Residual-Based RKF with Recursive Measurement Noise Covariance

Matching

CHINGIZ HAJIYEV1, ULVIYE HACIZADE2

1Faculty of Aeronautics and Astronautics,

Istanbul Technical University,

Maslak, 34469, Istanbul,

TURKEY

2Department of Computer Engineering,

Halic University, Güzeltepe Neighborhood, 15,

Temmuz Şehitler Street, No.15,

34060 Eyüp-Istanbul,

TURKEY

Abstract: - A new residual-based recursive measurement noise covariance estimation method is proposed. The

presented algorithm is used for Kalman filter tuning, as a result, the robust Kalman filter (RKF) against

measurement malfunctions is derived. The proposed residual-based RKF with recursive estimation of

measurement noise covariance is applied to the model of Unmanned Aerial Vehicle (UAV) dynamics.

Algorithms are examined for two types of measurement fault scenarios; constant bias at measurements

(additive sensor faults) and measurement noise increments (multiplicative sensor faults). The simulation results

show that the proposed RKF can accurately estimate UAV dynamics in real-time in the presence of various

types of sensor faults. Estimation accuracies of the proposed RKF and conventional KF are investigated and

compared.

Key-Words: - Kalman filter, residual, robust estimation, unmanned aerial vehicle, sensor faults, covariance

matching.

Received: May 9, 2024. Revised: November 4, 2024. Accepted: December 6, 2024. Published: December 31, 2024.

1 Introduction

The Kalman Filter can be used to estimate the states

of an Unmanned Aerial Vehicle (UAV). That is the

preferred method because it is crucial to exactly

know the characteristics, such as velocity, altitude,

attitude, etc. Successful aircraft control can be

attained when these UAV states are attained without

any issues. However, that procedure is contingent

on how accurate the measurements are. The filter

produces erroneous findings and diverges over time

if the measurements are unreliable due to any type

of sensor fault. Due to the significance of obtaining

fault tolerance in the design of a UAV flight control

system, filters should be constructed robustly to

overcome such issues.

The Kalman filter method of state estimation is

extremely sensitive to defects in the measurement

system. Changes in the measurement channels

significantly degrade the performance of the

estimating systems if the measurement system's

state of operation differs from the models used in

the filter's synthesis. The possible errors can be

recovered with adaptive Kalman filters.

A variety of alternative strategies can be used to

make the Kalman filter flexible and hence

insensitive to a priori measurements or system

uncertainties. Multiple-model adaptive estimation

(MMAE) [1], [2], innovation-based adaptive

estimation (IAE) [3], [4], [5], and residual-based

adaptive estimation (RAE) [5], [6] are all essential

techniques to addressing the adaptive Kalman

filtering problem. Changes in the innovation or

residual sequences cause rapid adjustments to the

measurement and/or process noise covariance

matrices.

The MMAE approach can only be utilized in

specific situations because it calls for a number of

parallel Kalman filters, and the faults should be

known. IAE and RAE methods must use the

innovation vectors or residual vectors of m epochs

in the moving window to estimate the covariance

matrices. The number, kind, and distribution of the

measurements for all epochs inside a window must

be consistent for IAE and RAE estimators. If not,

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DOI: 10.37394/23202.2024.23.45

Chingiz Hajiyev, Ulviye Hacizade

E-ISSN: 2224-2678

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Volume 23, 2024

neither the innovation nor the residual vectors can

be used to estimate the covariance matrices of the

measurement noises.

Another idea is to multiply the noise covariance

matrix by a time-dependent variable to scale it. This

algorithm is called adaptive fading Kalman filter

(AFKF). One approach to creating such an

algorithm is to multiply the process or measurement

noise covariance matrices by a single adaptive

factor [5], [7], [8]. The AFKF technique can be used

if there is a fault in the measurement system, and the

filter's insensitivity to present measurement faults

can be ensured by multiplying the measurement

noise scale factor by the measurement noise

covariance matrix. As a result, by applying an

adjustment to the filter gain, the filter's accurate

estimating behavior will be protected from being

affected by inaccurate measurements.

