# Kalman Filter Adaptation to Disturbances of the Observer’s Parameters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Synthesis of observers with the property of invariance to disturbances of their own parameters [10].
- Two-stage estimation of the system state parameters using the extended Kalman filter, which reduces its computational complexity when expanding the state vector [12].
- Extension of the observation vector based on an additional evaluating observer formed using nonlinear programming [17].
- Robust scaling of the observer’s transmission coefficient, which increases the stability of the evaluation process [18].
- Ensuring the invariance of the Kalman filtering process to parametric disturbances of the observer through the use of integrated neural networks [19], etc.

- Orientation and navigation systems of mobile robots, in which the navigation parameters of the robot are adjusted based on taking into account the zero speed of the lower point of the wheel (or the robot’s foot) at the moment of contact with the earth’s surface [24].
- Transport information and measurement systems of various types—railway, automobile, marine, unmanned aerial vehicle (UAV), etc., in which the orientation and navigation parameters of an object are corrected at the time of passing reference points with precisely known coordinates (for example, traffic lights, eurobalises, radio frequency tags, buoys, etc.) [25,26,27,28,29,30].
- Complexing orientation and navigation systems based on inertial sensing elements, allowing to solve the navigation problem inside confined rooms [31] and so on.

## 2. Theoretical Assumptions

#### 2.1. Task Definition

_{i}is the N-dimensional state vector at a discrete time i, ${\Phi}_{i+1/i}$ is the transition state matrix of N × N dimension, and W

_{i+}

_{1}is the N-dimensional vector of white Gaussian noise of an object with zero mean and a known intensity matrix G

_{i+}

_{1}·δ

_{i+}

_{1,j+1}(δ

_{i}

_{,j}is the Kronecker Delta function). For a discrete stochastic meter, the vector of the output signals of the meter is described by the equation

_{i+}

_{1}is the M-dimensional measurement vector, H

_{i+}

_{1}is the measurement matrix of M × N dimensions, and V

_{i+}

_{1}is the M-dimensional vector of white Gaussian noise of an object with zero mean and a known intensity matrix R

_{i+}

_{1}·δ

_{i}

_{+1,j+1}.

_{i}. In the direct formulation, calculating the true values of the H

_{i}matrix from accurate observations of the state vector is a solution to the inverse dynamics problem, computationally unrealizable in real time with existing iterative procedures for solving systems of nonlinear equations due to the essentially nonlinear dependence of the evaluation vector on the measurement matrix.

^{(2)}H can be neglected. These assumptions allow us to use for the development of the desired algorithm the method of studying disturbed multidimensional linear systems described by Chernov and Yastrebov [35]. For its application, it is predefined for an arbitrary matrix A of m × n dimension a column vector A

^{(ν)}formed from its elements as follows [35]:

#### 2.2. Task Solution

_{K(j)}is the j-th column,

^{(ν)}components is easily determined. At the same time, in order to be able to correctly apply such a procedure, the dimensions of the vectors $\delta {\widehat{\xi}}_{i+1},\delta {H}_{}^{\left(\nu \right)}$ must match. In the presence of a single accurate measurement of the state vector, this is possible only in the case of a scalar observer (see Equation (2))—when the measurement matrix has 1 × N dimension. Then, the vector δH

^{(ν)}is defined directly from Equation system (9):

_{1}, i + 1 + s

_{2}, i + 1 + s

_{3}, …, i + 1 + s

_{M}, where s

_{i}are random time intervals. Then, the Equation system (9) is converted to the following form:

^{(ν)}makes it possible to correct the measurement matrix, thereby increasing the accuracy and stability of the Kalman filtering process as a whole.

_{(i+}

_{1)H}and B

_{iH}require only three matrix multiplications and additions, which does not present any difficulties for modern estimator. To illustrate the effectiveness of the proposed approach, the following example was considered.

## 3. Results and Discussions

#### Numerical Solving the Adaptive Assessment of Navigation Parameters of an Unmanned Vehicle

_{y}, V

_{x}are the linear UV-velocity projections on the GCS-axes.

- Starting point coordinates of the UV-movement φ
_{0}= 0.76 rad and λ_{0}= 0.32 rad; - Time interval [0; 1000] s;
- UV-movement constant velocity V = 18 m s
^{−1}; - UV-trajectory along the Earth’s surface is loxodromic with an azimuthal angle A = 0.23 rad;
- Random height h changes generated by the trajectory relief are distributed according to the Gaussian law with zero mean and dispersion G
_{h}= (0.14 m)^{2}; - Type of UV navigation system is satellite-based.

_{0}= 0.

^{2}.

^{2}, which is much more accurate than the traditional filtering scheme.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Graphs of latitude φ (

**a**) and longitude λ (

**b**) estimation errors obtained when implementing the classical Kalman filter.

**Figure 2.**Graphs of latitude φ (

**a**) and longitude λ (

**b**) estimation errors obtained when implementing the adaptive filter.

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**MDPI and ACS Style**

Manin, A.A.; Sokolov, S.V.; Novikov, A.I.; Polyakova, M.V.; Demidov, D.N.; Novikova, T.P.
Kalman Filter Adaptation to Disturbances of the Observer’s Parameters. *Inventions* **2021**, *6*, 80.
https://doi.org/10.3390/inventions6040080

**AMA Style**

Manin AA, Sokolov SV, Novikov AI, Polyakova MV, Demidov DN, Novikova TP.
Kalman Filter Adaptation to Disturbances of the Observer’s Parameters. *Inventions*. 2021; 6(4):80.
https://doi.org/10.3390/inventions6040080

**Chicago/Turabian Style**

Manin, Alexander A., Sergey V. Sokolov, Arthur I. Novikov, Marianna V. Polyakova, Dmitriy N. Demidov, and Tatyana P. Novikova.
2021. "Kalman Filter Adaptation to Disturbances of the Observer’s Parameters" *Inventions* 6, no. 4: 80.
https://doi.org/10.3390/inventions6040080