Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results
Abstract
MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development Fund
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Recei ed May 10, 2018, accep ed June 7, 2018, da e o publica ion July 2, 2018, da e o cu en e sion July 19, 2018.
Digi al Objec Iden i ie 10.1109/ACCESS.2018.2851453
Spec um Analysis o LTI Con inuous-Time
Sys ems Wi h Cons an Delays: A Li e a u e
O e iew o Some Recen Resul s
LIBOR PEKAŘ 1AND QINGBIN GAO 2
1Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín, 76005 Zlín, Czech Republic
2Depa men o Mechanical Enginee ing, The Uni e si y o Alabama, Tuscaloosa, AL 35487, USA
Co esponding au ho : Libo Pekař (peka @u b.cz)
This wo k was suppo ed in pa by he Eu opean Regional De elopmen Fund h ough he P ojec CEBIA-Tech Ins umen a ion unde
G an CZ.1.05/2.1.00/19.0376 and in pa by he Na ional Sus ainabili y P og am P ojec unde G an LO1303 (MSMT-7778/2014).
ABSTRACT In ecen decades, inc easingly in ensi e esea ch a en ion has been gi en o dynamical
sys ems con aining delays and hose a ec ed by he a e -e ec phenomenon. Such esea ch co e s a wide
ange o human ac i i ies and he solu ions o ela ed enginee ing p oblems o en equi e in e disciplina y
coope a ion. The knowledge o he spec um o hese so-called ime-delay sys ems (TDSs) is e y c ucial
o he analysis o hei dynamical p ope ies, especially s abili y, pe iodici y, and dumping e ec . A g ea
olume o ma hema ical me hods and echniques o analyze he spec um o he TDSs ha e been de eloped
and u he applied in he mos ecen imes. Al hough a b oad amily o nonlinea , s ochas ic, sampled-
da a, ime- a ian o ime- a ying-delay sys ems has been conside ed, he s udy o he mos undamen al
con inuous linea ime-in a ian (LTI) TDSs wi h ixed delays is s ill he dominan esea ch di ec ion wi h
e e -inc easing new esul s and no el applica ions. This pape is p ima ily aimed a a (sys ema ic) li e a u e
o e iew o ecen (mos ly published be ween 2013 o 2017) ad ances ega ding he spec um analysis o
he LTI-TDSs. Speci ically, a o al o 137 collec ed a icles–which a e mos closely ela ed o he esea ch
a ea–a e e en ually e iewed. The e a e wo main objec i es o his e iew pape : Fi s , o p o ide he eade
wi h a de ailed li e a u e su ey on he selec ed ecen esul s on he opic and Second, o sugges possible
u u e esea ch di ec ions o be ackled by scien is s and enginee s in he ield.
INDEX TERMS Delay sys ems, eigen alues and eigen unc ions, li e a u e e iew, s abili y.
I. INTRODUCTION
Sys ems e incing delays o a e -e ec phenomenon eme ge
in a wide ange o human ac i i ies and can be obse ed in
many applica ions including bu no limi ed o enginee ing,
economy and biology. Fo ins ance, he cogni i e delays ep-
esen ed by pas s a es can be obse ed in p eda o -p ay mod-
els [1] which a e e y ea ly eal-li e-inspi ed ime-delay sys-
ems (TDSs). The human ne ous sys em ep esen s ano he
common biological example. The associa ed cogni ion- o-
eac ion delay may cause balancing p oblems [2], [3] o
a ic jams [4]. The e ec s o delays on he physiological sys-
ems we e analyzed in [5]. Enginee s om a ious indus ial
b anches mus deal wi h he a e -e ec phenomena in daily
p ac ice, such as delays in me allu gic p ocesses [6], delays
in he dis ibu ion o hea o powe [7], [8], delays incu ed by
he mass low in he p oduc ion o suga [9], e c. In machin-
ing p ocesses, he milling dynamics models a e usually
ep esen ed by delay di e en ial equa ions [10], and he
machine ool cha e s can be modeled by he so-called egen-
e a i e delays [11]. All in all, he mode n wo ld is ull o a -
ious ne wo ks, wi h hei di e en componen s communica -
ing wi h one ano he , he p ocesses o which a e a ec ed by
he so-called communica ion la encies ine i ably [12], [13].
Because o a e y high in e dependence wi h he e e yday
ou ine and p ac ice, TDSs ha e a ac ed esea che s and
enginee s since he nascence o mode n sys ems and con ol
heo y. This is suppo ed by he ac ha he exis ence o
delays may cause much wo se dynamical p ope ies o he
sys em (especially, hei s abili y and pe iodici y), and he
a ac i eness also inc eases because o ha he inclusion
o delays o en leads o mo e ealis ic (in ini e-dimensional)
models bu i signi ican ly complica es he analysis and syn-
hesis [14]. In he ecen decades, an inc easing numbe o
in e na ional mee ings and esea ch a icles ha e deal wi h
VOLUME 6, 2018
2169-3536 2018 IEEE. T ansla ions and con en mining a e pe mi ed o academic esea ch only.
Pe sonal use is also pe mi ed, bu epublica ion/ edis ibu ion equi es IEEE pe mission.
See h p://www.ieee.o g/publica ions_s anda ds/publica ions/ igh s/index.h ml o mo e in o ma ion.
35457
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
FIGURE 1. The o al numbe o ecen ly published esea ch a icles
dealing wi h delay(s) lis ed in Scopus and WoS indexing da abases.
TDSs; i is wo h no ing ha some impo an ecen books
summa ize he up- o-da e knowledge [15]–[22]. In addi ion,
he op wo ld’s leading scien i ic, esea ch and de elopmen
publica ion indexing da abases, Else ie ’s Scopus and Web
o Science (WoS) by Cla i a e Analy ics, would yield o e
200 housand esul s i delay is included in he sea ching
i le o keywo d. Fig. 1 shows ha he o al numbe o such
publica ions has been inc easing o e he pas en yea s, wi h
an app oxima e annual inc emen o 14,000 publica ions.
The e o e, i is almos impossible nowadays o add ess all he
mos ecen ad ances in he ield by only one pe son.
The knowledge o he loci o he cha ac e is ic oo s, i.e.
sys em poles, cons i u es c ucial in o ma ion abou he sys-
ems’ s abili y and dynamics. I is well-known ha a TDS
has an in ini e numbe o cha ac e is ic oo s, i.e. i s spec um
is in ini e, and he e o e he comple e image o hei loci
is e y ha d o analyze, i possible a all. The me hods and
echniques o he spec al analysis o he TDSs deal wi h
he a o emen ioned nega i e e ec s o he delays on he
o e all pe o mance including s abili y and dynamics, and
hey ha e yielded many p ac ical impac s and conclusions.
Al hough esea che s ha e ound many new esul s on a
a ie y o complex sys ems wi h di e en kinds o delays,
including bu no limi ed o ime-in a ian , ime- a ian , non-
linea , chao ic o disc e e- ime ones, e en mo e s udies ha e
been published in he ield o con inuous linea ime-in a ian
(LTI) TDSs which ep esen s he mos common ye s ill a e y
a ac i e class o TDSs.
In spi e o he ela i e simplici y o LTI-TDS models, an
eno mous numbe o books, con e ences and jou nal pape s
indica es ha he e a e many unsol ed p oblems ela ed
o hese sys ems. Fo example, he e y amous wo k o
Richa d [23] p o ided an o e iew o a ious modeling,
s abili y analysis and con ol app oaches o LTI-TDSs and
discussed some open p oblems as well, e.g. he con ol using
delayed in o ma ion.
In addi ion, Gu and Niculescu [24] p esen ed a b oad
o e iew o he s abili y and con ol o TDSs conce ning
p ac ical p oblems and enginee ing applica ions. Va ious
esea ch ela ed o li e a u e o e iew in he ield o TDSs
has been done unde di e en names wi h di e en aims.
Fo ins ance, a delay ji e p oblem in he packe ne wo k
based elephony was discussed in de ails in [25] whe e an
o e iew o a ious a emp ing me hods was also p o ided.
Wang e al. [26] p o ided a su ey on ecen p og ess in
he measu emen o wo ypes o he end- o-end la encies
( ound ip delay and one-way delay) o e he In e ne .
Chen e al. [27] p esen ed a sys ema ic o e iew o he ime-
delay-es ima ion algo i hms anging om he simple c oss-
co ela ion me hod o he ad anced blind channel iden i i-
ca ion based echniques. F ucha d and Schä ke [28] p o ided
an o e iew o he p oblem o bi u ca ion delay om i s
appea ance in F ance a he end o he eigh ies o he mos
ecen con ibu ions. A esea ch ocused on he pe o mance
o domes ic and in e na ional anspo and logis ics sys ems
as pe cei ed by Chinese impo e s and expo e s was p o-
ided by Zhang and Figliozzi [29] along wi h a b oad li e a-
u e e iew o he me eo ic logis ics indus ial de elopmen
in China. A su ey on he i ual-en i onmen -based and
bila e al eleope a ions o space obo s wi h ime delay e ec s
was add essed in [30]. Wang [31] p esen ed a e iew o he
ecen ad ances in delay- ime-based main enance modeling,
which is one o he ma hema ical echniques o op imiz-
ing inspec ion planning and ela ed p oblems. A s ep-by-
s ep in oduc ion o he no ion o ime-delay in classical
and quan um mechanics, aiming a cla i ying i s ounda ion
a a concep ual le el, was made by Sassoli de Bianchi [32].
