Robust Decentralized Controller Design for Descriptor-type

Systems with Distributed Time Delay

ALTU ˘

G˙

IFTAR

Department of Electrical and Electronics Engineering

Eskis¸ehir Technical University

26555 Eskis¸ehir, TURKEY

Abstract: - Robust decentralized controller design is considered for linear time-invariant large-scale

descriptor-type systems with distributed time delay which are composed of overlapping subsystems. A

robustness bound, that accounts for the interactions among the subsystems and for modeling uncertainties both

in the subsystem models and the interactions, is derived using overlapping decompositions and expansions. A

robust decentralized controller design approach using this bound is then proposed. Once the robustness bound

is derived, the proposed approach is decoupled for each subsystem and, for each subsystem, it is based on

a local nominal model, which is also derived using overlapping decompositions and expansions. Satisfying

a simple condition, involving the derived robustness bound, however, guarantees the robust stability of the

overall actual closed-loop system.

Key-Words: - Robust controller design, Decentralized control, Time-delay systems, Descriptor-type systems,

Distributed time delay, Large-scale systems, Overlapping decompositions

Received: April 12, 2023. Revised: February 7, 2024. Accepted: March 19, 2024. Published: April 18, 2024.

1 Introduction

Many practical systems, especially large-scale sys-

tems [1] may be subject to time delays [2]. Such

systems, which are typically named as time-delay

systems [3], can be described by delay-differential

equations [4]. For some time-delay systems, such as

telerobotic systems [5], however, delay-differential

equations must be coupled with delay-algebraic

equations to describe the dynamics of the system.

Such systems are known as descriptor-type time-

delay systems. Descriptor-type time-delay systems

impose a challenge since their response may be

discontinuous and even impulsive [6].

Time delays in a system may be pointwise or

distributed [7]. Distributed time delay may appear

in many application areas, such as neural networks

[8], biology [9], trafﬁc ﬂow [10], logistics [11], and

combustion control [12].

There are many examples of large-scale systems,

such as interconnected power systems [13], freeway

trafﬁc regulation systems [14], intelligent vehicle-

highway systems [15], and data-communication net-

works [16], which consist of subsystems with over-

lapping dynamics. To analyze and design controllers

for such systems, the approach of overlapping de-

compositions was ﬁrst introduced in [17]. Although,

initially, this approach was introduced for linear

time-invariant (LTI) delay-free systems, since then it

has also been extended to time-delay systems with

both pointwise and distributed time delays [18], [19],

[20], [21].

In general, it is not possible to obtain an ex-

act model of any practical system. Therefore, any

designed controller for a practical system must be

robust against modeling uncertainties [22]. To en-

sure such a robustness, a robustness bound, which

accounts for modeling uncertainties in centralized

descriptor-type time-delay systems were considered

in [23]. In the case of large-scale systems, however,

it may be necessary or practical to ignore the inter-

actions among subsystems during controller design

[24]. In this case, besides modeling uncertainties,

the designed controllers must also be robust against

the neglected interactions. Therefore, in the present

work, we consider large-scale LTI descriptor-type

distributed-time-delay systems consisting of subsys-

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tems which dynamically overlap and extend the

robustness bound also to account for the neglected

interactions between the subsystems and any model-

ing uncertainties in the interactions. We then propose

a robust decentralized controller design approach

using this bound. Once the robustness bound is

derived, the proposed approach is decoupled for

each subsystem and, for each subsystem, it is based

on a local nominal model, which is also derived

using overlapping decompositions and expansions.

Satisfying a simple condition, involving the derived

robustness bound, however, guarantees the robust

stability of the overall actual closed-loop system.

Throughout the paper, Rand Crespectively

denote the sets of real and complex numbers. For

s∈C,Re(s)denotes the real part of s. For positive

integers kand l,Rkand Rk×ldenote the spaces of,

respectively, k-dimensional real vectors and k×l-

dimensional real matrices. Ikdenotes the k×k-

dimensional identity matrix and Idenotes the iden-

tity matrix of appropriate dimensions. 0may denote

either the scalar zero, a zero matrix, or a matrix

function which is identically zero. det(·),rank(·),

¯σ(·), and σ(·)respectively denote determinant, the

rank, the maximum singular value, and the minimum

singular value of the indicated matrix. bdiag(···)

denotes a block diagonal matrix with the indicated

matrices on the main diagonal. For a matrix function

M(·) : [−τ, 0] →Rk×l,kMk:= R0

−τ¯σ(M(θ)) dθ.

