Abstract: - This paper is concerned with the design of switching observers for nonlinear positive systems,

where the basic parameters of the Metzler-Takagi-Sugeno fuzzy models are intervally dened. With

consideration of the measurable set of premise variables the associated structure of the switching fuzzy

interval observers is proposed for system state estimation to maintain stability and H∞performance level

under the inuence of norm-bounded additive disturbance. The design conditions take into account the

lower and upper bounds of nonnegative system state. A numerical example is included to demonstrate

the eectiveness of the developed theory for the condisered class of systems.

Key-Words: - Metzler matrices, Takagi-Sugeno models, switched systems, interval observers, state

estimation, linear matrix inequalities.

1 Introduction

Many complex systems are usually nonlinear and

the Takagi-Sugeno (T-S) fuzzy models, [1], ad-

dress many related practical purposes preferring

the sector nonlinearity concept and the system

state-space approach. Typically, the mathemat-

ical formalizations presented in [2], reflect

many practical design problems, where synthesis

is con-stituted on linear matrix inequalities

(LMI), [3].

Focusing on the systems with positive states,

[4], [5], their mathematical models should be able

to reflect strict constraints implying from the Met-

zler matrix theory, [6]. Their main reflections

mean that the design is conditioned by additional

constraints, [7], to tolerate the system positive-

ness. A strictly LMI-based approach for design

under Metzler constraints is given in [8], reflecting

the diagonal stabilization principle (DSP). Devel-

oping methods to provide system state estimation

of positive systems if only system matrix bounds

are known, [9], some approaches for interval ob-

server (IO) analysis are presented in [10], [11],

interpreting the above design problems for T-S

fuzzy IOs, [12]. The main limitations are mea-

surable premise variables when implementing the

such kind of observers.

The estimation performances of positive ob-

servers for switched fuzzy positive systems carry

out the equivalent difficulties, [13], and so

fuzzy approaches have to be formulated over the

system positivity and the premise variables

availability, [14]. The problem of positive

properties of the re-

sponse of IOs for switched T-S fuzzy positive sys-

tems remains relevant, where a new design goal is

the satisfactory IO dynamics, [15], [16]. It should

be pointed out that the measurable premise vari-

ables remain important issues.

The compatibility of an LMIs set in design

of positive Metzler-Takagi-Sugeno (M-T-S) fuzzy

switching IOs is solved in the paper, forming a new

algorithmic platform with relationships to switch-

ing IO stability, positivity, upper and lower state

bounds, H∞disturbance attenuation and the Met-

zler parametric constraints. Despite the specic

limitations from the concept of DSP and the struc-

tural constraints that are imposed on the observer

dynamics, the design is formulated as a parameter

feasibility problem. The resulting design condi-

tions are given in terms of LMIs, which by existing

programming tools can be easily solved.

The paper is structured in the following way.

In Section 2 the main characterization of M-T-

S positive fuzzy switching models are presented.

The properties of positive M-T-S fuzzy switching

IOs are analyzed in Section 3 and the observer

parameter design conditions for this class of ob-

servers are derived Section 4. The approach is

illustrated by application to a M-T-S model with

interval Metzler system matrices in Section 5 and

the conclusions with relation properties are pre-

sented in Section 6.

Throughout the paper xT,XTdenotes the

transpose of the vector x, and the matrix X, re-

spectively, diag [·]marks a (block) diagonal ma-

trix, for a square symmetric matrix X≺0means

Interval Observers for Switched Metzler-Takagi-Sugeno Fuzzy

Systems

DUŠAN KROKAVEC, ANNA FILASOVÁ

Department of Cybernetics and Artificial Intelligence

Faculty of Electrical Engineering and Informatics

Technical University of Košice

Letná 9, 042 00 Košice

SLOVAKIA

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WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.10

Dušan Krokavec, Anna Filasová

E-ISSN: 2224-2856

Volume 18, 2023

that Xis negative denite matrix, Inlabels the

n-th order unit matrix, R(R+) marks the set of

(nonnegative) real numbers, Rn×n(Rn×n

+) refers

to the set of (nonnegative) real matrices and Rn×n

−+

indicates the set of matrices with the strictly Met-

zler structure.

