This article explains how to design an observer-based feedback controller for multi-rate systems where sensors and actuators operate at different sampling periods. The key idea is to convert the multi-rate system into a periodically time-varying (LPTV) system and then apply cyclic reformulation to obtain a time-invariant representation. This allows the use of standard LMI (Linear Matrix Inequality) techniques to jointly design observer gains and feedback gains that minimize the $l_{2}$ -induced norm from disturbances to outputs.

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

This article is based on the following paper.

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) (Open Access)

MATLAB Code: GitHub — MATLAB_state_observer / 06_multirate_feedback · Code Ocean Capsule

The co-authors, Kenta Arinaga and Atsuo Hayashida, were graduate students at the Graduate School of Science and Technology, Kumamoto University at the time of publication. Arinaga received the B.E. (2021) and M.E. (2023) degrees from Kumamoto University. Hayashida received the B.E. degree (2023) from Kumamoto University.

Contents

Why Multi-Rate Observer-Based Feedback Matters
Plant Model Under Multi-Rate Environment
Observer-Based Feedback Control System
Cyclic Reformulation: From Time-Varying to Time-Invariant
LMI-Based Performance Analysis and Gain Design
Numerical Example
Connections to Related Research
MATLAB Code
Related Articles and Videos
Paper Information

Why Multi-Rate Observer-Based Feedback Matters

In practical control systems, sensors and actuators rarely operate at the same sampling rate. A camera sensor may capture images at 30 fps, while an IMU provides data at 200 Hz, and a force sensor updates at 50 Hz. On the actuation side, different actuators may accept commands at different rates depending on their physical characteristics and operating costs.

This mismatch in sampling periods creates a multi-rate environment that poses significant design challenges. The standard separation principle — which allows independent design of observer and controller — does not directly apply when the observation and control periods vary across channels. Lifting techniques, which are commonly used for multi-rate analysis, introduce products of design variables when multiple sensor periods are involved, making the design problem nonconvex.

This paper provides a unified design framework that handles both multi-rate sensing and multi-rate actuation simultaneously. The approach converts the multi-rate system into a periodically time-varying (LPTV) system, then applies cyclic reformulation to obtain an equivalent LTI system. This enables the use of LMI-based methods for jointly designing the observer gains $L_{k}$ and feedback gains $F_{k}$ while minimizing the worst-case disturbance amplification measured by the $l_{2}$ -induced norm.

Plant Model Under Multi-Rate Environment

LTI Plant

The underlying plant $P$ is a discrete-time linear time-invariant multi-input multi-output system:

$\displaystyle x(k+1) = Ax(k) + Bu_r(k) + B_2 d_u(k)$

$\displaystyle y_r(k) = Cx(k) + Dd_w(k)$

where $x(k) \in \mathbb{R}^{n}$ is the state, $u_{r}(k) \in \mathbb{R}^{m}$ is the control input, $d_{u}(k) \in \mathbb{R}^{m_{2}}$ is the process noise, $d_{w}(k) \in \mathbb{R}^{q}$ is the observation noise, and $y_{r}(k) \in \mathbb{R}^{q}$ is the observed output. It is assumed that $(A, B)$ is controllable and $(C, A)$ is observable.

Multi-Rate Sampling Structure

Each output channel $i$ is observed periodically with sensing period $\mathcal{M}_{i} \in \mathbb{N}$ , and each input channel $i$ is applied with input period $M_{i} \in \mathbb{N}$ . Let $M$ be the least common multiple of all sensing and input periods.

The multi-rate structure is described by two periodically time-varying diagonal matrices:

$\displaystyle S_k = \mathrm{diag}\left(s_1(k), s_2(k), \ldots, s_q(k)\right)$

$\displaystyle T_k = \mathrm{diag}\left(t_1(k), t_2(k), \ldots, t_m(k)\right)$

where $s_{i}(k) = 1$ if the $i$ -th output is observed at time $k$ (and 0 otherwise), and $t_{i}(k) = 1$ if the $i$ -th input is applied at time $k$ (and 0 otherwise). Both $S_{k}$ and $T_{k}$ have period $M$ .