Finally, jdenotes the imaginary unit (i.e., j:=

√−1).

2 Problem Statement

In this work, as mentioned in the introduction, we

consider descriptor-type large-scale LTI distributed-

time-delay systems, which are composed of overlap-

ping subsystems. In practice, such subsystems may

overlap in many different ways. For simplicity of

presentation, however, in here we consider the case

of only two subsystems which are interconnected

through another dynamic subsystem, which form the

overlapping part. Such a system can compactly be

described as:

E˙x(t) = A0x(t)

+Z0

−τ

(A(θ)x(t+θ) + B(θ)u(t+θ)) dθ (1)

y(t) = Z0

−τ

C(θ)x(t+θ)dθ (2)

where τ > 0is the maximum time-delay in the

system and tis the time variable. The state, the input,

and the output vectors are decomposed as:

x=





, u =u1

u2, y =y1

y2,(3)

where xi∈Rni,ui∈Rpi, and yi∈Rqiare, re-

spectively, the state, the input, and the output vectors

of the ith subsystem (i= 1,2), and xc∈Rncis the

state vector of the overlapping part. The matrices

E∈Rn×n,A0∈Rn×n, and the matrix functions

A(·) : [−τ, 0] →Rn×n,B(·) : [−τ, 0] →Rn×p, and

C(·):[−τ, 0] →Rq×n(where n:= n1+nc+n2,

p:= p1+p2, and q:= q1+q2) are also decomposed

as:

E=



E10 0

0Ec0

0 0 E2



, A0=



10 0

0A0

0 0 A0



,

A(·) = 



A1(·)A1c(·) 0

Ac1(·)Ac(·)Ac2(·)

0A2c(·)A2(·)



,

B(·) = 



B1(·) 0

0 0

0B2(·)



,

and

C(·) = C1(·) 0 0

0 0 C2(·),

where the partitionings are compatible with those

in (3). All the submatrix functions shown above are

assumed to be bounded, except that they are allowed

to include Dirac-delta terms, δ(θ+h),h∈[0, τ],

except that A(0) is assumed to be bounded (i.e.,

each submatrix function of A(θ)can include a term

δ(θ+h),h∈(0, τ], but not δ(θ)). Inclusion of these

terms allow representation of pointwise time delays,

together with distributed time delay. Furthermore, it

is assumed that the submatrix functions of A(·)and

B(·)are subject to uncertainties. Thus, representing

any one of these submatrix functions by M•(·), we

assume that M•(·) = Mn

•(·) + Mu

•(·), where Mn

•(·)

is the known nominal part and Mu

•(·)is the unknown

uncertain part. The uncertain parts, however, are

assumed to be bounded as follows:

kAu

1k ≤ α1,kAu

1ck ≤ α1c,kAu

c1k ≤ αc1,

kAu

ck ≤ αc,kAu

c2k ≤ αc2,kAu

2ck ≤ α2c,

kAu

2k ≤ α2,kBu

1k ≤ β1,kBu

2k ≤ β2,

(4)

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where α’s and β’s are known non-negative bounds.

Since the input-output uncertainties can in general be

represented either at the input or at the output, here

we assume that all such uncertainties are represented

at the input, hence there are no uncertainties in C(·).

It is also assumed that rank(E•) = ¯n•≤n•,

where •stands for 1,c, or 2, with ¯n•< n•, for at

least one of 1,c, or 2. This assumption means that

rank(E)< n, which means that the system (1)–

(2) is a descriptor-type system [6]. It is, however,

asuumed that:

Assumption 1: If ¯n•< n•, then rank(L•A0

•R•) =

n•−¯n•, where L•∈Rn•−¯n•×n•and R•∈

Rn•×n•−¯n•are such that the rows of L•span the

left null space of E•and the columns of R•span

the right null space of E•.

The above assumption implies that there exists

a unique solution to (1), for any suitable initial

condition, x(t+θ),θ∈[−τ, 0] [4].