2 M-T-S Fuzzy Switching Models

This paper considers a class of switching dynami-

cal structures with M-T-S fuzzy models

q(t) =

i=1

hσ

i(ϑ(t))Aσ

iq(t) + Bσu+Dd(t)(1)

y(t) = Cq(t)(2)

where q(t)∈Rn

+,y(t)∈Rm

+are nonnegative,

d(t)∈Rd

+is positive and bounded, C∈Rm×n

D∈Rn×d

+are nonnegative and Aσ

i∈Mn×n

−+

are strictly Metzler and Hurwitz. Index σ∈Σ

marks an active switching mode from the list

Σ = {1, . . . ns}while nsis an integer, hσ

i(θ(t)) is a

normalized membership function satisfying for all

items i∈ {1, . . . , s},σ∈ {1, . . . , ns}that

0≤hσ

i(ϑ(t)) ≤1,

i=1

hσ

i(ϑ(t)) = 1 (3)

to contain the number of fuzzy rules sand to

overlay the o-dimensional premise variables vec-

tor ϑ(t)=[θ1(t)θ2(t)· · · θo(t)]. It is supposed

that the premise variables are measurable.

A strictly Metzler matrix Aσ

iis character-

ized by its negative diagonal elements and by its

strictly positive o diagonal elements. Conse-

quently, a strictly Metzler Aσ

iis so limited by n2

parametric constraints

aσ

lh <0, l =h, aσ

lh >0, l 6=h, ∀l, h ∈ h1, ni(4)

To guaranty () in design task the DSPs have to

be used, [17].

Denition 1 [8] If a strictly Metzler A∈Mn×n

−+is

represented in the following rhombic form, where

the diagonal exactness are constructed by the col-

umn index dened multiple circular shifts of ele-

ments of the columns of Aas follows

AΘ=







a11

a21 a22

a31 a32 a33

....

an1an2an3· · · ann

a12 a13 · · · a1n

a23 · · · a2n

....

an−1,n







,(5)

then the following diagonal matrix structures

A(ν+h, ν)

=diag [a1+h,1

· · ·an,n−ha1,n−h+1

· · ·ah,n]0(6)

A(ν, ν) = diag [a11 a22 · · · ann]≺0(7)

represent Metzler parametric constraints (2).

Remark 1 The DSP leads to parameterizations of

a Metzler matrix Aas, [18],

n−1

h=0

A(ν+h, ν)LhT,L=0T1

In−10(8)

where L∈Rn×nis the circulant permutation ma-

trix, [19]. Applying (8) for any Ae=A−JC ∈

Rn×n

−+address the following parametrization (see,

for example, [18])

Ae=

n−1

h=0 Ai(ν+h, ν)−

k=0

JkhCdkLhT(9)

where, with relation to (7), (6), the representing

diagonal matrices Jkh,Cdk ∈Rn×n

+are dened as

follows:

CT= [c1· · · cm],Cdk =diag cT

k(10)

J= [j1

· · ·jm],Jk=diag [jik](11)

where Jkh =LhTJkLh. Note the above men-

tioned matrix Ae=A−JC has to be strictly

Metzler and Hurwitz.

3 M-T-S Switching Fuzzy IO

In this case there are considered that q(0) as

well as Aσ

iare unknown but bounded by known

constant bounding vectors and known constant

bounding matrices of appropriate dimensions in

such a way that for all i∈ h1, si,σ∈ {1, . . . , ns}

(these inequalities are understood elementwise)

0≤q(0) ≤q(0) ≤q(0) ,Aσ

i≤Aσ

i(12)

0≤ϑ(t)≤ϑ(t)≤ϑ(t),0≤qe(t)≤q(t)≤qe(t)

(13)

Due to by interval dened system parameters

and measurable premise variables, it can be used

the IO structure for an M-T-S switched strictly

positive system

qe(t) =

i=1

hσ

i(ϑ(t))(Aσ

i−Jσ

iC)qe(t)

i=1

hσ

i(ϑ(t))Jσ

iCq(t) + Bσu(t)

(14)

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DOI: 10.37394/23203.2023.18.10

