This article provides a detailed explanation of system identification for control systems with multiple sensors operating at different sampling rates, using cyclic reformulation and subspace identification methods. 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, R. Furukawa and N. Matsunaga, System Identification Under Multirate Sensing Environments, Journal of Robotics and Mechatronics, Vol. 37, No. 5, pp. 1102–1112 (2025) (Open Access)

This paper is co-work with Risa Furukawa (Master Course Student, Kumamoto University) and Prof. Nobutomo Matsunaga (Professor, Kumamoto University).

Why Multirate System Identification Matters

In modern control systems, multiple sensors with different sampling rates are commonly used. For example:

In mobile robot control and autonomous driving, cameras, LiDAR, IMUs, and wheel encoders all operate at different rates. System identification for model-based control must handle these heterogeneous rates.
In industrial process control, temperature sensors, pressure sensors, and flow meters may have very different observation periods.
In sensor fusion for IoT networks, diverse measurement devices provide data at distinct intervals.

When input and output signals are sampled at different rates, the datasets effectively have missing values, making system identification significantly more challenging than the standard single-rate case. Conventional methods often require specific periodic input signals or resort to lifting techniques that produce models where reverting to the original multirate system parameters is difficult.

This paper proposes an algorithm that identifies multirate systems without requiring special periodic inputs, by formulating the multirate system as a periodically time-varying system and applying cyclic reformulation with a novel state coordinate transformation.

Multirate System Formulation

Single-Rate Plant Model

The underlying plant is a discrete-time LTI system:

$\displaystyle x(k+1) = Ax(k) + Bu(k)$

$\displaystyle y(k) = Cx(k) + Du(k)$

where $x(k) \in \mathbb{R}^{n}$ is the state, $u(k) \in \mathbb{R}^{m}$ is the input, and $y(k) \in \mathbb{R}^{l}$ is the output. The system is assumed to be controllable and observable, and the matrix rank of $A$ is $n$ .

Multirate Sensing via Periodic Matrices

When different outputs have different observation periods, the multirate system is represented using a periodically time-varying diagonal matrix $V _{k}$ :

$\displaystyle y(k) = V_k C x(k) + V_k D u(k)$

$\displaystyle V_{k} = \mathrm{diag}[v_{1}(k), \ldots, v_{l}(k)$ ]

where $v _{i}(k) = 1$ when output $y _{i}$ is observed (i.e., when $k$ is a multiple of the observation period $M _{i}$ ), and $v _{i}(k) = 0$ otherwise. The frame period is $M = \mathrm{lcm}(M _{1}, \ldots, M _{l})$ , and $V _{k} = V _{k \bmod M}$ .

Key insight: By incorporating the multirate structure into $V _{k}$ , the parameters to be identified are reduced to just the four time-invariant matrices $A, B, C, D$ , rather than time-varying parameters at each step.

Example: For a system with $l = 2$ outputs, $M _{1} = 2$ and $M _{2} = 3$ , the frame period is $M = 6$ . The matrices $V _{0}, \ldots, V _{5}$ encode which sensors observe at each step. For instance, at $k = 0$ both sensors observe ( $V _{0} = I$ ), while at $k = 1$ neither observes ( $V _{1} = 0$ ).

Assumption 1 (Observability): There exists at least one $j$ such that the pair $(V _{j} C, A^{M})$ is observable.

Problem: Given input-output data from the multirate system, estimate $A, B, C, D$ up to state coordinate transformation.

Cyclic Reformulation of Multirate Systems

The cyclic reformulation transforms the multirate system into an equivalent LTI system by constructing cycled signals.

Cycled Signals

From the original input $u(k)$ , the cycled input $\check{u}(k) \in \mathbb{R}^{Mm}$ is constructed by placing $u(k)$ at the position corresponding to $k \bmod M$ and filling the rest with zeros. The cycled output is constructed similarly.

Cycled System

The cyclic reformulation yields an LTI system:

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

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

where $\check{A} \in \mathbb{R}^{Mn \times Mn}$ and $\check{B} \in \mathbb{R}^{Mn \times Mm}$ are cyclic matrices (each sub-diagonal block contains $A$ or $B$ , with one wrap-around block), and $\check{C} \in \mathbb{R}^{Ml \times Mn}$ and $\check{D} \in \mathbb{R}^{Ml \times Mm}$ are block diagonal matrices with entries $V _{i} C$ and $V _{i} D$ on the diagonal.

A notable difference from the LPTV case: in the multirate system, the cyclic matrices $\check{A}$ and $\check{B}$ have identical blocks $A$ and $B$ at every position (since the underlying plant is LTI), while the periodic structure appears only in $\check{C}$ and $\check{D}$ through the observation matrices $V _{k}$ .

Controllability and Observability of the Cycled System

The paper proves that:

Controllability of the cycled system is automatically satisfied when the pair $(A, B)$ is controllable.
Observability of the cycled system follows from Assumption 1 and the regularity of $A$ .

These conditions are easier to satisfy than the corresponding conditions for general LPTV systems, because the multirate structure has simpler requirements.

Properties of Markov Parameters

The Markov parameters of the cycled system are:

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

Using a permutation matrix $\check{S} _{q}$ that cyclically shifts block rows, the paper establishes key structural lemmas:

Lemma 1 (General): The matrix $\check{S} _{l}^{\,i} \check{H}(i+j) \check{S} _{m}^{\,j}$ is a block diagonal matrix for any non-negative integers $i, j$ .

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

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

These properties reveal that the Markov parameters of the cycled system have a hidden block diagonal structure when appropriately shifted — an essential property for recovering the original system parameters from an identified model.

