# Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thick Ellipsoids

**Definition**

**1**

**.**Define a thick ellipsoid $\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)=\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)\left(\mathit{\mu},\mathbf{\Gamma},\left[\underset{\xaf}{\rho}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}\overline{\rho}\right]\right)$ as a subset of the power set $\mathcal{P}\left({\mathbb{R}}^{n}\right)$ so that

**Definition**

**2.**

**.**A thick ellipsoid extension of the binary operators $\diamond \in \{+,-,\xb7,/,\cup ,\cap \}$ (For the case of division, the value zero is assumed not to belong to the denominator expression in analogy to classical interval arithmetic [11]) satisfies the relation

## 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

**Remark**

**1.**

**Theorem**

**1**

**.**The ellipsoid ${\mathcal{E}}_{k+1}^{\mathrm{O}}$ according to Equation (9) is a guaranteed outer bound for ${\mathcal{E}}_{k+1}^{*}$ if the matrix inequality

**Proof.**

**Remark**

**2.**

**Theorem**

**2**

**.**For a fixed combination of the shape and stretch parameters ${\mathcal{R}}_{k}$ and ${\alpha}_{\mathrm{O},k+1}$, respectively, ${\mathcal{E}}_{k+1}^{\mathrm{O}}$ is a guaranteed outer enclosure of the solution set ${\mathcal{E}}_{k+1}^{*}$, if

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

#### 3.1.2. Inner Bounds

**Theorem**

**3**

**.**The ellipsoid ${\mathcal{E}}_{k+1}^{\mathrm{I}}$ according to Equation (25) is a guaranteed outer bound for ${\mathcal{E}}_{k+1}^{*}$ if the matrix inequality

**Proof.**

**Remark**

**5.**

**Theorem**

**4**

**.**For a fixed combination of the shape and stretch parameters ${\mathcal{R}}_{k}$ and ${\alpha}_{\mathrm{I},k+1}$, respectively, ${\mathcal{E}}_{k+1}^{\mathrm{I}}$ is a guaranteed inner enclosure of the solution set ${\mathcal{E}}_{k+1}^{*}$, if

**Proof.**

**Remark**

**6.**

**Remark**

**7.**

#### 3.1.3. Illustrating Example

**case****1**- ${p}_{1}\in \left[-1\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}2\right]$, ${p}_{2}\in \left[-1\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}2\right]$;
**case****2**- ${p}_{1}\in \left[-0.25\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}0.5\right]$, ${p}_{2}\in \left[-0.25\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}0.5\right]$;
**case****3**- ${p}_{1}\in \left[-0.1\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}0.2\right]$, ${p}_{2}\in \left[-0.1\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}0.2\right]$.

**cases 2**and

**3**, the first setting is superior. This is not astonishing because it corresponds to that choice of matrix that would be employed for the covariance prediction in an Extended Kalman Filter (EKF). As it is well known, strong deviations from a point-valued system matrix ${\mathbf{A}}_{k}$ may turn an EKF quite inaccurate. Then, other choices of the forecasted covariance are often superior, as they could be determined by the Unscented Kalman Filter technique [27]. This approach was not investigated here. However, the almost symmetric interval bounds of $\left[{\mathbf{A}}_{k}\right]$ motivate the second choice of ${\mathcal{R}}_{k}$, which corresponds to ${\mathbf{Q}}_{k+1}={\mathbf{Q}}_{k}$.

**case 1**in Figure 2a,b corresponds to a scenario in which the inner bound is empty, the

**case 2**in Figure 2c,d to a scenario in which the inner bound just appears due to the matrix $\left[{\mathbf{A}}_{k}\right]$ containing purely regular realizations; finally, Figure 2e,f shows the case of small uncertainty with a clearly visible inner solution set, where the proposed one-parameter optimization task from the previous subsections provides results that are close to the solution of the robust linear matrix inequality design, where ${\mathcal{R}}_{k}$ is not predefined but rather optimized with the help of the matrices ${\mathcal{P}}_{k+1}^{\mathrm{O}}$ and ${\mathcal{P}}_{k+1}^{\mathrm{I}}$ to obtain independent outer and inner enclosures (dashed and dotted lines, respectively).