Dušan Krokavec, Anna Filasová

E-ISSN: 2224-2856

Volume 18, 2023

qe(t) =

i=1

hσ

i(ϑ(t))Aσ

i−Jσ

iCqe(t)

i=1

hσ

i(ϑ(t))Jσ

iCq(t) + Bσu(t)

(15)

where (2) yields together with

ye(t) = Cqe(t),ye(t) = Cqe(t)(16)

and for t≥0if qe(0) = q(0),qe(0) = q(0) it is

Aσ

ei =Aσ

i−Jσ

iC,Aσ

ei =Aσ

i−Jσ

iC(17)

Using the observation errors

e(t) = q(t)−qe(t),e(t) = q(t)−qe(t)(18)

it follows from (1), (14), (15), (18) that

e(t) = Dd(t)+

i=1

hσ

i(ϑ(t))Aσ

eie(t)−Λσ(ϑ(t))q(t)(19)

e(t) = Dd(t)+

i=1

hσ

i(ϑ(t))Aσ

eie(t) + Λσ(ϑ(t))q(t)(20)

where

Λσ(ϑ(t)) =

i=1

hσ

i(ϑ(t))(Aσ

ei −Aσ

ei)

i=1

hσ

i(ϑ(t))(Aσ

ei −Aσ

ei)

(21)

Λσ(ϑ(t)) =

i=1

hσ

i(ϑ(t))(Aσ

ei −Aσ

ei)

i=1

hσ

i(ϑ(t))(Aσ

ei −Aσ

ei)

(22)

owing to that (12) implies

Aσ

i−Jσ

iC≤Aσ

i−Jσ

iC≤Aσ

i−Jσ

iC(23)

Remark 2 Since Aσ

ei,Aσ

ei must be strictly Metzler

and Hurwitz and if q(t)is at its upper limit q(t),

then

e(t) =

i=1

hσ

i(ϑ(t))Aσ

eie(t) + Dd(t)(24)

e(t) =

i=1

hσ

i(ϑ(t))Aσ

eie(t) + Dd(t)+

i=1

hσ

i(ϑ(t))Aσ

i−Aσ

iq(t)

(25)

Otherwise, if the current state of the system

q(t)is at its lower limit q(t), then the IO error

dynamics e(t)is approximated by the equation

e(t) =

i=1

hσ

i(ϑ(t))Aσ

eie(t) + Dd(t)−

−

i=1

hσ

i(ϑ(t))Aσ

i−Aσ

iq(t)

(26)

while the dynamics of e(t)at the lower q(t)is

given as

e(t) =

i=1

hσ

i(ϑ(t))Aσ

eie(t) + Dd(t)(27)

The second elements of (24)-(27) express the ef-

fect of the disturbances (generalized disturbances)

on the dynamics of the estimation error. It can be

seen that all these components are non-negative

and bounded, so they cannot cause IO instability.

Assumption 1 Denoting the lower limit of q(t)as

q(t)and performing (27) as

q(t)−˙

qe(t)

i=1

hσ

i(ϑ(t))Aσ

ei(q(t)−qe(t)) + Dd(t)(28)

it follows from (28)

qe(t) = ˙

q(t)−

i=1

hσ

i(ϑ(t))Aσ

eiq(t)+

i=1

hσ

i(ϑ(t))Aσ

eiqe(t)−Dd(t)

(29)

Rewriting (1) for this limit case in the form

q(t)

i=1

hσ

i(ϑ(t))(Aσ

iq(t) + Bσu(t) + Dd(t)) (30)

then, substituting (30) in (29) it yields

qe(t)

i=1

hσ

i(ϑ(t))(Aσ

i−Aσ

i+JσC))q(t)+

i=1

hσ

i(ϑ(t))Aσ

eiqe(t) + Bσu(t)

(31)

Thus, for a positive M-T-S model, where C∈

Rm×n

+is nonnegative and q(t)∈Rn

+is positive,

the lower system state estimate by M-T-S fuzzy

positive IO is nonnegative if Jσ∈Rn×m

+is non-

negative, Aσ

i≤Aσ

iand Aσ

ei are Metzler and Hur-

witz.

