Robust Tracking and Disturbance Rejection for Decentralized

Neutral Distributed-Time-Delay Systems

ALTU ˘

G˙

IFTAR

Department of Electrical and Electronics Engineering

Eskis¸ehir Technical University

26555 Eskis¸ehir, TURKEY

Abstract: - An important problem in control engineering is to design a controller to achieve robust asymptotic

tracking of certain reference signals despite certain disturbance inputs. In the present work, this problem,

which is known as the robust servomechanism problem, is considered for decentralized linear time-invariant

(LTI) neutral systems with distributed time-delay. However, the system is also allowed to have discrete

time-delays besides distributed time-delays. The reference signals and the disturbance input are assumed to

satisfy a LTI neutral delay-differential equation with distributed and/or discrete time-delays. The necessary

and sufﬁcient conditions for the existence of a controller which solves this problem are derived. The structure

of this controller (when it exists) is also presented.

Key-Words: - Servomechanism problem, Decentralized control, Neutral time-delay systems, Distributed

time-delay, Robust control, Tracking, Disturbance rejection

Received: December 12, 2022. Revised: August 18, 2023. Accepted: September 24, 2023. Published: October 17, 2023.

1 Introduction

Controller design has, in general, two aims: stability

and performance. One of the most common per-

formance requirements is to track given reference

signals in the presence of certain disturbance inputs.

The conroller design problem to achieve robust

tracking and disturbance rejection, besides stability,

is known as the robust servomechanism problem.

This problem has been studied extensively for linear

time-invariant (LTI) ﬁnite-dimensional systems (e.g.,

see, [1], [2], [3], [4], [5]; a system is said to be

ﬁnite-dimensional if its state can be represented by

a ﬁnite-dimensional vector). This problem has also

been studied for ﬁnite-dimensional nonlinear, [6],

[7], [8], and discrete-event, [9], systems.

For many systems, generally known as large-

scale systems, it is not feasible, if not impossible,

to collect all the measured outputs in a centralized

place, calculate the control inputs centrally and then

let various actuators apply these inputs. Thus, one

has to use a decentralized control structure in such

a case, [10]. When such a structure is imposed,

the robust servomechanism problem is known as

the decentralized robust servomechanism problem,

which has also been extensively studied for ﬁnite-

dimensional LTI systems (e.g., [11], [12], [13], [14],

[15]).

Many systems, especially large-scale systems,

however, may involve time-delays which must be

taken into account during controller design. Such

systems are generally called as time-delay systems.

The state of a time-delay system can not be rep-

resented by a ﬁnite-dimensional vector. Thus, these

systems are in the class of inﬁnite-dimensional sys-

tems, [16]. The dynamics of time-delay systems

can, in general, be described by delay-differential

equations, [17]. If these equations do not involve

the delayed derivative of the state vector, then

the corresponding time-delay system is said to be

retarded. Otherwise, it is said to be neutral. It is

known that a retarded LTI time-delay system can

have only ﬁnitely many modes in any right-half

complex-plane, [18]. However, a neutral time-delay

system can have chains of inﬁnitely many modes

going to inﬁnity along a vertical axis. Thus, in

general, it is more difﬁcult to control a neutral

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system, compared to a retarded system.

The time-delays in a system can be discrete or

distributed, [19]. In fact, many systems may involve

both types of time-delays at the same time, [20].

However, it is possible to represent both discrete

and distributed time-delays together using Dirac-

delta functions in a distributed-time-delay formula-

tion (see Section 2 below). Thus, distributed-time-

delay systems are more general than discrete-time-

delay systems.

Robust servomechanism problem has recently

been considered for time-delay systems in [21], [22],

[23], among others. The necessary and sufﬁcient

conditions for the solvability of the (centralized)

robust servomechanism problem for LTI time-delay

systems, has ﬁrst been presented in [24]. In [24],

however, only retarded discrete-time-delay systems

were considered. Retarded systems with distributed

time-delay were later considered in [25]. Neutral

systems with discrete and distributed time-delays

were respectively considered in [26], [27]. Results of

[26], were then extended to descriptor-type systems

in [28] (a time-delay system described by delay-

algebraic-differential equations, rather than delay-

differential equations, is said to be of descriptor-

type).