An adaptation method based on the multiple

fading factor is provided in [5], and [9]. The

variation in the impacts of measurement noise

covariance change on estimating the performance of

each estimated state is the primary reason for

adopting several fading factors. It is critical to

carefully evaluate how modifying the measurement

noise covariance would affect each state,

particularly for complicated multivariable systems,

and to employ a matrix with many fading factors

rather than a single factor (so that the adaptation is

weighted differently for each state).

The measurement covariance matrix can be

modified with the help of the fuzzy inference

system, [10]. The results showed that the proposed

adaptive fuzzy extended Kalman filter is robust

against disturbances and outliers. Although adaptive

Kalman filter algorithms based on fuzzy logic work

well in some situations, they are knowledge-based

systems that function with linguistic variables and

cannot be widely applied to critical systems like

aircraft flight control systems since they are human

experience-based.

Because the noise estimator cannot be expressed

in a recursive form and each previous state vector

must be smoothed by the most recent measurements

at each point in time, the algorithm in the studies

mentioned cannot be used to directly estimate the

measurement noise covariance in practical

operations.

In this study, a residual-based robust Kalman

filter with a recursive measurement noise covariance

estimator is proposed and applied for the state

estimation process of a UAV platform. The results

of the proposed robust and conventional Kalman

filter algorithms are compared for different types of

measurement faults and recommendations about

their utilization are given.

The article is presented as follows. The UAV's

flight dynamics model is presented first. The

optimal Kalman filter for estimating UAV state is

then described in the following section. After that, a

novel recursive measurement noise covariance

estimation method for Kalman filter tuning is

proposed. Based on the introduced residual-based

recursive measurement noise covariance estimation

approach the robust Kalman filter (RKF) against

sensor faults is derived. The proposed RKF with

recursive measurement noise covariance estimation

algorithm is applied for the model of UAV

dynamics and the performance of the proposed filter

is tested via simulations for the state estimation

process of a UAV platform. The conclusion and the

results are briefly summarized in the final section.

2 Preliminaries

Consider the linear dynamic system represented by

the state equation:

( 1) ( ) ( ) ( )x j Ax j Bu j Gw j   

(1)

and measurement equation:

() ()() ()z j H j x j V j

, (2)

where

()xj

is the system state; A is the system

transition matrix; B is the control distribution

matrix;

()uj

is the control input;

()wj

is the random

system noise; G is the system noise transition

matrix;

()zj

is the measurement vector;

()Hj

is the

measurement matrix;

()Vj

is a random measurement

noise.

Assume that

()wj

and

()Vj

are Gaussian white

noise random vectors with zero mean and

covariances:

( ) ( ) ( ) ( );

( ) ( ) ( ) ( ).

T

E w j w k Q j jk

E V j V k R j jk















(3)

where

()δ jk

is the Kronecker delta symbol. Note

that

{ }

()wj

and

{ }

()Vj

are assumed mutually

uncorrelated.

The state vector (1) can be estimated via the

optimal linear Kalman filter (LKF), [11]. Equations

for the estimation value and gain matrix of the LKF

respectively are:

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ˆˆ

( / ) ( / 1) ( ) ( )x j j x j j K j j   

(4)

1

( ) ( / 1) ( )

T

K j P j j H j P





(5)

where

ˆˆ

( / 1) ( 1/ 1) ( 1)x j j Ax j j Bu j     

is the

extrapolation value,

()j

and

()Pj



are the

innovation and innovation covariance respectively.

The expressions for the

()j

and

()Pj



are:

ˆ

( ) ( ) ( ) ( / 1)j z j H j x j j   

(6)

( ) ( ) ( / 1) ( ) ( )P j H j P j j H j R j

  

(7)

Here

( / 1)P j j 

is the covariance matrix of the

extrapolation error.

The residual of Kalman filter is defined as:

ˆ

( ) ( ) ( ) ( / )j z j H j x j j





(8)

The residual covariance is [6]

( ) ( ) ( ) ( / ) ( )P j R j H j P j j H j





(9)

The residual sequence (8) will be white

Gaussian noise with zero-mean and covariance (9) if

the system is functioning normally [3], i.e.