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DOI: 10.37394/23203.2023.18.10

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Consequence 1 Adaptation of Remark 1 and (17)

entail the following matrix parameterizations

Aσ

i(ν, ν) = diag [aσ

i11 aσ

i22 · · · aσ

inn]≺0(32)

Aσ

i(ν, ν) = diag [aσ

i11 aσ

i22 · · · aσ

inn]≺0(33)

Aσ

i(ν+h, ν)

=diag aσ

i,1+h,1

· · ·aσ

i,n,n−haσ

i,1,n−h+1

· · ·aσ

ihn

0

(34)

Aσ

i(ν+h, ν)

=diag aσ

i,1+h,1

· · ·aσ

i,n,n−haσ

i,1,n−h+1

· · ·aσ

ihn

0

(35)

CT= [c1· · · cm],Cdk =diag cT

k(36)

Jσ

i= [jσ

i1· · · jσ

im],Jσ

ik =diag [jσ

ik](37)

Jσ

ikh =LhTJσ

ikLh(38)

Aei =

n−1

h=0 Aσ

i(ν+h, ν)−

k=0

Jσ

khCdkLhT(39)

Aei =

n−1

h=0 Aσ

i(ν+h, ν)−

k=0

Jσ

khCdkLhT(40)

To perform an M-T-S fuzzy switching positive

IO, the observer synthesis must be able to oer

strictly positive IO parameters as it is given by

the following armation.

4 Design of M-T-S Switched IO

A statement of positive M-T-S fuzzy switching IOs

is provided by the following theorem.

Theorem 1 Suppose that Aσ

i,Aσ

i∈Rn×n

−+are

strictly Metzler and C∈Rm×n

+is non-negative.

If there exist positive denite diagonal matrices

P,Vσ

ik ∈Rn×n

+and positive scalar η∈R+

such that for i= 1,2, . . . , s,h= 1,2, . . . , n −1,

σ∈ {1, . . . , ns}





Ωσ

i∗ ∗

DTP−ηId∗

Cσ0−ηIm

≺0,P0(41)





Ωσ

i∗ ∗

DTP−ηId∗

Cσ0−ηIm

≺0,Vik 0(42)

P Aσ

i(ν, ν)−

k=1

Vσ

ikCdk ≺0(43)

P Aσ

i(ν, ν)−

k=1

Vσ

ikCdk ≺0(44)

P LhAσ

i(ν+h, ν)LhT−

k=1

Vσ

ikLhCdkLhT0

(45)

P LhAσ

i(ν+h, p)LhT−

k=1

Vσ

ikLhCdkLhT0

(46)

where

Ωσ

i=P Aσ

i+AσT

iP−

−

k=1

Vσ

ikllTCdk −

k=1

CdkllTVσ

(47)

Ωσ

i=P Aσ

i+AσT

iP−

−

k=1

Vσ

ikllTCdk −

k=1

CdkllTVσ

(48)

and, if the task is feasible, the positive gains for

all i∈ {1, . . . , s},σ∈ {1, . . . , ns}are given as

Jσ

ik =P−1Vσ

ik,jσ

ik =Jσ

ikl(49)

Jσ

i= [jσ

i1· · · jσ

im],lT= [1· · · 1](50)

and Aσ

ei,Aσ

ei ∈Rn×n

−+are strictly Metzler and Hur-

witz.

Hereafter, ∗denotes the symmetric item in a

symmetric matrix.

Proof: Choosing Lyapunov function in the fol-

lowing form

v(e(t)) = eT(t)P e(t)−ηZt

dT(τ)d(τ)dτ+

+η−1Zt

y(τ)ey(τ)dτ

(51)

where P∈Rn×m

+is a diagonal positive denite

matrix (PDDM) and η∈R+is a positive scalar,

then the time derivative of (51) along all the ob-

server error trajectories is computed as follows

˙v(e(t)) = ˙

eT(t)P e(t) + eT(t)P˙

e(t)+

+η−1eT

y(t)ey(t)−ηdT(t)d(t)

(52)

When substituting the lower limit of e(t)given

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in (24), it can be xed that

˙v(e(t))

i=1

hσ

i(ϑ(t))eT(t)(AσT

ei P+P Aσ

ei)e(t)+

i=1

hσ

i(ϑ(t))(eT(t)P Dd(t)+

i=1

hσ

i(ϑ(t))eT(t)η−1CTCe(t)−

−ηdT(t)d(t)

(53)

Denoting as following

d(t) = eT(t)dT(t)(54)

then in terms of (54) one can easily have

˙v(ed(t)) =

i=1

hσ

i(ϑ(t))eT

d(t)Ωσ

ied(t)<0(55)

where, by the nomenclature,

Ωσ

i=AσT

ei P+P Aσ

ei +η−1CTC∗

DTP−ηId(56)

is negative denite.

