# Formulation of the Alpha Sliding Innovation Filter: A Robust Linear Estimation Strategy

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Sliding Innovation Filter (SIF)

## 3. The Proposed Alpha Sliding Innovation Filter (ASIF) Strategy

- The range of acceptable $\delta $ is infinitely wide, as it can be any positive value $\left(0,\infty \right)$.
- The number of boundary layers required is equal to the number of measurements, which means exhaustive trials for high-dimensional systems (e.g., AI output layers).
- In most cases, $\delta $ is selected to be the maximum allowable error in the system. For small innovation amplitudes, the filter will have a very small convergence rate, and for large amplitudes, the estimates may chatter.

- If $\alpha =1$, ASIF collapses to the SIF with a very small boundary layer.
- If $\alpha \to 0$, the filter depends more on the system and less on the measurement. This can be used to reduce the effect of the measurement noise. This helps significantly when $R$ is larger than $Q$.
- If $\alpha $ is large, the filter depends more on the measurement, which makes the ASIF sensitive to the measurement noise but less sensitive to modeling uncertainties. This helps when $Q$ is larger than $R$.

## 4. Proof of Stability

## 5. Computer Experiments and Results

#### 5.1. Mercury Thermometer

#### 5.2. Spring-Mass-Damper

#### 5.3. Electrohydrostatic Actuator

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Comments |

${}^{-1}$$,{}^{+}$$,{}^{T}$ | Notation denoting an inverse, a pseudo inverse and a transpose, respectively. |

|A| | Absolute value of A. |

$A,B$ | The system, and input matrices, respectively. |

$C$ | The measurement matrix, which is the identity matrix in all examples. |

${w}_{k},{v}_{k}$ | The system and measurement noise vectors at time $k$, respectively. |

${x}_{k},{z}_{k},{u}_{k}$ | The states, measurements and imput vectores, respectively. |

$\widehat{A}$ | The estimate value of $A$. |

${K}_{k}$ | The correction matrix at time k. |

$\delta $ | The Boundary Layer |

${A}_{k|k-1},{A}_{k|k}$ | The a priori and the a posteriori estimate of $A$ at time $k$, respectively. |

$\alpha $ | The fading memory coefficient. |

$Q,R$ | The system and measurement noise covariance matrices, respectively. |

$\eta $ | The uncertainity vector |

$\overline{A},\tilde{A}$ | The error in $A$. |

$E\left(a\right)$ | The expectation operator of the element $a$. |

$h,{A}_{p},{C}_{p},m$ | The thermometer parameters; convective heat transfer coefficient, surface area, specific heat and mass, respectively. |

$T,\tau ,{T}_{s}$ | The tempretaure, time constant, and sampling time, respectively. |

$M,b,k$ | The spring-mass-damper parameters; mass, damping coefficient, and spring constant, respectively. |

${I}_{n\times n}$ | The identity matrix with dimensions of $n\times n$. |

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**Figure 1.**SIF estimation concept showing the sliding mode [29].

**Figure 3.**An example of the outputs from the different filters applied to the thermometer system with (

**a**) no modeling and (

**b**) modeling uncertainties.

**Figure 4.**RMSE and MAE for the S-M-D system’s (

**a**) first and (

**b**) second states with no modeling uncertainties, and (

**c**) first and (

**d**) second states with modeling uncertainties.

**Figure 5.**An example of the (

**a**) position and (

**b**) velocity estimates from the different filters applied to the S-M-D system with no modeling uncertainties, and (

**c**) position and (

**d**) velocity estimates from the different filters applied to the S-M-D system with modeling uncertainties.

**Figure 6.**RMSE and MAE for the EHA system’s (

**a**) first, (

**b**) second, and (

**c**) third states with no modeling uncertainties, and the system’s (

**d**) first, (

**e**) second, and (

**f**) third states with modeling uncertainties.

**Figure 7.**An example of the (

**a**) position, (

**b**) velocity, and (

**c**) acceleration estimates from the different filters applied to the aerospace actuator system with no modeling uncertainties, and the (

**d**) position, (

**e**) velocity, and (

**f**) acceleration estimates from the different filters applied to the aerospace actuator system with modeling uncertainties.