Cui e al. [33] ga e a su ey on se e al majo sys ema ic
app oaches in dealing wi h delay-awa e con ol p oblems,
namely he equi alen a e cons ain app oach, he Lyapuno
s abili y d i app oach, and he app oxima e Ma ko decision
p ocess app oach using s ochas ic lea ning. The esea ch o
Wang e al. [34] ocused on a e iew o some design and un-
ing me hods o ac i e dis u bance ejec ion con ol me hodol-
ogy o TDS wi h i s applica ions as well. Flunke e al. [35]
p o ided an o e iew o he e ec o delayed coupling and
eedback on dynamical sys ems. A comp ehensi e and excel-
len su ey on s ochas ic hyb id sys ems by Teel e al. [36]
add essed esul s on he s abili y o delayed ones as well.
Simila ly, Tao [37] included TDSs in o his o e iew o some
undamen al heo e ical aspec s and echnical issues o he
mul i a iable adap i e con ol. Doudou e al. [38] p o ided a
comp ehensi e e iew and axonomy o he s a e-o - he-a
synch onous medium access con ol p o ocols wi h espec
o sende - ecei e la encies. Lu and Shen [39] p o ided an
o e iew o he de elopmen in he a ea o scaling laws
o he h oughpu capaci y and he delay in wi eless ne -
wo ks. Liu and Yang [40] summa ized he p og ess o g ey
sys em esea ch om 2004 o 2014 including he p oblems
o obus s abili y o g ey s ochas ic ime-delay sys ems o
neu al ype, dis ibu ed-delay ype and neu al dis ibu ed-
delay ype. An o e iew o he basic esul s and me hods o
he s abili y in es iga ions o highe -o de au onomous linea
di e ence equa ions, wi h a special emphasis on delay di e -
enceequa ionswaspublishedbyČe mák [41].Zhu e al. [42]
p esen ed a a ie y o s abili y condi ions o TDSs wi h
35458 VOLUME 6, 2018
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
a ying delays in he o m o scaled small-gain con-
di ions. In he b oad in oduc ion in [43], Liao e al.
summa izes he ecen ad ances in op imal, obus and p e-
iew con ol o sys ems wi h inpu delays. Mahmoud [44]
p o ided an o e iew o he esea ch in es iga ions in he
ield o ne wo ked con ol sys ems (NCSs) ha include
discussions on add essing communica ion delays. A simi-
la opic was e iewed in [45], whe e he au ho s ocused
on ime-delay uzzy-model-based nonlinea NCSs as well.
To name jus one publica ion example om medicine,
Wechkunanukul e al. [46] p o ided a summa ized e iew on
a ange o coun ies desc ibing he di e ences in ime aken
o seek medical ca e o ches pain and he ac o s which
con ibu e o delay imes. Besides jou nal e iew a icles,
se e al books summa izing he up- o-da e knowledge abou
TDSs [15], [16], [19], [47], and hose ep esen ing a com-
piled se o selec ed pape s om he mos impo an con ol
con e ences and specialized wo kshops wi hin he esea ch
a ea [17], [18], [20]–[22] ha e been published as well.
Cu en ly, a amily o LTI-TDSs is lacking a holis ic
li e a u e o e iew ha co e s and summa izes mos o
he spec al analysis echniques o sys ems wi h cons an
(known o unknown) delays. This su ey pape aims o p o-
ide he eade wi h a li e a u e o e iew o such ecen ly
published esul s. The exigency o such an up- o-da e su ey
can be conside ed as he g ea es con ibu ion o his wo k.
Howe e , because o a huge numbe o published esul s in
his a ea i does no allow o co e all he ideas, he e o e
his s udy does no claim o be exhaus i e. I should be
no ed ha his s udy is p ima ily no in e es ed in con ol-
o ien ed s udies, and ha he eade a e e e ed o he ci ed
li e a u e o mo e de ails – especially, ega ding comple e
ma hema ical issues.
Th oughou he pape , C,R,Zand Ndeno e he se o ,
espec i ely, complex numbe s, eal numbe s, in ege s and
non-nega i e in ege s. The closed uni disk, he uni ci cle and
he imagina y axis a e deno ed as D,T, and C0, espec i ely.
Rndeno es he n-dimensional Euclidean space (a column ec-
o ), Rn×mis he se o all eal- alued ma ices o he dimen-
sion n×m. Fo s∈C, Re(s)and Im(s)deno e, espec i ely,
he eal and imagina y pa s o s;C−
0:= {s∈C|Re(s)<0},
C+=C C−
0, he se o eal polynomials is deno ed as
R[s]. The supe sc ip T s ands o he ma ix anspose. Fo
F(s):s∈C7→ F∈C, de ine se s
H2:= F(s):Z∞
−∞
F(jω)F(jω)dω < ∞,
H∞:= (F(s):kF(s)k∞:= sup
s∈C+|F(s)|<∞),
and le
Lp:= ( ( ):Z
0| ( )|pd 1/p
<∞,p∈N)
o ( ): ∈R7→ ∈R. We use L(·) o he Laplace
ans o m o (·). The uni ma ix o he dimension nis deno ed
as In.
The es o he pape is o ganized as ollows: In he p elim-
ina y Sec ion II, he esea ch ield o LTI-TDSs is speci ied
and hei basic spec al and s abili y p ope ies a e in oduced.
Sec ion III concisely explains he me hodology o he li e a-
u e e iew used he ein he s udy. Sec ion IV p o ides a b ie
o e iew o some ea lie impo an and amous me hods in
he ield o he spec um analysis o LTI-TDSs and ela ed
s abili y issues. The main con ibu ion o he pape is gi en in
Sec ions V and VI. In Sec ion V, a de ailed e iew o ecen ly
published pape s wi h a heo e ical con ibu ion o pole loci
is p esen ed. Sec ion VI includes selec ed ecen esul s on
he s abili y analysis based on he knowledge o he spec um.
In Sec ion VII ecen pape s on ela ed p ac ical p oblems and
enginee ing applica ions a e ou lined. Sec ion VIII lis s ou
esea ch ques ions and poin s ou unexplo ed a eas in he ield
o spec um analysis o LTI-TDSs aiming a p o iding di ec-
ions o he u u e esea ch. Finally, Sec ion IX concludes
he pape .
II. PRELIMINARIES
In his sec ion, LTI-TDSs a e de ined o de e mine he ield
o s udy; hen hei essen ial spec al p ope ies and s abili y
issues ela ed o he eigen alue spec um a e in oduced.
A. LTI-TDS MODEL
A gene al con inuous- ime LTI-TDS wi h cons an delays
can be o mula ed by s a e and ou pu unc ional di e en ial
equa ions (FDEs) as
˙
x( )+XnH
i=1Hi˙
x −τH,i+ZL
0
Hd(τ)˙
x( −τ)dτ
=A0x( )+XnA
i=1Aix −τA,i+B0u( )
+XnB
i=1Biu −τB,i+ZL
0[Ad(τ)x( −τ)
+Bd(τ)u( −τ)]dτ
y( )=Cx ( )+XnC
i=1Cix −τC,i
+ZL
0
Cd(τ)x( −τ)dτ(1)
whe e x( )∈Rn,u( )∈Rm,y( )∈Rl, s and o he ec o
o s a e a iables, inpu s and ou pu s, espec i ely, ˙
x( )=
dx( )/d ,Ai,Ad(τ),Bi,Bd(τ),C,Cd(τ),H,Hd(τ)a e
eal- alued ma ices o compa ible dimensions, 0 < τ·,1<
τ·,2< . . . ≤Lexp ess lumped (poin -wise) delays and
con olu ion in eg als cha ac e ize dis ibu ed delays. I τ·,i=
n·,ih o all iwhe e n·,i∈N, wi h he base delay, h∈R, delays
a e called commensu a e. The ini ial condi ion is a unc ion
segmen x(θ)=ϕ(θ), θ ∈[−L,0],ϕ∈`([−L,0],Rn),
whe e `means he Banach space o con inuous- ime unc-
ions mapping θ∈[−L,0]in o Rnequipped wi h he
sup eme no m k·ks. Fo ≥0, deno e x =x (θ, ϕ)=
x( +θ), θ ∈[−L,0] o he ini ial da a ϕ. I ∃iso ha
VOLUME 6, 2018 35459
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
Hi6= 0o Hd(τ)6= 0, he sys em is o neu al ype;
o he wise, i is o e a ded ype. Fo he sake o his e iew,
e a ded and neu al sys ems wi hou dis ibu ed delays a e
abb e ia ed as RTDSs and NTDSs, espec i ely, and hose
wi h (non-lumped) delay dis ibu ion le be dRTDSs and
dNTDSs, espec i ely.
No e ha an LTI-TDS can also be ep esen ed in he con-
ex o he Hilbe space as ollows. The homogeneous s a e
equa ion has he o m ˙
x( )=_
`( )x , ≥0 whe e _
`( ):
R+×X→Rn,X=([−L,0],Rn)wi h Xbeing he
Hilbe space o con inuous unc ions on he in e al [−L,0].
Fo u he de ails, he eade is e e ed e.g. o [48].