The characteristic function of the system (1) is

given by

ψ(s) := det(sE − A(s)) ,(5)

where

A(s) := A0+Z0

−τ

A(θ)esθdθ . (6)

The modes of the system (1) are the roots of ψ(s) =

0. It is known that the system (1) has inﬁnitely many

modes, in general [25]. However, under Assumption

1, it has only ﬁnitely many modes with real part

greater than

νf:= sup{Re(s)|det( ¯

A(s)) = 0},(7)

where ¯

A(s) := LA(s)R, where A(s)is as given in

(6),

L:= bdiag(L1,Lc,L2)(8)

and

R:= bdiag(R1,Rc,R2),(9)

where L•and R•are as in Assumption 1, if ¯n•<

n•, and they are missing in (8) and (9), otherwise

[26].

It is known that (1)–(2) can not be stabilized

by a ﬁnite-dimensional proper LTI controller unless

νf<0[27]. Thus, since our aim is to stabilize (1)–

(2) for all uncertainties satisfying (4), we make the

following assumption:

Assumption 2: νf<0for any uncertainties satis-

fying (4).

This assumption implies that the system (1) has

only ﬁnitely many unstable modes. A mode is called

as an unstable mode if it has a non-negative real part.

Our aim is to design decentralized (possibly time-

delay) LTI controllers:

Ui(s) = −Ki(s)Yi(s), i = 1,2,(10)

where Ui(s)and Yi(s)are the Laplace transforms

of, respectively, ui(t)and yi(t), and Ki(s)is the

transfer function matrix (TFM) of the ith controller.

These controllers are to be designed such that

•each local nominal closed-loop system is stable,

•a local performance criteria is satisﬁed for each

local nominal closed-loop system, and

•the actual overall closed-loop system is ro-

bustly stable for all uncertainties that satisfy the

bounds (4).

Furthermore, the design of each decentralized con-

troller is to be based on a local nominal model,

which is to be derived next.

3 Local Nominal Models

In order to obtain a local nominal model, we expand

[21] the above overlappingly decomposed system by

using the transformation

T=





In10 0

0Inc0

0 0 In2







.(11)

The expanded system is described as:

E˙

ˆx(t) = ˆ

A0ˆx(t)

+Z0

−τˆ

A(θ)ˆx(t+θ) + ˆ

B(θ)ˆu(t+θ)dθ (12)

ˆy(t) = Z0

−τ

C(θ)ˆx(t+θ)dθ (13)

where

E:= bdiag( ˆ

E1,ˆ

E2),

where

E1:= bdiag (E1, Ec),ˆ

E2:= bdiag (Ec, E2),

A0:= bdiag( ˆ

1,ˆ

2),

where

1:= bdiag A0

1, A0

c,ˆ

2:= bdiag A0

c, A0

2,

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A(θ) := ˆ

A1(θ)ˆ

A12(θ)

A21(θ)ˆ

A2(θ),

where

A1(θ) := A1(θ)A1c(θ)

Ac1(θ)Ac(θ),

A12(θ) := 0 0

0Ac2(θ),

A21(θ) := Ac1(θ) 0

0 0 ,

and

A2(θ) := Ac(θ)Ac2(θ)

A2c(θ)A2(θ),

B(θ) := bdiag ˆ

B1(θ),ˆ

B2(θ),

where

B1(θ) := B1(θ)

0,ˆ

B2(θ) := 0

B2(θ),

and

C(θ) := bdiag ˆ

C1(θ),ˆ

C2(θ),

where,

C1(θ) := C1(θ) 0 ,

and

C2(θ) := 0C2(θ).

Note that relations ˆ

ET =T E,ˆ

A0T=T A0,

A(θ)T=T A(θ),ˆ

B(θ) = T B(θ), and ˆ

C(θ)T=

C(θ)are satisﬁed. This implies that the expanded

system (12)–(13) is an extension of the original sys-

tem (1)–(2) [21]. Thus, the expanded system includes

the original system, and hence the two systems have

the same input-output map [20].

