H. Okajima, Y. Fujimoto, H. Oku, and H. Kondo, "Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems," IEEE Access, Vol. 13, pp. 26483–26493, 2025.
- Paper (IEEE Xplore, Open Access): https://doi.org/10.1109/ACCESS.2025.3537086
- Preprint (arXiv): https://arxiv.org/abs/2411.00318
Extended Abstract
Background: Identifying Time-Varying Plants
System identification for linear time-invariant (LTI) plants is a mature field, with well-established subspace methods (e.g., N4SID, MOESP) providing reliable state-space models from input-output data. However, many practical systems exhibit periodically time-varying (LPTV) dynamics — systems whose internal parameters repeat with a known period $M$. Examples include rotating machinery, power converters operating under periodic switching, seasonal processes, and multi-rate sampled-data systems.
An LPTV system in discrete time is described by
$$
x(k+1) = A_k, x(k) + B_k, u(k), \quad y(k) = C_k, x(k) + D_k, u(k)
$$
where $A_k = A_{k+M}$, $B_k = B_{k+M}$, $C_k = C_{k+M}$, and $D_k = D_{k+M}$, meaning the system matrices are $M$-periodic.
Identifying such systems is fundamentally harder than the LTI case. The parameter matrices change at every time step within a period, so a single LTI model cannot capture the dynamics accurately. Conventional approaches often require specially designed periodic input signals or rely on approximating the LPTV system as LTI, sacrificing accuracy.
The Structural Gap: From Cycled System to LPTV Parameters
A classical tool for handling LPTV systems is to convert them into equivalent LTI representations of higher dimension. Two principal methods exist: the lifting (or time-lifted) reformulation and the cyclic reformulation, both systematically studied by Bittanti and Colaneri (Automatica, 2000; Springer, 2009).
The cyclic reformulation constructs "cycled" input and output signals by interleaving the original signals according to their position within each period. The resulting cycled system is LTI and can therefore be identified using standard subspace methods. However, the state-space model obtained from subspace identification is expressed in an arbitrary coordinate system — it does not directly reveal the original LPTV parameters $A_k, B_k, C_k, D_k$. This is because subspace identification determines the system only up to a similarity transformation.
The central challenge addressed in this paper is: how to recover the LPTV parameters from the identified cycled system through an appropriate state coordinate transformation.
Proposed Algorithm
The identification algorithm proceeds in four steps:
- Construct cycled signals from the measured input-output data according to the known period $M$.
- Apply standard subspace identification to the cycled signals, obtaining a state-space model $(A_, B_, C_, D_)$ in an arbitrary coordinate basis.
- Design a coordinate transformation matrix $T$ that converts $(A_, B_, C_, D_)$ into cyclic reformulation form — i.e., the form where the matrices exhibit the block-cyclic and block-diagonal sparsity structure that characterizes a valid cyclic reformulation.
- Extract the LPTV parameters $A_k, B_k, C_k, D_k$ for $k = 0, \ldots, M-1$ from the components of the transformed cyclic reformulation matrices.
The key theoretical contribution lies in Step 3. When subspace identification is applied to the cycled signals, the resulting model $(A_* , B_* , C_* , D_* )$ is mathematically equivalent to the true cyclic reformulation but expressed in an unknown coordinate basis. The paper shows that the Markov parameters of this identified model — the sequence of matrices $C_* A_* ^i B_*$ that characterize the impulse response — carry structural information that can be exploited to determine $T$.
Specifically, the Markov parameters of a cyclic reformulation possess a characteristic sparsity pattern: certain products of these matrices, when arranged appropriately, exhibit block-cyclic or block-diagonal structure. This sparsity is not an accident but a direct consequence of the periodic structure encoded in the cyclic reformulation. Even though subspace identification scrambles the coordinate basis, the Markov parameters themselves are coordinate-invariant (they depend only on the input-output behavior, not on the internal state representation). The paper constructs auxiliary matrices $\check{S}_i \check{H}(i)$ from these Markov parameters and proves that, under a rank condition (Assumption 1 in the paper), their sparsity pattern uniquely determines the column structure of $T$.