By using he Laplace ans o m, one can cons uc he
cha ac e is ic unc ion
1(s)=de
sI+PnH
i=1Hiexp−sτH,i
+RL
0Hd(τ)exp(−sτ)dτ
−A0−PnA
i=1Aiexp−sτA,i
−RL
0Ad(τ)exp(−sτ)dτ
(2)
Fo RTDSs and NTDSs, (2) is a quasipolynomial; howe e ,
due o dis ibu ed delays i akes a o m o a quasipolyno-
mial ac ion whe e some nume a o /denomina o oo s can
be mu ually algeb aically canceled. Sys empoles (eigen al-
ues o cha ac e is ic alues) cons i u ing he spec um 6o
(1) sa is y
6:= {s:1(s)=0}.(3)
Le he nume a o (i applicable) o 1(s)be deno ed as
¯
1(s):= Pn1
i=0di(s)siwhe e dia e exponen ial polynomials
and n1≥n. Then dn1(s)=dn1,0+P˜n
i=1dn1,iexp(−s˜τi)
wi h dn1,·∈R,˜n≥nH,˜τi≤L+PnH
i=1τH,icons i u es
he associa ed cha ac e is ic exponen ial polynomial. The
essen ial spec um o a (d)NTDS eads
6ess := s:dn1(s)=0.(4)
B. BASIC SPECTRAL PROPERTIES
Le us in oduce some e y basic p ope ies o 6,6ess as
ollows, lea ing p oo s o e e ences:
P oposi ion 1 [14]–[16]: Fo a (d)RTDS i can be deduced
ha :
(i) I he nume a o o ¯
1(s)is a quasipolynomial no a
polynomial, hen |6|= ∞;
(ii) All sk∈6a e isola ed;
(iii) The e a e only ini ely many cha ac e is ic oo s sk∈
6in he s ip {β1<Re(sk)< β2}⊂C;
(i ) Fo any β∈Rwi h β < 0, only ini ely many poles
a e loca ed in he hal -plane Res > β, while in ini ely many
ones a e loca ed o he le -hand side o Res =β;
( ) Isola ed poles beha e con inuously and smoo hly wi h
espec o τand pa ame e s on C.
P oposi ion 2 [14], [24], [48], [51]: Fo a (d)NTDS he
ollowing s a emen s a e ue:
(i) Le sk∈˘
6⊆6,se,l∈˘
6ess ⊆6ess wi h |sk−1|<|sk|,
se,l−1<se,l. Then o e e y ε > 0, he e exis K,L, such
ha sk−se,l< ε o k>K,l>L. I means ha bo h
sys em and essen ial spec a cons i u e e ical s ips a high
equencies which con e ge o each o he ;
(ii) De ine γ:= sup{Re(6ess)}, hen limk→∞ Im(sk)=
∞ o |sk−1|<|sk|,sk∈˘
6as limk→∞ Re(sk)=γ;
(iii) In he hal -plane wi h Re(s)> γ , he e may lie
in ini ely many sys em poles;
(i ) The alue o γis no con inuous wi h espec o τ
(whe e τ∈Rnτ>0 ep esen s he ec o o all sys em
delays;
( ) De ine ˜γ:= sup{γ(τ+δτ): ∀kδτk< ε, ε > 0},
he sa e uppe bound es ima ion ¯c( ha is con inuous wi h
espec o delays) on ˜γcan be calcula ed om
¯c=c∈R:X˜n
i=1
dn1,i
dn1,0
exp(c˜τi)=1,(5)
see e.g. [52]. Only a ini e numbe o poles is loca ed in he
hal -plane wi h Re(s)>¯c≥ ˜γand hey a e isola ed.
The spec al abscissa is he unc ion
α(p):= p7→ sup {Re6}(6)
whe e pis a sys em pa ame e (including delay).
P oposi ion 3 [53]: Fo α(τ)o a (d)NTDS (1) he ollow-
ing wo claims a e alid:
(i) The unc ion may be non-smoo h and hence no di e -
en iable; e.g. in poin s wi h mo e han one eal pole o conju-
ga e pai s wi h he same maximum eal pa ;
(ii) I is non-Lipschi z; o ins ance, a poin s whe e he
maximum eal pa has mul iplici y g ea e han one.
C. LTI-TDS STABILITY
In his subsec ion, s abili y issues o delayed sys ems, which
a e closely ela ed o hei eigen alue spec um, a e con-
cisely add essed. Tasks o s abili y analysis as well as hose
a emp ing o gua an ee he s able con ol sys em cons i u e
he p ima y p oblems sol ed when dealing wi h he sys em
spec um.
De ini ion 1 [14]–[16], [48], [54]: The sys em (1) (o
mo e p ecisely, i s null solu ion) is said o be s able, i o
any ε > 0, he e exis s δ(ε)>0 such ha kϕks:=
maxθ∈[−L,0]kϕ(θ)k< δ implies ha kx( )k< ε o any
≥0 whe e k·kdeno es he Euclidean no m. I , mo eo e ,
o any ϕ∈`([−L,0],Rn)holds ha lim →∞ x( )=0,
he sys em is asymp o ically s able. Sys em (1) is said o
be exponen ially s able i he e exis a>0, µ > 0 such
ha kx (θ, ϕ)ks≤aexp(−µ )kϕks,∀ ≥0 o all ϕ∈
`([−L,0],Rn).
P oposi ion 4 [14], [15], [24], [48]: A (d)RTDS (1) is
(i) asymp o ically and exponen ially s able i and only i
α(·)<0,
(ii) s able i and only i α(·)≤0 and o any pole sk∈C0
i holds ha
ankskI−A0−PnA
i=1Aiexp−skτA,i
−RL
0Ad(τ)exp(−skτ)dτ=n−qk(7)
whe e qkis he algeb aic mul iplici y o sk.
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L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
Whe eas asymp o ic and exponen ial s abili y coincide o
a RTDS, i is no he same case o a NTDS. The mos delica e
is he case o poles in one chain o neu al- ype gi en by (ii) in
P oposi ion 2 shaped asymp o ically o he imagina y axis ye
loca ed in C−
0(wi h no coun e pa in C+)
P oposi ion 5 [14], [48], [51], [55]: Fo a (d)NTDS gi en
by (1), he ollowing s a emen s hold:
(i) I is exponen ially s able i and only i he e exis s ε > 0
such ha α(·)<−ε;
(ii) I α(·)<0 and he e is a c i ical chain o poles close o
C0, asymp o ic s abili y may o may no occu . Fo ins ance,
i H16= 0,Hi=Hd=0,i>1, i depends on geome ic and
algeb aic mul iplici ies o eigen alues o H1.
The amily o (d)NTDSs is equipped wi h he ollowing
wo speci ic s abili y issues.
De ini ion 2 [16], [56]: A (d)NTDS is said o be o mally
s able i
ank"I+XnH
i=1Hiexp−sτH,i
+RL
0Hd(τ)exp(−sτ)dτ#=n,∀s∈C+.(8)
I is said o be s ongly s able i ˜γ < 0.
To eph ase he de ini ion, o mal s abili y means ha sys-
em (1) has only a ini e numbe o poles in C+, whe eas he
s ong one exp esses ha i emains o mally s able unde
small delay de ia ions (see i ems (i ) and ( ) o P oposi-
ion 1 and P oposi ion 2).
O he commonly conce ned s abili y issues, such as BIBO
(Bounded-Inpu Bounded-Ou pu ) and H∞s abili ies canno
be in es iga ed solely om pole loci. No e ha , in he single-
inpu single-ou pu (SISO) case – o he simplici y – BIBO
means ha |u( )|<M1,M1>0 implies |y( )|<M2,
M2>0 o ≥0; o equi alen ly, he sys em impulse
esponse g( )∈L1. The sys em is H∞s able i i s ans e
unc ion sa is ies G(s)∈H∞(i.e. i is a bounded analy ic
unc ion in C+); o equi alen ly, u( )∈L2implies y( )∈L2.
Fo ins ance, le G1(s)=1/(s+sexp(−s)+1),G2(s)=
G1(s)/(s+1),G3(s)=G1(s)/(s+1)4. All he h ee
ans e unc ions ha e poles asymp o ically app oaching C0
om he le a in ini y. I can be p o ed ha G1/∈H∞,
ye G2,G3∈H∞, and G1,G2a e no BIBO s able, unlike
G3[57].
De ini ion 3: Sys em (1) is said o be (weakly) delay-
independen s able (DIS) i i emains s able o any τ∈
Rnτ>0. I is said o bedelay-dependen s able (DDS) i
i is s able o a disjoin se Iτ,io egions in he delay
space.
No e ha DIS and DDS acco ding o De ini ion 3 a e
usually conside ed in e ms o exponen ial o asymp o ic
s abili y.
III. LITERATURE REVIEW METHODOLOGY
The pu pose o a sys ema ic e iew is o summa ize he bes
a ailable esea ch on a pa icula opic. I should anspa -
en ly collec , analyze and syn hesize he esul s o ele an
esea ch. As s a ed abo e, he au ho s do no da e o call his
su ey as sys ema ic due o he eno mous numbe o esul s
on he add essed opic. Howe e , hey a emp ed o ollow he
p inciples o he sys ema ic e iew. Fou s eps o he li e a u e
e iew a e implemen ed in his esea ch: planning, sea ching,
sc eening, and ex ac ion.
In he planning phase, he p oblem o be add essed is
speci ied in he o m o clea esea ch ques ions. Based on
he p elimina y sec ion he ollowing esea ch ques ions a e
conside ed he e: Wha is he cu en s a us o esea ch on he
spec um analysis o LTI-TDSs wi h cons an delays? Wha
a e he open p oblems in his ield?
In ea ly 2018, esea ch pape s ela ed o he spec um anal-
ysis o LTI-TDSs wi h cons an delays we e sea ched and
collec ed by he au ho s. No e ha nonlinea , ime- a ying,
s ochas ic, non-in ege -o de sys ems, e c., a e no wi hin he
scope o his s udy.