We note that, because of the block diagonal

structure of ˆ

Eand of ˆ

A0, Assumption 1 implies

that a unique solution to (12), for any suitable initial

condition, ˆx(t+θ),θ∈[−τ, 0], is guaranteed. Here,

we also make the following assumption:

Assumption 3:

sup nRe(s)|det ¯

A(s)= 0o<0,(14)

for any uncertainties satisfying (4), where ¯

A(s) :=

Lˆ

A(s)ˆ

R, where

A(s) := ˆ

A0+Z0

−τ

A(θ)esθdθ , (15)

L:= bdiag(L1,Lc,Lc,L2),(16)

and

R:= bdiag(R1,Rc,Rc,R2),(17)

where L•and R•are as in Assumption 1, if ¯n•<

n•, and they are missing in (16) and (17), otherwise.

We note that, since the expanded system includes the

original system, Assumption 3, in particular, implies

Assumption 2. Thus, under this assumption, both the

expanded system and the original system have only

ﬁnitely many unstable modes.

We note that the expanded system (12)–(13) is

composed of two disjoint subsystems with only

weak interactions (through ˆ

A12(·)and ˆ

A21(·)). Thus,

alocal model can be obtained by ignoring these

interactions as:

Ei˙

ˆxi(t) = ˆ

iˆxi(t) + Z0

−τˆ

Ai(θ)ˆxi(t+θ)

+ˆ

Bi(θ)ˆui(t+θ)dθ (18)

ˆyi(t) = Z0

−τ

Ci(θ)ˆxi(t+θ)dθ (19)

for i= 1,2. A local nominal model, for i= 1,2,

can then be obtained by taking the nominal part of

(18)–(19):

Ei˙

ˆxi(t) = ˆ

iˆxi(t) + Z0

−τˆ

i(θ)ˆxi(t+θ)

+ˆ

i(θ)ˆui(t+θ)dθ (20)

ˆyi(t) = Z0

−τ

Ci(θ)ˆxi(t+θ)dθ (21)

where

1(θ) := An

1(θ)An

1c(θ)

c1(θ)An

c(θ),

2(θ) := An

c(θ)An

c2(θ)

2c(θ)An

2(θ),

1(θ) := Bn

1(θ)

0,

and

2(θ) := 0

2(θ).

Our ﬁnal two assumptions are as follows:

Assumption 4: For i= 1,2,

sup nRe(s)|det ¯

i(s)= 0o<0,(22)

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where ¯

i(s) := ˆ

Liˆ

i(s)ˆ

Ri, where

i(s) := ˆ

i+Z0

−τ

i(θ)esθdθ , (23)

L1:= bdiag(L1,Lc),ˆ

L2:= bdiag(Lc,L2),(24)

and

R1:= bdiag(R1,Rc),ˆ

R2:= bdiag(Rc,R2),(25)

where L•and R•are as in Assumption 3.

Assumption 5: ˆm=m1+m2, where ˆmdenotes the

number of unstable modes of the expanded system

(12)–(13) (which is ﬁnite by Assumption 3) and mi

(i= 1,2) denotes the number of unstable modes of

the ith local nominal model (20)–(21) (which is also

ﬁnite by Assumption 4).

4 A Robustness Bound

In this section, we derive a robustness bound, to

account for the

•uncertainties in the subsystem models,

•neglected interactions between the subsystem

models, and

•uncertainties in the interactions between the

subsystem models.

For this, from (20)–(21), we obtain the TFM of the

ith local nominal model (i= 1,2) as:

Γn

i(s) = ˆ

Ci(s)sˆ

Ei−ˆ

i(s)−1ˆ

i(s)(26)

where ˆ

Ci(s) := R0

−τˆ

Ci(θ)esθdθ,ˆ

i(s)is deﬁned in

(23), and ˆ

i(s) := R0

−τˆ

i(θ)esθdθ. The TFM for

the overall design model is then given as

Γn(s) = bdiag (Γn

1(s),Γn

2(s)) .(27)

On the other hand, since the expanded system

(12)–(13) includes the original system (1)–(2), the

TFM, Γ(s), of the actual original system is equal to

the TFM, ˆ

Γ(s), of the expanded system:

Γ(s) = ˆ

C(s)sˆ

E−ˆ

A(s)−1ˆ

B(s)(28)

where ˆ

C(s) := R0

−τˆ

C(θ)esθdθ,ˆ

A(s)is deﬁned in

(15), and ˆ

B(s) := R0

−τˆ

B(θ)esθdθ.