Using these matrices, the multi-rate plant is written as:

$\displaystyle x(k+1) = Ax(k) + BT_k u(k) + B_2 d_u(k)$

$\displaystyle y(k) = S_k C x(k) + S_k D d_w(k)$

This is an $M$ -periodic time-varying system where the periodic matrices $S_{k}$ and $T_{k}$ encode the sampling schedule of all sensors and actuators.

Step-Like Input Extension

The paper also considers the case where control inputs are held constant between update times (step-like inputs). This is handled by augmenting the state to include a memory $u_{m}(k)$ that holds the most recent input value:

$\displaystyle x_*(k) = \begin{pmatrix} x(k) \cr u_m(k) \end{pmatrix}$

The augmented system matrices $\mathcal{A}\bar{T}_{k}$ and $\mathcal{B}T_{k}$ switch between two modes depending on whether the input is being updated or held. This extension is important for practical implementation where actuators maintain their command between update instants.

Observer-Based Feedback Control System

Periodically Time-Varying Observer

For the multi-rate plant, a periodically time-varying observer with gains $L_{k}$ (where $k = 0, 1, \ldots, M-1$ ) is defined as:

$\displaystyle x_{ob}(k+1) = (A - L_k S_k C) x_{ob}(k) + Bu(k) + L_k y(k)$

The observer gain $L_{k}$ has period $M$ , i.e., $L_{k+M} = L_{k}$ . The gain varies periodically because the available output information changes at each time step depending on which sensors are active.

State Feedback with Periodic Gains

Using the estimated state $x_{ob}$ , the state feedback controller is:

$\displaystyle u(k) = -F_k x_{ob}(k)$

where $F_{k}$ (with $k = 0, 1, \ldots, M-1$ ) is a periodically time-varying feedback gain. The feedback gain also varies with period $M$ because not all input channels are active at every time step.

Closed-Loop Error System

Defining the estimation error $e(k) = x(k) - x_{ob}(k)$ , the observer-based feedback control system is:

$\displaystyle \begin{pmatrix} x(k+1) \cr e(k+1) \end{pmatrix} = \begin{pmatrix} A - BT_k F_k & BT_k F_k \cr O_{n,n} & A - L_k S_k C \end{pmatrix} \begin{pmatrix} x(k) \cr e(k) \end{pmatrix} + \begin{pmatrix} B_2 & O_{n,q} \cr B_2 & -L_k S_k D \end{pmatrix} \begin{pmatrix} d_u(k) \cr d_w(k) \end{pmatrix}$

$\displaystyle y(k) = \begin{pmatrix} S_k C & O_{q,n} \end{pmatrix} \begin{pmatrix} x(k) \cr e(k) \end{pmatrix} + \begin{pmatrix} O_{q,m_2} & S_k D \end{pmatrix} \begin{pmatrix} d_u(k) \cr d_w(k) \end{pmatrix}$

The design objective is to find $F_{k}$ and $L_{k}$ that stabilize this system and minimize the $l_{2}$ -induced norm from the disturbance $d(k)$ to an evaluation signal $z(k)$ .

Cyclic Reformulation: From Time-Varying to Time-Invariant

The Cycled Signal

The key technique in this paper is cyclic reformulation, which converts the $M$ -periodic time-varying system into an equivalent LTI system while preserving the original sampling rate (unlike lifting, which reduces it by a factor of $M$ ).

For a signal $u(k)$ , the cycled signal $\check{u}(k)$ is a column vector with $M$ elements, where only one element is non-zero at each time step and the non-zero position shifts cyclically:

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

All signals in the system — states, inputs, outputs, disturbances — are cycled in the same way.