Inclusion and exclusion c i e ia a e de ined and applied
o he ound esul s in he sc eening phase. To p esen up-
o-da e esul s, he publica ion pe iod was de e mined o be
wi hin he las i e yea s (i.e., om 2013 o 2017). I is
no ewo hy ha he au ho s o his e iew a e by no means
in ended o claim ha ea lie esul s a e less impo an – ye ,
his pape is p ima ily aimed a gi ing an o e iew o some
mos ecen esul s. Howe e , some o he s (ou o his pe iod)
o c i ical impo ance (i.e. hose no wi hin he pe iod) a e
also included when app op ia e, and a b ie o e iew o he
amous ea lie me hods is p esen ed as well. A o al amoun
o 137 esul s co e ed by he a o emen ioned pe iod ha e
been inally selec ed o p o ide some signi ican insigh s in o
he conside ed esea ch ques ions, based on he sc eening o
abs ac s and he linkage o pa icula ci ed pape s and ci ing
sou ces; he ele ance o he opic o his e iew has been
he mos impo an selec ion c i e ion. Fo ins ance, me hods
based on Lyapuno s abili y heo y and linea ma ix inequal-
i ies (LMIs) a e mos ly ou o his scope o his s udy since
hey usually p o ide only he es ima e o he exponen ial
decay, i.e. he spec al abscissa, no a deepe insigh in o he
spec um.
Finally, in he ex ac ion phase, open esea ch ques ions
a ising om he analysis o selec ed pape s a e concisely
in oduced and discussed (see Sec ion VIII).
IV. SOME WELL-ESTABLISHED METHODS OF LTI-TDS
SPECTRUM ANALYSIS
In his sec ion, we b ie ly summa ize selec ed signi ican and
amous well-es ablished di ec me hods on LTI-TDS spec-
um analysis and ela ed s abili y issues.
A. SPECTRUM APPROXIMATION BY SPECTRAL AND
PSEUDOSPECTRAL METHODS
This me hodology a emp s o es ima e he in ini e-
dimensional spec um by a su icien ly accu a e ini e-
dimensional one by means o a disc e ized s a e-space
o mula ion (i.e., he disc e iza ion o he so-called solu ion
ope a o o he in ini esimal gene a o ).
VOLUME 6, 2018 35461
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
The solu ion ope a o ( )is de ined by he ela ion
( )ϕ=x (ϕ), ≥0.(9)
The in ini esimal gene a o :D( ) ⊆`→`o ( )
has domain
D( )=ϕ∈`, ϕ0:=dϕ(θ)/dθ∈`:Fl(ϕ, 0)=F (ϕ, 0)
(10)
whe e Fl(ϕ, 0),F (ϕ, 0)exp ess le -hand and igh -hand
sides, espec i ely, o he i s equali y in (1) whe e xis
subs i u ed by ϕ, =0 and u(·)≡0. The gene a o ac s
as ϕ=ϕ, ϕ ∈D( ). Then (1) can be ea ed as an abs ac
Cauchy p oblem gi en by he ollowing ope a o (o dina y)
di e en ial equa ion
˙
x =x , ≥0
x0=ϕ. (11)
I holds ha
sk=1
lnµ, µ ∈σ( )/{0}
sk∈σ( ).(12)
whe e sk∈6, and σ(·)deno es he ma ix spec um.
Hence, he p oblem can be ans o med in o a sui able ma ix
disc e iza ion o he solu ion ope a o o i s in ini esimal
gene a o .
Many o hese me hods disc e ize x by a ini e-
dimensional app oxima ion in he o m o a block ec-
o X ∈`Nwi h componen s ˜
x θN,iwhe e `Nis he
space o disc e e unc ions de ined on he g id N=
θN,i,i=0,1,...,N,θN,0=0, θN,i> θN,i+1,θN,N=
−L, and ˜
x is he app oxima ion o x . Then (11) can be
app oxima ed as
˙
X =ANX
AN∈Rn(N+1)×n(N+1).(13)
and he solu ion ope a o (9) is usually compu ed by he
ollowing one-s ep app oxima ion
X +1 =TN(1 )X ,TN∈Rn(N+1)×n(N+1)
1 =θN,i+1−θN,i.(14)
whe e di e en echniques a e used o de e mine ANo
TN(1 ), see e.g. [14], [58]–[60].
B. FREQUENCY-DOMAIN APPROACHES TO GET
FINITE-DIMENSIONAL MODEL REDUCTION
Whene e an LTI model (1) is ep esen ed by he ans e
unc ion, one may apply a a ional app oxima ion o expo-
nen ial e ms o ob ain a ini e-dimensional model, he spec-
um o which can be easily compu ed [61], [62]. Howe e ,
some a i icial oo s appea a e he app oxima ion. Fo
ins ance, he n h o de Padé, diagonal Padé, Lague e and
Kau z shi app oxima ions can be exp essed as exp (−τs)≈
p(−s)/p(s)whe e, espec i ely,
p(s)=Xn
k=0n
k(2n−k)!
(2n)!(τs)k,
p(s)=Xn
k=0
(2n−k)!
k!(n−k)!(−sτ)k,
p(s)=lim
n→∞1+τs
2nn
.
p(s)=1+τs
2n+1
2τs
2n2n
.
Theses app oxima ions play an essen ial ole in some o he
delay app oxima ion concep s. Fo ins ance, a amous esul
based on he T o e -Ka o app oxima ion heo em [63] o
s ongly con inuous semig oups was de eloped in [64]. Using
his gene al amewo k, wo amilies o pa icula app oxima-
ion schemes we e cons uc ed. App oxima ion o he s a e
is done by unc ions which a e piecewise polynomials on a
mesh (m- h o de splines o de iciency m).
C. LAMBERT W FUNCTION
The use o he Lambe W unc ion yields he analy ical
solu ion o he pole loci. Howe e , i is only applicable o
sys ems wi h commensu a e delays. The Lambe unc ion
W(z)is a mul i alued complex unc ion de ined as
z=W(z)exp(W(z)) (15)
whe e i s solu ion cons i u es an in ini e se o b anches and
zcan be a scala o a ma ix [67]. We e e o Wk(z)as he
k- h b anch o he Lambe W unc ion o z. The main idea
is o exp ess he solu ion o (3) by means o he Lambe
W unc ion. Fo ins ance, le he scala sys em be ˙x( )=
a0x( )+a1x( −L), hen he cha ac e is ic equa ion can
be w i en as (s−a0)exp(sL)=a1which is equi alen
o he iden i y L(s−a0)=W(a1Lexp(−a0L)), i.e., z=
a1Lexp(−a0L), and hence he solu ion o he cha ac e is ic
equa ion eads s=W(a1Lexp(−a0L)) /L+a0.
D. ARGUMENT PRINCIPLE TECHNIQUE
I he ask is o decide on he numbe NDo poles loca ed
inside a egion D⊂Cgi en by he closed posi i e Jo dan
cu e ϕ+, i holds ha
ND=1
2πjZ
φ+
10(s)
1(s)ds=1
2π1φ+a g1(s),(16)
whe e 1(s)is a e a ded cha ac e is ic quasipolynomial, and
10(s)=d1(s)/ds[65]. F om (16) i ollows ha , i 1(0)>
0, 1(s)6= 0 o any s=jω, ω ≥0, hen
=n
2−1
π1a g1(s)
s=jω,ω∈[0,∞)
whe e means he numbe o poles in C+. In [66], he esul
was ex ended o NTDSs as ollows: Conside a s ongly s able
35462 VOLUME 6, 2018
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
sys em wi h 1(0)>0, 1(s)6= 0 o any s=jω, ω ≥0,
hen (1) is asymp o ically s able i and only i
nπ
2−8≤1a g1(s)
s=jω,ω∈[0,∞)≤nπ
2+8,
8=a csinX˜n
i=1dn1,i.
As clea om P oposi ion 4 and P oposi ion 5, exponen-
ial s abili y can e ec i ely be s udied by he de e mina ion
o sys em (o e en delay) pa ame e s such ha i s spec-
um c osses imagina y axis. A bunch o well-es ablished
echniques ollows. The echniques ely on wo impo -
an undamen al heo e ical esul s; namely, he associa ed
D-decomposi ion/τ-decomposi ion heo em [68], [69],
and he con inui y p ope y o sys em eigen alues wi h
espec o sys em pa ame e s [70]. By D-decomposi ion/
τ-decomposi ion heo em, one can ind coun ably many
egions in he pa ame e space, whe e in each egion he
sys em can possess only a ini e numbe o . The sys em
is s able o all he poin s in ha egion whene e =0, and
he bounda ies ha sepa a e hese egions a e o med by some
c i ical pa ame e alues ha impa imagina y-axis poles.
E. REKASIUS SUBSTITUTION
Rekasius [71] p oposed he ollowing exac (unlike he Padé
one) ans o ma ion implemen ed o calcula e he s able-
uns able egions o an LTI-TDS
exp(−τijω)→1−jTiω
1+jTiω,i=1,2, . . . nτ(17)
whe e Ti∈Ra e so-called pseudo-delays. As a consequence,
a new cha ac e is ic polynomial in ωpa ame e ized by Tiis
ob ained.Thesubs i u ion(17)is alid and exac i τcomplies
wi h
τi,k=2ω−1 an−1(ωTi)+kπ,k∈Z.(18)
Tha is, e e y pai sa is ying (17) wi h s=jωis mapped
o delays acco ding o (18). These poin s bisec he delay
pa ame e space in o in e als, whe e in each in e al he
sys em is ei he s able o uns able.
