Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design
Abstract
:1. Introduction
2. Thick Ellipsoids
3. Union and Intersection of Thick Ellipsoid Enclosures
3.1. Mapping of Thick Ellipsoidal Domains via (Quasi-)Linear System Models
3.1.1. Outer Bounds
3.1.2. Inner Bounds
3.1.3. Illustrating Example
- case 1
- , ;
- case 2
- , ;
- case 3
- , .
3.2. Dikin Ellipsoids for the Intersection of Ellipsoids
3.2.1. Intersection of Ellipsoids with Identical Midpoints
3.2.2. Generalization to the Intersection of Ellipsoids with Different Midpoints
- Step 1
- Determine the common center point for the desired inner and outer bounds of the intersection that must be included in all ellipsoids to be intersected;
- Step 2
- Determine initial approximations of the shape matrices for the inner and outer bounds according to Section 3.2.1;
- Step 3
- For non-empty inner bounds, correct the outer enclosure so that the inner and outer ellipsoid surfaces become parallel to each other and, thus, form a thick ellipsoid .
- Step 3*
- As an alternative to Step 3, the initial outer enclosure remains fixed and the inner ellipsoid surface is adapted to become parallel to each other to form a thick ellipsoid .
3.2.3. Illustrating Example
3.3. Thick Ellipsoid Union of Two Ellipsoids with Different Midpoints
3.3.1. General Solution Procedure
3.3.2. Illustrating Example
4. Thick Ellipsoid State Estimation Algorithm
4.1. Thick Ellipsoid Prediction Step
- Propagate the inner bound of the thick ellipsoid (68) and extract the inner hull of the image set that is obtained by applying the mapping
- Compute the updated ellipsoid midpoint as
- Finally, determine the predicted thick ellipsoid set
4.2. Thick Ellipsoid Correction Step
- Determine the inner shape matrix on the basis of Equation (47) according to
- Finally, Equation (49) yields the thick Dikin ellipsoid as the result of the measurement-based correction step.
4.3. Visualization of the Thick Ellipsoid State Estimation Procedure
5. Application Scenario: State and Disturbance Estimation for an Underactuated Hovercraft
5.1. Modeling
5.2. Simulation Results
6. Conclusions and Outlook on Future Work
Author Contributions
Funding
Conflicts of Interest
References
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Rauh, A.; Bourgois, A.; Jaulin, L. Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design. Algorithms 2021, 14, 88. https://doi.org/10.3390/a14030088
Rauh A, Bourgois A, Jaulin L. Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design. Algorithms. 2021; 14(3):88. https://doi.org/10.3390/a14030088
Chicago/Turabian StyleRauh, Andreas, Auguste Bourgois, and Luc Jaulin. 2021. "Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design" Algorithms 14, no. 3: 88. https://doi.org/10.3390/a14030088