Robust servomechanism problem for decentral-

ized time-delay systems was ﬁrst considered in [29]

for retarded and in [30], for neutral systems. In

[29], [30], however, only systems with discrete time-

delays were considered. Retarded distributed-time-

delay systems were recently considered in [31]. In

the present work, we extend the results of [31], to

neutral systems with distributed time-delay. How-

ever, we use a formulation in which we can represent

discrete and distributed time-delays together. We

consider references and disturbances which satisfy a

LTI delay-differential equation of neutral type with

distributed and/or discrete time-delays. As special

cases, this formulation also allows references and

disturbances which satisfy a LTI delay-differential

equation of retarded type and/or a simple LTI dif-

ferential equation. We ﬁrst present the necessary and

sufﬁcient conditions for the existence of a controller

which solves this problem. We then present the

structure of this controller when it exists.

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×lrespectively denote

the spaces of kdimensional real vectors and k×l

dimensional real matrices. Ikand 0k×lrespectively

denote the k×kdimensional identity matrix and the

k×ldimensional zero matrix. When the dimensions

are apparent, Iand 0are used to denote, respectively,

the identity and the zero matrices of appropriate

dimensions. For ξ:I → Rk, where Iis an interval

of the real line, ˙

ξ,¨

ξ, and ξ(l)respectively denote the

ﬁrst, the second, and the lth derivative of ξ.det(·)

and rank(·)respectively denote the determinant and

the rank of the indicated matrix. bdiag(· · ·)denotes

a block diagonal matrix with indicated matrices on

the main diagonal, ⊗denotes the Kronecker product,

and ·Tdenotes the transpose.

2 Problem Statement

Consider a decentralized LTI neutral distributed-

time-delay system with µcontrol agents, which is

described by:

˙x(t) + Z0

−¯τ

F(τ) ˙x(t+τ)dτ

=Z0

−¯τ"A(τ)x(t+τ) +

k=1

Bk(τ)uk(t+τ)

+E(τ)w(t+τ)#dτ (1)

zk(t) = Z0

−¯τhCk(τ)x(t+τ)

l=1

Dk,l(τ)ul(t+τ)

+Gk(τ)w(t+τ)idτ , k = 1, . . . , µ (2)

yk(t) = Z0

−¯τhHk(τ)x(t+τ)

l=1

Jk,l(τ)ul(t+τ)

+Lk(τ)w(t+τ)idτ , k = 1, . . . , µ (3)

Here x(t)∈Rnis the state vector and w(t)∈Rris

the disturbance input at time t. Furthermore, uk(t)∈

Rpkis the control input,zk(t)∈Rqkis the output,

and yk(t)∈Rmkis the measurement at time t, for

the kth control agent, k= 1, . . . , µ. Finally, ¯τ > 0

is the maximum time-delay in the system. Moreover,

A: [−¯τ, 0] →Rn×n,F: [−¯τ, 0] →Rn×n,

Bk: [−¯τ, 0] →Rn×pk,Ck: [−¯τ, 0] →Rqk×n,

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Dk,l : [−¯τ , 0] →Rqk×pl,Hk: [−¯τ , 0] →Rmk×n,

and Jk,l : [−¯τ , 0] →Rmk×pl(k, l = 1, . . . , µ) are

matrix functions which are known nominally, and

E: [−¯τ, 0] →Rn×r,Gk: [−¯τ, 0] →Rqk×r, and

Lk: [−¯τ, 0] →Rmk×r(k= 1, . . . , µ) are arbitrary

matrix functions, which are not necessarily known.

It is assumed that all of these matrix functions are

bounded, except that they may involve Dirac-delta

functions, δ(τ+hi), in order to represent ﬁnitely

many discrete time-delays hi∈[0,¯τ](i= 1, . . . , l,

for some positive integer l; including zero delay

when hi= 0). However, it is assumed that F(τ)

does not involve δ(τ+hi)for hi= 0, but may

involve δ(τ+hi)for hi>0. This assumption

guarantees that the given system is not of descriptor-

type. Otherwise, however, it does not impose any

restrictions, since the system dynamics (1) already

includes ˙x(t).

Here, δ(τ+hi)is deﬁned as

−¯τ

δ(τ+hi)χ(t+τ)dτ =χ(t−hi)(4)

for any χ: [−¯τ, ∞)→R.