#### 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 $\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)$.
**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 $\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)$.

**Step 1**of two ellipsoids that are described by the midpoints ${\mathit{\mu}}_{1,k}$ and ${\mathit{\mu}}_{2,k}$ as well as the shape matrices ${\mathbf{Q}}_{1,k}$ and ${\mathbf{Q}}_{2,k}$, respectively, the first one is interpreted as the prior knowledge in the innovation step of a Kalman filter [25,55] and the second one as the corresponding information of the state measurement.

**Steps 2**and

**3**listed above.

**Step 3**according to the following theorem so that it contains the outer ellipsoid with certainty.

**Theorem**

**5**

**.**The thick Dikin ellipsoid

**Proof.**

**Step 3***is given by the thick ellipsoid enclosure

**Remark**

**8.**

#### 3.2.3. Illustrating Example

#### 3.3. Thick Ellipsoid Union of Two Ellipsoids with Different Midpoints

#### 3.3.1. General Solution Procedure

**Remark**

**10.**

#### 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$${\stackrel{\u02c7}{\mathbf{x}}}_{k+1}=\mathbf{A}\left({\mathbf{x}}_{k},{\mathbf{p}}_{k}\right)\xb7{\stackrel{\u02c7}{\mathbf{x}}}_{k}\phantom{\rule{5.0pt}{0ex}}.$$$${\stackrel{\u02c7}{\mathbf{Q}}}_{k+1}^{\mathrm{I}}={\alpha}_{\mathrm{I},k+1}^{2}\xb7{\rho}_{k}^{2}\xb7{\mathbf{\Gamma}}_{k}\xb7{\mathbf{\Gamma}}_{k}^{T}\phantom{\rule{5.0pt}{0ex}}.$$
- Propagate the outer bound of the thick ellipsoid (68) and extract the outer hull of the image set that is obtained by applying the mapping (71). Denote the corresponding shape matrix of the outer ellipsoid by ${\stackrel{\u02c7}{\mathbf{Q}}}_{k+1}^{\mathrm{O}}$. This matrix is related to the shape matrix ${\alpha}_{\mathrm{O},k+1}^{2}\xb7{\mathbf{Q}}_{k+1}$ in Equation (9) by$${\stackrel{\u02c7}{\mathbf{Q}}}_{k+1}^{\mathrm{O}}={\alpha}_{\mathrm{O},k+1}^{2}\xb7{\overline{\rho}}_{k}^{2}\xb7{\mathbf{\Gamma}}_{k}\xb7{\mathbf{\Gamma}}_{k}^{T}\phantom{\rule{5.0pt}{0ex}}.$$
- Compute interval bounds for the term$${\mathbf{b}}_{k}=\left(\mathbf{A}\left({\mathbf{x}}_{k},{\mathbf{p}}_{k}\right)-{\tilde{\mathbf{A}}}_{k}\right)\xb7{\mathit{\mu}}_{k}\in \left[{\mathbf{b}}_{k}\right]$$$$\begin{array}{cc}\hfill {\mathbf{Q}}_{k+1}^{\mathrm{I}}& ={\left(1-{\rho}_{\mathrm{I},k+1}\right)}^{2}\xb7{\stackrel{\u02c7}{\mathbf{Q}}}_{k+1}^{\mathrm{I}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\rho}_{\mathrm{I},k+1}=sup\left\{\u2225{\alpha}_{\mathrm{I},k+1}^{-1}\xb7{\underset{\xaf}{\rho}}_{k}^{-1}\xb7{\mathbf{\Gamma}}_{k}^{-1}\xb7\left[{\mathbf{b}}_{k}\right]\u2225\right\}\hfill \end{array}$$$$\begin{array}{cc}\hfill 1-1{\mathbf{Q}}_{k+1}^{\mathrm{O}}& ={\left(1+{\rho}_{\mathrm{O},k+1}\right)}^{2}\xb7{\stackrel{\u02c7}{\mathbf{Q}}}_{k+1}^{\mathrm{O}}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\rho}_{\mathrm{O},k+1}=sup\left\{\u2225{\alpha}_{\mathrm{O},k+1}^{-1}\xb7{\overline{\rho}}_{k}^{-1}\xb7{\mathbf{\Gamma}}_{k}^{-1}\xb7\left[{\mathbf{b}}_{k}\right]\u2225\right\}\phantom{\rule{5.0pt}{0ex}}.\hfill \end{array}$$
- Compute the updated ellipsoid midpoint as$${\mathit{\mu}}_{k+1}={\tilde{\mathbf{A}}}_{k}\xb7{\mathit{\mu}}_{k}\phantom{\rule{5.0pt}{0ex}}.$$
- Finally, determine the predicted thick ellipsoid set$${\mathbf{x}}_{k+1}\in {\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)}_{k+1}\left({\mathit{\mu}}_{k+1},{\mathbf{\Gamma}}_{k+1},\left[{\rho}_{k+1}\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}{\overline{\rho}}_{k+1}\right]\right)\phantom{\rule{5.0pt}{0ex}},$$$$\begin{array}{cc}\hfill {\rho}_{k+1}& ={\rho}_{k}\xb7{\alpha}_{\mathrm{I},k+1}\xb7\left(1-{\rho}_{\mathrm{I},k+1}\right)\phantom{\rule{5.0pt}{0ex}},\hfill \\ \hfill {\overline{\rho}}_{k+1}& ={\overline{\rho}}_{k}\xb7{\alpha}_{\mathrm{O},k+1}\xb7\left(1+{\rho}_{\mathrm{O},k+1}\right)\phantom{\rule{5.0pt}{0ex}},\phantom{\rule{1.em}{0ex}}\mathrm{and}\hfill \\ \hfill {\mathbf{\Gamma}}_{k+1}& ={\tilde{\mathbf{A}}}_{k}\xb7{\mathbf{\Gamma}}_{k}\phantom{\rule{5.0pt}{0ex}}.\hfill \end{array}$$

