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

Hiroshi Okajima

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.

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

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.
Paper Structure (15 sections, 9 theorems, 44 equations, 3 figures)

This paper contains 15 sections, 9 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cyclic Reformulation of Periodically Time-Varying Systems
  4. Inverse Systems for LTI Plants
  5. Main Results
  6. Problem Formulation
  7. Stability Analysis
  8. Extension to Higher Relative Degree Systems
  9. Numerical Examples
  10. Relative Degree Zero: Scalar LPTV System
  11. Relative Degree Zero: Non-Scalar LPTV System
  12. Periodic Relative Degree $r = 1$
  13. Discussion
  14. Conclusion
  15. Declaration of Generative AI and AI-assisted technologies in the writing process

Key Result

Lemma 1

Let $(x(k), u(k), y(k))$ and $(\check{x}(k), \check{u}(k), \check{y}(k))$ be the trajectories of the LPTV system (eq:lptv_state)--(eq:lptv_output) and the cycled system (eq:cycled_state)--(eq:cycled_output), respectively, with the same initial condition and corresponding cycled inputs. Then:

Figures (3)

  • Figure 1: Input reconstruction for the scalar stable inverse (Example 4.1). Reference input $u_{\rm ref}(k) = \sin(0.1\pi k)$ (dashed) and reconstructed input $\hat{u}(k)$ (solid) with mismatched initial condition $\zeta(0) = 0 \neq x(0) = 1$. The reconstruction error vanishes rapidly due to $\rho(\Phi_{\rm inv}) \approx 0.0067$.
  • Figure 2: Input reconstruction for the non-scalar stable inverse (Example 4.2, $N = 3$, $n = 2$). Top: reference input $u_{\rm ref}(k)$ (dashed) and reconstructed input $\hat{u}(k)$ (solid). Bottom: reconstruction error $\hat{u}(k) - u_{\rm ref}(k)$. The error vanishes rapidly due to $\rho(\Phi_{\rm inv}) \approx 0.032$.
  • Figure 3: Input reconstruction for periodic relative degree $r = 1$ (Example 4.3, $N = 3$, $n = 2$). Top: reference input $u_{\rm ref}(k) = \sin(0.15\pi k)$ (dashed) and reconstructed input $\hat{u}(k)$ (solid). Bottom: reconstruction error $\hat{u}(k) - u_{\rm ref}(k)$. The reconstruction is exact (error at machine precision) since $\zeta(0) = x(0)$.

Theorems & Definitions (20)

  • Lemma 1: bittanti2009
  • Remark 1
  • Lemma 2: LTI inverse, relative degree zero moylan1977tacchen1984tac
  • Lemma 3: LTI inverse, relative degree $r$ silverman1969tackono1981ijc
  • Definition 1: Stable inverse system
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Definition 2: Periodic minimum phase
  • Corollary 1
  • ...and 10 more