Also consider a LTI delay-differential operator of

neutral type with differential degree ν, deﬁned as

Dχ(t) := dν

dtνχ(t) +

l=0

dtlZ0

−¯τ

αl(τ)χ(t+τ)dτ

(5)

where χ: [−¯τ, ∞)→Ris any at least νtimes

differentiable function. Here, αl: [−¯τ , 0] →R,

l= 0, . . . , ν, are bounded functions, except that

they may involve Dirac-delta functions δ(τ+γi)for

ﬁnitely many γi∈[0,¯τ], which represent discrete

time-delays. It is assumed, however, that αν(0) is

bounded, i.e., αν(τ)does not involve δ(τ), but may

involve δ(τ+γi), for γi>0.

In our problem, the kth control input, uk(t), is

to be determined by the kth control agent based on

the kth measurement, yk(t), and the kth reference,

rk(t)(k= 1, . . . , µ), so that the overall closed-loop

system is asymptotically stable and the kth tracking

error

ek(t) := zk(t)−rk(t)(6)

satisﬁes

lim

t→∞

ek(t)=0, k = 1, . . . , µ (7)

None of the references is necessarily known in

advance, however, it is assumed that they satisfy

Drk(t)=0, k = 1, . . . , µ (8)

The disturbance input, w(t), which is not generally

available, is also assumed to satisfy

Dw(t)=0 (9)

Remark 1: It is of course possible that the distur-

bance input w(t)and each reference rk(t)satisfy

different delay-differential equations. That is, there

may exist µ+ 1 different LTI delay-differential

operators of neutral type, D0,D1,. . .,Dµ, such that

D0w(t)=0,D1r1(t)=0,. . ., and Dµrµ(t)=0. In

such a case, however, it is possible to ﬁnd LTI delay-

differential operators, D0,D1,...,Dµ, such that (8)

and (9) are satisﬁed, where D:= D0D0=D1D1=

. . . =DµDµ. It is also possible that the maximum

time-delay in the system and in Dare different. In

such a case, however, maximum of the two delays

can be taken as ¯τand the functions in (5) can be

extended by zero functions (in case the system has

a larger maximum delay) or the matrix functions in

(1)–(3) can be extended by zero matrix functions (in

case the operator (5) has a larger maximum delay).

Remark 2: In (5) we assumed the most general form

of LTI neutral delay-differential operators. However,

as special cases, this operator also includes LTI

retarded delay-differential operators (in which case

αν(τ)=0,τ∈[−¯τ , 0]), as well as simple LTI

differential operators. In fact, simple LTI differential

operators may be more common in practice. As an

example, if the the references and the disturbance are

constant signals, then we should have D=d

dt . This,

however, can be written as in (5) with ν= 1 and

α0(τ) = α1(τ)=0,τ∈[−¯τ, 0]. If any one of these

signals also involve a sinusoidal signal of frequency

fo, then D=d3

dt3+ω2

dt , which can be written as in

(5) with ν= 3,α0(τ) = α2(τ) = α3(τ) = 0, and

α1(τ) = ω2

0δ(τ),τ∈[−¯τ, 0], where ω0= 2πfo.

Our problem can now be stated as follows:

Problem P: Find µdecentralized LTI feedback

controllers (from ykto uk,k= 1, . . . , µ) for the

system (1)–(3), such that the closed-loop system is

asymtotically stable and, for all initial condtions of

the system (1), for all disturbances w(t)satisfying

(9), for all references rk(t), satisfying (8), and for

all non-destabilizing matrix functions appearing in

(1)–(3), (7) is satisﬁed.

3 Preliminaries

Let ¯

F(s) := I+R0

−¯τF(τ)esτ dτ and

A(s) := R0

−¯τA(τ)esτ dτ . The function

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φ(s) := det s¯

F(s)−¯

A(s)is then known as

the characteristic function of (1). A mode of (1) is

any λ∈Cwhich satisﬁes φ(λ) = 0. If Re(λ)≥0,

then such a mode is known as an unstable mode

of (1). Asymptotic stability of (1) is equivalent to

the condition that (1) has no unstable modes, [32].

In general, there exist inﬁnitely many modes of

(1). However, there exist only ﬁnitely many modes

(if any) λof (1) with Re(λ)> γf, for some ﬁnite

γf∈R, [32].