Cycled Plant

The cyclic reformulation of the multi-rate plant gives an LTI system:

$\displaystyle \check{x}(k+1) = \check{A}\check{x}(k) + \check{B}\check{u}(k) + \check{B}_2 \check{d}_u(k)$

$\displaystyle \check{y}(k) = \check{C}\check{x}(k) + \check{D}\check{d}_w(k)$

where the matrices $\check{A}$ , $\check{B}$ , $\check{C}$ , $\check{D}$ are time-invariant block matrices of size $Mn \times Mn$ (etc.). For example, $\check{A}$ has the structure of a cyclic permutation matrix with $A$ on the sub-diagonal and $A$ in the top-right corner. The observation and input timing information from $S_{k}$ and $T_{k}$ is absorbed into the block structure of $\check{B}$ and $\check{C}$ .

Cycled Observer-Based Feedback System

The cyclic reformulation of the entire observer-based feedback control system gives an LTI system $\check{G}$ from the cycled disturbance $\check{d}(k)$ to the cycled evaluation signal $\check{z}(k)$ . The system matrices involve the block-diagonal observer gain matrix $\check{L}$ and feedback gain matrix $\check{F} = \mathrm{diag}\lbrack F_{0}, \ldots, F_{M-1} \rbrack$ .

A crucial property is that the $l_{2}$ -induced norms of the original periodic system and the cycled LTI system are equal:

$\displaystyle \|G\|_{l_2/l_2} = \|\check{G}\|_{l_2/l_2}$

This means that analyzing or optimizing the performance of the cycled LTI system is equivalent to doing so for the original multi-rate system.

LMI-Based Performance Analysis and Gain Design

Performance Analysis (Theorem 5.1)

For given $\check{L}$ and $\check{F}$ , the $l_{2}$ -induced norm of $\check{G}$ is less than $\gamma$ if and only if there exists $\check{X} \gt 0$ (or $\check{P} \gt 0$ ) satisfying a standard LMI condition for discrete-time systems. Since $\check{G}$ is LTI, these are standard LMI feasibility problems solvable by SDP solvers.

Initial Gain Design

The initial observer gain $\check{L}$ is obtained by Theorem 5.2: minimizing $\gamma_{L}^{2}$ subject to an LMI with variables $\check{P}_{L}$ and $\check{Y} = -\check{P}_{L}\check{L}$ . Once $\check{Y}$ and $\check{P}_{L}$ are found, the observer gains are recovered as $L_{k} = -P_{L,k}^{-1}Y_{k}$ .

Similarly, the initial feedback gain $\check{F}$ is obtained by Theorem 5.3: minimizing $\gamma_{F}^{2}$ subject to an LMI with variables $\check{X}_{F}$ and $\check{Y} = -\check{F}\check{X}_{F}$ . The feedback gains are recovered as $\mathcal{F}_{k} = -Y_{k}X_{F,k}^{-1}$ .

Iterative Algorithm (Algorithm 1)

Since the joint design of $\check{F}$ and $\check{L}$ in the full system is not a convex problem, an iterative algorithm is proposed:

Compute initial $\check{L}$ using Theorem 5.2 (observer-only LMI).
Compute initial $\check{F}$ using Theorem 5.3 (state feedback LMI).
Loop for :
- Fix $\check{L}$ and $\check{F}$ , solve for $\gamma^{2}$ and $\check{P}$ via Theorem 5.1.
- Fix $\check{P}$ , solve for $\check{L}$ and $\check{F}$ via Problem 1 (LMI).
Return the final $L_{k}$ , $F_{k}$ .

The value of $\gamma$ decreases monotonically with each iteration. For the numerical example in the paper, $i_{\max} = 60$ iterations were used.

Numerical Example

Setup

A discrete-time LTI plant with 3 states, 2 inputs, and 2 outputs is considered. The plant matrices are:

$\displaystyle A = \begin{pmatrix} 0.92 & 0.1 & 0.2 \cr -0.15 & 1.1 & -0.25 \cr -0.1 & 0.3 & 0.95 \end{pmatrix}, \quad B = B_2 = \begin{pmatrix} 1 & -2 \cr -1 & 1.5 \cr 2 & 0.5 \end{pmatrix}$

$\displaystyle C = \begin{pmatrix} 1 & 0.5 & 0.2 \cr -0.3 & 1 & 1 \end{pmatrix}, \quad D = \begin{pmatrix} 1 & 0 \cr 0 & 1 \end{pmatrix}$

Note that this plant is unstable. The input periods are $M_{1} = 2$ and $M_{2} = 3$ ; the output sensing periods are $\mathcal{M}_{1} = 2$ and $\mathcal{M}_{2} = 4$ . The LCM is $M = 12$ , resulting in a periodic time-varying system with period 12.