 

( ) ~ 0, ( )j N P j



. On the other hand, when there are

abnormal changes occurring in the system or

measurement channels, it can be assumed that

 

( ) ~ ( ), ( ) ,

f

fj N j P j





where either

( ) 0j





or

( ) ( )

f

P j P j





or both. Note that faults that only

result in

( ) 0j





are generally called additive or

bias type faults. They can be denoted as

( ) ( ) ( )

fj j f j





and satisfy

 

( ) ~ ( ), ( )

fj N j P j





, where

 

( ) ( )E f j j





. Those

faults that lead to changes in innovation covariance

()Pj



are called multiplicative or noise increment

type faults, which can be denoted as

( ) ( ) ( )

fj F j j





with

 

( ) ~ 0, ( ) ( ) ( )

T

fj N F j P j F j



.

3 The Influence of Sensor Faults on

Kalman Fılter Residual

The statistical properties of the Kalman filter

residual will alter as a result of measurement bias

and sensor noise increase type sensor faults. This

section examines the impact of these types of sensor

faults on the Kalman filter's residual sequence.

3.1 Influence of Sensor Biases on the

Kalman Filter Residual

Theorem 1: In the event that measurements are

processed using LKF (4)–(7) and a measurement

bias arises at an iteration step

j





, then at all

j





steps the residual bias will be equal to the difference

between the measurement bias and the estimated

observation bias.

Proof: At the first step following the bias occurring

at iteration

j





, the extrapolation value can be

expressed as

ˆˆ

( 1/ ) ( / ) ( ) ( )

ˆˆ

( / ) ( / ) ( ) ( )

ˆˆ

( 1/ ) ( 1/ )

bb

x j j Ax j j Bu j Gw j

Ax j j A x j j Bu j Gw j

x j j x j j

    

   

    

(10)

where

ˆˆ

( 1/ ) ( / )x j j A x j j   

is the extrapolation

value bias.

Residual of Kalman filter is:

ˆ

( 1) ( 1) ( 1) ( 1) ( 1/ 1)

ˆ

( 1) ( 1) ( 1/ 1) ( 1)

ˆ

( 1) ( 1/ 1) ( 1) ( 1)

b z b

z

j z j b j H j x j j

z j H j x j j b j

H j x j j j j





        

        

       

(11)

where

ˆ

( 1) ( 1) ( 1) ( 1/ 1)

z

j b j H j x j j



       

(12)

is the residual bias.

The residual bias is equal to the difference

between the measurement bias and estimated

observation bias, as may be observed from

expression (12), as shown. For all

j





steps, this

situation applies. As a result, Theorem 1 is proven.

Consequently, measurement bias type sensor faults

will cause a bias in the residual of the Kalman filter.

3.2 Influence of Measurement Noise

Increment to the Residual

Let the measurements are processed by the LKF (4)-

(7) and a measurement noise increment occurs at the

iteration step

j





. Measurement noise increment

can be simulated by multiplying the measurement

noise vector with the diagonal matrix

()Fj

, which

diagonal elements meet the following condition:

( ) 1

ii j





, (

1,in

) for

j





. Here n is the

dimension of the measurement vector. As it is clear,

for the noise increment type sensor fault in the

i

th

measurement channel, the appropriate diagonal

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element of

()Fj

will be larger than 1, i.e.

( ) 1

ii j





for

j





and rest of the measurement channels

become 1. Consequently, the diagonal elements of

()Fj

can be presented in the following form:

1: no measurement fault

>1: measurement fault

ii









The measurement model in this case can be written

in the form:

() ()() () ()z j H j x j F j V j

, (13)

where

 

11 22

1 1 . . . 1 , for j

( ) . . . if 1, ,

where 1 for j

nn

ii

diag F j i n



  









  









(14)

Theorem 2: In the event that measurements are

processed using LKF (4)–(7) and a measurement

noise increment occurs at an iteration

j





, then at

all

j





steps the measurement noise increment

leads to increment in the residual covariance (7).

Proof. The innovation covariance at the iteration

steps

j





can be expressed as

( ) ( ) ( ) ( ) ( ) ( / ) ( )

ni

TT

P j F j R j F j H j P j j H j





(15)

The residual covariance increment is:

( ) ( ) ( ) ( ) ( )

ni

T

P j F j R j F j R j



  

(16)

Since the matrices

()Fj

and

()Rj

are assumed to

be diagonal, the expression (16) can be rewritten in

the following form:

   

 

22

( ) ( ) ( ) ( ) ( ) ( )

ni

P j F j R j R j F j I R j



    

(17)

where

I

is the

nn

identity matrix. Because

()Rj

and

()Fj

are positive definite diagonal matrices and

()Fj

has diagonal elements

( ) 1

ii j





, (

1,is

) for



j





, then the matrix

 

 

2

( ) ( )F j I R j

is also

positive definite. Since the innovation covariance

increment is a positive definite matrix, the Theorem

2 is valid, therefore, the noise increment type sensor

fault leads to an increment of residual covariance

(7).

