTV system into an equivalent LTI representation; (ii) the inverse of the resulting LTI system is constructed using standard LTI inversion theory; and (iii) the periodically time-varying inverse matrices are recovered from the block structure of the cycled inverse through parameter extraction. For the fundamental case of relative degree zero, where the output depends directly on the current input, the inverse system is obtained as an explicit closed-form time-varying matrix expression. For systems with periodic relative degree r >= 1, the r-step-delayed inverse is similarly obtained in explicit closed form via the periodic Markov parameters. The stability of the resulting inverse system is characterized by the transmission zeros of the cycled plant, generalizing the minimum phase condition from the LTI case. Numerical examples for both relative degree zero and higher relative degree systems confirm the validity of the stability conditions and demonstrate the effectiveness of the proposed framework, including exact input reconstruction via causal real-valued inverse systems.