Sampling Schedule

The input timing matrices $T_{k}$ (for $k = 0, \ldots, 11$ ) show that input $u_{1}$ is applied at even time steps and input $u_{2}$ is applied at multiples of 3. The output observation matrices $S_{k}$ show that $y_{1}$ is observed at odd time steps and $y_{2}$ is observed at multiples of 4 (with an offset). This creates a challenging scenario where different channels have interleaved, non-synchronized sampling schedules.

Results

Algorithm 1 with $i_{\max} = 60$ produces $\gamma = 31.8283$ , characterizing the $l_{2}$ -induced norm from disturbance $d(k)$ to the plant state $x(k)$ .

The resulting gains are 12 periodic feedback gains $F_{0}, \ldots, F_{11}$ (each of size $2 \times 3$ ) and 12 periodic observer gains $L_{0}, \ldots, L_{11}$ (each of size $3 \times 2$ ). Several of these gains are zero matrices, corresponding to time steps where neither input channel is active or neither output channel is observed. The remaining gains have non-trivial entries that vary with the position within the period.

Simulations with initial state $x(0) = \lbrack 3, 3, 3 \rbrack^{T}$ , initial observer state $x_{ob}(0) = \lbrack 0, 0, 0 \rbrack^{T}$ , process noise $d_{u}(k) \sim \mathcal{N}(0, 0.1^{2})$ , and observation noise $d_{w}(k) \sim \mathcal{N}(0, 0.05^{2})$ demonstrate that:

All plant states converge near zero (stabilization achieved despite the unstable open-loop dynamics).
All estimation errors converge near zero (accurate state estimation despite multi-rate observations).
The simulation-based $l_{2}$ -norm ratio is $G_{\mathrm{sim}} = 9.9733$ , which is well below the theoretical bound $\gamma = 31.8283$ .

This paper is the third in a sequence of works on multi-rate systems using cyclic reformulation. It builds directly on the multi-rate observer (Spoke #5) and extends it to the full observer-based feedback control problem.

Multi-Rate State Observer (IEEE Access 2023) — The predecessor paper H. Okajima, Y. Hosoe, and T. Hagiwara, "State observer under multi-rate sensing environment and its design using l2-induced norm," IEEE Access, 2023 addresses the observer design problem alone. It shows how to design multi-rate observer gains $L_{k}$ using cyclic reformulation and LMI techniques when only the sensing periods vary. The present paper extends this to include multi-rate actuation and state feedback gain design.

Multi-Rate State Feedback Control (Trans. SICE 2022) — The paper by H. Okajima, "Design of a disturbance reduction control system for multi-rate systems with long input periods," Trans. SICE, 2022, addresses the state feedback problem for systems with multiple control input periods. The present paper combines the observer design from the 2023 paper with the state feedback approach from this 2022 work into a unified observer-based feedback framework.

Multi-Rate System Identification (JRM 2025) — H. Okajima, R. Furukawa, and N. Matsunaga, "System Identification Under Multirate Sensing Environments," Journal of Robotics and Mechatronics, 2025 provides the system identification algorithm that obtains the plant model in multi-rate environments. The identified model serves as the basis for the observer and feedback design described in this paper.

LPTV System Identification (IEEE Access 2025) — H. Okajima and K. Arinaga, "Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems," IEEE Access, 2025 extends the cyclic reformulation approach to system identification for general LPTV systems, including the multi-rate case.

Multirate Kalman Filter (IEEE Access 2026) — H. Okajima, "LMI Optimization Based Multirate Steady-State Kalman Filter Design," IEEE Access, 2026 provides an alternative observer design approach based on the Kalman filter framework for multi-rate environments.

