# Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Discrete Variable Fractional Order State-Space System

**Definition**

**1.**

## 3. Triple Estimation Algorithm Based on UFKF Filter

#### 3.1. Dual Estimation Scheme

#### 3.2. Triple Estimation Scheme—The Main Result

#### 3.2.1. Order Estimation Filter KFo

**Proposition**

**1.**

#### 3.2.2. State Estimation Filter KFx

**Proposition**

**2.**

#### 3.2.3. Parameters Estimation Filter KFw

**Proposition**

**3.**

## 4. Numerical Results

- Noises parameters$$\mathrm{E}\left[\omega {\omega}^{T}\right]=2.5\times {10}^{-5},$$$$\mathrm{E}\left[\nu {\nu}^{T}\right]={10}^{-4},$$
- Parameters of the KFx filter$${P}_{0}=\left[\begin{array}{c}1\end{array}\right],{Q}_{0}=\left[\begin{array}{c}2.5\times {10}^{-5}\end{array}\right],$$$${x}_{0}=\left[0\right],R=\left[{10}^{-4}\right],$$
- Parameters of the KFo filter$${P}_{0}^{o}=\left[\begin{array}{c}0.01\end{array}\right],{Q}_{0}^{o}=\left[\begin{array}{c}0.001\end{array}\right],$$$${\alpha}_{0}=\left[1\right],{R}^{o}=[1.25\times {10}^{-4}],\mathfrak{A}=1,\mathfrak{B}=2,{\delta}^{o}=0.5.$$
- Parameters of the KFw filter$${P}_{0}^{w}=\left[\begin{array}{c}0.01\end{array}\right],{Q}_{0}^{w}=\left[\begin{array}{c}0.001\end{array}\right],$$$${w}_{0}=\left[0\right],{R}^{w}=[1.25\times {10}^{-4}],\mathfrak{A}=1,\mathfrak{B}=2,{\delta}^{w}=0.5.$$

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Sierociuk, D.; Macias, M.
Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter. *Sensors* **2021**, *21*, 8159.
https://doi.org/10.3390/s21238159

**AMA Style**

Sierociuk D, Macias M.
Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter. *Sensors*. 2021; 21(23):8159.
https://doi.org/10.3390/s21238159

**Chicago/Turabian Style**

Sierociuk, Dominik, and Michal Macias.
2021. "Triple Estimation of Fractional Variable Order, Parameters, and State Variables Based on the Unscented Fractional Order Kalman Filter" *Sensors* 21, no. 23: 8159.
https://doi.org/10.3390/s21238159