H. Okajima, Y. Kaneda, and N. Matsunaga, "State estimation method using median of multiple candidates for observation signals including outliers," SICE Journal of Control, Measurement, and System Integration, Vol. 14, No. 1, pp. 257–267, 2021.
- Paper (Open Access): https://doi.org/10.1080/18824889.2021.1985702
- Support Page: https://sites.google.com/view/state-estimation-for-outlier
- MATLAB Code (GitHub): Hiroshi-Okajima/MATLAB_state_estimation
- MATLAB Code (MATLAB Central): Outlier-Robust State Estimator: JCMSI 2021
- MATLAB Code (Code Ocean): https://doi.org/10.24433/CO.4062942.v1
- YouTube (Numerical Example): https://youtu.be/Nx29cbfQwBs
Extended Abstract
Background: Why Outliers Break Standard Observers
State estimation is one of the most fundamental tasks in control engineering. In state feedback control, all state variables must be available at each time step, but in practice many of them are not directly measurable. State observers — including the Kalman filter — reconstruct unmeasured states from sensor outputs. These classical methods work well under the assumption that measurement noise follows a known distribution (typically Gaussian).
However, real-world sensors occasionally produce outliers: measurement values that deviate drastically from the true signal. Outliers can arise from sensor malfunction, temporary occlusion, communication glitches, or environmental interference. Even a single outlier can cause a standard observer's estimate to diverge significantly, because the observer gain amplifies the innovation (the difference between the measured and predicted output) — including the erroneous part. The Kalman filter, optimized for Gaussian noise, has no built-in mechanism to detect or reject such non-Gaussian disturbances.
Existing Approaches and Their Limitations
Several strategies have been proposed to handle outlier-corrupted measurements. Robust statistics-based filters (e.g., those using Huber's M-estimators) down-weight suspicious observations, but they require tuning thresholds and may still be affected by consecutive outliers. Switching-type detectors that discard observations flagged as outliers risk discarding valid information. The fundamental challenge is: how can we maintain estimation accuracy when we do not know in advance which measurements are corrupted?
The MCV Observer: Core Idea
The Median of Candidate Vectors (MCV) observer, proposed by Okajima et al., addresses this problem with an elegant structural approach rather than a statistical one. The key insight is:
Instead of using a single observer that processes all measurements, generate multiple candidate state estimates, each based on a different sensing time instant. Then select the best candidate using a median operation.
The procedure works as follows:
-
Generate $N$ candidate estimates. Each candidate observer uses the output measurement from a different past time step $k, k{-}1, \ldots, k{-}N{+}1$. All candidates share the same system model but differ in which observation they incorporate.
-
Apply median selection. At each time step, the observer computes the element-wise median (or weighted median) across the $N$ candidate state vectors:
$$
\hat{x}(k) = \mathrm{med} \bigl( \hat{x}^{(1)}(k),, \hat{x}^{(2)}(k),, \ldots,, \hat{x}^{(N)}(k) \bigr)
$$
If outliers occur infrequently (i.e., fewer than half of the $N$ candidates are affected), the median naturally excludes the corrupted candidates without any explicit detection or thresholding.
- Design observer gains via Lyapunov-based LMIs. The observer gains for each candidate are designed by bounding the reachable set of the estimation error. A Lyapunov-based inequality formulation provides a systematic design procedure that guarantees estimation accuracy for the outlier-free candidates.
Why Median Selection Works
The median is a well-known robust statistic: it is unaffected by up to 50% of corrupted data points. By exploiting this property in the state-estimation context, the MCV observer achieves a structural guarantee — the selected estimate is always one of the "clean" candidates, provided outliers are sufficiently rare. This is fundamentally different from trying to filter or down-weight outliers after the fact.
The design also supports a weighted median variant, where candidates associated with more recent (and presumably more informative) observations receive higher priority. This allows the designer to balance outlier rejection capability against estimation responsiveness.
Applications
The MCV observer has been applied to practical systems beyond the original numerical validation:
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Personal mobility vehicle navigation: In a system where an electric wheelchair follows a cluster of pedestrians using RGB-D cameras, pedestrian positions are tracked via YOLO and DeepSORT. Occlusions and sensing errors generate outliers in the cluster position estimate. The MCV observer removes these outliers, enabling stable crowd-following navigation (JRM 2023, SICE 2021).
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Packet loss environments: The MCV structure has been adapted for networked control systems where data dropouts create observation gaps analogous to outliers (SICE 2018).
Related Works
| Year | Title | Key Contribution | Ref |
|---|---|---|---|
| 2019 | State Estimation by Observer Using Median Operation for Observed Output with Outliers | Conference precursor — initial MCV concept presented at SICE Annual Conference (Finalist of poster award) | IEEE Xplore |
| 2019 | Analysis and Design of MCV Observer Based on Robust Invariant Set for Overcoming Observed Outputs with Outliers and Noises | Robust invariant set-based analysis of MCV observer (in Japanese) | Trans. SICE, Vol. 55, No. 12 |
| 2021 | State estimation method using median of multiple candidates for observation signals including outliers (this paper) | Full journal paper — systematic gain design via Lyapunov-based LMIs, median and weighted median variants | JCMSI (OA) |
| 2021 | Crowd Tracking of Electric Wheelchair using RGB-D Camera with MCV Observer | Application: pedestrian cluster tracking with outlier rejection | IEEE Xplore |
| 2023 | Navigation System for Personal Mobility Vehicles Following a Cluster of Pedestrians Using MCV Observer | Application: full navigation system integrating YOLO + DeepSORT + MCV observer | JRM (OA) |
Connections to Other Work by the Authors
The MCV observer shares a design philosophy with the broader research program at Kumamoto University:
- Robust invariant set estimation ([15] JCMSI 2021) provides the theoretical tools (Lyapunov-based reachable set analysis) used for MCV observer gain design.
- Model Error Compensator (MEC) reduces the impact of modeling errors on the plant side, while the MCV observer addresses measurement corruption on the sensor side. Together, they form complementary robustness layers.
Reproducibility
MATLAB codes are available from multiple sources for immediate use:
- GitHub: Hiroshi-Okajima/MATLAB_state_estimation
- MATLAB Central: Outlier-Robust State Estimator: JCMSI 2021
- Code Ocean (reproducible capsule): https://doi.org/10.24433/CO.4062942.v1
A YouTube video demonstrating the numerical example is also available: https://youtu.be/Nx29cbfQwBs
For full derivations, proofs, and additional numerical results, see the original paper.