4 Resudual-Based Recursive

Measurement Noise Covariance

Estimator

The statistical properties of the Kalman filter residual

will change as a result of measurement bias and

measurement noise increment. For the compensation

of measurement bias or measurement noise

increment, the real and theoretical values of the

residual covariance matrices must be compared.

In the absence of measurement fault in the

estimation system, the real innovation covariance

()Cj

is equal to the theoretical one, [6]

( ) ( ) ( ) ( / ) ( )

T

C j R j H j P j j H j

(18)

The real covariance matrix of

()j



is an average of

( ) ( )

T

kk



within a moving window

M

:

1

( ) ( ) ( )

jT

k j M

C j k k

M



  



(19)

Substituting Eq. (19) into (18) we have

1

1( ) ( ) ( ) ( ) ( / ) ( )

jTT

k j M

k k R j H j P j j H j

M



  





(20)

The real residual covariance matrix

()Cj

can be

estimated by

( ) ( )

T

jj



at the current epoch in order

to avoid the smoothness of the average of

( ) ( )

T

jj



within

M

epochs, which does not adequately reflect

the uncertainty of the model errors at the current

step

( ) ( ) ( )

T

C j j j





(21)

Taking into account (18) and (21), the

expressions for the measurement noise covariances

for

1j

and

j

iterations can be written in the

following form:

( 1) ( 1) ( 1) ( 1) ( 1/ 1) ( 1)

TT

R j j j H j P j j H j



        

(22)

( ) ( ) ( ) ( ) ( / ) ( )

TT

R j j j H j P j j H j





(23)

Therefore

( 1)Rj

minus

()Rj

equals:

( 1) ( ) ( 1) ( 1)

( ) ( ) ( 1) ( 1/ 1) ( 1)

( ) ( / ) ( )

T

TT

T

R j R j j j

j j H j P j j H j

H j P j j H j



     

    



(24)

The equation (24) can be written as:

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( 1) ( ) ( 1) ( 1)

( ) ( ) ( 1) ( 1/ 1) ( 1)

( ) ( / ) ( )

T

TT

T

R j R j j j

j j H j P j j H j

H j P j j H j



     

    



(25)

If measurements are linear, than

( 1) ( )H j H j

and

the expression (25) can be written in simple form as:

 

( 1) ( ) ( 1) ( 1) ( ) ( )

( / ) ( 1/ 1)

TT

T

R j R j j j j j

H P j j P j j H

   

     

   

(26)

The resulting expression (26) makes it possible

to recursively estimate the measurement noise

covariance for the Kalman filter tuning. Below the

RKF with recursive estimation of measurement

noise covariance is applied for the UAV dynamics

model.

If a measurement bias occurs at the iteration

step

j





, and the biased residual sequence is

denoted by

()

bj



, then the biased residual is

defined as:

( ) ( )

bjj





j=1,2,...



-1 (27)

( 1) ( 1) ( 1)

bj j j

  

    

j=



+1,



+2,… (28)

When j<



, the mathematical expectation of the real

residual covariance matrix (28) can be determined

by the following equation:

 

( ) ( ) ( ) ( ) ( / ) ( )

T

E C j P k R j H j P j j H j



  

(29)

In the case of j





, in the real residual

covariance, a biased values

( 1) ( 1) ( 1)

bj j j

  

    

is used instead of an

unbiased value

( 1)j





, where

( 1)j





is the residual

bias

( ) ( ) ( )

T

b b b

C j j j





(30)

Remark. Note that the expected value of the residual

()

bj



in this case is not zero, therefore the formula

(30) is not a real covariance. This is the square of

the residual. Bias type measurement fault may be

converted to the square of residual and such types of

faults can be compensated using covariance

matching techniques.

Statement. For iteration steps

j





, measurement

bias leads to an increase in the mathematical

expectation of the square of residual.

