# Jordan Canonical Form for Solving the Fault Diagnosis and Estimation Problems

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## Abstract

**:**

## 1. Introduction

## 2. Diagnostic Observer Design

## 3. Virtual Sensor Design

## 4. Interval Observer Design

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Sliding Mode Observer Design

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

## 6. Nonlinear Systems

## 7. Robust Solution

## 8. Example

## 9. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ICF | identification canonical form |

JCF | Jordan canonical form |

SMO | sliding mode observers |

## References

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**Figure 2.**Graphs of the functions ${x}_{1}\left(t\right)$, ${\underline{x}}_{1}\left(t\right)$, and ${\overline{x}}_{1}\left(t\right)$ with ${k}_{*}{w}_{*1}=0.3$ and $D\left[\rho \right]=0.6$.

**Figure 3.**Graphs of the functions ${x}_{1}\left(t\right)$, ${\underline{x}}_{1}\left(t\right)$, and ${\overline{x}}_{1}\left(t\right)$ with ${k}_{*}{w}_{*1}=0.6$ and $\rho =0$.

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**MDPI and ACS Style**

Sergiyenko, O.; Zhirabok, A.; Mercorelli, P.; Zuev, A.; Filaretov, V.; Tyrsa, V.
Jordan Canonical Form for Solving the Fault Diagnosis and Estimation Problems. *Technologies* **2023**, *11*, 72.
https://doi.org/10.3390/technologies11030072

**AMA Style**

Sergiyenko O, Zhirabok A, Mercorelli P, Zuev A, Filaretov V, Tyrsa V.
Jordan Canonical Form for Solving the Fault Diagnosis and Estimation Problems. *Technologies*. 2023; 11(3):72.
https://doi.org/10.3390/technologies11030072

**Chicago/Turabian Style**

Sergiyenko, Oleg, Alexey Zhirabok, Paolo Mercorelli, Alexander Zuev, Vladimir Filaretov, and Vera Tyrsa.
2023. "Jordan Canonical Form for Solving the Fault Diagnosis and Estimation Problems" *Technologies* 11, no. 3: 72.
https://doi.org/10.3390/technologies11030072