Reecting (17) for all i∈ {1, . . . , s}and σ∈

{1, . . . , ns}, then

P(Aσ

i−Jσ

iC)+(Aσ

i−Jσ

iCTP

=PAσ

i−

k=1

jσ

ikcσT

k+

+Aσ

i−

k=1

jσ

ikcσT

kTP

=PAσ

i−

k=1

Jσ

ikllTCdk+

+Aσ

i−

k=1

Jσ

ikllTCdkTP

(57)

and setting

Vσ

i=P Jσ

i(58)

then (57) denes (47) and (41) can be constructed

from (56) by the Schur complement property.

According to the parametrization of Aσ

ei, (32),

(39) it has to yield

Aσ

i(ν, ν)−

k=0

Jσ

ikCdk ≺0(59)

Aσ

i(ν+h, ν)LhT−

k=0

Jσ

ikhCσ

dkLhT0(60)

Multiplying by PDDM Pthe left side of (59)

in turn this implies that

P Aσ

i(ν, ν)−

k=0

P Jσ

ikCσ

dk ≺0(61)

and using (58) then (61) implies (43).

Multiplying by P Lhthe left side of (60) it

yields

P LhAσ

i(ν+h, ν)LhT−

k=0

P Jσ

ikLhCσ

dkLhT0

(62)

and with (58) then (62) implies (45), since

LhLhT=In.

Since analogously can be set the LMIs working

on Aσ

ei, this closes the proof.

Note, in the given sense (43)-(46) enforce Met-

zler parametric constraints in the observer gains

design problem.

5 Illustrative Example

The considered M-T-S system (1), (2) is built on

the parameters

1="−0.2580 2.0160 1.5570

0.1420 −3.6480 0.0720

0.2060 0.0730 −2.5540#

2="−0.2580 2.0660 1.5530

0.1420 −3.6450 0.2010

0.2120 0.0510 −2.5560#

1="−0.2410 2.1600 1.4450

0.1450 −3.6420 0.1170

0.1830 0.0970 −2.5950#

2="−0.2680 2.1640 1.5560

0.1570 −3.6390 0.1720

0.2020 0.0810 −2.5750#

1="−0.2720 1.9380 1.4540

0.0580 −3.9610 0.0650

0.1100 0.0580 −2.9080#,C=100

001

2="−0.2730 1.9440 1.4510

0.0590 −3.9610 0.1070

0.1090 0.0510 −2.9180#,D="0.045

0.080

0.053#

1="−0.2760 2.0940 1.4450

0.0520 −3.9510 0.0920

0.1250 0.0840 −2.9380#,L="001

100

010#

2="−0.2720 2.1020 1.4150

0.0570 −3.9510 0.1200

0.1000 0.0770 −2.9420#,l="1

Tt is not hard to attest that Aσ

i,Aσ

iare strictly

Metzler and Hurwitz for all i,σand Aσ

i≤Aσ

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To reect DSP, it is necessary the diagonal rep-

resentations of the system parameters such that

Cd1=diag [100],Cd2=diag [001]

2(ν, ν) = diag [−0.2680 −3.6450 −2.5560]

2(ν+ 1, ν) = diag [0.1420 0.0510 1.5530]

2(ν+ 2, ν) = diag [0.2120 2.0660 0.2010]

where only the set of desired diagonal representa-

tions of the matrix A1

2is presented.

