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

Jul 16, 2026

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Inverse systems for discrete-time linear periodically time-varying (LPTV) plants underpin feedforward control and iterative learning control in multirate and periodic settings. This work extends classical cyclic reformulation by deriving an explicit closed-form N-periodic state-space realization of the inverse for any uniform periodic relative degree r≥0. A structure-preservation result shows that, after absorbing a phase shift when r≥1, the LTI inverse of the cycled plant keeps the cyclic block structure, enabling block-by-block recovery, with real-valued causality or delayed implementation and stability characterized by invariant zeros.

LanguageEnglish

CategoryResearch & Report

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Pages22

Reading Time55 min

Abstract

1 Introduction

arXiv :2603 .23147v2 [ ee ss . SY] 10 Jul 2026 Stable Inversion of Discrete-Time Linear Periodically Time-Varying Systems via Cyclic Reformulation ∗ Hiroshi Okajima Faculty of Advanced Science and Technology, Kumamoto University, Japan Keywords: Inverse systems; Periodically time-varying systems; Cyclic reformulation; Stable inversion; Multirate systems. Abstract 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 key technical result 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. 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. Its stability is characterized by the invariant zeros of the cycled plant, generalizing the minimum phase condition of the LTI case. Numerical examples illustrate the construction, the stability characterization, and the implementation as an online periodic ﬁlter. 1 Introduction The inverse of a dynamical system plays a fundamental role in feedforward control and output tracking [1, 2], iterative learning control (ILC) [3], and invertibility analysis [4] . For linear time-invariant (LTI) systems, stable inversion theory is classical: if all zeros lie strictly inside the unit disk, a causal stable inverse exists and can be constructed explicitly in state-space ∗ Corresponding author: H. Okajima. Email: [okajima@cs.kumamoto-u.ac.jp](okajima@cs.kumamoto-u.ac.jp) form [5, 6], and a bounded noncausal inverse is available otherwise via exponential dichotomy [7, 8], at the cost of requiring future output information (preview) . Many practical systems, however, exhibit periodically time-varying dynamics that cannot be captured by an LTI model. Linear periodically timevarying (LPTV) systems arise naturally in multirate sampled-data control [9], position-dependent and non-equidistantly sampled systems [8, 10], and high-precision positioning stages [11] . Because the system matrices vary periodically, the transfer function framework underlying LTI inversion theory is no longer directly available. Invertibility conditions for general linear time-varying discrete-time systems were established by Kono [12] . For periodic systems speciﬁcally, the inversion problem was ﬁrst addressed by Perdon, Conte, and Longhi [13], who adapted Silverman’s structure algorithm [1] to the periodic setting (with Dk = 0) and obtained necessary and suﬃcient conditions for left/right invertibility together with a synthesis procedure for the inverse; the resulting inverse is expressed through the output matrices of a Periodic Structure Algorithm and applies a time-varying bank of forward shift operators to future outputs, rather than being given in closed form as functions of the plant matrices. On the computational side, Varga [14] developed the computation of generalized inverses of periodic systems via lifted pencil methods, noting explicit formulas only for the square case with invertible feedthrough asa starting point. The delay-inverse construction for systems with Dk = 0 was listed therein as an open computational problem; see [15] for a survey of computational paradigms for linear periodic systems. More recently, stable inversion of LPTV syst

FAQs

How is stability of the constructed inverse characterized?

Stability is characterized by the invariant zeros of the cycled plant, which generalizes the minimum phase condition known for the LTI case.

Jul 16, 2026

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