**Figure 8.**The simulation time for (

**a**) thermometer, (

**b**) spring-mass-damper, and (

**c**) EHA examples for the different filters.

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $9.5\times {10}^{-1}$ | $2.2\times {10}^{-2}$ | $5.8\times {10}^{-1}$ | $3.7\times {10}^{-3}$ | $4.5\times {10}^{-1}$ | $4.6\times {10}^{-3}$ |

2 | ${\mathrm{x}}_{1}$ | $2.9\times {10}^{1}$ | $7.1\times {10}^{0}$ | $1.1\times {10}^{0}$ | $2.5\times {10}^{-2}$ | $6.6\times {10}^{0}$ | $1.5\times {10}^{0}$ |

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $3.3\times {10}^{0}$ | $2.2\times {10}^{-1}$ | $1.1\times {10}^{0}$ | $6.8\times {10}^{-3}$ | $1.3\times {10}^{0}$ | $4.4\times {10}^{-2}$ |

2 | ${\mathrm{x}}_{1}$ | $9.998\times {10}^{-1}$ | $4.99\times {10}^{1}$ | $1.4\times {10}^{0}$ | $3.6\times {10}^{-3}$ | $1.0\times {10}^{0}$ | $1.0\times {10}^{0}$ |

Parameter | Case 1 | Case 2 |
---|---|---|

$\mathrm{M}\left(\mathrm{k}\mathrm{g}\right)$ | $500$ | $500$ |

$\mathrm{k}\left(\mathrm{k}\mathrm{N}/\mathrm{m}\right)$ | $1$ | $0.5$ |

$\mathrm{b}\left(\mathrm{N}\mathrm{s}/\mathrm{m}\right)$ | $5$ | $0$ |

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $2.0\times {10}^{-4}$ | $6.1\times {10}^{-5}$ | $1.0\times {10}^{-3}$ | $3.2\times {10}^{-5}$ | $3.0\times {10}^{-4}$ | $3.8\times {10}^{-5}$ |

${\mathrm{x}}_{2}$ | $3.0\times {10}^{-4}$ | $6.4\times {10}^{-5}$ | $5.4\times {10}^{-3}$ | $4.5\times {10}^{-5}$ | $3.0\times {10}^{-4}$ | $4.0\times {10}^{-5}$ | |

2 | ${\mathrm{x}}_{1}$ | $7.9\times {10}^{-1}$ | $4.0\times {10}^{-3}$ | $1.0\times {10}^{-3}$ | $0.0\times {10}^{0}$ | $9.0\times {10}^{-4}$ | $0.0\times {10}^{0}$ |

${\mathrm{x}}_{2}$ | $8.9\times {10}^{-1}$ | $1.3\times {10}^{-3}$ | $5.4\times {10}^{-3}$ | $0.0\times {10}^{0}$ | $1.2\times {10}^{-2}$ | $1.0\times {10}^{-4}$ |

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $5.8\times {10}^{-3}$ | $2.3\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $3.0\times {10}^{-4}$ | $8.0\times {10}^{-4}$ | $1.0\times {10}^{-4}$ |

${\mathrm{x}}_{2}$ | $5.8\times {10}^{-3}$ | $2.2\times {10}^{-3}$ | $1.0\times {10}^{-2}$ | $0.0\times {10}^{0}$ | $9.0\times {10}^{-4}$ | $1.0\times {10}^{-4}$ | |

2 | ${\mathrm{x}}_{1}$ | $1.7\times {10}^{0}$ | $8.1\times {10}^{-3}$ | $3.5\times {10}^{-3}$ | $3.0\times {10}^{-4}$ | $3.3\times {10}^{-3}$ | $3.0\times {10}^{-4}$ |

${\mathrm{x}}_{2}$ | $1.4\times {10}^{0}$ | $2.0\times {10}^{-3}$ | $1.1\times {10}^{-2}$ | $3.0\times {10}^{-4}$ | $2.5\times {10}^{-2}$ | $5.0\times {10}^{-4}$ |

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $2.0\times {10}^{-2}$ | $1.7\times {10}^{-4}$ | $5.4\times {10}^{-2}$ | $4.4\times {10}^{-4}$ | $5.8\times {10}^{-2}$ | $3.6\times {10}^{-4}$ |

