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

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.

💡 Research Summary

The paper addresses the long‑standing problem of constructing stable inverse systems for discrete‑time linear periodically time‑varying (LPTV) plants, which are ubiquitous in multirate sampling, non‑uniform sampling, and high‑precision positioning applications. Existing solutions either rely on complex‑valued Floquet factorizations and non‑causal preview (requiring future output data) or employ lifting techniques that operate on an N‑step block time scale, thereby losing the original step‑by‑step input‑output relationship and complicating real‑time implementation.

The authors propose a three‑step framework that eliminates these drawbacks by exploiting cyclic reformulation. First, the N‑periodic LPTV system
x(k+1)=A_k x(k)+B_k u(k), y(k)=C_k x(k)+D_k u(k)
is transformed into an equivalent linear time‑invariant (LTI) system of dimension N n, N m, N p by stacking the state, input, and output vectors according to the phase i = k mod N. The resulting “cycled” system has a 1‑shift block‑circulant  matrix, a similarly structured B̂, and block‑diagonal Ĉ and D̂. Lemma 1 guarantees a one‑to‑one correspondence between trajectories of the original LPTV plant and the cycled LTI plant, and shows that the spectral radius of the monodromy matrix Φ = A_{N‑1}…A_0 is related to the spectral radius of  by ρ(Â)=ρ(Φ)^{1/N}.

Second, the inverse of the cycled LTI system is obtained using standard LTI inversion theory. For the fundamental case of relative degree zero (all D_k are square and nonsingular), Lemma 2 yields the explicit inverse matrices:
Â⁻¹ = Â – B̂ D̂⁻¹ Ĉ, B̂⁻¹ = B̂ D̂⁻¹, Ĉ⁻¹ = –D̂⁻¹ Ĉ, D̂⁻¹ = D̂⁻¹.
Because B̂ D̂⁻¹ Ĉ preserves the 1‑shift block‑circulant pattern, Â⁻¹ also has the same structure, with each block given by A_i⁻¹ = A_i – B_i D_i⁻¹ C_i.

Third, the periodic inverse matrices for the original LPTV system are extracted from the block‑wise structure of Â⁻¹, B̂⁻¹, Ĉ⁻¹, D̂⁻¹. This “parameter extraction” step yields N‑periodic matrices Γ_k, Λ_k, Ω_k, Π_k that define a causal, real‑valued inverse system operating at the original sampling rate:
ζ(k+1)=Γ_k ζ(k)+Λ_k y(k+r), u(k)=Ω_k ζ(k)+Π_k y(k+r).

For systems with periodic relative degree r ≥ 1 (some D_k may be zero), the authors extend the approach by invoking Lemma 3, which characterizes the r‑step delayed inverse for LTI systems with relative degree r. By applying the same cyclic reformulation, the required periodic Markov parameters (C A^{j} B) are computed, and a closed‑form r‑step delayed inverse is derived:
u(k)=−(C A^{r‑1} B)⁻¹ C A^{r} ζ(k)+(C A^{r‑1} B)⁻¹ y(k+r).

Stability analysis is a central contribution. Theorem 2 and Corollary 2 prove that the poles of the inverse (i.e., the eigenvalues of the monodromy matrix of the inverse, Φ_inv = Γ_{N‑1}…Γ_0) coincide with the transmission zeros of the original plant. Consequently, the inverse is Schur stable if and only if all transmission zeros of the cycled plant lie strictly inside the unit disk—exactly the minimum‑phase condition familiar from LTI theory, now generalized to periodic systems. This condition can be verified using only real‑valued matrices, without resorting to complex Floquet factors.

Two numerical examples illustrate the theory. The first example, a relative‑degree‑zero plant with all transmission zeros inside the unit circle, demonstrates exact reconstruction of a reference input using the derived causal inverse. The second example, a relative‑degree‑two plant, shows how the periodic Markov parameters lead to a stable two‑step delayed inverse that perfectly recovers the input. In both cases, when a non‑minimum‑phase zero is introduced, the inverse becomes unstable, confirming the theoretical stability criterion.

Overall, the paper makes four major contributions:

1. It introduces a cyclic‑reformulation‑based inversion framework that reduces the LPTV inversion problem to a standard LTI inversion problem, eliminating the need for complex arithmetic and preview.
2. It provides explicit closed‑form inverse matrices for both relative‑degree‑zero and higher‑relative‑degree periodic systems, enabling direct state‑space implementation at the original sampling rate.
3. It establishes a clear stability condition based on the transmission zeros of the cycled plant, thereby extending the minimum‑phase concept to periodic systems.
4. It validates the methodology with numerical simulations, showing exact input reconstruction and highlighting the practical relevance for feed‑forward control, iterative learning control, and multirate controller design.

The proposed technique thus offers a practical, theoretically sound tool for engineers dealing with periodic or time‑varying dynamics, bridging a gap between rigorous inverse system theory and real‑time control applications.