It can be found that for the number of matrix

inequalities N= 43 the feasible variables which

provide by using SeDuMi, [20], a solution for the

problem are

P=diag [3.1300 2.7628 3.0234], η = 5.8844

11 =diag [4.1263 0.0530 0.1279]

12 =diag [2.0099 0.0630 1.3823]

21 =diag [4.1493 0.0533 0.1280]

22 =diag [2.0078 0.1239 1.3821]

11 =diag [4.2250 0.0472 0.1350]

12 =diag [1.9133 0.0936 1.3743]

21 =diag [4.2174 0.0515 0.1185]

22 =diag [1.9764 0.1274 1.3777]

These results full the diagonal positiveness

criterion on the LMI variables and it can be ob-

served that such parameters produce strictly pos-

itive observer matrix gains

1="1.3183 0.6421

0.0192 0.0228

0.0423 0.4572#,J1

2="1.3257 0.6415

0.0193 0.0448

0.0424 0.4571#

1="1.3499 0.6113

0.0171 0.0339

0.0447 0.4546#,J1

2="1.3474 0.6314

0.0186 0.0461

0.0392 0.4557#

It is worth to mention that the positive ob-

server gains have to be directly used in the IO

fuzzy switched IOs. This limitation is due to the

M-T-S fuzzy switched system positiveness.

From those, it can be presented for comparison

that

e2="−1.6259 2.0940 0.8337

0.0349−3.9510 0.0581

0.0803 0.0840−3.3926#

ρ(A1

e2) = {−1.5560 −3.4304 −3.9831}

e2="−1.5909 2.1600 0.8337

0.1279−3.6420 0.0831

0.1383 0.0970−3.0496#

ρ(A1

e2) = {−1.3889 −3.1266 −3.7669}

where, evidently, A1

e2<A1

e2and Aσ

ei ≤Aσ

ei for

all i= 1,2and σ= 1,2.

The proposed LMI conditions with diagonal

matrix variables and measurable premise variable

vector in design of M-T-S fuzzy positive switched

IOs are illustrated in this example.

However, if the switched M-T-S fuzzy IO ob-

server needs to be designed for purely Metzler sys-

tems with interval-specied parameters, it requires

the use of structured non-negative diagonal matrix

variables Vσ

ik. This manifests itself in the fact that

the stable matrices Aσ

ei,Aσ

ei of the switched inter-

val observer will be purely Metzler and Hurwitz,

since the matrix gains Jσ

iwill be non-negative.

Unfortunately, the design of the structure of non-

negative diagonal matrix variables Vσ

ik may not

be unambiguous, [21].

6 Concluding Remarks

The main objectives in this paper are the synthesis

conditions of IOs design for positive M-T-S fuzzy

switched systems with interval-specied dynam-

ics of the form of strictly Metzler matrices and

bounded system disturbances. The DSP, and the

proposed LMI structures reect the key idea to ob-

tain the Metzler and Hurwitz matrix structures,

while Lyapunov function and the related LMIs

form the base of the observer stability. The re-

sults are presented to the case for the underlying

system under arbitrary switching and under in-

uence of the bounded unknown disturbance, the

used Lyapunov function guaranties quadratic sta-

bility of the observer in all switched modes. De-

spite its design conditions complexity, estimation

using positive switched M-T-S fuzzy IOs is robust

to the changes covered in plant dynamics by given

interval bounds, taking into account that the pos-

itivity of the lower state estimation need to be

keep.

In future works, the proposed method will be

extended to the issue of M-T-S fuzzy switched sys-

tems with ostensible Metzler interval-specied pa-

rameters, also parameterizing the M-T-S model

performance requirements for partly unmeasur-

able premise variables.

Acknowledgement

The work presented in this paper was supported by

VEGA, the Grant Agency of the Ministry of

Education and Academy of Science of Slovak

Republic, under Grant No. 1/0483/21. This support

is very gratefully acknowledged.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.10

Dušan Krokavec, Anna Filasová

E-ISSN: 2224-2856

100

Volume 18, 2023

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Contribution of Individual Authors to the

Creation of a Scientific Article

(Ghostwriting Policy)

Anna Filasová elaborated the principles of matrix

constraints representation in the positive fuzzy IO

gain synthesis and implemented their linear matrix

structures, Dušan Krokavec addressed the incidence

of diagonal stabilisation principle into set of LMIs

for stability of positive fuzzy IOs and converted the

lower bound system state limit to an LMI problem.

Both authors have read and agreed to the proposed

version of the manuscript.

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

that are relevant to the content of this article.

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

_US

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.10

Dušan Krokavec, Anna Filasová

E-ISSN: 2224-2856

101

Volume 18, 2023