F. ELIMINATION OF TRANSCENDENTAL TERMS IN THE
CHARACTERISTIC EQUATION (DIRECT METHOD)
The co e idea o he di ec me hod [72] lies in he i e a i e
elimina ion o exponen ial e ms in 1(s)wi h commensu a e
delays based on he ac ha whene e 1(jω)=0, i also
holds ha 1(−jω)=0 ( he complex conjuga e symme y o
complex oo s). Le 1(s)=P2
i=0di(s)siexp(−ihs),h>0,
hen cons uc
1(1)(s):= d0(−s)1(s)−d2(s)1(−s)exp(−2hs)
=d0(−s)d0(s)−d2(−s)d2(s)
+(d0(−s)d1(s)−d1(−s)d2(s)) exp(−hs)
=: d(1)
0(s)+d(1)
1(s)exp(−hs).(19)
The second i e a ion eads
1(2)(s):= d(1)
0(s)d(1)
0(−s)−d(1)
1(s)d(1)
1(−s)
=: d(2)
0(s).(20)
Since (20) has no exponen ial e ms and i holds ha he
elimina ion p ocedu e p ese es he imagina y solu ions sc=
±jωco he o iginal cha ac e is ic equa ion, one can w i e he
ollowing polynomial equa ion o ge he imagina y poles
Wω2
c=1(2)(jωc)=0.(21)
The se o po en ial base delays is hen compu ed by
hk=ω−1
c
an−1
Red(1)
0(jωc)/d(1)
1(jωc)
Im−d(1)
0(jωc)/d(1)
1(jωc)
+2kπ
,
k∈N0.(22)
The c ossing p ope y is hen checked by a nonze o RT
alue calcula ed ia RT =sgn nd(1)
0(jωc)dWω2
c/dω2
co.
G. FREQUENCY SWEEPING
This is e y simple ye e ec i e echnique o de e mine s a-
bili y bo de in he pa ame e (o delay) space see e.g. [73].
Howe e , i can be used only i 1(s|p)is linea wi h espec
o he unknown pa ame e s p. The leading idea was based on
he pa i ion o 1(s,p)|s=jωin o eal and imagina y pa s,
i.e., Re(1(jω, p)) and Im(1(jω, p)), espec i ely. Then,
he common solu ion o Re (1(jω, p)) =Im(1(jω, p)) =0
can be plo ed in he pa ame e space o ω∈[0, ωmax],
whe e ωmax means a pa icula ly selec ed maximum e-
quency. The gene a ed plo in he pa ame e space ep esen s
a po en ial s abili y bounda ies and hey de e mine egions
ha can be u he es ed in o de o iden i y he s able ones
( o ins ance, ia he D-decomposi ion p ocedu e).
H. SCHUR-COHN PROCEDURE
This p ocedu e can be used o compu e he s abili y ma gin
o LTI-TDSs wi h commensu a e delays [15], [74]. The
o iginal Schu -Cohn p ocedu e compu es he de e minan o
a pa i ioned ma ix
S=31(s)32(s)
3∗
2(s)3∗
1(s)
whe e 31, 32∈Cn×n[s]a e app op ia e ma ices o e he
ing o polynomial in s, and he as e isk deno es he pa icula
He mi ian.
In o de o de e mine all he imagina y oo s o he cha -
ac e is ic quasipolynomial 1(s)wi h he base delay τ0and
he commensu acy deg ee nC, he a iable q=exp(−τ0s)
is in oduced and 1(s)is ew i en as a polynomial p(s,q)
in wo unknowns o e R. Then, he Schu -Cohn c i e ion
sol es he p oblem o compu ing he alues o ωsuch
ha p(jω, q)=0 by mul iplying p(jω, q)by q−i,i=
0,1, . . . nC−1, and he complex conjuga e ¯p(jω, q)by qi,i=
1, . . . nC. This yields a sys em o 2nChomogenous equa ions
wi h he unknowns qi.
VOLUME 6, 2018 35463
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
I. KRONECKER SUM AND MATRIX PENCIL METHOD
To in oduce hese echniques concisely, le A,B∈Rn×n,
hen he spec um o he K onecke sum A⊕B=A⊗I+I⊗B
consis s o all sums in he o m o λ+µ, whe e λ, µ belong
o he spec um o A,B, espec i ely, and Iis he iden i y
ma ix [15], [74]–[76].
Fo ins ance, conside he ollowing simple sys em
˙
x( )=A0x( )+A1x( −L).(23)
I some sk,−ska e ze os o 1(s) o (23), hen skbelongs
o he spec um o A0+A1z, while −sklies in he spec um
o A0+A1z−1, whe e z=exp(−sL); hence
1ass,1(z):= de (A0+A1z)⊕A0+A1z−1=0.(24)
I skis a pu ely imagina y sys em pole, (24) mus hold.
The ansc ip ion o he K onecke sum in o he ma ix
pencil o m educes he p oblem o he exis ence o he
uni ci cle o gene alized eigen alues o he co esponding
ma ix pencil. C ossing delays a e hen gi en by τk=
ω−1(a g(z)+2kπ),k∈Z.
J. KRONECKER MULTIPLICATION METHOD
The main esul ha cha ac e izes his amewo k can be
exp essed by he ollowing heo em.
Theo em 1 [77]: Le 5be he se o all eigen alues o he
ma ix
5=000
−32−31∈Rn2×n2(25)
whe e
00=I⊗I,01=I⊗A0−A0⊗I,
02=A1⊗A1−A0⊗A0,
31=0−1
001,32=0−1
002.(26)
Then ⊆5whe e := Im6∩C0.
Las bu no leas , le us in oduce wo well-es ablished
amewo ks om he ield ha enables o decide abou he
exponen ial s abili y, i.e., whe he all cha ac e is ic oo s a e
loca ed in C−
0.
K. EXTENDED HERMITE-BIEHLER THEOREM
The amous wo k o Bellman and Cooke [78] applied he
ex ension o he He mi e-Biehle heo em o RTDSs based on
he ea lie wo k o Pon yagin [79]. This heo em yields he
necessa y and su icien condi ions o he s abili y o 1(s).
Le 1∗(s)=1(s)exp(sL)(ha ing he same oo s as 1(s)),
1∗
R(s)=Re(1∗(s)),1∗
I(s)=Im(1∗(s)), hen he sys em
is s able i and only i 1∗
R(s)and 1∗
I(s)ha e only simple
eal oo s, hese oo s in e lace, and 1∗
I(ω0)01∗
R(ω0)−
1∗
I(ω0)1∗
R(ω0)0>0 o some ω0∈(−∞,∞). The
c ucial p oblem is o make su e ha 1∗
R(s),1∗
I(s)ha e only
eal oo s, see e.g. [80] o de ails.
L. CLUSTER TREATMENT OF CHARACTERISTIC ROOTS
The amous Clus e T ea men o Cha ac e is ic
Roo s (CTCR) pa adigm was de eloped mo e han a decade
ago, see e.g. [81], and i o iginally consis s o he ollow-
ing basic s eps: Fi s , he cha ac e is ic quasipolynomial
is ans o med o he co esponding polynomial p(ω, T)
including pseudo-delays Ti( ia he Rekasius subs i u ion).
Second, he e en ual polynomial is subjec ed o he Rou h
a ay scheme o ge he so-called po en ial s abili y swi ching
hype su aces (o , he ke nel cu es in he 2D delay space)
de ined as
℘0(τ):= ττ:1(s, τ)=0,s=jω, ω ∈c(27)
whe e τ ep esen s all ke nel poin s in he delay space
and cmeans he se o all possible co esponding c ossing
equencies. The necessa y c ossing condi ion is ha he oo
endency RT := sgnReds/dτi|s=jω,i=1,2, .., nτ,
is nonze o, whe e nτdeno es he numbe o independen
delays in 1(s). As he hi d s ep, he ke nel cu es (o , po en-
ial s abili y swi ching hype su aces) oge he wi h he c oss-
ing equencies gi e ise o he so-called o sp ing hype su -
aces ℘o (τ)which a e gene a ed based on he knowledge
ha whene e he e exis s an imagina y pole sc= ±jωc
o some τ0, he same pole exis s o τi=τ0,i+2πki/ωc,
τ0,i−2π/ωc≤0, i=1,2, .., nτ;ki∈N, as well. Finally,
he D-subdi ision me hod [68] is deployed o de e mine he
numbe o uns able poles on he igh hal -plane, s a ing om
he delay- ee case.
V. OVERVIEW OF RECENT THEORETICAL RESEARCH ON
POLE LOCI CALCULATION, COMPUTATION AND
APPROXIMATION
This sec ion is dedica ed o he e iew o ecen heo e ical
esea ch on he analysis o he eigen alue (cha ac e is ic
oo s) spec um 6o LTI-TDSs wi h cons an pa ame e s and
delays. Me hods o he oo loci compu a ion o app oxima-
ion and hose de e mining a pa o 6wi hin he speci ied
egion in Ca e p esen ed.
Me hods o he compu a ion o he cha ac e is ic oo s
co e ed he ein can be di ided in o he ollowing ca ego ies:
(i) Nume ical me hods based on he app oxima ion o
he solu ion ope a o associa ed wi h he sys em (1) o i s
in ini esimal gene a o ia spec al (o pseudospec al)
me hod;
(ii) Semidisc e iza ion and ull-disc e iza ion me hods;
(iii) Nume ical in eg a ion and di e en ial quad a u e
me hods;
(i ) Con ou in eg al me hod;
( ) Lambe W unc ion;
( i) Special nume ical, semi-analy ic and analy ic
me hods.