Next, suppose that the following decentralized

LTI static feedbacks:

uk(t) = Kkyk(t), k = 1, . . . , µ (10)

are applied to the system (1,3). Here

Kk∈Rpk×mkare arbitrary, except that

they satisfy det (I−KJ0)6= 0, where

K:= bdiag(K1, . . . , Kµ)and J0:= lims→∞ ¯

J(s),

where the limit is taken along the positive real axis

and

J(s) := 





J1,1(s)· · · ¯

J1,µ(s)

Jµ,1(s)· · · ¯

Jµ,µ(s)







where ¯

Jk,l(s) := R0

−¯τJk,l(τ)esτ dτ. This condi-

tion guarantees the well-posedness of the closed-

loop system, [33]. A mode of (1) which remains

a mode of the closed-loop system under all con-

trols of the form (10) is known as a decentralized

ﬁxed mode (DFM) of (1,3), [34]. A necessary and

sufﬁcient condition for λ∈Cto be a DFM

of (1,3) is that for some κ∈ {0, . . . , µ}and

{k1, . . . , kκ}⊂{1, . . . , µ}, where k1, . . . , kκare

distinct ({k1, . . . , kκ}=∅if κ= 0),

rank 





A(λ)−s¯

F(λ)¯

Bk1(λ)· · · ¯

Bkκ(λ)

Hkκ+1 (λ)¯

Jkκ+1,k1(λ)· · · ¯

Jkκ+1,kκ(λ)

Hkµ(λ)¯

Jkµ,k1(λ)· · · ¯

Jkµ,kκ(λ)







< n (11)

where {kκ+1, . . . , kµ}:= {1, . . . , µ} \ {k1, . . . , kκ},

Bk(s) := R0

−¯τBk(τ)esτ dτ , and ¯

Hk(s) :=

−¯τHk(τ)esτ dτ , [33]. A necessary and sufﬁcient

condition for the existence of a (possibly dynamic)

decentralized LTI feedback controller which asymp-

totically stabilizes (1,3) is that (1,3) must not have

any unstable DFMs, [33].

4 Main Results

In this section we will present the necessary and

sufﬁcient conditions for the existence of a controller

which solves Problem P. We will then present the

structure of this controller when these conditions are

satisﬁed. For this purpose, we ﬁrst deﬁne

A(τ) := 





0 0 · · · 0−α0(τ)

δ(τ) 0 −α1(τ)

0δ(τ)−α2(τ)

.....

0δ(τ)−αν−1(τ)







and

F(τ) := 0ν−1×ν−10ν−1×1

01×ν−1αν(τ)

where αl(·),l= 0, . . . , ν, are the functions appear-

ing in (5).

Next, let us deﬁne a ﬁctitious system:

˙p(t) + Z0

−¯τ

F(τ) ˙p(t+τ)dτ =Z0

−¯τ

A(τ)p(t+τ)dτ

(12)

Then, we can write any rk(t)and any w(t), respec-

tively satisfying (8) and (9), as

rk(t) = Cr

kp(t), k = 1, . . . , µ, w(t) = Cwp(t)

(13)

for some arbitrary constant matrices Cr

k,k=

1, . . . , µ, and Cw. These matrices are arbitrary, since,

apart from the fact that they respectively satisfy (8)

and (9), rk(t)and w(t)are assumed to be unknown.

Because of the same reason, the initial condition of

the system (12) is also arbitrary.

Next, let us state the following assumptions:

Assumption 1: System (12) together with the output

r(t)

w(t)=Cr

Cwp(t)(14)

where Cr:= 









, is observable.

Assumption 2: 



E(τ)

G(τ)

L(τ)

χ= 0,∀τ∈[−¯τ, 0],

where χis a constant vector, implies that χ= 0,

where G(·) := 





G1(·)

Gµ(·)





and L(·) := 





L1(·)

Lµ(·)





.

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Assumption 3: For any solution p(t)of (12),

limt→∞ p(t)=0, only if p(τ)=0,∀τ∈[−ˆτ, 0],

where ˆτ≥0is the maximum time-delay in D

(which may be smaller than ¯τ- see Remark 1).

Assumption 4: B(τ)χ= 0,∀τ∈[−¯τ, 0], where

χis a constant vector, implies that χ= 0, where

B(·) := B1(·)· · · Bµ(·).

Assumption 5: χTC(τ) = 0,∀τ∈[−¯τ , 0], where

χis a constant vector, implies that χ= 0, where

C(·) := 





C1(·)

Cµ(·)





.

