State Estimation

Overview

State estimation is a fundamental problem in control engineering: given a mathematical model of a dynamical system and measured input-output data, estimate the internal state variables that cannot be directly measured. Accurate state estimation enables state feedback control, monitoring, fault detection, and prediction.

The basic approach is the Luenberger observer, which constructs a copy of the plant model and uses the output estimation error to drive the state estimate toward the true state. In the stochastic setting, the Kalman filter provides the optimal state estimate that minimizes the estimation error covariance under Gaussian noise assumptions. When statistical characterization of disturbances is unavailable or unreliable, the H-infinity filter provides robust state estimation by minimizing the worst-case estimation error measured by the l2-induced norm — designed using Linear Matrix Inequality (LMI) optimization.

In practical control systems, sensors often operate at different sampling rates. Our research addresses this multi-rate sensing problem using cyclic reformulation, which converts the periodically time-varying observer into an equivalent LTI framework while preserving the original sampling rate. We have also developed the MCV (Median of Candidate Vectors) observer, which achieves robustness against sensor outliers by creating multiple estimation candidates along the time axis and selecting the best estimate using a median operation.

In our research, state estimation methods play a central role across multiple topics, including multi-rate observer and controller design using cyclic reformulation and LMI optimization, outlier-robust estimation using the MCV observer, and integration with the Model Error Compensator (MEC) for robust control under both model uncertainty and sensor imperfections.

Blog Articles on State Estimation

For detailed explanations with mathematical formulations, MATLAB code examples, and references to key literature, see the following blog articles:

Comprehensive guide:

• State Observer and State Estimation: A Comprehensive Guide — Covers Luenberger observers (continuous-time and discrete-time), Kalman filters, H-infinity filters, multi-rate state estimation, and outlier-robust observers. Includes links to all research papers, MATLAB codes, and videos.

Individual topics:

• State Observer: Understanding the Basic Mechanism — Introduction to the Luenberger observer with intuitive explanations and MATLAB simulations.

• State Observer for State Space Model — Observer-based feedback control and the separation principle.

• Kalman Filter: From Basic Algorithm to Multi-Rate Extensions — Standard Kalman filter algorithm, derivation, and connections to multi-rate extensions.

• H-infinity Filter: Robust State Estimation Using LMI Optimization — H-infinity filter design using LMI, handling bounded disturbances without statistical assumptions.

• Multi-Rate Observer — State observer design for systems with sensors operating at different sampling rates. Based on IEEE Access 2023.

• MCV Observer — Outlier-robust state estimation using the median of candidate vectors. Based on JCMSI 2021.

Key Papers

• H. Okajima, Y. Hosoe and T. Hagiwara, "State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm," IEEE Access, 2023. Paper

• 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. Paper

• H. Okajima, Y. Kaneda and N. Matsunaga, "State estimation method using median of multiple candidates for observation signals including outliers," SICE JCMSI, Vol. 14, No. 1, pp. 257–267, 2021. Paper (Open Access)

• H. Okajima, "LMI Optimization Based Multirate Steady-State Kalman Filter Design," arXiv:2602.01537, 2026.

MATLAB Code

• State Estimation under Multi-Rate Sensing: IEEE ACCESS 2023 — MATLAB File Exchange

• Multi-Rate System Code — Code Ocean

• Outlier-Robust State Estimator: JCMSI 2021 — MATLAB File Exchange

• MATLAB State Estimation — GitHub

• MATLAB Fundamental Control LMI — GitHub

Related Pages

• Multi-rate System — Multi-rate observer and controller design

• MCV Observer for Overcoming Outliers — MCV observer research page

• Linear Matrix Inequality — LMI optimization used in observer gain design

• Model Error Compensator — Combining observer-based estimation with model error compensation

• System Identification — Obtaining dynamical models for observer design

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