Proof. It is proven in Theorem 1 that the

measurement bias will cause bias in the residual of

the Kalman filter.

The mathematical expectation of the square of

innovation (30) for

j





can be written as:

    

 

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

T

b b b

T T T T

E C j E j j E j j j j

E j j j j j j j j

     

       



   





   



(31)

Taking into account

 

( ) 0Ej





, and the absence

of correlation between the parameters

()j



and

()j



,

we have

   

( ) ( ) ( ) ( )

T

b

E C j E C j E j j









(32)

Expressions (11) and (32) prove the Statement

1. Consequently, the measurement bias will increase

the mathematical expectation of the square of

residual. It can be seen from the Theorem 1 and the

Statement above that the measurement bias is

transferred to the residual bias and changes the

mathematical expectation of the square of residual

(21). As a result, the measurement bias is transferred

to the mathematical expectation of the square of

residual. Thus, the square of residual can be used to

compensate of measurement bias.

Therefore, the measurement bias will increase

the mathematical expectation of the square of

residual (23). As a result, according to formulas (23)

and (26), the measurement noise covariance matrix

R

will increase, resulting in a smaller Kalman gain,

which will reduce the influence of measurements on

the state update process and increase the influence

of the mathematical model of the system. As a

result, the robustness of the filter against the

measurement bias fault is ensured and the

deterioration of the estimation procedure caused by

the measurement bias fault is prevented.

5 Results of Simulation

The proposed innovation-based adaptive KF

algorithm is applied to the UAV platform dynamics

model. As the experimental platform, the ZAGI

UAV was selected, and Kalman filter applications

were carried out while taking into consideration its

dynamics and characteristics, [12]. In order to

estimate the UAV state vector

 

() T

k

x k u w q h p r

  

         

the proposed residual-based RKF with R-adaptation

and conventional LKF (4)-(7) is used. Here,

,u

,v

w

are the velocities along

,x

y

and

z

directions in

the body frame,

,p

,q

r

are the angular rates

around

,x

y

and

z

axes,





is the pitch angle,





is

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the roll angle,





is the sideslip angle,

h

is the

height.

Simulations are carried out in 1000 steps over a

time frame of 100 seconds with a sampling time of

0.1 seconds. Two different types of measurement

fault scenarios—constant bias in measurements and

measurement noise increment—are taken into

consideration during simulations to test the

proposed residual-based RKF with recursive

estimate of measurement noise covariance.

5.1 Constant Bias in Measurements

A constant bias term is added to the measurements

of the pitch angle gyro after the 30th second of

the simulation

( ) ( ) ( ) 0.5z j z j v j

  

  

,

( 300)j

(33)

The residual-based RKF with recursive R-

adaptation simulation results for the pitch angle in

the presence of pitch angle gyro bias are presented

in Figure 1. The findings of the RKF's state

estimation are compared to the actual values in the

first section of the figure. The estimation error based

on the actual values of the UAV states is depicted in

the second portion of the picture. The estimation

error variance is shown in the final section.

Figure 1 shows that the proposed residual-based

RKF with recursive estimation of measurement

noise covariance achieves estimation of the states

accurately in the presence of bias at the pitch angle

gyro. In this case, RKF gives sufficiently good

estimation results by totally eliminating the

estimation error caused by the bias in the pitch angle

gyro.

Fig. 1: Pitch angle estimation results using residual-

based RKF with recursive R-adaptation in the

presence of bias at the pitch angle gyro

Figure 2 displays the results of the conventional

KF estimation for the pitch angle in the presence of

pitch gyro bias. As can be seen, the conventional KF

estimates shift after the 30th second of simulation

(after the pitch angle gyroscope fails), and the

estimation results are erroneous.

Fig. 2: Pitch angle estimation results using

conventional KF in the presence of bias at the pitch

angle gyro

5.2 Measurement Noise Increment

In the second measurement malfunction scenario,

the measurement fault is defined as the pitch angle

gyro measurement noise's standard deviation

multiplied by a constant term after the 30th second:

( ) ( ) ( ) 3z j z j v j

  

  

, (

300)j

. (34)

The proposed residual-based RKF with

recursive R-adaptation estimation results for the

pitch angle in the presence of measurement noise

increment at the pitch angle gyro are presented in

Figure 3. As seen from the graphs presented in

Figure 3, the proposed residual-based RKF with

recursive estimation of measurement noise

covariance gives sufficiently good estimation results

in the presence of measurement noise increment at

the pitch angle gyro.