A. SPECTRAL AND PSEUDOSPECTRAL METHODS
In he li e a u e, Wu and Michiels [60] summa ized he
me hodology in oduced in (9)-(14) o ge ANconce ning he
35464 VOLUME 6, 2018
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
compu a ion o all cha ac e is ic oo s in a gi en igh hal -
plane and hey also p o ided a p ocedu e o he au oma ic
selec ion o N; Chebyshe polynomials o he i s kind we e
u ilized o app oxima e x by ˜
x – he so-called pseudospec al
colloca ion (PsC) me hod is hen ob ained ( he e m pseu-
dospec al means ha he solu ion is app oxima ed in a ini e
dimensional subspace).
Ano he poin o iew was p esen ed by Leho zky and
Inspe ge [48] whe e a de ailed compa ison o a amily o
weigh ed esidual- ype me hods o app oxima e he ope a o
di e en ial equa ion (11) in he Hilbe space (see a e (1))
was p o ided. In mo e de ail, le he solu ion o (11) be
app oxima ed by ˜
x (θ)=PNa
j=1φj(θ)aj( )whe e θ∈
[−L,0], and aj( )a e unknown a iables spanned by he
basis φjNa
j=1, hen he esidual unc ion eads (θ)=˙
˜
x −
˜
x0 . The objec i e is o de e mine aj( ) o ge ˜
x →x as close
as possible. Me hods o weigh ed esiduals yield , ψj:=
R0
−L (θ)ψj(θ)dθ=0, j=1,2,...,Na[48], [82], whe e
ψj(θ)a e es unc ions, which can also be ep esen ed in a
ma ix o m as
N˙
a( )=Ma ( ).(28)
whe e a( )=aj( )Na
j=1,N,M∈RnNa×nNa, and en ies o
hese ma ices include inne p oduc s φj, ψj. In he so-called
pseudospec al au app oxima ion (PsT), he ough solu ion
segmen is gi en by
˜
x (θ)=XNa
j=1φj(θ)x (θ).(29)
whe e φj(θ)a e Lag ange base polynomials, i.e. φjθN,i=
1 o i=j, else φjθN,i=1. The ba ycen ic o mula
o Lag ange polynomials used in [48] is mo e nume ically
s able and de i a i es o φja e hen less complica ed. The
PsC me hod in oduced abo e (also called he Chebyshe
spec al con inuous- ime app oxima ion) educes he e o
˜
x (θ)−x (θ)by means o he selec ion o Chebyshe nodes
o he nodes o in e pola ion in (29). The spec al Legend e
au (SLT) me hod employs Legend e polynomials as base
unc ions in (29). The au app oxima ion (TA) uses (29) o ge
(28) in he o m o (13) whe e a ime-dependen ma ix G( )
appea s ins ead o AN. This ma ix con ains φj, ψjwhich
can be app oxima ed by using quad a u e me hods.
Unlike he a o emen ioned me hods (PsC, PsT, SLT, TA),
he spec al elemen (SE) me hod (called he ime ini e ele-
men s me hod as well), which was also included in he com-
pa a i e s udy [48], app oxima es ( ) o ob ain he disc e e
mapping. The idea lies in he spli ing o he his o y segmen
[−L,0]in o Enumbe o empo a y elemen s wi h he leng h
1 . Each elemen con ains Nsinne poin s. An app oxima e
solu ion is hen sough in each elemen acco ding o
˜
xk( )=XNs
j=1φj( )xk k,j, ∈[−k1 ,−(k−1)1 ],
k=0,1,...,E(30)
whe e k,j∈[−k1 ,−(k−1)1 ]a e inne ime ins an s
and φj ep esen ial es unc ions. Then (12) is used o
compu e he sys em spec um. No e ha in [48], φja e
Lag ange base polynomials, and he abo emen ioned p in-
ciples we e compa ed in e alia by using he loci o he
igh mos poles he e.
Vyasa ayani e al. [83] compa ed he spec al au (ST)
and he spec al leas -squa e (SLS) me hods. The o -
me one compu es N,Min (28) simply using N=
R0
−Lϕ(θ)ϕT(θ)dθ, M=R0
−Lϕ(θ)ϕ0T(θ)dθwhe e ϕ=
φ1, φ2,...φNaT; whe eas, he la e one a emp s o sol e
he ollowing cons ained op imiza ion p oblem
min 0.5˙
a( )Z0
−L
2
(θ)dθ
s. . FlϕT,0˙
a( )=F ϕT,0a( ).(31)
Base unc ions φjwe e conside ed as mixed Fou ie basis,
shi ed Legend e polynomials and shi ed Chebyshe poly-
nomials, in [83]. The au ho s compa ed he me hods ia
pole loci and s a ed ha he ST me hod is easy o code and
unde s and, and pe o ms be e han he SLS me hod.
To o e come compu a ional bu dens o he eigen alues
associa ed wi h he spa se AN, an i e a i e PsC me hod o
RTDSs was p esen ed by Ye e al. [84]. The spa si y o ANis
explo ed by e o mula ing i s blocks in o K onecke p oduc s
as ollows
AN=RN
MN⊗In(32)
whe e RN=PnA
i=0Li⊗Aiwi h LT
i∈RN+1being
cons an Lag ange ec o s. Then, he shi ope a ion ˜s=
s−ssh,ssh >0, is u ilized o ge eigen alues o ma ix
˜
ANwi h he la ges modulus, ollowed by he compu a ion
o ˜
AN−1. This shi -in e p econdi ioning ans o ma ion
echnique is used o spa se eigen alue compu a ions wi h
a educed dimension by means o he implici ly es a ed
A noldi algo i hm (IRA) [85] o ge K ylo sequences {qk}
ia qk+1=(AN)−1qk. A di e en echnique was p esen ed
by Ye e al. [86] whe e ˜
AN−1=(0N)−15N, in which
5Nis a highly spa se companion- ype cons an ma ix and
˜
0N=eT
1⊗In−PnA
i=0Li⊗Ai. In a simila manne , a pseu-
dospec al disc e iza ion o T( )was published in [87]. The
au ho s u ilize a echnique in oduced in [88]; howe e , ma i-
ces o ming T·(1 )a e e o mula ed by using K onecke
p oduc s o educe hei dimensions. Then, he o a ion-and-
ampli ica ion ope a ion ˜s=αexp (−θj)s, α > 1, θ > 0,
is made p io o he gene a ion o K ylo ec o s ia he IRA,
o accele a e i s con e gence a e. This ope a ion implies ha
sys em poles a e i s o a ed by θand hen ampli ied by α.
Fabiano [89] sugges ed he app oxima ion o A( he ein
called a semidisc e e app oxima ion) o a NTDS ia linea
spline unc ions whe e he p o ed T o e -Ka o ype semi-
g oup con e gence [63] o his scheme as well. The same
au ho used his scheme o in es iga e he DIS p oblem in
[90]. In bo h pape s, he exac eigen alues o ANwe e used
o measu e he accu acy o he app oxima ion.
VOLUME 6, 2018 35465
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
FIGURE 2. G-sec o and S-sec o [159]–[161].
done ia ad anced compu e algeb a echnique, he so-called
a ional uni a ia e ep esen a ion [158], which is a one- o-one
mapping be ween he solu ions o he polynomial sys em and
he oo s o a uni a ia e polynomial. I is wo h no ing ha
he ollowing o m o he Möbius ans o ma ion (mapping
ω∈R∪{∞} o he uni ci cle T:= {x∈C:|x|=1}) was
used
exp(−τjω)→x−j
x+j.(48)
The c i ical pai s can be compu ed by sol ing he ollowing
iden i ies Re{p(ω, x)}=Im {p(ω, x)}=0 whe e p(ω, x)is
he bi a ia e polynomial co esponding o 1(s). The c i ical
delays a e hen ob ained as ollows
τk=ω−1 an−12x
x2−1+kπ,k∈Z.(49)
No e also ha he well-known Rekasius subs i u ion (29)
is ob ained by he se ing x= −Tω, T∈R. An al e na i e
Möbius mapping eads exp(−τjω)→z=u+j ∈T. The
la e sub- esul o [157] included an e icien algo i hm o
compu e he di e en e ms in he Puiseux se ies.
The mo emen o double, iple and quad uple imagi-
na y poles when wo delays a e subjec ed o small de i-
a ions was analy ically s udied wi hou using he Puiseux
se ies in [159]–[161], espec i ely. Two sec o s, g ea (G) and
small (S), we e in oduced in he neighbo hood o he poin in
he delay pa ame e space ha causes he mul iple imagina y
oo , see Fig. 2. This c i ical poin cons i u es a cusp in he
s abili y c ossing cu e. When he delay pa ame e s mo e
in o he G-sec o , one oo ( wo oo s) mo e(s) o C+, and
he o he one ( wo o he s) mo e(s) o C− o he double
(quad uple) imagina y pole. I he pa ame e s mo e in o he
S-sec o , hen one ( h ee) o he oo s mo e(s) o one hal -
plane, and he emaining oo mo es o he o he hal -plane.
Fo he iple pole, i was p o ed in he ci ed pape ha he
s abili y c ossing cu es a e smoo h - wo oo s mo e o one
hal -plane and one oo goes o he o he hal -plane.
B. H∞AND BIBO STABILITY
As men ioned abo e, Bonne e al. [51] p o ided a ho ough
H∞analysis by means o es ima ing he pole loci asymp o ic
o C0which was hen implemen ed by using YALTA so wa e
package by A anesso e al. [120]. H∞s abili y o some
classes o TDSs wi h mul iple chains o poles asymp o ic o
he same se o poin s on C0was s udied in [117] and [118].
Simila app oxima ion ools we e also u ilized o analyze
poles o a ‘‘small’’ modulus and he co esponding BIBO and
H∞s abili y o a NTDS wi h a single delay in [161].