**Remark**

**11.**

#### 4.2. Thick Ellipsoid Correction Step

- Determine the inner shape matrix on the basis of Equation (47) according to$${\mathbf{Q}}_{k+1}^{{\mathrm{I}}^{\prime}}=2\xb7{\left(2\xb7\left({\left({\xi}_{1,k+1}^{2}\xb7{\mathbf{Q}}_{k+1}^{\mathrm{I}}\right)}^{-1}+{\left({\xi}_{2,k+1}^{2}\xb7{\mathbf{Q}}_{\mathrm{m},k+1}\right)}^{-1}\right)\right)}^{-1}\phantom{\rule{5.0pt}{0ex}},$$
- Use Equation (48) to determine the intermediate outer bound$${\mathbf{Q}}_{k+1}^{{\mathrm{O}}^{\prime}}=4\xb7{\left(2\xb7\left({\left({\zeta}_{1,k+1}^{2}\xb7{\mathbf{Q}}_{k+1}^{\mathrm{O}}\right)}^{-1}+{\left({\zeta}_{2,k+1}^{2}\xb7{\mathbf{Q}}_{\mathrm{m},k+1}\right)}^{-1}\right)\right)}^{-1}$$
- 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

- Kletting, M.; Rauh, A.; Aschemann, H.; Hofer, E.P. Interval Observer Design for Nonlinear Systems with Uncertain Time-Varying Parameters. In Proceedings of the 12th IEEE International Conference on Methods and Models in Automation and Robotics MMAR, Miedzyzdroje, Poland, 28–31 August 2006; pp. 361–366. [Google Scholar]
- Kletting, M.; Rauh, A.; Aschemann, H.; Hofer, E.P. Interval Observer Design Based on Taylor Models for Nonlinear Uncertain Continuous-Time Systems. In Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN, Duisburg, Germany, 26–29 September 2006. [Google Scholar]
- Nedialkov, N.S. Interval Tools for ODEs and DAEs. In Proceedings of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN, Duisburg, Germany, 26–29 September 2006. [Google Scholar]
- Nedialkov, N.S. Computing Rigorous Bounds on the Solution of an Initial Value Problem for an Ordinary Differential Equation. Ph.D. Thesis, Graduate Department of Computer Science, University of Toronto, Toronto, ON, Canada, 1999. [Google Scholar]
- Lin, Y.; Stadtherr, M.A. Validated Solutions of Initial Value Problems for Parametric ODEs. Appl. Numer. Math.
**2007**, 57, 1145–1162. [Google Scholar] [CrossRef][Green Version] - Berz, M.; Makino, K. Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models. Reliab. Comput.
**1998**, 4, 361–369. [Google Scholar] [CrossRef] - Makino, K.; Berz, M. Suppression of the Wrapping Effect by Taylor Model-Based Validated Integrators; Technical Report MSU HEP 40910; Michigan State University: East Lansing, MI, USA, 2004. [Google Scholar]
- Goubault, E.; Mullier, O.; Putot, S.; Kieffer, M. Inner Approximated Reachability Analysis. In Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control, Berlin, Germany, 15–17April 2014; pp. 163–172. [Google Scholar] [CrossRef][Green Version]
- Goubault, E.; Putot, S. Robust Under-Approximations and Application to Reachability of Non-Linear Control Systems With Disturbances. IEEE Control Syst. Lett.
**2020**, 4, 928–933. [Google Scholar] [CrossRef] - Djeumou, F.; Vinod, A.; Goubault, E.; Putot, S.; Topcu, U. On-The-Fly Control of Unknown Systems: From Side Information to Performance Guarantees through Reachability. 2020. preprint. Available online: www.researchgate.net/publication/345756893On-The-FlyControlofUnknownSystemsFromSideInformationtoPerformanceGuaranteesthroughReachability (accessed on 13 March 2021).
- Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, É. Applied Interval Analysis; Springer: London, UK, 2001. [Google Scholar]
