Algorithms for Adaptive Estimation of Dynamic Objects Under the

Influence of Additive Noise

Oripjon Zaripov

, Jasur Sevinov

, Fuzayl Odilov

, Furkat Odilov

and Shaxlo Zaripova

Tashkent State Technical University, 100057, Tashkent, Uzbekistan

Karshi Institute of Engineering and Economics, 180100, Karshi, Uzbekistan

Keywords: Adaptive Filtering, Kalman Filter, Nonlinear Systems.

Abstract: Algorithms for adaptive evaluation of dynamic object control systems under the influence of additive noise

are considered. Algorithms of the Kalman filter are given. An adaptive filtering approach for nonlinear

systems with additive noise is also considered. The developed adaptive estimation algorithm can calculate the

square root of the covariance matrix in a simple way in such a way that positive semi-certainty is guaranteed,

which significantly increases the stability and accuracy of the filter.

1 INTRODUCTION

The Kalman filter is widely used in numerous tasks

of synthesizing and designing systems for managing

dynamic objects of various functional purposes

(Krasovsky, 1987; Zaripov & Khamrakulov, 2021).

The Kalman filter provides an unbiased estimate with

minimal variance about the state of a discrete linearly

varying dynamic system, the input and output of

which are distorted by Gaussian white noise with an

additive character. This approach was extended to

continuous dynamical systems by Kalman and Bucy

with a linear character (Leondes, 1980; Sinitsyn et al.,

2006).

The Kalman filter has one major drawback

(Ogarkov, 1990; Sinitsyn et al., 2006; Olimovich et

al., 2024). The equations used in the optimal filter

require precise knowledge of the dynamic equations

of the system and the statistics of random variables,

including the need to know the transition matrices of

the system and the covariance of disturbances such as

additive white noise. However, usually only their

estimates are available. Recently, Kalman filter

schemes (Krasovsky, 1987; Ogarkov, 1990) have

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appeared in order to circumvent this problem. These

schemes are commonly referred to as "adaptive

filters". Various adaptive filters can be grouped

according to the principle of identifying undefined

parameters, heuristic weighting coefficients, or the

absence of correlation of residual terms.

Theoretically and practically, the synthesis of

control systems for dynamic objects very often

addresses estimation issues in which measurement

uncertainty is represented as an additive purely

random sequence or white noise. At the same time,

there are estimation problems (Zaripov et al., 2021;

Peltsverger, 2004; Kolos, 2000; Bryson et al., 1972)

when an additive Markov sequence, i.e. sequentially

correlated or non-white noise, is a more accurate

model of uncertainty in measurements.

When using systems used to control dynamic

objects, the structural and parametric data of the

controller do not have a dependence associated with

the structure and parameters of the observer. This, in

turn, makes it possible to use well-known control

laws, and in the future, the possibility of adapting

them using the evaluation contour. It should be borne

in mind that the use of the given structure will be

convenient in the event that there is a need to

Zaripov, O., Sevinov, J., Odilov, F., Odilov, F. and Zaripova, S.

Algorithms for Adaptive Estimation of Dynamic Objects Under the Inﬂuence of Additive Noise.

DOI: 10.5220/0014268300004738

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 4th International Conference on Research of Agricultural and Food Technologies (I-CRAFT 2024), pages 321-324

ISBN: 978-989-758-773-3; ISSN: 3051-7710

321

modernize or adapt the existing management system.

When studying the evaluation algorithm with

relatively high performance, the qualitative indicators

of the adaptive system are not much inferior to the

non-adaptive system in terms of minimum indicators.

But even at the same time, during the period of

transients carried out in the observer, the qualitative

indicators of a closed system may have deteriorations

reaching the loss of asymptotic stability. Such

disadvantages can be eliminated by increasing the

speed of estimation algorithms in areas with large

deviations.

In general, considering the formulation of the

problem, multiple observations may contain certain

white noises, while non-white Markov-type noises

may not contain noise as a whole or be considered as

some combination of the three studied possibilities.

Following from this, in further descriptions, under the

terminology "non-white noise" we will understand

the presence of Markov-type noise or the non-

participation of noise as a whole in one or more

dimensions.