Now, let ∆(s)be such that:

Γ(s) = Γn(s) (I+ ∆(s)) .(29)

Then, we can use a frequency-dependent upper

bound on the norm of ∆(jω)as a robustness bound.

Such a bound can be obtained as follows:

Lemma 1: Suppose that δd(ω)>0,∀ω∈R. Then

¯σ(∆(jω)) ≤δ(ω) := δn(ω)

δd(ω),∀ω∈R,(30)

where

δd(ω) := σ(H(jω)) −max δ1

d(ω), δ2

d(ω)

−max nδ1,2

d(ω), δ2,1

d(ω)o,(31)

where

H(s) := ˆ

1(s)H12(s)

H21(s)ˆ

2(s),(32)

where

H12(s) := 0 0

0−R0

−τAn

c2(θ)esθdθ G2(s)

and

H21(s) := −R0

−τAn

c1(θ)esθdθ 0

0 0 G1(s)

where, for i= 1,2,

Gi(s) := sˆ

Ei−ˆ

i(s)−1ˆ

i(s),

and

δ1

d(ω) := ˆα1g1(ω), δ1,2

d(ω) := αc2g2(ω),

δ2

d(ω) := ˆα2g2(ω), δ2,1

d(ω) := αc1g1(ω),

where, for i= 1,2,gi(ω) := ¯σ(Gi(jω)) and ˆαi:=

max(αi, αc) + max(αci, αic), and

δn(ω) := max δ1

n(ω), δ2

n(ω)

+ max δ1,2

n(ω), δ2,1

n(ω),(33)

where

δ1

n(ω) := β1+ ˆα1g1(ω), δ2

n(ω) := β2+ ˆα2g2(ω),

δ1,2

n(ω) := ¯σ(H12(jω)) + αc2g2(ω),

and

δ2,1

n(ω) := ¯σ(H21(jω)) + αc1g1(ω).

Proof: ∆(s)in (29) can be chosen to satisfy

sˆ

E−ˆ

A(s)−1ˆ

B(s)

=sˆ

E−ˆ

An(s)−1ˆ

Bn(s) (I+ ∆(s)) ,(34)

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where ˆ

An(s) := bdiag ˆ

1(s),ˆ

2(s)and

Bn(s) := bdiag ˆ

1(s),ˆ

2(s). By premultiplying

both sides of (34) by (sˆ

E−ˆ

A(s)) and rearranging

terms, we obtain N(s) = D(s)∆(s), where

D(s) := sˆ

E−ˆ

A(s)G(s)

and

N(s) := ˆ

B(s)−D(s),

where G(s) := bdiag (G1(s), G2(s)). The result

then follows by noting that ¯σ(N(jω)) ≤δn(ω)and

σ(D(jω)) ≥δd(ω).

Remark: The bound (30) is not deﬁned if δd(ω)≤

0, i.e., if σ(H(jω)) ≤max{δ1

d(ω), δ2

d(ω)}+

max{δ1,2

d(ω), δ2,1

d(ω)}, for some ω∈R. However,

note that the matrix H(jω)contains the nominal

input terms ˆ

i(jω)on the diagonal blocks (which

typically have large norm for any ω) and inter-

action terms (which typically have smaller norm)

on the off-diagonal blocks. Furthermore, the terms

δ•

d(ω)contain bounds on the uncertainties (which are

typically small). Therefore, typically, σ(H(jω)) >

max{δ1

d(ω), δ2

d(ω)}+ max{δ1,2

d(ω), δ2,1

d(ω)}, and

hence δd(ω)>0.

5 Controller Design

We now consider the problem of designing de-

centralized controllers (10) so that the following

requirements are satisﬁed:

(i) each local nominal closed-loop system, (20)–

(21) under the local controller Ki(s), is stable,

(ii) a local performance criteria (if any) is satisﬁed

for each local nominal closed-loop system, and

(iii) the actual overall closed-loop system, (1)–(2)

under the overall decentralized controller (10) is

robustly stable for all uncertainties that satisfy

(4).

