This article provides a detailed explanation of system identification for linear periodically time-varying (LPTV) systems using cyclic reformulation and subspace identification. Related articles, related papers, and MATLAB links are placed at the bottom.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

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) (Open Access)

This paper is co-work with Prof. Yusuke Fujimoto (Associate Professor, The University of Osaka), Prof. Hiroshi Oku (Professor, Osaka Institute of Technology), and Haruto Kondo (Kumamoto University).

Contents

Why Identification of LPTV Systems Matters
LPTV System Model
Cyclic Reformulation: Converting LPTV to LTI
Properties of Markov Parameters for Cycled Systems
Subspace Identification Using Cycled Signals
State Coordinate Transformation for Recovering LPTV Parameters (Theorem 1)
Identification Algorithm
Numerical Examples
Connections to Related Research
MATLAB Code
Related Articles and Videos
Paper Information

Why Identification of LPTV Systems Matters

In many practical systems, internal parameters change periodically over time. Such systems are called linear periodically time-varying (LPTV) systems. Examples include:

Multi-rate control systems, where sensors and actuators operate at different sampling rates, can be treated as LPTV systems.
Space tether systems and optical disc systems involve periodic variations in their dynamics.
Multi-agent systems with periodically switching network topologies exhibit periodic time-varying behavior.
Communication systems using periodically time-varying quantizers for bit-rate control.

While analysis and design theories for linear time-invariant (LTI) systems are well established, obtaining accurate models of LPTV systems from data remains a challenging problem. Conventional identification methods for LPTV systems often rely on specific periodic input signals, which limits their practical applicability.

This paper proposes a novel identification algorithm that does not require any special periodic input signals. The method uses cyclic reformulation to convert the LPTV identification problem into an equivalent LTI identification problem, then recovers the LPTV parameters through a carefully designed state coordinate transformation.

LPTV System Model

Consider an $n$ -order discrete-time LPTV system:

$\displaystyle x(k+1) = A_k x(k) + B_k (u(k) + w(k))$

$\displaystyle y(k) = C_{k} x(k) + D_{k} u(k) + v(k)$

where $x(k) \in \mathbb{R}^{n}$ is the state, $u(k) \in \mathbb{R}^{m}$ is the control input, $y(k) \in \mathbb{R}^{l}$ is the output, and $w(k)$ , $v(k)$ are disturbances. The system matrices $A _{k}$ , $B _{k}$ , $C _{k}$ , $D _{k}$ are periodic with period $M$ :

$\displaystyle A_k = A_{k \bmod M}, \quad B_k = B_{k \bmod M}, \quad C_k = C_{k \bmod M}, \quad D_k = D_{k \bmod M}$

The pairs $(C _{k}, A _{k})$ and $(A _{k}, B _{k})$ are assumed to be observable and controllable, respectively. The period $M$ is assumed to be known.

The system identification problem (Problem 1 in the paper) is: given input-output data, estimate $A _{k}$ , $B _{k}$ , $C _{k}$ , $D _{k}$ for $k = 0, \ldots, M-1$ up to state coordinate transformation.

Cyclic Reformulation: Converting LPTV to LTI

The key mathematical tool is the cyclic reformulation, which transforms an LPTV system into an equivalent LTI system. The idea is to rearrange the signals into a cycled structure.

Cycled Input Signals

The cycled input $\check{u}(k) \in \mathbb{R}^{Mm}$ is constructed by placing the original input $u(k)$ into the appropriate block position that cycles through the $M$ blocks:

$\displaystyle \check{u}(0) = \begin{pmatrix} u(0) \cr O_{m,1} \cr \vdots \cr O_{m,1} \end{pmatrix}, \quad \check{u}(1) = \begin{pmatrix} O_{m,1} \cr u(1) \cr \vdots \cr O_{m,1} \end{pmatrix}, \quad \ldots$

The non-zero sub-vector $u(k)$ cyclically shifts along the block positions.

Cycled System

The cyclic reformulation of the $M$ -periodic system yields an LTI system:

$\displaystyle \check{x}(k+1) = \check{A}\check{x}(k) + \check{B}(\check{u}(k) + \check{w}(k))$

$\displaystyle \check{y}(k) = \check{C}\check{x}(k) + \check{D}\check{u}(k) + \check{v}(k)$

where $\check{A} \in \mathbb{R}^{Mn \times Mn}$ , $\check{B} \in \mathbb{R}^{Mn \times Mm}$ , $\check{C} \in \mathbb{R}^{Ml \times Mn}$ , $\check{D} \in \mathbb{R}^{Ml \times Mm}$ .

The matrices $\check{A}$ and $\check{B}$ have a cyclic structure: the original matrices $A _{0}, \ldots, A _{M-1}$ (and $B _{0}, \ldots, B _{M-1}$ ) appear on shifted sub-diagonals. The matrices $\check{C}$ and $\check{D}$ are block diagonal.