- Moore, R. Interval Arithmetic; Prentice-Hall: Englewood Cliffs, NJ, USA, 1966. [Google Scholar]
- Moore, R.; Kearfott, R.; Cloud, M. Introduction to Interval Analysis; SIAM: Philadelphia, PA, USA, 2009. [Google Scholar]
- Mayer, G. Interval Analysis and Automatic Result Verification; De Gruyter Studies in Mathematics, De Gruyter: Berlin, Germany; Boston, MA, USA, 2017. [Google Scholar]
- Piegat, A.; Dobryakova, L. A Decomposition Approach to Type 2 Interval Arithmetic. Int. J. Appl. Math. Comput. Sci.
**2020**, 30, 185–201. [Google Scholar] - Desrochers, B.; Jaulin, L. Thick Set Inversion. Artif. Intell.
**2017**, 249, 1–18. [Google Scholar] [CrossRef][Green Version] - Măceş, D.; Stadtherr, M.A. Computing Fuzzy Trajectories for Nonlinear Dynamic Systems. Comput. Chem. Eng.
**2013**, 52, 10–25. [Google Scholar] [CrossRef][Green Version] - Neumaier, A. The Wrapping Effect, Ellipsoid Arithmetic, Stability and Confidence Regions. In Validation Numerics: Theory and Applications; Albrecht, R., Alefeld, G., Stetter, H.J., Eds.; Springer: Vienna, Austria, 1993; pp. 175–190. [Google Scholar]
- Kurzhanskii, A.B.; Vályi, I. Ellipsoidal Calculus for Estimation and Control; Birkhäuser: Boston, MA, USA, 1997. [Google Scholar]
- Rauh, A.; Jaulin, L. A Computationally Inexpensive Algorithm for Determining Outer and Inner Enclosures of Nonlinear Mappings of Ellipsoidal Domains. Appl. Math. Comput. Sci. AMCS
**2021**. under review. [Google Scholar] - Wang, Y.; Puig, V.; Cembrano, G.; Zhao, Y. Zonotopic Fault Detection Observer for Discrete-Time Descriptor Systems Considering ${\mathcal{H}}_{\_}$ Fault Sensitivity. Int. J. Syst. Sci.
**2020**, 1–15. [Google Scholar] [CrossRef] - Zhang, W.; Wang, Z.; Raïssi, T.; Wang, Y.; Shen, Y. A State Augmentation Approach to Interval Fault Estimation for Descriptor Systems. Eur. J. Control
**2020**, 51, 19–29. [Google Scholar] [CrossRef] - Valero, C.; Paulen, R. Set-Theoretic State Estimation for Multi-output Systems using Block and Sequential Approaches. In Proceedings of the 22nd International Conference on Process Control (PC19), Strbske Pleso, Slovakia, 11–14 June 2019; pp. 268–273. [Google Scholar]
- Rauh, A.; Jaulin, L. A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems. In Proceedings of the 19th IFAC Symposium System Identification: Learning Models for Decision and Control, Padova, Italy, 14–16 July 2021. under review. [Google Scholar]
- Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME J. Basic Eng.
**1960**, 82, 35–45. [Google Scholar] [CrossRef][Green Version] - Daum, F. Nonlinear Filters: Beyond the Kalman Filter. IEEE Aerosp. Electron. Syst. Mag.
**2005**, 20, 57–69. [Google Scholar] [CrossRef] - Julier, S.; Uhlmann, J.; Durrant-Whyte, H. A New Approach for the Nonlinear Transformation of Means and Covariances in Filters and Estimators. IEEE Trans. Autom. Control
**2000**, 45, 477–482. [Google Scholar] [CrossRef][Green Version] - Rauh, A.; Briechle, K.; Hanebeck, U.D. Nonlinear Measurement Update and Prediction: Prior Density Splitting Mixture Estimator. In Proceedings of the 2009 IEEE Control Applications, (CCA) & Intelligent Control, (ISIC), St. Petersburg, Russia,, 8–10 July 2009. [Google Scholar]
- Hanebeck, U.D.; Briechle, K.; Rauh, A. Progressive Bayes: A New Framework for Nonlinear State Estimation. In Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications 2003; Dasarathy, B.V., Ed.; International Society for Optics and Photonics, SPIE: Orlando, FL, USA, 2003; Volume 5099, pp. 256–267. [Google Scholar]