System Identification Algorithm

The complete identification procedure (Algorithm 1 in the paper) consists of four steps:

Step 1: Determine the frame period $M$ from each output rate $M _{i}$ and prepare cycled input $\check{u}(k)$ and cycled output $\check{y}(k)$ signals from the measured data.

Step 2: Apply a standard subspace identification method (e.g., N4SID) to the cycled signals to obtain $\mathcal{A} _{\ast}, \mathcal{B} _{\ast}, \mathcal{C} _{\ast}, \mathcal{D} _{\ast}$ .

Step 3: Apply the state coordinate transformation using the matrix $T$ to recover the cyclic reformulation structure.

Step 4: Extract the identified parameters $A _{m}, B _{m}, C _{m}, D _{m}$ from the cyclic reformulation matrices.

All steps require minimal computation time, comparable to identifying a standard LTI system.

Key advantage: The algorithm works with arbitrary (non-periodic) inputs, eliminating the need for special input signal design.

State Coordinate Transformation (Theorem 1)

The identified parameters $\mathcal{A} _{\ast}, \mathcal{B} _{\ast}, \mathcal{C} _{\ast}, \mathcal{D} _{\ast}$ are generally dense matrices. A coordinate transformation matrix $T$ is defined as:

$\displaystyle T = \sum_{i=0}^{n-1} \sum_{j=0}^{M-1} \mathcal{A}_\ast^{Mi+j} \mathcal{B}_\ast \check{S}_m^{j+1} \check{G}_j$

where $\check{G} _{j}$ are block diagonal matrices constructed from selection matrices.

Assumption 2: The shifted Markov parameters of the identified system satisfy the block diagonal property: $\check{S} _{l}^{\,i} \check{\mathcal{H}}(i+j) \check{S} _{m}^{\,j}$ is a block diagonal matrix for any $i, j \geq 0$ . This assumption is verified numerically.

Theorem 1: Under Assumption 2, if the identified system is controllable and observable, the state coordinate transformation using $T$ produces matrices with the cyclic reformulation structure. Specifically, $T^{-1}\mathcal{B} _{\ast}$ is a cyclic matrix, $\mathcal{C} _{\ast} T$ is a block diagonal matrix, and $T^{-1}\mathcal{A} _{\ast} T$ is a cyclic matrix.

Important note: The coordinate transformation matrix in this paper differs from the LPTV case in the earlier paper. In general LPTV systems, observability and controllability are required at each time instant. In multirate systems, this condition is typically not satisfied because observability varies across sampling instants. The new transformation matrix is specifically designed for multirate systems to ensure regularity.

Numerical Examples

System Setup

The paper uses a 3rd-order SISO plant ( $n = 3, m = 1, l = 2$ ) with output rates $M _{1} = 2$ and $M _{2} = 3$ , giving frame period $M = 6$ . The plant has the transfer functions:

$\displaystyle \mathrm{TF}_1 = \frac{z^2 + 0.9z}{z^3 + 0.4z^2 - 0.5z - 0.8}$

$\displaystyle \mathrm{TF}_{2} = \frac{0.1z^{2} + 0.34z + 0.77}{z^{3} + 0.4z^{2} - 0.5z - 0.8}$

Results

The input is randomly generated (not periodic). After applying Algorithm 1:

The identified matrices $A _{m0} = A _{m1} = \cdots = A _{m5}$ are all equal, confirming that the underlying system is time-invariant.
Similarly, $B _{m0} = \cdots = B _{m5}$ are all equal.
The matrices $C _{mi}$ correctly reflect the multirate observation structure: rows corresponding to unobserved outputs at step $i$ are zero.
The matrices $D _{mi}$ are all zero, matching the true plant.

The transfer functions computed from the identified parameters are:

$\displaystyle \mathrm{TF}_{m1} = \frac{z^2 + 0.9z + 1.29 \times 10^{-15}}{z^3 + 0.4z^2 - 0.5z - 0.8}$

$\displaystyle \mathrm{TF}_{m2} = \frac{0.1z^{2} + 0.34z + 0.77}{z^{3} + 0.4z^{2} - 0.5z - 0.8}$

The identified transfer functions match the true transfer functions to machine precision, successfully solving the multirate identification problem without requiring periodic inputs.

This work builds on and extends a broader research program on multi-rate control systems and cyclic reformulation:

Cyclic Reformulation for LPTV System Identification — H. Okajima, Y. Fujimoto, H. Oku and H. Kondo, Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems, IEEE Access (2025). The predecessor paper that develops cyclic identification for general LPTV systems. The present JRM paper extends those results specifically to multirate systems, with a new coordinate transformation that handles the structural differences.

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 multi-rate sensing environments using LMI optimization. This observer design relies on knowing the plant model, which can now be obtained using the identification algorithm proposed in 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 (2023). Extends to a complete observer-based feedback controller for multi-rate systems using cyclic reformulation.

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

Model Error Compensator (MEC) — The identified multirate model can be combined with the Model Error Compensator to achieve robust control in multirate environments, compensating for model inaccuracies in the identified model.

MATLAB Code

MATLAB File Exchange (Multi-Rate Observer): State Estimation under Multi-Rate Sensing: IEEE ACCESS 2023
Code Ocean (Multi-Rate System): Multi-Rate System 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, R. Furukawa and N. Matsunaga, "System Identification Under Multirate Sensing Environments", Journal of Robotics and Mechatronics, Vol. 37, No. 5, pp. 1102–1112, 2025. DOI: 10.20965/jrm.2025.p1102 (Open Access)

Co-authors: Risa Furukawa (Master Course Student, Kumamoto University), Nobutomo Matsunaga (Professor, Kumamoto University)

arXiv preprint: arXiv:2503.12750

Self-Introduction

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

www.control-theory.com / Blog / YouTube / GitHub

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