C. STRONG AND ROBUST STABILITY
S ong s abili y has been in oduced in De ini ion 2. By he
no ion o obus s abili y we mean he abili y o a sys em
o emain s able (in some sense) wi h espec o pa ame-
e s luc ua ions, o unde model o pa ame e unce ain-
ies. A s ong s abili y c i e ion o NTDSs was p esen ed
in [103], he de i a ion o which was made by means o he
DQ me hod. Rabah e al. [119], in e alia s udied condi ions
unde which a mixed RTDS/NTDS is s ongly s able.
Du e al. [163] p esen ed necessa y and su icien con-
di ions o exponen ial s abili y o TDS go e ned by
di e en ial-algeb aic equa ions. In pa icula , he obus ness
o his ype o s abili y was s udied when he equa ion is
subjec o s uc u ed pe u ba ions. A compu able o mula o
he s uc u ed s abili y adius was also de i ed.
O en and Mönnigmann [164] p oposed an op imiza ion
me hod o pa ame ically unce ain delay di e en ial equa-
ions wi h s a e-dependen delays. The cen al idea o he
op imiza ion is o s ay o he s abili y bounda ies in he
pa ame e space. As a esul , dynamical p ope ies such as
s abili y can be gua an eed in spi e o pa ame ic unce ain-
ies in he model unde he op imiza ion. The so-called old
bi u ca ion exp essing he si ua ion when a eal pole c osses
he imagina y axis played a c ucial ole in his esea ch.
D. DDS
S abili y issues depended on he alue o τcan be in es iga ed
using se e al me hods and echniques. Two basic amilies
o DDS me hods o compu ing he delay s abili y ma gins
p e ail in he li e a u e; namely, ime-domain indi ec and
equency-domain di ec me hods. In his su ey (dealing
inhe en ly wi h he la e g oup), esea ch esul s u ilize he
ollowing me hods o he delay-ma gin compu a ion:
(i) CTCR;
(ii) Di ec me hod;
(iii) A gumen p inciple (Cauchy heo em) me hod;
(i ) Schu -Cohn c i e ion;
( ) K onecke sum and ma ix pencil me hods;
( i) Lyapuno ma ix (K onecke mul iplica ion)
app oaches;
( ii) O he nume ical, semi-analy ic and analy ic me hods.
The key idea lies in he de e mina ion o all s abili y
swi ching sys em poles (i.e. he cha ac e is ic quasipolyno-
mial ze os) loca ed exac ly on C0, which can be used o
de e mine he s abili y ma gin. In ac , only he igh mos
subse o he spec um makes he sys em swi ch om s abili y
o ins abili y o ice e sa. Techniques included in all he
abo e i ems (excep o (iii)) a e based on he elimina ion o
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L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
exponen ial e ms om 1(s); howe e , only bi a ia e poly-
nomials include he in o ma ion o he c i ical delay alues
explici ly. This can be done by he di ec eplacemen o
he exponen ial e ms by using, e.g., he Möbius o Reka-
sius ans o ma ion (subs i u ion) acco ding o (48) and (49).
A common al e na i e way is o apply he hal -angle angen
subs i u ion as ollows:
exp(−τjω)→cos(υ)−jsin(υ), υ =jω,
cos(υ)=1−z2
1+z2,sin(υ)=2z
1+z2,z= anυ
2,
τk=ω−12 an−1(z)+kπ,
k∈Z, an1(·)∈[0, π).(50)
1) CTCR
Some esea ch esul s ha e ex ended o imp o ed he o iginal
CTCR concep . Sipahi and Delice [165] ocused on he so-
called co e hype su aces and showed some hei ea u es
o he case when nτ>0. The co e hype su aces mean
he image o ℘0(τ)compu ed om he co esponding mul i-
a ia e algeb aic polynomial p(ω, T)in he pa ame e space
o pseudo-delays. The au ho s we e conce ned wi h he iden-
i ica ion o he asymp o ic di ec ions o he delays on he
po en ial s abili y swi ching hype su aces app oaching in in-
i y. These esul s can also be used in connec ion wi h [166] o
s udy s ong DIS by co e ing bo h ini e and in ini e delays.
Jesin ha Ma y and Ranga ajan [167] applied he esul an
heo y in oduced in [165] in o de o in es iga e a new
lexible me hodology o s abili y analysis o in load e-
quency con ol scheme wi h delays in he ansmission o
con ol signals om he con ol cen e o gene a ing uni . The
p oposed me hod o e ed la ge delay ma gin and akes less
compu a ion ime compa ed o some exis ing me hods.
A ecen wo k o Sipahi’s [168] u ilized he knowledge
(based on he abo e-in oduced esea ch) ha i is possible
o compu e he exac ange o he imagina y spec um o
such sys ems o design imagina y poles wi h he objec i e o
manipula e s abili y egions in he delay space. In addi ion,
Kamme and Olgac [169] s udied s abili y o dRTDSs ia he
CTCR pa adigm by means o he equi alence o a gene al
class o dis ibu ed delay sys em o a disc e e-delay sys em
wi h mul iple independen delays.
A compa ison be ween delay space ( ep esen ed by ℘0(τ),
℘o (τ)) and he spec al delay space was p esen ed in [170].
The la e domain con ains poin wise equency in o ma ion
as well as he delay and i was p e e ed he e o i s ad an a-
geous boundedness p ope ies and he simple cons uc ion o
s abili y ansi ion bounda ies.
Gao and Olgac [171]–[173] in es iga ed he bounds o he
imagina y spec a ia he subs i u ion (49) and by deploy-
ing he Dixon esul an heo y [174] o a RTDS wi h an
a bi a y numbe o delays. Consequen ly, he p oo o he
di e en iabili y o he c ossing- equency a ia ions dω/dzi
was p o ided o in es iga e he bounds. As a esul , 2D c oss-
sec ions o he hype su aces we e ex ac ed. The concep
o he so-called 3D building blocks in he spec al delay
space [175] was u ilized o mee his objec i e.
Theexac delaybound o aconsensuso linea mul i-agen
sys ems wi h a ixed and uni o m communica ion ime delay
was de e mined by Cepeda-Gomez in [176] in an e icien
manne by using he CTCR me hodology. A s a e ans o -
ma ion was pe o med o decouple he sys em and simpli y
he p oblem p io o he s abili y analysis.
2) DIRECT METHOD
The ‘‘di ec ’’ e e s o he me hod in oduced by (19)-(22).
Sönmez e al. [177] s udied he DDS p oblem o load e-
quency con ol sys ems wi h cons an communica ion delays
o he commensu a e deg ee o one and wo. Howe e ,
he comple e s abili y windows we e no conside ed because
only he minimum posi i e alue o (22) wi h RT = +1 was
aken as he unique delay ma gin.
3) ARGUMENT PRINCIPLE METHOD
This de ini e in eg al s abili y me hod, o igina ed om he
a gumen p inciple (o he Cauchy heo em), is e ec i e
because i only equi es a ough es ima ion o he es ing
in eg al o e a ini e in e al o judge DDS. Conside he so-
called es ing in eg al F(, a)de ined in Theo em 2. Xu and
Wang [106] p o ed ha i 1(s)o a NTDS has no imagina y
oo s and he condi ion (36) is sa is ied, hen he e exis s a
su icien ly la ge 0>0 so ha o all > 0i holds
ha
∈−F(, 0)
π+n−1
2,−F(, 0)
π+n+1
2(51)
whe e is he numbe o poles in C+. Two DDS algo-
i hms, o inding he pa ame e (delay)-dependen c i i-
cal uppe limi and a pa ame e (delay)-independen uppe
limi wi hou any es ic ion on he numbe o ime delays,
we e p esen ed by Xu e al. [178] who p o ed he ollowing
heo em.
Theo em 4: Assume ha 1(s)has no oo s on C0and (36)
holds. Le 0(τ)=max(ωR,0)whe e ωRs ands o he
maximal posi i e oo o R(ω):= Rej−n11(jω). Then
(37) and (51) a e ue o all >0.
4) SCHUR-COHN METHOD
Mule o-Ma ínez [179] p esen ed a modi ied Schu -Cohn
c i e ion o RTDSs wi h commensu a e delays ha equi es
seeking eal oo s only, which is compa able o he Rekasius
subs i u ion c i e ion. In con as o he classical Schu -Cohn
c i e ion, he app oach is based on he applica ion o ian-
gula ma ices o e a polynomial ing in a simila way as in
he Ju y es o s abili y o disc e e sys ems, and i hal es he
dimension o he subjec ed polynomial. I s a s wi h he con-
s uc ion o a bi a ia e polynomial ∈C[ω, z], (ω, z)=
PnC
i=1b(ω)zi om 1(s)=PnC
i=0d(s)exp(−shi)whe e
b(ω)=d(jω),z=exp(−sh),nCis he commensu acy
deg ee, and hs ands o he base delay. Then, wo asso-
cia ed iangula ma ices a e assembled, om which he
VOLUME 6, 2018 35473
L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
de e minan polynomial ζ(ω)is calcula ed. As a co e o he
app oach, he ollowing heo em holds
Theo em 5: Le ωc>0 be a eal oo o ζ√s, hen
±j√ωcis a pai o poles o he RTDS.
5) KRONECKER SUM AND MATRIX PENCIL METHODS
Ma e al. [180] s udied DDS o a NTDS wi h a single delay
H0˙
x( )+H1˙
x( −τ)=A0x( )+A1x( −τ)(52)
by applying he ma ix pencil and he linea ope a o me hods.
The main esul o he me hod ega ding he DDS p oblem
was ensh ined in he ollowing heo em.