- Jambawalikar, S.; Kumar, P. A Note on Approximate Minimum Volume Enclosing Ellipsoid of Ellipsoids. In Proceedings of the 2008 International Conference on Computational Sciences and Its Applications, Perugia, Italy, 30 June–3 July 2008; pp. 478–487. [Google Scholar] [CrossRef][Green Version]
- Halder, A. On the Parameterized Computation of Minimum Volume Outer Ellipsoid of Minkowski Sum of Ellipsoids. In Proceedings of the IEEE Conference on Decision and Control (CDC), Miami, FL, USA, 17–19 December 2018; pp. 4040–4045. [Google Scholar] [CrossRef][Green Version]
- Yildirim, E.A. On the Minimum Volume Covering Ellipsoid of Ellipsoids. SIAM J. Optim.
**2006**, 17, 621–641. [Google Scholar] [CrossRef][Green Version] - Bourgois, A.; Jaulin, L. Interval Centred Form for Proving Stability of Non-Linear Discrete-Time Ssystems. In Proceedings of the 6th International Workshop on Symbolic-Numeric methods for Reasoning about CPS and IoT, Online, 31 August 2020; 2021; pp. 1–17. [Google Scholar] [CrossRef]
- Chapoutot, A. Interval Slopes as a Numerical Abstract Domain for Floating-Point Variables. In Static Analysis; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2010; pp. 184–200. [Google Scholar]
- Cornelius, H.; Lohner, R. Computing the Range of Values of Real Functions with Accuracy Higher Than Second Order. Computing
**1984**, 33, 331–347. [Google Scholar] [CrossRef] - Scherer, C.; Weiland, S. Linear Matrix Inequalities in Control. In Control System Advanced Methods, 2nd ed.; Levine, W.S., Ed.; The Electrical Engineering Handbook Series; CRC Press: Boca Raton, FL, USA, 2011; pp. 24-1–24-30. [Google Scholar]
- Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]
- Sturm, J. Using SeDuMi 1.02, A MATLAB Toolbox for Optimization over Symmetric Cones. Optim. Methods Softw.
**1999**, 11–12, 625–653. [Google Scholar] [CrossRef] - Löfberg, J. YALMIP: A Toolbox for Modeling and Optimization in MATLAB. In Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design, Taipei, Taiwan, 2–4 September 2004; pp. 284–289. [Google Scholar]
- Cichy, B.; Gałkowski, K.; Dąbkowski, P.; Aschemann, H.; Rauh, A. A New Procedure for the Design of Iterative Learning Controllers Using a 2D Systems Formulation of Processes with Uncertain Spatio-Temporal Dynamics. Control Cybern.
**2013**, 42, 9–26. [Google Scholar] - Barmish, B.R. New Tools for Robustness of Linear Systems; Macmillan: New York, NY, USA, 1994. [Google Scholar]
- Rump, S. IntLab—INTerval LABoratory; Developments in Reliable Computing; Csendes, T., Ed.; Kluwer Academic Publishers: New York, NY, USA, 1999; pp. 77–104. [Google Scholar]
- Durieu, C.; Polyak, B.T.; Walter, E. Trace Versus Determinant in Ellipsoidal Outer-Bounding, with Application to State Estimation. IFAC Proc. Vol.
**1996**, 29, 3975–3980. [Google Scholar] [CrossRef] - Dabbene, F.; Henrion, D. Set Approximation via Minimum-Volume Polynomial Sublevel Sets. In Proceedings of the 2013 European Control Conference (ECC), Zurich, Switzerland, 17–19 July 2013; pp. 1114–1119. [Google Scholar]
- Tarbouriech, S.; Garcia, G.; Gomes da Silva, J.; Queinnec, I. Stability and Stabilization of Linear Systems with Saturating Actuators; Springer: London, UK, 2011. [Google Scholar]
