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Robust State Estimation with Set-Valued Estimators

This paper discusses the application of set-valued estimators for robust state estimation and system identification in linear systems. It highlights the limitations of the Kalman filter framework and introduces alternative approaches such as guaranteed cost estimators and adaptive estimators. The set-valued estimation method is described, including its main characteristics and the challenges posed by discrete-time measurements. The paper concludes with the advantages and considerations of implementing set-valued estimators for system identification.

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Robust State Estimation with Set-Valued Estimators

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  1. Online Model Identification for Set-valued State Estimators With Discrete-Time Measurements João V. Messias Institute for Systems and Robotics Instituto Superior Técnico Lisbon, Portugal

  2. Contents • Introduction / Motivation • Robust State Estimation • Set-Valued Estimators • Implementation: HTS • Results • Conclusions

  3. Introduction/ Motivation The Kalman Filter framework is a powerful tool for state estimation in linear systems; • The main assumption is that the system model is known with sufficient accuracy; • Lack of robustness to parameter variations (performance loss); • The goal of this work is to exemplify the application of a robust filter, based on the KalmanFilter; • State estimation and system identification.

  4. Robust State Estimation Parameter variations have to be taken explicitly into account.

  5. Robust State Estimation Robust Estimation Methods: - Guaranteed Cost Estimators; - Set-Valued Estimators; - Adaptive Estimators (eg. MMAE); Approaches differ in the way that parameter uncertainty is described.

  6. Set-Valued Estimation (Petersen & Savkin, 1995) • Main characteristics: • Estimates the set of possible states for a given instant; • Deterministic interpretation of system uncertainty; • Allows for non-linear, time-varying uncertainties; • Allows for continuous / discrete sensors; • Allows for missing data; • - Model validation can be performed as a dual problem.

  7. Set-Valued Estimation Integral Quadratic Constraint:

  8. Set-Valued Estimation Ricattijump equations:

  9. Set-Valued Estimation Solution isn’t smooth due to discrete-time measurements.

  10. Set-Valued Estimation Set estimation:

  11. System Identification • This method allows for dynamic re-evaluation of system uncertainty. • However, this only checks if a given model is feasible or not; • No explicit methodology for system identification is given; • MMAE is used to compensate;

  12. Implementation Physical System MMAE SVE

  13. Implementation MMAE pmf 99%

  14. Implementation: HTS

  15. Results

  16. Results

  17. Results M2 = 75kg M2 = 65kg

  18. Results β α

  19. Results

  20. Conclusions • Main advantage of set-valued estimators: deals with a large class of uncertainties; • Not straightforward to implement; • Performance is dependant on the quality of the norm bounds on the parameters; • If some of these bounds are configured simply by user design, the results are overly conservative; • It is possible to perform system identification to reduce these norm bounds on-line; • It may be feasible to implement some form of particle-filter based parameter search.

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