# Observer Based Multi-Level Fault Reconstruction for Interconnected Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description and Problem Formulation

## 3. On Condition of Fault Reconstructability Locally and Globally

#### 3.1. Fault Reconstructability Condition

**Definition**

**1.**

**(Local Algebraic observability).**For subsystem (2), a fault element${\omega}_{2}\in \kappa \left(u,{\omega}_{2}\right)$is said to be locally algebraically observable if${\omega}_{2}$satisfies a differential algebraic equation with coefficients over$\kappa \left(u,{u}_{1},{\omega}_{2}\right)$.

**Definition**

**2.**

**(Global Algebraic observability).**For interconnected systems depicted by (1) and (2), a fault element${\omega}_{2}\in \kappa \left(u,{\omega}_{2}\right)$is said to be globally algebraically observable if${\omega}_{2}$satisfies a differential algebraic equation with coefficients over$\kappa \left(u,y,{\omega}_{2}\right)$.

**Definition**

**3.**

**(Local reconstructability).**For system (2), it is said to be locally reconstructable if the system is invertible. In this way, it is capable of estimating the unknown input${\omega}_{2}$from local system information u and${u}_{1}$.

**Definition**

**4.**

**(Global reconstructability).**For the interconnected nonlinear system described by (1) and (2), it is said to be globally reconstructable if the interconnected system is invertible, in this way, it is capable of estimating the unknown input${\omega}_{2}$from global system information u and$y$.

#### 3.2. Minimum Number of Measurements and Reconstructable Unknown Inputs

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Remark**

**3.**

## 4. Observer Design for Unknown Input Reconstruction

#### 4.1. Asymptotic Reduced-Order Observer Design with Auxiliary Output

^{1}real-valued function $\mathsf{\gamma}$ exists, such that a proportional asymptotic reduced-order unknown input observer exists, for system (6) it can be written as:

**Theorem**

**1.**

**Remark**

**4.**

**Remark**

**5.**

#### 4.2. Auxiliary Output Estimation

#### 4.3. Reconstruction of the Unknown Inputs by Asymptotic Reduced-Order Observer with Auxiliary Output

**Proposition**

**2.**

**Theorem**

**2.**

**Proof.**

## 5. Numerical Simulation Implementation on a Pilot Intensified Heat Exchanger

#### 5.1. Interconnected System Modelling

#### 5.2. Observer Design for Unknown Input Reconstruction

#### 5.2.1. Reduce-Order Observer Design

#### 5.2.2. System Inversion Based Interconnection Reconstruction

#### 5.3. Simulation Results and Discussion

**Case**

**1.**

**Case**

**2.**

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

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Zhang, M.; Dahhou, B.; Wu, Q.; Li, Z.
Observer Based Multi-Level Fault Reconstruction for Interconnected Systems. *Entropy* **2021**, *23*, 1102.
https://doi.org/10.3390/e23091102

**AMA Style**

Zhang M, Dahhou B, Wu Q, Li Z.
Observer Based Multi-Level Fault Reconstruction for Interconnected Systems. *Entropy*. 2021; 23(9):1102.
https://doi.org/10.3390/e23091102

**Chicago/Turabian Style**

Zhang, Mei, Boutaïeb Dahhou, Qinmu Wu, and Zetao Li.
2021. "Observer Based Multi-Level Fault Reconstruction for Interconnected Systems" *Entropy* 23, no. 9: 1102.
https://doi.org/10.3390/e23091102