- Rohn, J. Bounds on Eigenvalues of Interval Matrices. Zamm J. Appl. Math. Mech./Z für angew. Math. und Mech.
**1998**, 78, 1049–1050. [Google Scholar] [CrossRef][Green Version] - Paradowski, T.; Lerch, S.; Damaszek, M.; Dehnert, R.; Tibken, B. Observability of Uncertain Nonlinear Systems Using Interval Analysis. Algorithms
**2020**, 13, 66. [Google Scholar] [CrossRef][Green Version] - Weinmann, A. Uncertain Models and Robust Control; Springer: Wien, Austria, 1991. [Google Scholar]
- Henrion, D.; Tarbouriech, S.; Arzelier, D. LMI Approximations for the Radius of the Intersection of Ellipsoids: Survey. J. Optim. Theory Appl.
**2001**, 108, 1–28. [Google Scholar] [CrossRef] - John, F. Extremum Problems with Inequalities as Subsidiary Conditions. In Studies and Essays Presented to R. Courant on His 60th Birthday; Interscience Publishers, Inc.: New York, NY, USA, 1948; pp. 187–204. [Google Scholar]
- Roos, C.; Terlaky, T.; Nemirovski, A.; Roos, K. On Maximization of Quadratic form over Intersection of Ellipsoids with Common Center. Math. Program.
**1999**, 86. [Google Scholar] [CrossRef][Green Version] - Gilitschenski, I.; Hanebeck, U.D. A Robust Computational Test for Overlap of Two Arbitrary-dimensional Ellipsoids in Fault-Detection of Kalman Filters. In Proceedings of the15th International Conference on Information Fusion, Singapore, 9–12 July 2012. [Google Scholar]
- Gilitschenski, I.; Hanebeck, U.D. A Direct Method for Checking Overlap of Two Hyperellipsoids. In Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications, Bonn, Germany, 8–10 October 2014. [Google Scholar]
- Becis-Aubry, Y. Ellipsoidal Constrained State Estimation in Presence of Bounded Disturbances. 2020. Available online: arxiv.org/pdf/2012.03267v1.pdf (accessed on 13 March 2021).
- Stengel, R. Optimal Control and Estimation; Dover Publications, Inc.: New York, NY, USA, 1994. [Google Scholar]
- Kurzhanskiy, A.; Varaiya, P. Ellipsoidal Toolbox (ET). In Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 13–15 December 2006; pp. 1498–1503. [Google Scholar]
- Rauh, A.; Grigoryev, V.; Aschemann, H.; Paschen, M. Incremental Gain Scheduling and Sensitivity-Based Control for Underactuated Ships. In Proceedings of the IFAC Conference on Control Applications in Marine Systems, CAMS, Rostock-Warnemunde, Germany, 15–17 September 2010. [Google Scholar]
- Rauh, A. Sensitivity Methods for Analysis and Design of Dynamic Systems with Applications in Control Engineering; Shaker: Düren, Germany, 2017. [Google Scholar]
- Fossen, T. Guidance and Control of Ocean Vehicles; John Wiley & Sons: Chichester, UK, 1994. [Google Scholar]
- Reyhanoglu, M. Exponential Stabilization of an Underactuated Autonomous Surface Vessel. Automatica
**1997**, 33, 2249–2254. [Google Scholar] [CrossRef][Green Version] - Pfaff, F.; Noack, B.; Hanebeck, U.D. Data Validation in the Presence of Stochastic and Set-membership Uncertainties. In Proceedings of the 16th International Conference on Information Fusion, Istanbul, Turkey, 9–12 July 2013. [Google Scholar]

