This article provides a detailed explanation of state observer design for systems with multiple sensors operating 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, 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) (Open Access)

MATLAB Code (GitHub) / MATLAB Code (Code Ocean)

This paper is co-work with Prof. Yohei Hosoe (Associate Professor, Kyoto University) and Prof. Tomomichi Hagiwara (Professor, Kyoto University).

Contents

Why Multi-Rate Sensing Matters

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

In hard disk drive track-following systems, the position error sensing rate is limited and slower than the control period of the head arm.
In autonomous driving and SLAM (Simultaneous Localization and Mapping), camera sensors, 3D-LiDAR, and IMU have very different feasible sensing periods.
In robotic systems, encoders (fast), vision sensors (slow), and force sensors (medium) coexist.

If we treat such a system as a single-rate system, the common period must match the slowest sensor, leading to deteriorated control performance. The challenge is to design a state observer that effectively integrates sensor information arriving at different rates.

This paper provides a systematic solution based on a periodically time-varying observer with LMI (Linear Matrix Inequality) optimization of the $l _{2}$ -induced norm.

Plant and Observed Outputs with Different Sampling Periods

Plant Model

Consider the discrete-time LTI MIMO plant:

$\displaystyle x(k+1) = Ax(k) + B_u u(k) + B_d d(k)$

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

where $x(k) \in \mathbb{R}^{n}$ is the state, $u(k) \in \mathbb{R}^{m _{u}}$ is the input, $d(k) \in \mathbb{R}^{m _{d}}$ is the process noise, and $y _{r}(k) \in \mathbb{R}^{q}$ is the output vector. The pair $(C, A)$ is assumed to be observable.

Multi-Rate Sensing via Periodic Matrices

The key idea is that the $i$ -th sensor has its own sensing period $N _{i} \in \mathbb{N}$ . To describe which sensors are active at each time step, we introduce the periodically time-varying diagonal matrix:

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

where $s _{i}(k) = 1$ if sensor $i$ observes at time $k$ , and $s _{i}(k) = 0$ otherwise. The period of $S _{k}$ is $N = \mathrm{lcm}(N _{1}, \ldots, N _{q})$ .

Example: For a two-output plant with $N _{1} = 3$ and $N _{2} = 6$ , the frame period is $N = 6$ . If both sensors observe at $k = 0$ (i.e., $\theta _{1} = \theta _{2} = 0$ ):

$S _{0} = I$ , $S _{1} = S _{2} = 0$ , $S _{3} = \mathrm{diag}(1,0)$ , $S _{4} = S _{5} = 0$

The observation timing parameter $\theta _{i}$ allows for asynchronous sensing — two sensors can have the same period but observe at different times.

The observed output at time $k$ is then:

$\displaystyle y(k) = C_k x(k) + D_k w(k)$

where $C _{k} = S _{k} C$ , $D _{k} = S _{k} D$ , and $w(k)$ is the measurement noise.

Periodically Time-Varying State Observer

The proposed observer has the structure:

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

where $L _{k}$ ( $k = 0, 1, \ldots$ ) are N-periodic observer gains. When $S _{k} = 0$ (no sensor active), the observer simply runs open-loop using the plant model. When sensors are active, the observer corrects using the available measurements weighted by $L _{k} S _{k}$ .

Note: Setting $S _{0} = I, S _{1} = 0, \ldots, S _{N-1} = 0$ recovers the dual-sampling-rate observer studied in earlier work.

Error System

The state estimation error $e(k) = x(k) - x _{\mathrm{ob}}(k)$ satisfies:

$\displaystyle e(k+1) = A_{ek} e(k) - L_k S_k D w(k) + B_d d(k)$

where $A _{ek} = A - L _{k} S _{k} C$ . With the combined disturbance $d _{\ast}$ consisting of $d$ and $w$ , and evaluation output $z(k) = We(k)$ (with weighting matrix $W$ ), the design goal is to minimize the l2-induced norm of the system $G _{z}$ from $d _{\ast}$ to $z$ :

$\displaystyle \lVert G_z \rVert_{l_2/l_2} = \sup_{d_\ast \in l_2} \frac{\lVert z \rVert_2}{\lVert d_\ast \rVert_2}$

Setting $W = I$ directly evaluates the norm from disturbances to the state estimation error.

Performance Analysis (Theorem 1)

For given observer gains $L _{k}$ , the $l _{2}$ -induced norm can be analyzed using a periodically time-varying energy supply function:

$\displaystyle V_k(\chi) = \chi^{T} P_{k-1} \chi$

where $P _{k} > 0$ are N-periodic Lyapunov matrices ( $P _{-1} = P _{N-1}$ ).

Theorem 1: For given $\gamma > 0$ , the error system is stable and the l2-induced norm of $G _{z}$ is less than $\gamma$ if there exist N-periodic matrices $P _{k} > 0$ such that the matrix inequality $\Theta _{k} > 0$ holds for all $k = 0, \ldots, N-1$ . The matrix $\Theta _{k}$ involves $P _{k}$ , $A _{ek}$ , $B _{d}$ , $L _{k} S _{k} D$ , $W$ , and $\gamma$ . For the explicit form, see Eq. (20) in the paper.