2 MATERIALS AND METHODS

Consider a linear dynamical system described by the

equation

iiiiiii

wГxAx

|1|11 +++

, (1)

1111 ++++

iiii

vxHz , (2)

for

,...2,1,0=i

with an initial condition

x and a

measurement matrix

1+i

H .

The measurement error

1+i

v is identified with the

state vector of some additional linear dynamic system

(forming filter) with a transition matrix

Ψ and a

perturbation vector

iiii

+Ψ=

for

,...2,1=i

with an initial condition

v .

It is assumed that perturbations {

,...2,1,0, =iw

}

and {

,...2,1,0, =i

} – are sequences of random

vectors with known correlation matrices

iii

REQwwE == ][,][

ξξ

, where E – is the

averaging operator and "т" is the transposition

operation. These two successive equations do not

depend on each other and also do not depend on the

initial conditions

x ,

v .

With mutually uncorrelated errors {

,...2,1,0, =iv

}

kii

VvvE

=][

, where

–

is the symbol called Kronecker, is optimal in the form

of a minimum of the variance estimate

of the

state vector

x of the system (1), based on

measurements {

ikz

,...,2,1, =

} of the form (2), it

is formed according to the recurrent Kalman

algorithm [1-11]:

)

(

ˆˆ

1|1|| −−

−+=

iiiiiiiii

xHzKxx

(3)

1|11|1|

ˆˆ

−−−−

iiiiii

xAx

, (4)

1|1|

)(

−

−−

iiii

VHPHHPK

, (5)

iiiii

iiiiiiii

ГQГAPAP

1|11|1|1|11|1| −−−−−−−−

, (6)

1||

)(

−

−=

iiiiii

PHKIP

(7)

for

,...2,1=i

, where I – is a unit matrix, and the

initial conditions for equations (4) and (6) are set,

respectively, by an a priori estimate

0|0

of the initial

state vector

x and the correlation matrix

0|0

of its

error

00|00|0

xxx −=

, uncorrelated with {

,...2,1,, =iw

}. In this case,

is a correlation

matrix of the error of the optimal estimate

of the

current state

x , calculated using formulas (5) – (7)

without using measurements

z .

To solve the estimation problem under the

influence of additive noise, the following algorithms

can be proposed (Sinitsyn et al., 2006, Zhou et al.,

2013, Zhou et al., 2015). It is assumed that the

statistical property of system noise is known.

However, in real time, the process noise covariance

matrix Q or the observation noise variance matrix R

are often unknown. In addition, these parameters may

change over time. Therefore, an adaptive Kalman

filter must be designed to adjust Q and R, where it is

important to increase the accuracy and stability of

filtration.

Thus, we assume an adaptive filtering algorithm

based on the maximum a posteriori estimate (Zhou et

al., 2013). Subsequently, the algorithm has the ability

to evaluate unknown time-varying noise. The specific

calculation process is as follows:

[],

Ex=

00000

ˆˆ

[( )( ) ],

PExxxx=− −

(0), (0)QQ RR==

during initialization

0i =

I-CRAFT 2024 - 4th International Conference on Research of Agricultural and Food Technologies

322

|1 |1 1|1 |1 |1 1|1 |1 1

ˆˆ

ii ii i i ii ii i i ii i

Ax P AP A Q

− − −− − − −− − −

==−

during iteration

1, 2,...i =

, that is, the time update.

Let's evaluate the statistical properties of

measurement noise when updating measurement

results:

1|1

ˆˆ

(1 ) ( ).

ii iii

iiiiiiiiii

zzHx

RdRdzzHPH

−

−−

=−

=− + −





Let's estimate the value of the state, after correction

we will calculate the a posteriori variance of the state:

|1 |1

||1

(),

ˆˆ

().

iiiiiiii i

ii ii i i

iiiii

KPHHPH R

xx Kz

PIKHP

−

−−

−

=−



Then we will evaluate the statistical properties of the

process noise in accordance with

1||11|1|1

ˆˆ

(1 ) ( ),

TT T

i i i i iii i ii ii i i ii

QdQdKzzKPAPA

−−−−−

=− + + −



where

(1 ) / (1 )

dbb

=− −

it is a factor of

forgetting, and

01b<<

Now let's consider the adaptive filtering approach

for nonlinear systems with additive noise. Both the

process equations and the measurement equations are

nonlinear according to

() ,

iii

fx w

zhx v

−−

where

()



and

()h 

they are non-linear functions.

