This article provides a detailed explanation of observer-based feedback controller design for systems where both sensors and actuators operate at different sampling rates. 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, 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)

This paper is co-work with Kenta Arinaga (Master Course Student, Kumamoto University) and Atsuo Hayashida (Bachelor Student, Kumamoto University).

Contents

Why Multi-Rate Feedback Control Matters

In practical control systems, sensors and actuators often operate at different sampling rates. For example:

In IoT-based control systems, various sensors and actuators connected to the network have different specifications and operate with different sampling periods.
In autonomous driving and SLAM, camera sensors, 3D-LiDAR, and IMU have very different feasible sensing periods, while the actuators also have their own update rates.
In multi-vehicle systems, management of time-series information for multiple sensors with different rates is critical for accurate control.

While previous multi-rate research focused primarily on the difference between sensing and control periods (i.e., a single observation rate vs. a single input rate), real systems involve multiple sensors and multiple actuators, each with its own period. This paper addresses the full observer-based feedback control problem where both the observation and actuation periods can differ for each sensor/actuator channel.

The design is based on cyclic reformulation and LMI optimization of the $l_{2}$ -induced norm.

Plant Under Multi-Rate Environment

LTI Plant Model

Consider the discrete-time LTI MIMO plant:

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

$\displaystyle y_{r}(k) = Cx(k) + Dd_{w}(k)$

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. The pair $(A, B)$ is controllable and $(C, A)$ is observable.

Multi-Rate Sensing and Actuation via Periodic Matrices

The key extension over the multi-rate observer paper is that both inputs and outputs have different periods. Two periodically time-varying diagonal matrices are introduced:

Sensing matrix $S_{k}$ — characterizes which sensor outputs are observed:

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

where $s_{i}(k) = 1$ if the $i$ -th output is observed at time $k$ , and $s_{i}(k) = 0$ otherwise. The period of $s_{i}(k)$ is $\mathcal{M}_{i}$ .

Actuation matrix $T_{k}$ — characterizes which control inputs are applied:

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

where $t_{i}(k) = 1$ if the $i$ -th input is applied at time $k$ , and $t_{i}(k) = 0$ otherwise. The period of $t_{i}(k)$ is $M_{i}$ .

The frame period is $M = \mathrm{lcm}(M_{1}, \ldots, M_{m}, \mathcal{M}_{1}, \ldots, \mathcal{M}_{q})$ . The multi-rate plant is then:

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

$\displaystyle y(k) = S_{k} C x(k) + S_{k} D d_{w}(k)$

This representation encompasses any combination of different input and output periods by appropriate choice of $S_{k}$ and $T_{k}$ .

The paper also describes a step-like input extension, where inputs are held between application timings using an augmented state $x_{\ast}(k) = \lbrack x(k)^{T}, u_{m}(k)^{T} \rbrack^{T}$ .

Observer-Based Feedback Controller

Periodically Time-Varying Observer

The observer for the multi-rate plant has the same form as in the multi-rate observer paper:

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

where $L_{k}$ ( $k = 0, 1, \ldots, M-1$ ) are M-periodic observer gains. When $S_{k} = 0$ , the observer runs open-loop.

Periodically Time-Varying State Feedback

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

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

where $F_{k}$ ( $k = 0, 1, \ldots, M-1$ ) are M-periodic feedback gains. When $T_{k} = 0$ (no input applied), the feedback has no effect on the plant.

Closed-Loop System

Defining the estimation error $e(k) = x(k) - x_{\mathrm{ob}}(k)$ , the closed-loop system is:

$\displaystyle \begin{pmatrix} x(k+1) \cr e(k+1) \end{pmatrix} = \begin{pmatrix} A - BT_kF_k & BT_kF_k \cr 0 & A - L_kS_kC \end{pmatrix} \begin{pmatrix} x(k) \cr e(k) \end{pmatrix} + \begin{pmatrix} B_2 & 0 \cr B_2 & -L_kS_kD \end{pmatrix} \begin{pmatrix} d_u(k) \cr d_w(k) \end{pmatrix}$

An evaluation signal $z(k)$ is introduced for design:

$\displaystyle z(k) = E_1 \begin{pmatrix} x(k) \cr e(k) \end{pmatrix} + E_2 \begin{pmatrix} d_u(k) \cr d_w(k) \end{pmatrix}$

For example, setting $E_{1} = \lbrack I_{n} \; O_{n,n} \rbrack$ and $E_{2} = 0$ evaluates the effect of disturbances on the plant state $x(k)$ .

The design goal is to minimize the l2-induced norm of the system from disturbances $d(k) = \lbrack d_{u}(k)^{T}, d_{w}(k)^{T} \rbrack^{T}$ to $z(k)$ .

Cyclic Reformulation of the Control System

The entire observer-based feedback control system is transformed into an LTI system using cyclic reformulation:

$\displaystyle \begin{pmatrix} \check{x}(k+1) \cr \check{e}(k+1) \end{pmatrix} = \check{A}_G \begin{pmatrix} \check{x}(k) \cr \check{e}(k) \end{pmatrix} + \check{B}_G \begin{pmatrix} \check{d}_u(k) \cr \check{d}_w(k) \end{pmatrix}$

where:

$\check{A}_{G}$ contains the cyclic reformulations of the plant, observer, and feedback controller.
The observer gain matrix $\check{L}$ has a cyclic structure with blocks $L_{0}, L_{1}, \ldots, L_{M-1}$ .
The feedback gain matrix $\check{F}$ has a block diagonal structure: $\check{F} = \mathrm{diag}\lbrack F_{0}, \ldots, F_{M-1} \rbrack$ .