The construction proceeds by analyzing how the sparsity propagates through the Markov parameter sequence. The transformation matrix $T$ is designed so that $T^{-1} A_* T$ becomes a block-cyclic matrix (where the blocks cycle through positions along the block-superdiagonal) and $C_* T$ becomes a block-diagonal matrix. These two structural properties together guarantee that the transformed system is a valid cyclic reformulation, from which the original LPTV parameters $A_k, B_k, C_k, D_k$ can be read off directly from the block components.
Why Not Lifting?
Both lifting and cyclic reformulations convert an LPTV system to an LTI system of dimension $Mn$ (where $n$ is the original state dimension). The lifting approach packages $M$ consecutive samples into a single vector, producing an LTI system that operates at a rate $M$ times slower. While this is straightforward, the lifted signals have a different sampling rate from the original, which complicates the connection back to the original time-domain parameters.
The cyclic reformulation preserves the original sampling rate: the cycled system operates at the same rate as the original LPTV system. This makes the relationship between the cycled system parameters and the original LPTV parameters more transparent and structurally exploitable — a property that this paper leverages to design the coordinate transformation.
Significance
A distinctive feature of this approach is that it does not require specific periodic input signals. Conventional LPTV identification methods often assume that the input signal has certain periodic properties aligned with the system period. The proposed method instead leverages the inherent periodic structure of the system itself through the cyclic reformulation, accepting general (e.g., random) excitation signals. This significantly improves flexibility in practical applications where input design may be constrained.
The algorithm is implementable using standard MATLAB subspace identification routines, with the coordinate transformation step being an additional algebraic computation. Numerical examples in the paper demonstrate accurate recovery of LPTV parameters both in noise-free and noisy settings.
Related Works Using Cyclic Reformulation
This paper is part of a broader research program that applies cyclic reformulation to various problems in multi-rate and periodically time-varying systems. The development trajectory is as follows:
| Year | Topic | Key Contribution | Ref |
|---|---|---|---|
| 2023 | Multi-rate state observer | Periodically time-varying observer for multi-rate sensing; $\ell_2$-induced norm design via LMI | IEEE Access (OA) |
| 2023 | Multi-rate feedback controller | Observer-based controller for multi-rate sensing/actuating; cyclic reformulation + LMI synthesis | IEEE Access (OA) |
| 2025 | LPTV system identification (this paper) | Subspace identification via cyclic reformulation + state coordinate transformation; no periodic input required | IEEE Access (OA) |
| 2025 | Multi-rate system identification | Extension of cyclic identification to systems with different sensor/actuator sampling rates | JRM |
| 2026 | Polytopic uncertainty construction (submitted) | Cyclic reformulation applied to LTI systems for robust model construction | arXiv:2601.12695 |
Development Arc
The research evolved from analysis and control design (2023) to system identification (2025–), using cyclic reformulation as a unifying framework. The multi-rate observer and controller papers established the cyclic reformulation as a practical tool for handling systems with different sampling rates. The identification paper then addressed the inverse problem: given input-output data, how to recover the LPTV model through the cyclic structure. The most recent work (arXiv, 2026) explores an application of cyclic reformulation to robust model construction for LTI systems.
Reading Guide
The theoretical foundation for the cyclic reformulation is developed in Bittanti and Colaneri's comprehensive treatment (Automatica, 2000; Springer, 2009), which systematically compares four reformulations of periodic systems: time-lifted, cyclic, frequency-lifted, and Fourier. For subspace identification methods applied to cycled signals, standard references such as Van Overschee and De Moor (1996) and Verhaegen and Verdult (2007) provide the necessary background. The coordinate transformation technique in this paper builds on these established frameworks but introduces new algebraic results on the sparsity structure of Markov parameters specific to cyclic reformulations.
For the full derivation of the sparsity conditions and the transformation matrix construction, see Sections III–IV of the original paper.