Example: For a 2nd-order SISO system with period $M = 3$ :

$\displaystyle \check{A} = \begin{pmatrix} O_{2,2} & O_{2,2} & A_2 \cr A_0 & O_{2,2} & O_{2,2} \cr O_{2,2} & A_1 & O_{2,2} \end{pmatrix}, \quad \check{C} = \begin{pmatrix} C_0 & O_{1,2} & O_{1,2} \cr O_{1,2} & C_1 & O_{1,2} \cr O_{1,2} & O_{1,2} & C_2 \end{pmatrix}$

This conversion is exact and preserves the input-output behavior of the original LPTV system.

Properties of Markov Parameters for Cycled Systems

The paper derives important structural properties of the Markov parameters of the cycled system. The Markov parameters are defined as:

$\displaystyle \check{H}(i) = \begin{cases} \check{D}, & i = 0 \cr \check{C}\check{A}^{i-1}\check{B}, & i = 1, 2, \ldots \end{cases}$

A shift matrix $\check{S} _{q}$ is introduced, which cyclically permutes block rows. Using this matrix, the paper establishes three lemmas:

Lemma 1: The matrix $\check{S} _{l}^{i} \check{H}(i)$ is a block diagonal matrix for any $i \geq 0$ .

Lemma 2: The matrix $\check{H}(i) \check{S} _{m}^{i}$ is a block diagonal matrix for any $i \geq 0$ .

Lemma 3 (General form): The matrix $\check{S} _{l}^{i} \check{H}(i+j) \check{S} _{m}^{j}$ is a block diagonal matrix for any $i, j \geq 0$ .

These properties are fundamental for the proposed identification algorithm. They reveal that the Markov parameters of the cycled system have a hidden block-diagonal structure when appropriately shifted — a property unique to cyclic reformulations.

Subspace Identification Using Cycled Signals

The identification strategy proceeds as follows. The cycled input and output signals $\check{u}(k)$ and $\check{y}(k)$ are constructed from the original input-output data. Then, a standard subspace identification method (such as N4SID) is applied to the cycled signals, yielding an LTI state-space model with parameters $\mathcal{A} _{\ast}$ , $\mathcal{B} _{\ast}$ , $\mathcal{C} _{\ast}$ , $\mathcal{D} _{\ast}$ .

The obtained model has the correct Markov parameters but, in general, the matrices $\mathcal{A} _{\ast}$ , $\mathcal{B} _{\ast}$ , $\mathcal{C} _{\ast}$ , $\mathcal{D} _{\ast}$ are dense matrices — they do not exhibit the cyclic/block-diagonal structure of the original cyclic reformulation. The key challenge is to recover this structure.

The paper introduces Assumption 1: the block-diagonal property of Lemma 3 is also satisfied by the Markov parameters of the identified system. This assumption is verified numerically in the simulations.

State Coordinate Transformation for Recovering LPTV Parameters (Theorem 1)

The central theoretical contribution is a state coordinate transformation that converts the dense identified model into the cyclic reformulation form.

The transformation matrix $T$ is defined through its inverse:

$\displaystyle T^{-1} = \sum_{j=1}^{n} \check{F}_j \check{S}_l^{j-1} \mathcal{C}_{\ast} \mathcal{A}_{\ast}^{j-1}$

where $\check{F} _{j}$ are block diagonal matrices constructed from a matrix $F$ that satisfies certain rank conditions related to observability.

Theorem 1: Under Assumption 1 and the controllability/observability conditions, the state coordinate transformation using $T$ from the above equation converts the identified model into the cyclic reformulation form:

$\displaystyle \check{\mathcal{A}} = T^{-1}\mathcal{A}_{\ast}T, \quad \check{\mathcal{B}} = T^{-1}\mathcal{B}_{\ast}, \quad \check{\mathcal{C}} = \mathcal{C}_{\ast}T, \quad \check{\mathcal{D}} = \mathcal{D}_{\ast}$

The proof shows that $T^{-1}\mathcal{B} _{\ast}$ is a cyclic matrix, $\mathcal{C} _{\ast}T$ is a block diagonal matrix, and $T^{-1}\mathcal{A} _{\ast}T$ is a cyclic matrix. These structural properties follow from the Markov parameter properties established in the lemmas. For the complete proof, see Section IV of the paper.

Identification Algorithm

The complete identification procedure is summarized as Algorithm 1 in the paper:

Prepare cycled signals: Transform the input-output data into cycled signals $\check{u}(k)$ and $\check{y}(k)$ .
Subspace identification: Apply a standard subspace identification method (e.g., N4SID) to the cycled signals to obtain $\mathcal{A} _{\ast}$ , $\mathcal{B} _{\ast}$ , $\mathcal{C} _{\ast}$ , $\mathcal{D} _{\ast}$ .
State coordinate transformation: Compute the transformation matrix $T$ and apply the coordinate transformation to obtain the cyclic reformulation $\check{\mathcal{A}}$ , $\check{\mathcal{B}}$ , $\check{\mathcal{C}}$ , $\check{\mathcal{D}}$ .
Extract LPTV parameters: Read off $A _{k}$ , $B _{k}$ , $C _{k}$ , $D _{k}$ from the block elements of the cyclic reformulation.