Remark 3: Assumptions 1–5 can be made without

loss of generality. They are made only to avoid

triviality. Assumption 1 implies that the ﬁctitious

system does not include any dynamics whose ef-

fect do not appear in both the reference and the

disturbance. If so, these dynamics can be deleted.

Assumption 2 implies that the shown matrix function

does not have any linearly dependent columns. If it

does, these columns, as well as the corresponding

elements of w, can be deleted. Assumption 3 implies

that the ﬁctitious system does not include any stable

dynamics. Since our problem is asymptotic tracking,

stable dynamics are irrelevant and can be deleted.

Assumption 4 implies that B(·)does not have

any linearly dependent columns. If it does, these

columns, together with the corresponding elements

of u:= 





uµ





can be deleted. Finally, Assump-

tion 5 implies that C(·)does not have any linearly

dependent rows. If it does, these rows, together with

the corresponding elements of z:= 





zµ





can be

deleted.

Next, let ˆ

A(·) := A(·)⊗Iq,ˆ

F(·) := F(·)⊗

Iq, and B:= Iq

0ˆq×q, where q:= Pµ

k=1 qkand

ˆq:= (ν−1)q. Also, for k= 1, . . . , µ, let Dk(·) :=







D1,k(·)

Dµ,k(·)





and Ck:= 0qk×q−

kIqk0qk×q+

k,

where q−

k:= Pk−1

l=1 qk(q−

k= 0 if k= 1) and q+

k:=

Pµ

l=k+1 qk(q+

k= 0 if k=µ). Finally, let ˆ

Ck:=

Iν⊗ Ck.

Now, we can present the necessary and sufﬁcient

conditions for the existence of a controller which

solves Problem P:

Theorem 1: Under Assumptions 1–5, there exists a

controller which solves Problem P if and only if the

following hold:

1) For k= 1, . . . , µ, the output zk, given in (2),

is contained in the measurement yk, given in

(3). That is zk(t) = Tkyk(t),∀t≥0, for some

Tk∈Rqk×mk.

2) The system

ξ(t) + Z0

−¯τF(τ) 0

0ˆ

F(τ)˙

ξ(t+τ)dτ

=Z0

−¯τ A(τ) 0

BC(τ)ˆ

A(τ)ξ(t+τ)

k=1 Bk(τ)

BDk(τ)˜uk(t+τ))!dτ (15)

˜vk(t) = Z0

−¯τ Hk(τ) 0

0δ(τ)ˆ

Ckξ(t+τ)

l=1 Jk,l(τ)

0˜ul(t+τ)!,

k= 1, . . . , µ (16)

has no unstable DFMs.

Proof: Let us ﬁrst prove the only if part. Since the

problem requires robust tracking (i.e., (7) must be

satisﬁed robustly), ek(t)must be available to the

kth control agent for each k= 1, . . . , µ. However,

since the kth control agent can only access rk(t)and

yk(t), in order to calculate ek(t)from (6), zk(t)must

be contained in yk(t). This implies the necessity of

Condition 1.

To prove the necessity of Condition 2, let

˜e(t) := 





eν−1(t)

e1(t)

e(t)







(17)

where e(t) := 





e1(t)

eµ(t)





,

e1(t) := ˙e(t) + Z0

−¯τ

αν(τ) ˙e(t+τ)dτ

+Z0

−¯τ

αν−1(τ)e(t+τ)dτ

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e2(t) := ¨e(t) + d

dt Z0

−¯τ

αν(τ) ˙e(t+τ)dτ

dt Z0

−¯τ

αν−1(τ)e(t+τ)dτ

+Z0

−¯τ

αν−2(τ)e(t+τ)dτ

eν−1(t) := e(ν−1)(t) + dν−2

dtν−2Z0

−¯τ

αν(τ) ˙e(t+τ)dτ

+dν−2

dtν−2Z0

−¯τ

αν−1(τ)e(t+τ)dτ

+dν−3

dtν−3Z0

−¯τ

αν−2(τ)e(t+τ)dτ

+. . . +Z0

−¯τ

α1(τ)e(t+τ)dτ

Then, we obtain

˙e(t) + Z0

−¯τ

αν(τ) ˙e(t+τ)dτ =e1(t)

−Z0

−¯τ

αν−1(τ)e(t+τ)dτ , (18)