${\mathrm{x}}_{2}$ | $3.3\times {10}^{-2}$ | $4.2\times {10}^{-4}$ | $2.9\times {10}^{-2}$ | $6.3\times {10}^{-4}$ | $5.8\times {10}^{-2}$ | $3.6\times {10}^{-4}$ | |

${\mathrm{x}}_{3}$ | $5.8\times {10}^{-2}$ | $3.9\times {10}^{-4}$ | $5.8\times {10}^{-2}$ | $4.3\times {10}^{-4}$ | $5.99\times {10}^{-2}$ | $4.3\times {10}^{-4}$ | |

2 | ${\mathrm{x}}_{1}$ | $8.7\times {10}^{-1}$ | $6.7\times {10}^{-2}$ | $5.4\times {10}^{-2}$ | $5.0\times {10}^{-4}$ | $5.8\times {10}^{-2}$ | $4.0\times {10}^{-4}$ |

${\mathrm{x}}_{2}$ | $5.5\times {10}^{-2}$ | $6.5\times {10}^{-3}$ | $2.9\times {10}^{-2}$ | $6.0\times {10}^{-4}$ | $5.8\times {10}^{-2}$ | $4.0\times {10}^{-4}$ | |

${\mathrm{x}}_{3}$ | $7.1\times {10}^{-2}$ | $3.7\times {10}^{-3}$ | $5.8\times {10}^{-2}$ | $4.0\times {10}^{-4}$ | $6.6\times {10}^{-2}$ | $1.3\times {10}^{-3}$ |

Case | State | KF | SIF | ASIF | |||
---|---|---|---|---|---|---|---|

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | ||

1 | ${\mathrm{x}}_{1}$ | $4.9\times {10}^{-2}$ | $7.0\times {10}^{-4}$ | $1.2\times {10}^{-1}$ | $2.5\times {10}^{-3}$ | $9.99\times {10}^{-2}$ | $0.0\times {10}^{0}$ |

${\mathrm{x}}_{2}$ | $1.0\times {10}^{-1}$ | $4.5\times {10}^{-3}$ | $1.1\times {10}^{-1}$ | $9.2\times {10}^{-3}$ | $9.99\times {10}^{-2}$ | $0.0\times {10}^{0}$ | |

${\mathrm{x}}_{3}$ | $1.0\times {10}^{-1}$ | $4.0\times {10}^{-4}$ | $2.1\times {10}^{0}$ | $1.1\times {10}^{-2}$ | $1.3\times {10}^{-1}$ | $0.0\times {10}^{0}$ | |

2 | ${\mathrm{x}}_{1}$ | $2.95\times {10}^{0}$ | $3.4\times {10}^{-1}$ | $1.1\times {10}^{-1}$ | $2.7\times {10}^{-3}$ | $9.99\times {10}^{-2}$ | $0.0\times {10}^{0}$ |

${\mathrm{x}}_{2}$ | $1.98\times {10}^{-1}$ | $2.8\times {10}^{-2}$ | $1.1\times {10}^{-1}$ | $9.1\times {10}^{-3}$ | $9.99\times {10}^{-2}$ | $0.0\times {10}^{0}$ | |

${\mathrm{x}}_{3}$ | $2.2\times {10}^{-1}$ | $2.4\times {10}^{-2}$ | $2.1\times {10}^{-1}$ | $1.3\times {10}^{-2}$ | $1.9\times {10}^{-1}$ | $1.3\times {10}^{-2}$ |

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**MDPI and ACS Style**

AlShabi, M.; Gadsden, S.A.
Formulation of the Alpha Sliding Innovation Filter: A Robust Linear Estimation Strategy. *Sensors* **2022**, *22*, 8927.
https://doi.org/10.3390/s22228927

**AMA Style**

AlShabi M, Gadsden SA.
Formulation of the Alpha Sliding Innovation Filter: A Robust Linear Estimation Strategy. *Sensors*. 2022; 22(22):8927.
https://doi.org/10.3390/s22228927

**Chicago/Turabian Style**

AlShabi, Mohammad, and Stephen Andrew Gadsden.
2022. "Formulation of the Alpha Sliding Innovation Filter: A Robust Linear Estimation Strategy" *Sensors* 22, no. 22: 8927.
https://doi.org/10.3390/s22228927