Steps 1 and 3 require minimal computation time. Step 2 uses conventional subspace identification, so the overall computational cost is comparable to identifying an LTI system.

A key advantage: no specific periodic input signals are required. The input can be any signal with sufficient excitation.

Numerical Examples

Verification of Assumption 1

The paper considers a 2nd-order SISO LPTV system with period $M = 3$ . The plant $P _{ex}$ is given in observability companion form, with the following parameters:

$\displaystyle A_0 = \begin{pmatrix} 0 & 1 \cr 0.5 & 1 \end{pmatrix}, \quad A_1 = \begin{pmatrix} 0 & 1 \cr 0.9 & -0.95 \end{pmatrix}, \quad A_2 = \begin{pmatrix} 0 & 1 \cr 1 & 0.5 \end{pmatrix}$

$\displaystyle B_{0} = \begin{pmatrix} 1 \cr 2 \end{pmatrix}, \quad B_{1} = \begin{pmatrix} 1.5 \cr 2 \end{pmatrix}, \quad B_{2} = \begin{pmatrix} 1 \cr 0.5 \end{pmatrix}$

$\displaystyle C_{0} = C_{1} = C_{2} = \begin{pmatrix} 1 & 0 \end{pmatrix}, \quad D_{0} = D_{1} = D_{2} = 0.5$

Although $A _{0}$ , $A _{1}$ , and $A _{2}$ individually have unstable poles, the system is stable as a periodic system. The paper first verifies that Assumption 1 holds by computing $\check{S} _{1}^{i} \check{\mathcal{H}}(i)$ for the identified model and confirming that diagonal matrices are obtained for all $i$ .

Noise-Free Case

Using randomly generated (non-periodic) input and 1000 data points, the subspace identification (N4SID) is applied to the cycled signals. After the state coordinate transformation, the recovered LPTV parameters match the true parameters of $P _{ex}$ accurately. This confirms that the algorithm successfully solves the identification problem without requiring periodic inputs.

Case with Process Noise

The input is generated from a standard normal distribution, and process noise $w(k)$ with variance 1/5 is added. Even with 20 percent noise, the algorithm recovers parameters close to the true values, with a mean square error of 0.0691. Assumption 1 is also confirmed to hold in the noisy case.

This work is part of a broader research program on periodic time-varying systems and multi-rate control:

Multi-Rate State Observer — H. Okajima, Y. Hosoe and T. Hagiwara, State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm, IEEE Access (2023). Designs a periodically time-varying state observer for systems with multi-rate sensors using LMI optimization. Multi-rate systems can be treated as LPTV systems, and cyclic reformulation is the core mathematical tool shared with the present paper.

Multi-Rate Observer-Based Feedback Control — H. Okajima, K. Arinaga and A. Hayashida, Design of observer-based feedback controller for multi-rate systems with various sampling periods using cyclic reformulation, IEEE Access, Vol. 11, pp. 121956–121965 (2023). Extends to a complete observer-based feedback controller for multi-rate systems using cyclic reformulation.

Multi-Rate System Identification — H. Okajima, R. Furukawa and N. Matsunaga, System Identification Under Multi-rate Sensing Environment, Journal of Robotics and Mechatronics (2025). Applies cyclic reformulation-based identification to multi-rate systems where input and output have different sampling rates.

Multi-Rate Kalman Filter — H. Okajima, LMI Optimization Based Multirate Steady-State Kalman Filter Design, IEEE Access (2026, submitted) arXiv:2602.01537. Extends multi-rate estimation to Kalman filter design.

Model Error Compensator (MEC) — The identified LPTV model can be used in model-based control with the Model Error Compensator to achieve robust control for systems with periodic time-varying dynamics.

MATLAB Code

GitHub (LMI basics): MATLAB Fundamental Control LMI

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Multi-rate System / Publications / LMI / MEC

Video

YouTube: Control Engineering Channel / Portal

Paper Information

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. DOI: 10.1109/ACCESS.2025.3537086 (Open Access)

Co-authors: Yusuke Fujimoto (Associate Professor, The University of Osaka), Hiroshi Oku (Professor, Osaka Institute of Technology), Haruto Kondo (Kumamoto University)

arXiv preprint: arXiv:2411.00318

Self-Introduction

Hiroshi Okajima — Associate Professor, Graduate School of Science and Technology, Kumamoto University. Member of SICE, ISCIE, and IEEE.