**Figure 1.**Definition of a thick ellipsoid ${\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)}_{k}$ enclosing the domain ${\mathcal{A}}_{k}$ and its mapping via the system model (2).

**Figure 2.**Computation of the thick ellipsoid enclosure ${\left(\phantom{\rule{-5.55542pt}{0ex}}\left(\mathcal{E}\right)\phantom{\rule{-5.55542pt}{0ex}}\right)}_{k+1}$ for the example (35). The exact inner solution sets are visualized in light gray color, while the dark gray domains consist of the union of ellipsoid bounds that result from a numerical mapping of a $20\times 20$ grid for the interval parameters $\left[{p}_{1}\right]$ and $\left[{p}_{2}\right]$.

**Figure 3.**Intersection of ellipsoids ${\mathcal{E}}_{1,k}$ and ${\mathcal{E}}_{2,k}$ with identical and different midpoints. Inner bound of the resulting thick ellipsoid: ${\mathcal{E}}_{\mathrm{D},k}^{\mathrm{I}}$; outer bound: ${\mathcal{E}}_{\mathrm{D},k}^{\mathrm{O}}$ (∗ symbols denote the solution enclosure according to Equation (53)). (

**a**) Case according to Section 3.2.1 with the shape matrices (55). (

**b**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (55), (56), enclosure definition (53). (

**c**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (49). (

**d**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (53).

**Figure 4.**Intersection of an ellipsoid ${\mathcal{E}}_{1,k}$ with the degenerate ellipsoid ${\mathcal{E}}_{2,k}$ (vertical strip) with identical and different midpoints. Inner bound of the resulting thick ellipsoid: ${\mathcal{E}}_{\mathrm{D},k}^{\mathrm{I}}$; outer bound: ${\mathcal{E}}_{\mathrm{D},k}^{\mathrm{O}}$ (∗ symbols denote the solution enclosure according to Equation (53)). (

**a**) Case according to Section 3.2.1 with ${\mathbf{Q}}_{2,k}$ according to (55). (

**b**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (49). (

**c**) Case according to Section 3.2.2 with the initial ellipsoid parameterization (56), (57), enclosure definition (53).