˙ek(t) = ek+1(t)−Z0

−¯τ

αν−k−1(τ)e(t+τ)dτ ,

k= 1, . . . , ν −2,(19)

and

˙eν−1(t) = De(t)−Z0

−¯τ

α0(τ)e(t+τ)dτ (20)

Now, let ˜x(t) := Dx(t),˜uk(t) := Duk(t),k=

1, . . . , µ, and ξ(t) := ˜x(t)

˜e(t). Then, use (1), (2),

(6), (8), and (9) in (18)–(20) to obtain (15) and

ek(t) = 0qk×(n+ˆq)Ckξ(t), k = 1, . . . , µ

(21)

Therefore, in order to achieve (7), the part of

the system (15) which is observable through (21)

must be stabilized using inputs ˜uk(t). Since the

part of (15) which corresponds to ˜e(t), however,

is observable through (21) and the remaining part,

which corresponds to ˜x(t)(which is simply the

given system (1)) must also be stabilized as a

problem requirement, the whole system (15) must

be stabilized. Furthermore, this stabilization must be

achieved by decentralized LTI feedback, where the

kth control agent can access yk(t)and rk(t). By

Condition 1, however, using yk(t), the kth control

agent can obtain zk(t)and, using zk(t)and rk(t),

can also obtain ek(t). Thus, both ˜yk(t) := Dyk(t)

and ˜ek(t) := ˆ

Ck˜e(t)can be obtained by the kth

control agent. However, since ˜yk(t)

˜ek(t)= ˜vk(t),

given in (16), this means that the kth control agent,

which is to determine ˜uk(t), can access ˜vk(t). There-

fore, to solve Problem P under Assumptions 1–5,

a decentralized LTI controller which asymptotically

stabilizes the system (15,16) must exist. This, how-

ever, is equivalent to Condition 2. This proves the

only if part.

We will prove the if part constructively. As re-

marked above, due to Condition 1, the kth controller

can use

ek(t) = Tkyk(t)−rk(t)(22)

Then, the kth control agent can build:

˙sk(t) + Z0

−¯τ

Fk(τ) ˙sk(t+τ)dτ

=Z0

−¯τ

Ak(τ)sk(t+τ)dτ +Bkek(t)(23)

which is to be called as the kth servocompensator.

Here, ˆ

Ak(·) := A(·)⊗Iqk,ˆ

Fk(·) := F(·)⊗Iqk,

and Bk:= Iqk

0ˆqk×qk, where ˆqk:= (ν−1)qk, and

sk(t)∈Rνqkis the state vector at time t.

Let s(t) := 





s1(t)

sµ(t)





and ˆs(t) := M s(t), where

M:= 





bdiag(M1,1, . . . , M1,µ)

bdiag(Mν,1, . . . , Mν,µ)







where Mi,k := 0qk×qi−

kIqk0qk×qi+

k, where

qi−

k:= (i−1)qkand qi+

k:= (ν−i)qk,k= 1, . . . , µ,

i= 1, . . . , ν. Note that M∈Rνq×νq is invertible.

Let η(t) := x(t)

ˆs(t). Then, we can describe the

augmented dynamics of the plant (1) and the µ

decentralized servocompensators (23) as:

˙η(t) + Z0

−¯τF(τ) 0

0ˆ

F(τ)˙η(t+τ)

=Z0

−¯τ A(τ) 0

BC(τ)ˆ

A(τ)η(t+τ)

k=1 Bk(τ)

BDk(τ)uk(t+τ)

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+E(τ)

BG(τ)w(t+τ)dτ −0

Br(t)(24)

where r(t) := 





r1(t)

rµ(t)





.

From this augmented system, the kth control

agent, k= 1, . . . , µ, can measure

vk(t) := yk(t)

sk(t)

=Z0

−¯τ Hk(τ) 0

0δ(τ)ˆ

Ckη(t+τ)

l=1 Jk,l(τ)

0ul(t+τ)

+Lk(τ)

0w(t+τ)dτ (25)

Note that the systems (24,25) and (15,16) are

equivalent except for the existence of external inputs

wand rin (24,25) (which stay outside the loop when

the control loop is closed). Therefore, by Condition

2, there exist µdecentralized controllers (to be called

as the stabilizing compensators) that asymptotically

stabilize the system (24,25). Furthermore, the same

controllers also asymptotically stabilize the system

(15,16). Thus, when these controllers are applied, we

obtain limt→∞ ξ(t)=0(this is because, (15)–(16)

has no external inputs). By (21), however, this means

(7). Thus, these controllers achieve both asymptotic

stability and tracking. This completes the proof. 

