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.
- Paper (Open Access): https://ieeexplore.ieee.org/document/10054014
- MATLAB Code (MATLAB Central): State Estimation for Multi-Rate Sampled Systems: IEEE ACCESS
- MATLAB Code (Code Ocean): https://doi.org/10.24433/CO.2360629.v1
Extended Abstract
Background and Motivation
In practical control systems, different sensors operate at different sampling rates. A GPS receiver may update at 1 Hz, a vision system at 30 Hz, and an IMU at 200 Hz. Although the plant itself is linear and time-invariant, the sensing environment becomes multi-rate — and this mismatch makes conventional observer design methods difficult to apply directly.
Most existing observer designs assume either a single common sampling period for all sensors, or that all measurements arrive synchronously. When these assumptions are violated, the standard approach is to artificially slow everything down to the slowest sensor rate, discarding the information available from faster sensors between the slow updates. This clearly wastes valuable measurement data.
The question addressed in this paper is: can we design a state observer that fully exploits all sensors at their native sampling rates, with guaranteed worst-case estimation performance?
Problem Setting
Consider a discrete-time LTI plant driven by process disturbance:
$$
x_{k+1} = A x_k + w_k
$$
Multiple sensors observe the state, but each has its own sampling period. At any given time step $k$, only a subset of sensors may provide a measurement. This is modeled as:
$$
y_k = C_k x_k + v_k
$$
where $C_k$ is periodically time-varying: it changes depending on which sensors are active at time $k$. The sensing periods are assumed to have mutually rational ratios, so the overall pattern repeats with a known frame period $N$.
Approach: Periodic Observer with LMI-Based Design
Rather than lifting or reformulating the system into a higher-dimensional time-invariant form, this paper directly designs a periodically time-varying observer:
$$
\hat{x}_{k+1} = A \hat{x}_k + L_k (y_k - C_k \hat{x}_k)
$$
The observer gain $L_k$ is periodic with the same period $N$ as the sensing pattern. At time steps where a sensor provides no measurement, the corresponding rows of $C_k$ are absent, and the observer simply propagates the model prediction.
The design criterion is the $\ell_2$-induced norm from the disturbance input $(w_k, v_k)$ to the estimation error $e_k = x_k - \hat{x}_k$. The goal is to minimize the worst-case amplification:
$$
\sup_{|d|{\ell_2} \leq 1} |e|{\ell_2} \leq \gamma
$$
This is formulated as a set of coupled periodic Lyapunov inequalities — one for each time step within the frame period — which are solved simultaneously as an LMI feasibility/optimization problem. The LMI formulation naturally handles the fact that the measurement matrix changes across the period, and it provides a systematic, convex design procedure.
Key Insights from the Paper
Observation timing matters. An important finding is that even when all sensors share the same sampling period, the estimation performance depends on the relative timing of their measurements. Two sensors updating at the same rate but at staggered time instants can yield significantly better or worse performance than two sensors updating simultaneously. This is numerically demonstrated in the paper and has practical implications for sensor scheduling.
No structural conservatism from lifting. By working directly with the periodic system rather than lifting it into a larger time-invariant system, the approach avoids introducing structural zeros or artificial constraints that can make the design conservative.
Extension: Observer-Based Feedback Control
The observer design framework has been extended to the full output feedback problem:
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. (Open Access) | MATLAB Code
This companion paper addresses the natural next step: once the multi-rate observer provides state estimates, how should we close the loop? The paper designs observer-based feedback controllers for multi-rate systems where not only the sensor periods but also the actuator periods may differ. Using cyclic reformulation, the multi-rate system is reduced to a time-invariant form, and both the observer gains and feedback gains are co-designed based on the $\ell_2$-induced norm from disturbances to controlled outputs. This provides a complete observer–controller framework for multi-rate environments.
Related Works
| Year | Title | Role | Ref |
|---|---|---|---|
| 2023 | State Observer under Multi-rate Sensing Environment (this paper) | Multi-rate observer design via periodic LMIs and $\ell_2$-induced norm | IEEE Access (OA) |
| 2023 | Design of observer-based feedback controller for multi-rate systems using cyclic reformulation | Extension to observer-based output feedback — co-design of observer + controller | IEEE Access (OA) |
For full derivations, proofs, and additional numerical results including the sensor timing analysis, see the original paper.