Figure 4 displays the results of the conventional

KF estimation for the pitch angle and roll rate in the

presence of pitch gyro bias. As seen, the accuracy of

conventional KF estimates deteriorates after the

30th second of simulation (after the pitch angle

gyroscope fails).

WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.45

Chingiz Hajiyev, Ulviye Hacizade

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Fig. 3: Pitch angle estimation results using residual-

based RKF with recursive R-adaptation in the

presence of measurement noise increment at the

pitch angle gyro

Fig. 4: Pitch angle estimation results using

conventional KF in the presence of measurement

noise increment at the pitch angle gyro

5.3 RMS Errors of the Innovation-based

RKF

RMS errors of the residual-based RKF with

recursive estimation of measurement noise

covariance and conventional KF estimates in the

presence of pitch angle gyro bias are presented in

Table 1. As can be seen from the results presented

in Table 1, the proposed RKF is superior for both

longitudinal and lateral parameters in the presence

of pitch angle gyro bias. RMS errors of

conventional KF are sufficiently greater than the

RMS errors of the proposed robust filter.

RMS errors of the proposed residual-based RKF and

conventional KF estimates in the presence of

measurement noise increment at the pitch angle

gyro are presented in Table 2.

Table 1. RMS errors of the proposed RKF and

conventional KF estimates in the presence of pitch

angle gyro bias

Method

RKF

Conv.KF

u

0.1134

0.5309

w

0.0300

0.1738

q

0.0156

0.0651





0.0208

0.1357

h

0.8210

0.3068





0.0164

0.0286

p

0.0124

0.0386

r

0.0124

0.0285





0.0434

0.0475

Table 2. RMS errors of the proposed RKF and

conventional KF estimates in the presence of

measurement noise increment at the pitch angle

gyro

Method

RKF

Conv.KF

u

0.1505

0.1782

w

0.0285

0.0847

q

0.0222

0.0648





0.0158

0.0749

h

1.4480

0.1523





0.0148

0.0289

p

0.0091

0.0413

r

0.0135

0.0284





0.0277

0.0497

The presented in Table 2 results show that the

residual-based RKF with recursive R-adaptation

gives better results for both longitudinal and lateral

parameters in the presence of measurement noise

increment at the pitch angle gyro. The RMSE results

of conventional KF are worse compared to the

robust filter.

In all investigated sensor fault sceneries, the

proposed residual-based RKF gives better

estimation results than the conventional KF.

6 Conclusion

This study proposes a novel recursive method for

estimating measurement noise covariance for

Kalman filter tuning. Based on the covariance

difference approach to recursively estimate the

measurement noise covariance, a residual-based

robust Kalman filter against sensor faults is

presented. The sensor fault compensation in this

filter is accomplished with a simple change to the

conventional KF.

The proposed residual-based RKF with

recursive estimation of measurement noise

covariance is applied to the UAV dynamics model.

Two alternative scenarios of measurement error are

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Volume 23, 2024

evaluated on algorithms; constant bias at

measurements (additive sensor faults) and

measurement noise increments (multiplicative

sensor faults). The simulation results show that the

proposed residual-based RKF with recursive R-

adaptation can accurately estimate the UAV

dynamics in real-time in the presence of various

types of sensor faults.

Estimation accuracies of the proposed residual-

based RKF and conventional KF are compared. In

all investigated sensor fault sceneries, the results of

the proposed RKF are superior. The conventional

KF gives the worst estimation results in the presence

of sensor faults.

The residual-based RKF with recursive

estimation of measurement noise covariance can be

recommended as the reliable UAV state estimator in

the flight control system in the presence of sensor

faults.

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ISSN: 2572-4479.

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Volume 23, 2024

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Chingiz Hajiyev carried out the conceptualization,

methodology, formal analysis, writing, review &

editing.

-Ulviye Hacizade has organized and executed the

investigation, validation, simulation, data

curation, formal analysis.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on SYSTEMS

DOI: 10.37394/23202.2024.23.45

Chingiz Hajiyev, Ulviye Hacizade

E-ISSN: 2224-2678

443

Volume 23, 2024