Since the cycled system is LTI, standard l2-induced norm analysis tools can be applied. A key property is that the l2-induced norms of the original periodic system and the cycled system are equal.

Performance Analysis (Theorem 5.1)

For given observer gains $\check{L}$ and feedback gains $\check{F}$ , the l2-induced norm of the cycled system $\check{G}$ can be analyzed using standard LMI conditions.

Theorem 5.1: The following conditions are equivalent:

(i) The closed-loop system is stable and its l2-induced norm is less than $\gamma$ .

(ii) There exists $\check{X} > 0$ satisfying a matrix inequality involving $\check{A}_{G}, \check{B}_{G}, \check{E}_{1}, \check{E}_{2}$ , and $\gamma$ .

(iii) There exists $\check{P} > 0$ satisfying a dual matrix inequality.

Since the conditions (ii) and (iii) are LMIs when $\gamma^{2}$ is given, the minimum $\gamma$ can be found using standard SDP solvers. For the explicit forms of these LMIs, see Eqs. (44) and (45) in the paper.

Observer and Feedback Gain Design (Algorithm 1)

When both $\check{L}$ and $\check{F}$ are decision variables, the product terms make the LMI conditions non-convex. The paper resolves this through an iterative algorithm:

Initial Observer Gain (Theorem 5.2): The observer gains are designed separately by minimizing the l2-induced norm of the error system from disturbances to the estimation error. The change of variables $\check{Y} = -\check{P}_{L} \check{L}$ linearizes the problem. The observer gains are recovered as $L_{k} = -P_{L,k}^{-1} Y_{k}$ .

Initial Feedback Gain (Theorem 5.3): The feedback gains are designed separately by minimizing the l2-induced norm from disturbances to the state. The change of variables $\check{Y} = -\check{F} \check{X}_{F}$ linearizes the problem. The feedback gains are recovered as $F_{k} = -Y_{k} X_{F,k}^{-1}$ .

Algorithm 1 (Iterative Design):

Compute initial $\check{L}$ using Theorem 5.2.
Compute initial $\check{F}$ using Theorem 5.3.
Iterate:
- Fix $\check{L}$ and $\check{F}$ , solve for optimal $\gamma^{2}$ and $\check{P}$ (analysis step).
- Fix $\check{P}$ , solve for $\check{L}$ and $\check{F}$ that minimize $\gamma$ (synthesis step).
Repeat until convergence (the value of $\gamma$ decreases monotonically).

Numerical Examples

The paper considers an unstable plant with $n = 3$ states, $m = 2$ inputs, and $q = 2$ outputs. The input periods are $M_{1} = 2$ and $M_{2} = 3$ , and the output periods are $\mathcal{M}_{1} = 2$ and $\mathcal{M}_{2} = 4$ . The frame period is $M = \mathrm{lcm}(2, 3, 2, 4) = 12$ .

The plant matrices are:

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

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

With $i_{\max} = 60$ iterations in Algorithm 1, the resulting performance is $\gamma = 31.8283$ .

Simulation Results

The initial states are $x(0) = \lbrack 3, 3, 3 \rbrack^{T}$ and $x_{\mathrm{ob}}(0) = \lbrack 0, 0, 0 \rbrack^{T}$ . Process noise $d_{u}(k)$ has standard deviation 0.1, and observation noise $d_{w}(k)$ has standard deviation 0.05.

Key observations:

All states of the plant are stabilized (driven near zero) despite the plant being unstable.
All estimation errors converge near zero, indicating accurate state estimation.
The l2-norm ratio from simulation is $G_{\mathrm{sim}} = 9.9733$ , which is smaller than the theoretical bound $\gamma = 31.8283$ .
The feedback and observer gains vary across the 12-step period, with gains being zero at times when the corresponding input/output channel is inactive.

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

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). The predecessor paper that handles the observer-only design problem with multiple sensing periods. The present paper extends this to include state feedback with multiple actuation periods.

Multi-Rate System Identification — H. Okajima, R. Furukawa and N. Matsunaga, System Identification Under Multirate Sensing Environments, Journal of Robotics and Mechatronics (2025). Addresses how to obtain accurate plant models in multi-rate environments — a prerequisite for applying the observer-based feedback controller proposed here.

Cyclic Reformulation for LPTV Systems — H. Okajima, Y. Fujimoto, H. Oku and H. Kondo, Cyclic Reformulation-Based System Identification for Periodically Time-Varying Systems, IEEE Access (2025). Develops the cyclic reformulation framework for system identification, which uses the same mathematical tool employed in this feedback controller design.

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.

Model Error Compensator (MEC) — The multi-rate observer-based feedback controller can be combined with the Model Error Compensator to achieve robust control when the plant model contains uncertainties.

MATLAB Code

MATLAB File Exchange (Multi-Rate Observer): State Estimation under Multi-Rate Sensing: IEEE ACCESS 2023
Code Ocean (Multi-Rate System): State Estimation for Multi-Rate Sampled Systems
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, 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)

Co-authors: Kenta Arinaga (Kumamoto University), Atsuo Hayashida (Kumamoto University)

Related Japanese paper: H. Okajima, "マルチレート系の長い入力周期をもつ外乱低減制御系の設計," 計測自動制御学会論文集, Vol. 58, No. 10, pp. 451–458 (2022)

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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