3 RESULTS AND DISCUSSION

The adaptive assessment algorithm is as follows:

[],

Ex=

{

}

00000

ˆˆ

[( )( ) ] ,

ScholExxxx=−−

,QS=

{

}

00000

ˆˆ

[( )( ) ] .

R chol E z z x z=−−

111 11 1

ˆˆ ˆ

[]

iii ii i

xSxS

χγγ

−−− −− −

=+ −, при

1, 2,...,i =∞

Let's write the time update equations as follows:

(),

χχ

−

() *

|1 ,

ii k i k

−



{

}

() *

|1 1 ,1:2 |1 1

[( )],

ii i L ii i

SqrW xQ

−−−

=−

{

}

*()

|1 |1 ,0 |1 0

[, , ].

ii ii i ii

S cholupdate S x W

−−−

=−

Then we calculate the square root of the measured

noise matrix:

(),

−−

*()*

11,

ikik

zWZ

−−



111

iii

zzz

−−−

=−



{

}

** *

1,||,,

ii i i

cholupdate d R z d

−−

=−



(8)

{

}

*****()

1|0: 2 1

iLi ik

R cholupdate R Z z d W

−−

=−−

{

}

diag R=

where

|1 |1 |1 |1 |1 |1

ˆˆ ˆ

[],

ii ii ii ii ii ii

xx Sx S

χγγ

−−− −− −

=+ −

|1 |1

(),

ii ii

−−

()

|1 |1,

ii k ii k

zWZ

−−



11|1

iiii

zzz

−− −

=−



The measurement update equations are as follows:

()

( ) |1, |1 |1, |1

()(),

xz i k ii k ii ii k ii

PW xZz

−−−−

=−−



{

}

()

() 1 | 1,1:2 | 1

ˆˆ

(),

zi ii L ii i

SqrWZ zR

−−





=−









{

}

()

() () |1,0 |1 0

[, , ]

zi zi ii ii

S cholupdate S Z z W

−−

=−

()

() ()

()

xz i z i

ˆˆ

iii ii

xKz

−



()

izi

UKS=

() | 1

{,,1}.

zi ii

S cholupdate S U

−

=−

Let's estimate the square root of the process noise

matrix in accordance with

{

}

1|1

ˆˆ

,| |,

iiiii

Q cholupdate Q x x d

−−

=−

{

}

***

Q cholupdate Q U d=−

Algorithms for Adaptive Estimation of Dynamic Objects Under the Inﬂuence of Additive Noise

323

(

)

{

}

Q diag diag Q=

where

(1 ) / (1 )

dbb

=− −

and

it is a factor

of forgetting, as a rule

01b<<

The weights (

()m

and

()c

) of the mean value

and covariance are given by the formula

()

() 2

αβ

=+−+

() ()

,1,...,2,

2( )

WW k L

== =

where

()L

λα κ

it is a scaling parameter.

The constant

defines the spread of sigma points

around the average value, which is usually set to a

small positive value (for example,

10 1

−

≤≤

Constant

≥

is a secondary scaling parameter.

≥

is used to account for prior knowledge of the

distribution (for Gaussian distributions, the optimal

value is

) [4-10]. In addition,

Here we used three powerful linear algebra methods:

QR decomposition (

{}qr

), updating the Xoleski

coefficient (

{}cholupdate

) and efficient least

squares methods (/), which are briefly discussed in

(Sage et al., 1969, Julier et al., 1995).

4 CONCLUSIONS

Algorithms for adaptive evaluation of dynamic object

control systems under the influence of additive noise

are considered. Algorithms of the Kalman filter are

given. An adaptive filtering approach for nonlinear

systems with additive noise is also considered. The

developed adaptive estimation algorithm can

calculate the square root of the covariance matrix in a

simple way in such a way that positive semi-certainty

is guaranteed, which significantly increases the

stability and accuracy of the filter.

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