MATLAB Code

GitHub: MATLAB_state_observer / 06_multirate_feedback — MATLAB implementation of the observer-based feedback controller design for multi-rate systems.
Code Ocean: Capsule 3611894 — Reproducible code capsule (shared with the multi-rate observer paper).
MATLAB File Exchange: State Estimation for Multi-Rate Sampled Systems: IEEE ACCESS (ID: 182941) — Includes the multi-rate observer code that forms the basis of this work.

Blog Articles (blog.control-theory.com)

Observer and State Estimation: - State Observer and State Estimation: A Comprehensive Guide — Hub article covering the full landscape of observer design - State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm — The predecessor paper on multi-rate observer design (Spoke #5) - LMI Optimization Based Multirate Steady-State Kalman Filter Design — Multirate Kalman filter approach - Kalman Filter: From Basic Algorithm to Multi-Rate Extensions — Kalman filter tutorial - H-infinity Filter: Robust State Estimation Using LMI Optimization — Robust estimation using LMI

State Feedback and Control Design: - State Feedback Control and State-Space Design: A Comprehensive Guide — Hub article on state feedback control - State Feedback Control: Design Gains using Optimal Regulators (LQR) — LQR design tutorial - State Observer for State Space Model — Observer-based feedback basics

System Identification: - System Identification: From Data to Dynamical Models — A Comprehensive Guide — Hub article on system identification - System Identification Under Multirate Sensing Environments — Multi-rate system identification - Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems — LPTV system identification

Supporting Topics: - Linear Matrix Inequalities (LMIs) and Controller Design — LMI fundamentals - Model-Based Compensation Methods in Control Engineering: IMC, DOB, 2-DOF, and MEC — Model-based compensation overview - Model Error Compensator (MEC): Enhance the Robustness of Existing Control Systems — MEC for robust control

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

Multi-rate System — Overview of multi-rate system research
System Identification — System identification research
Linear Matrix Inequality — LMI-based design
Publications — Full publication list

Video

Control Engineering Channel (YouTube) — 制御工学に関する講義動画（in Japanese）

Video Portal

制御工学チャンネル — 動画ポータルサイト: 状態推定（in Japanese）

Paper Information

Full Citation: 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. DOI: 10.1109/ACCESS.2023.3329117 (Open Access, CC BY-NC-ND 4.0)

Co-authors: - Kenta Arinaga — Graduate School of Science and Technology, Kumamoto University (B.E. 2021, M.E. 2023) - Atsuo Hayashida — Graduate School of Science and Technology, Kumamoto University (B.E. 2023)

Related Papers by the Same Authors: - H. Okajima, Y. Hosoe, and T. Hagiwara, "State observer under multi-rate sensing environment and its design using l2-induced norm," IEEE Access, Vol. 11, pp. 20079–20087, 2023. DOI: 10.1109/ACCESS.2023.3249188 - H. Okajima, "Design of a disturbance reduction control system for multi-rate systems with long input periods," Trans. SICE (in Japanese), Vol. 58, No. 10, pp. 451–458, 2022.

Key References Cited in the Paper: - S. Bittanti and P. Colaneri, Periodic Systems: Filtering and Control, Springer, 2009 — Comprehensive reference on cyclic reformulation. - Y. Ebihara, System Control With LMI, Morikawa Publishing, 2012 — LMI-based control design.

Self-Introduction

I am Hiroshi Okajima, Associate Professor in the Graduate School of Science and Technology at Kumamoto University, Japan. I received my M.E. and Ph.D. degrees from Osaka University in 2004 and 2007, respectively. My research focuses on robust control, multi-rate systems, model error compensation, and state estimation for networked control systems.

Research Website: www.control-theory.com
Blog: blog.control-theory.com
YouTube: Control Engineering Channel
Video Portal: portal.control-theory.com
GitHub: Hiroshi-Okajima
MATLAB File Exchange: Author Profile

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