Requirements (i) and (ii) are local requirements

which can be satisﬁed independently for each chan-

nel. To satisfy requirement (iii), we propose to

design the ith controller (i= 1,2), Ki(s), to satisfy

¯σ(Tn

i(jω)) <1

δ(ω),∀ω∈R,(35)

where Tn

i(s) := Γn

i(s)Ki(s) [I+ Γn

i(s)Ki(s)]−1is

the complementary sensitivity matrix for the ith

local nominal closed-loop system and δ(ω)is given

by (30). The following theorem shows that when

requirement (i) is satisﬁed for each local nominal

closed-loop system, satisfying (35) guarantees robust

stability of the actual overall closed-loop system.

Theorem 1: Suppose that Assumptions 1, 3, 4, and

5 hold. Also suppose that each local nominal closed-

loop system is stable and that (35) is satisﬁed. Then,

the actual overall closed-loop system is robustly

stable for all uncertainties that satisfy (4).

Proof: Assumption 1 is needed for the validity of

the model (1)–(2). This also implies the validity of

the models (12)–(13), (18)–(19), and (20)–(21). The

block diagonal structure (27) of the overall design

model, together with Assumption 4, implies that

there are no modes of the overall design model

approaching the imaginary axis from left and that

the number of unstable modes of the overall design

model is ﬁnite. Assumption 3, on the other hand,

implies that there are no modes of the expanded

open-loop system approaching the imaginary axis

from left and that the number of unstable modes

of the expanded open-loop system is ﬁnite. The

complementary sensitivity matrix for the overall

closed-loop design model is

Tn(s)=Γn(s)K(s) [I+ Γn(s)K(s)]−1

= bdiag (Tn

1(s), T n

2(s)) (36)

where K(s) := bdiag (K1(s), K2(s)) is the TFM

for the overall controller. Due to the block diagonal

structure of the overall closed-loop design model,

stability of the local nominal closed-loop systems

implies the stability of the overall closed-loop design

model. The block diagonal structure of the overall

design model, on the other hand, implies that the

number of unstable modes of the overall design

model is equal to the sum of the number of the

unstable modes of the local nominal models. Then,

by Assumption 5, the number of the unstable modes

of the expanded system (12)–(13) is equal to the

number of unstable modes of the overall design

model. Furthermore, by (36),

¯σ(Tn(jω)) = max

i∈{1,2}{¯σ(Tn

i(jω))}.

Thus, when (35) is satisﬁed, by (30), we have

¯σ(Tn(jω)) <1

¯σ(∆(jω)) ,∀ω∈R.(37)

This, however, implies that the actual expanded

closed-loop system is robustly stable [22]. However,

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since the actual expanded system includes the actual

original system, this implies the robust stability of

the actual original closed-loop system [20]. 

Note that, once the robustness bound (30) is ob-

tained, (35) is in fact a local requirement. This means

that each local controller Ki(s)can be designed

independently of others based on the local nominal

model (20)–(21). For this, any centralized controller

design approach developed for descriptor-type LTI

time-delay systems (e.g., [28], [29]) can be used.

6 Conclusions

Robust decentralized controller design has been con-

sidered for large-scale LTI descriptor-type systems

with distributed time delay which are composed

of overlapping subsystems. Using overlapping de-

compositions and expansions, a frequency dependent

robustness bound, (30), has been derived. A robust

decentralized controller design approach using this

bound has then been proposed. Once the robustness

bound is derived, the proposed approach is decou-

pled for each subsystem and, for each subsystem, it

is based on a local nominal model. Satisfying a sim-

ple condition, (35), however, guarantees the robust

stability of the overall actual closed-loop system.

Since the derived bound is frequency dependent,

the approach is, in general, less conservative than

an approach in which a constant bound is used.

Furthermore, it also allows frequency shaping [30].

Although, for simplicity of presentation, we have

considered only the case of two overlapping subsys-

tems, the proposed approach can be extended to such

cases as Nsubsystems with a common overlapping

part or a string of Noverlapping subsystems.

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Contribution of Individual Authors to

the Creation of a Scientiﬁc Article

(Ghostwriting Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

ﬁnal ﬁndings and solution.

Sources of Funding

This work has been supported by the Scientiﬁc Re-

search Projects Commission of Eskis¸ehir Technical

University.

Conﬂicts of Interest

The author has no conﬂicts of interest to declare that

are relevant to the content of this article.

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(Attribution 4.0 International , CC BY 4.0)

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