The key insight is that the periodically time-varying Lyapunov matrices $P _{k}$ enable less conservative analysis by adapting to the periodic structure, compared with a single constant Lyapunov matrix.

Observer Synthesis (Theorem 2)

When $L _{k}$ are also decision variables, products $P _{k} L _{k}$ make the problem non-convex. This is resolved by the change of variables:

$\displaystyle Y_k = P_k L_k$

Theorem 2: For given $\gamma > 0$ , there exist observer gains achieving the l2-induced norm less than $\gamma$ if there exist $P _{k} > 0$ and $Y _{k}$ satisfying $\hat{\Theta} _{k} > 0$ for all $k = 0, \ldots, N-1$ , where $\hat{\Theta} _{k}$ is a 4-by-4 block matrix:

$\displaystyle \hat{\Theta}_k = \begin{pmatrix} P_k & P_k A - Y_k S_k C & P_k B_d & -Y_k S_k D \cr (P_k A - Y_k S_k C)^T & P_{k-1} - W^T W & 0 & 0 \cr (P_k B_d)^T & 0 & \gamma^2 I & 0 \cr -(Y_k S_k D)^T & 0 & 0 & \gamma^2 I \end{pmatrix} > 0$

with $P _{-1} = P _{N-1}$ . The observer gains are recovered as:

$\displaystyle L_k = P_k^{-1} Y_k$

Since $\hat{\Theta} _{k} > 0$ is linear in $P _{k}$ and $Y _{k}$ , the minimization of $\gamma$ is a standard SDP (semidefinite programming) problem, solvable with standard LMI solvers.

Design Procedure

Model the plant as $(A, B _{u}, B _{d}, C, D)$
Set the sensing schedule $S _{k}$ based on sensor periods $N _{i}$ and timing $\theta _{i}$
Compute frame period $N = \mathrm{lcm}(N _{1}, \ldots, N _{q})$
Solve the LMI minimization of $\gamma$ subject to $\hat{\Theta} _{k} > 0$
Recover observer gains $L _{k} = P _{k}^{-1} Y _{k}$

Numerical Examples

The paper demonstrates the method with a 3-state, 2-output plant where $N _{1} = 2$ and $N _{2} = 3$ , giving frame period $N = 6$ . The proposed multi-rate observer achieves $\gamma _{\mathrm{prop}} = 1.33$ .

For comparison, three dual-rate observer designs are considered:

Method	Description	Performance
Proposed	Multi-rate observer using both outputs	1.33
Case A	Dual-rate using only y1 (N=2)	1.67
Case B	Dual-rate using only y2 (N=3)	1.68
Case C	Both outputs treated as 6-periodic	2.10

The proposed method outperforms all three cases, confirming that properly combining multi-rate sensor information yields better estimation performance.

The paper also shows results for various $(N _{1}, N _{2})$ combinations. The key finding: higher observation frequency (shorter sensing period) leads to better estimation performance. For example, $(N _{1}, N _{2}) = (1,1)$ gives $\gamma = 1.00$ , while $(3,3)$ gives $\gamma = 1.43$ .

The observer can also be designed for unstable plants, as demonstrated in Section IV-C of the paper.

Effect of Observation Timing

An important finding: even when sensors have the same period, the observation timing matters.

With $N _{1} = N _{2} = 3$ , three timing cases are compared:

Case	Timing	Performance
Case 1	Synchronous: both at k=0	1.43
Case 2	Staggered by 1 step	1.41
Case 3	Staggered by 2 steps	1.42

Case 2 (staggered by 1 step) achieves the best performance. In Case 1, both sensors observe simultaneously, leaving 2 consecutive steps with no observation at all. In Case 2, observations are more evenly distributed over time.

This has practical implications: when designing a multi-sensor system, scheduling sensors to observe at staggered timings can improve estimation performance without adding any hardware.

This work forms the foundation of a broader research program on multi-rate control systems:

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 systems where both sensors and actuators operate at different rates.

Multi-Rate System Identification — H. Okajima, R. Furukawa and N. Matsunaga, System Identification Under Multi-rate Sensing Environment, Journal of Robotics and Mechatronics (2025). Addresses how to obtain accurate models when input and output signals have different sampling rates.

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 identification for LPTV systems using cyclic reformulation — the core mathematical tool also used in this multi-rate observer.

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

Model Error Compensator (MEC) — The multi-rate observer can be combined with the Model Error Compensator to achieve robust control in multi-rate environments.

MATLAB Code

GitHub: MATLAB_state_observer / 05_multirate_observer
MATLAB File Exchange: State Estimation under Multi-Rate Sensing: IEEE ACCESS 2023
Code Ocean: 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, 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.2023.3249187 (Open Access). MATLAB Code

Co-authors: Yohei Hosoe (Associate Professor, Kyoto University), Tomomichi Hagiwara (Professor, Kyoto University)

Earlier Japanese paper: H. Okajima, Y. Hosoe and T. Hagiwara, "マルチレート系の状態推定のための周期時変状態オブザーバのl2誘導ノルム評価による設計," 計測自動制御学会論文集, Vol. 55, No. 12, pp. 792–799 (2019)

Self-Introduction

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