From the if part of the above proof, it is evident

that each of the µthe decentralized controllers (when

they exist) which solve Problem P are composed of

two parts:

i) A servocompensator which is descried by (23)

with its input (22).

ii) A stabilizing compensator with inputs yk(t)and

sk(t)and output uk(t).

The servocompensators are responsible for

achieving the tracking and rejecting the disturbance.

The dynamics of these compensators are determined

by the delay-differential equation (5) (equivalently

by the ﬁctitious system (12)) by which the references

and the disturbance is satisﬁed.

The stabilizing compensators, on the other hand,

are designed to asymptotically stabilize the aug-

mented system (24,25). These compensators can be

designed using any decentralized stabilizing con-

troller design method developed for time-delay sys-

tems (e.g., see, [35], [36], [37], [38], and references

therein). A recently developed software package,

[39], may in particular be useful for this purpose.

Remark 4: It was shown in [40], that a LTI

time-delay (centralized) system can be stabilized

by LTI time-delay controllers if and only if it can

be stabilized by LTI ﬁnite-dimensional controllers.

This result was extended to the decentralized case

in [33]. More speciﬁcally, it was shown in [33],

that, a decentralized LTI time-delay system can be

stabilized by LTI time-delay controllers if and only

if it can be stabilized by LTI ﬁnite-dimensional

controllers (however, time-delay controllers may be

advantageous in some cases - see, [37], [41], [42]).

Therefore, when a solution to Problem P exists, the

stabilizing compensators can be designed as ﬁnite-

dimensional systems. However, since the servocom-

pensators are determined by the delay-differential

operator (5), these compensators must be time-

delay systems if (5) in fact involves time-delays.

As stated in Remark 2, however, references and

disturbance usually satisfy simple LTI differential

equations, rather than delay-differential equations.

In this case, then, when conditions of Theorem 1

hold, Problem P can be solved using LTI ﬁnite-

dimensional controllers.

5 Conclusions

Robust tracking and disturbance rejection problem

has been considered for LTI neutral systems with

distributed-time-delay. The formulation, however, al-

lows the representation of discrete time-delays to-

gether with distributed time-delay. Thus, the present

results can also be used for systems with discrete

time-delays, and/or for systems which involve both

kinds of delays. The necessary and sufﬁcient condi-

tions for the existence of a controller which solves

this problem have been derived. Although certain

assumptions (1–5) have been made, as discussed

in Remark 3, these assumptions were made only

to avoid triviality and can be made without loss

of generality. Furthermore, these assumptions are

needed only for the necessity of the conditions given

in Theorem 1. Thus, even if any one of these as-

sumptions do not hold, there still exist decentralized

LTI controllers which solve Problem P, as long as

the conditions given in Theorem 1 hold.

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It has also been shown that the controller which

solves the problem (when exists) is composed of

µdecentralized LTI controllers each of which con-

sists of a servocompensator and a stabilizing com-

pensator. This structure, as well as the conditions

given in Theorem 1, are in fact, generalizations

of the structure and the conditions given in [11],

for LTI ﬁnite-dimensional systems with references

and the disturbance satisfying a LTI differential

equation. This structure and the conditions are also

generalizations of the structure and the conditions

given in [30], for neutral discrete-time-delay systems

with references and the disturbance satisfying a

LTI delay-differential equation of neutral type with

discrete time-delays and of the structure and the

conditions given in [31], for retarded distributed-

time-delay systems with references and the distur-

bance satisfying a LTI delay-differential equation of

retarded type with distributed and/or discrete time-

delays. Furthermore, when µ= 1, both the structure

of the controller and the conditions of Theorem 1

reduce to the structure and the conditions given in

[27], for the centralized case.

Finally. as explained in Remark 4, when the

conditions of Theorem 1 hold and the references

and the disturbance satisfy simple LTI differential

equations, rather than delay-differential equations

(which is the more common case as discussed in

Remark 2), the problem can be solved using LTI

ﬁnite-dimensional controllers.

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Sources of Funding

This work has been supported by the Scientiﬁc

Research Projects Commission of Eskis¸ehir Techni-

cal University under grant numbers 22ADP301 and

23ADP033.

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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The author contributed in the present research, at all

stages from the formulation of the problem to the

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