**Figure 5.**Union of ellipsoids ${\mathcal{E}}_{1,k}$ and ${\mathcal{E}}_{2,k}$ with identical and different midpoints. Inner bound of the resulting thick ellipsoid: ${\mathcal{E}}_{\mathrm{U},k}^{\mathrm{I}}$; outer bound: ${\mathcal{E}}_{\mathrm{U},k}^{\mathrm{O}}$ (∗ symbols denote the solution enclosure according to Equation (65)). (

**a**) Union of ellipsoids with identical midpoints, cf. (55). (

**b**) Union of ellipsoids with the initial ellipsoid parameterization (55), (56), enclosure definition (65). (

**c**) Union of ellipsoids with the initial ellipsoid parameterization (56), (57), enclosure definition (60). (

**d**) Union of ellipsoids with the initial ellipsoid parameterization (56), (57), enclosure definition (65).

**Figure 6.**Thick ellipsoid state estimation algorithm consisting of prediction and correction stages.

**Figure 7.**Separation of the state equations according to (66)–(70) into the mapping of an origin-centered ellipsoid and the verified treatment of non-zero offset terms.

**Figure 8.**Visualization of the thick ellipsoid state estimation procedure for the example system in (82).

**Figure 9.**Definition of the velocity components in a body-fixed coordinate systems, velocity-proportional damping as well as disturbance forces and torques.

**Figure 10.**State estimation for the case of large uncertainties (91). (

**a**) Outer ellipsoidal enclosures for the velocities ${x}_{1,k}$ and ${x}_{2,k}$. (

**b**) Enlarged view of the ellipsoidal enclosures of ${x}_{1,k}$ and ${x}_{2,k}$. (

**c**) Outer ellipsoidal enclosures for the velocities ${x}_{1,k}$ and ${x}_{3,k}$. (

**d**) Enlarged view of the ellipsoidal enclosures of ${x}_{1,k}$ and ${x}_{3,k}$.

**Figure 11.**State estimation for the case of small uncertainties (92). (

**a**) Outer ellipsoidal enclosures for the velocities ${x}_{1,k}$ and ${x}_{2,k}$. (

**b**) Enlarged view of the ellipsoidal enclosures of ${x}_{1,k}$ and ${x}_{2,k}$. (

**c**) Outer ellipsoidal enclosures for the velocities ${x}_{1,k}$ and ${x}_{3,k}$. (

**d**) Enlarged view of the ellipsoidal enclosures of ${x}_{1,k}$ and ${x}_{3,k}$.

**Figure 12.**Comparison of the estimation results with noisy measurements and true system states for the case of large uncertainties (91). (

**a**) Estimated bounds $\left[{x}_{1,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}1,k}$ and ${x}_{1,k}$. (

**b**) Estimated bounds $\left[{x}_{2,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}2,k}$ and ${x}_{2,k}$. (

**c**) Estimated bounds $\left[{x}_{3,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}3,k}$ and ${x}_{3,k}$. (

**d**) Estimated bounds $\left[{d}_{3,k}\right]$ vs. true disurbance ${d}_{3,k}=0.05$.

**Figure 13.**Comparison of the estimation results with noisy measurements and true system states for the case of small uncertainties (92). (

**a**) Estimated bounds $\left[{x}_{1,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}1,k}$ and ${x}_{1,k}$. (

**b**) Estimated bounds $\left[{x}_{2,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}2,k}$ and ${x}_{2,k}$. (

**c**) Estimated bounds $\left[{x}_{3,k}\right]$ vs. measured and true velocities ${y}_{\mathrm{m}3,k}$ and ${x}_{3,k}$. (

**d**) Estimated bounds $\left[{d}_{3,k}\right]$ vs. true disurbance ${d}_{3,k}=0.05$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Rauh, 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