Theo em 6: I skis a pu ely complex oo o 1(s), i is also
a ze o o
1ass,2(s):= de (sH0−A0)⊗(sH0+A0)
−(sH1−A1)⊗(sH1+A1),(53)
see also Louisell [77].
6) LYAPUNOV MATRIX APPROACHES
Conside he sys em (23) again, one app oach is based on he
ac ha any pu ely imagina y oo o 1(s)is also a oo o
he polynomial
1ass,3(s):= de sI+AT
0⊗(sI−A0)−AT
1⊗A1
(54)
ha is also he cha ac e is ic polynomial o he sys em
X00(θ)=X0(θ)A0+X−1(θ)A1
X01(θ)= −AT
1X0(θ)−AT
0X−1(θ)(55)
see (53) o he compa ison. Sys em (55) can be hen
subjec ed o he compu a ion o Lyapuno ma ices. Once
he spec um o (55) is compu ed, c i ical alues o he
delay can be ob ained by subs i u ing hese oo s in o 1(s).
Ochoa e al. [181] adop ed he abo e idea o de i e explici
ela ions be ween he spec um o an o iginal dRTDS and
NTDS, and ha he delay- ee sys em (55), which cons i-
u ed a b idge be ween ime-domain and spec al app oaches.
Delay-dependen s abili y egions we e de e mined as well.
Since 1ass,3(s)has only e en powe s o s, he sea ching o
imagina y poles was educed o he compu a ion o eal oo s
o 1ass,3(λ)λ=s2. To sol e his ask, he au ho s u ilized
S u m‘s heo em ha is based on he compu a ion o sign
changes o he S u m sequence.
Ano he echnique based on he Lyapuno –K aso skii
me hodology o in es iga e delay-dependen ( obus ) expo-
nen ial s abili y o a RTDS was de i ed by Cao [182]. The
au ho used LMIs and slack ma ices o ge he uppe bound
o he exponen ial decay a e. The gi en c i e ion p o ides
he compu a ion me hod o he alue o Lmax, so ha he
sys em is ( obus ly) exponen ial s able o L∈(0,Lmax],
i.e. no o he s abili y windows we e conside ed. A compa -
ison wi h some o he me hods was also gi en o he eade .
Sun e al. [183] de i ed he su icien condi ion o he delay-
dependen asymp o ic s abili y o he closed-loop powe sys-
em wi h p esc ibed deg ee o s abili y α(i.e., he decay
a e o a spec al abscissa) based on he Lyapuno s abili y
heo y and ans o ma ion ope a ion in complex plane, and
p esen ed a me hod based on LMIs o calcula e he delay
ma gin o he closed-loop sys em conside ing he p esc ibed
alue o α.
7) OTHER METHODS
Rega ding he esea ch esul s al eady in oduced in his
su ey, Wang e al. [146] calcula ed mu ual delay alues sa -
is ying exponen ial s abili y o a RTDS wi h example case
s udies suppo ing hei me hod. Comple e s abili y in e als
o he base delay o a sys em wi h commensu a e delays we e
de e mined by means o he equency sweeping me hod and
he Puiseux se ies in he wo ks o Li e al. [151]–[154].
The singula alue decomposi ion echnique was used by
Ramachand an and Ram [184] o de e mine c i ical delays o
a single-inpu mul i-ou pu (SIMO) sys em. The leading idea
was as ollows: Le 1(s)=de (A−sB+exp(−sτ)H)
whe e A,B∈R2n×2n, and His a ank-one ma ix subjec
o he singula alue decomposi ion H=U6V,6=
diag σ0. . . 0. A e some algeb aic ope a ions, he con-
di ion 1(s)|s=jω=0 can be exp essed as
N(sk)¯
N(sk)−D(sk)¯
D(sk)=0 (56)
whe e N(sk)=de Q,D(sk)=σde Q1,Q=
UT(A−skB)V,Q1s ands o he (2n−1)×(2n−1) ail-
ing subma ix o Q1,skis a pu ely imagina y oo , and he ba
exp esses he complex conjuga e, i.e. he p oblem is educed
o he ask o inding he oo s o a polynomial. Ne e heless,
he echnique canno be used o a mul i-inpu mul i-ou pu
(MIMO) sys em. In his case, he au ho s sepa a ed he ma ix
eigen alue p oblem in o i s eal and imagina y componen s,
so ha he p oblem o de e mining he c i ical delay was
ans o med o
P(τ, sk)z=0 (57)
whe e skis a pu ely imagina y epea ed eigen alue wi h a
mul iplici y la ge han one. Since he Jacobian ma ix asso-
cia ed wi h (57) is singula in he neighbo hood o he solu-
ion, he con e gence o New on’s me hod is linea ; hence,
a bisec ion algo i hm o sol ing he p oblem was de eloped.
Pon es Du e al. [185] sol ed he model educ ion p ob-
lem (45) o RTDSs wi h mul iple delays ia he so-called TF-
IRKA algo i hm [186] gi ing ise o he ini e-dimensional
model
E˙
x( )=Ax ( )+Bu ( ),y( )=Cx ( ).(58)
The ob ained model was hen used o es ima e s abili y
egions.
An in e es ing compa a i e s udy on he s abili y anal-
ysis o DCPPS was p esen ed by Gao e al. [187] whe e
h ee me hods we e adop ed; namely, a Padé app oxima-
ion based me hod [123], he explici in ini esimal gene a o
disc e iza ion-based me hod [84] and a DDS echnique [188]
ha allows o he de e mina ion o he maximal delay such
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ha a DCPPS emain s able by using an LMI echnique
(see [182] o he compa ison). The conse a i eness o he
DDS me hod was analyzed ia an example and i was in e
alia obse ed ha he Padé app oxima ion based me hod is
su icien ly accu a e e en o a low app oxima ion o de .
A simply implemen able g idding delay-disc e iza ion
DDS echnique was published in [189] and [190]. This
nume ical concep is based on he ecu si e app oxima ion
o 1(s)by he associa e polynomial in he icini y o he
cu en es ima ion o he igh mos pole s0in e e y node
o he g id in he delay space. While in [189], he asso-
cia ed polynomial 1ass,4(s|s0)was ob ained om he Tay-
lo se ies expansion, he bilinea ans o ma ion oge he
wi h he p e-wa ping echnique we e u ilized in [190] o ge
1ass,5(z|s0,Ts)whe e Tsexp esses he sampling ime. The
leading ze o o 1ass,4(s|s0)o 1ass,5(z|s0,Ts) hen yielded
he e en ual igh mos pole es ima ion o he nex g id node.
The es ima ion o he swi ching poles was u he enhanced
by using he a e age o RT alues (a combined New on’s
echnique) and by he linea in e pola ion, espec i ely. The
concep clea ly u ilizes he oo con inui y p ope y (see i em
( ) o P oposi ion 1); howe e , one has o be ca e ul in he
neu al delay case (P oposi ion 2 (i ), P oposi ion 3).
A combina ion o h ee echniques o de e mine he
delay s abili y ma gin o wide-a ea measu emen sys ems
(WAMS) – modeled by RTDSs wi h mul iple delays – was
p oposed in [191]. Namely, ma ices Aiin (1) a e s anda d-
ized in o he Jo dan o m i s , yielding a new s a e ec o
z( ). Second, he Taylo expansion is applied o sepa a e he
connec ion be ween z( )and z( −τi). Finally, he Schu
simpli ica ion [192] is implemen ed o educe he numbe o
s a e a iables.
Roales and Ród iguez [193] s udied he exis ence o s a-
bili y swi ches and Hop bi u ca ions (i.e. he pe iodic s abil-
i y bounda y) o he second-o de scala delay di e en ial
equa ion ¨x( )+a˙x( −τ)+bx ( )=0, , τ > 0, in which
a,b∈C. The p esen ed analy ic de i a ions we e based on
he heo em es ablished in [194] ha cha ac e izes, o he
c i ical alues τisuch ha 1(jω, τi)=0, he a ia ion o
he numbe o ze os wi h nonnega i e eal pa s o 1(s, τ)
in e ms o he o de and sign o he i s nonze o de i a e o
F(ω):= |Re(1(jω))|2−|Im(1(jω))|2.
E. DIS
F equency-based DIS me hods a e gene ally buil on he
e i ica ion o he non-exis ence o pu ely imagina y sys em
poles o a bi a y delay alues. This ask is usually achie ed
by ans o ming 1(s)in o associa ed (auxilia y) polynomial
1ass (s)o 1ass (z), which is comple ely ee o delays and
can be uni-, bi- o e en mul i a ia e, and hen by p o ing ha
he e is no ze o o 1ass (s)lying exac ly on he imagina y
axis, o no ze o zko 1ass (z)such ha zk∈T.
Delice and Sipahi [166] used he echnique o compu ing
he esul an and consequen ly ha o he i e a ed disc imi-
nan [68] o elimina e pseudo-delays om p(ω, T)(see he
desc ip ion o he CTCR concep abo e), which allowed one
o cons uc a single- a iable unc ion D(ω) o be equal o
ze o. Then, he non-exis ence o any posi i e eal oo o
D(ω)- which is a su icien DIS condi ion - was p o ed by
he Désca es ules o signs. Howe e , in ini e delays we e
omi ed in his echnique. Asymp o ic di ec ions o he delays
on he po en ial s abili y swi ching hype su aces app oach-
ing in ini y de i ed by Sipahi and Delice [165] linked DDS
and he s ong DIS p oblem (he e, he au ho s used he e m
‘‘s ong’’ o DIS including in ini y delays).
Conce ning mul iple-delay cases, he comp ehensi e s udy
by Nia and Sipahi [195] also u ilized he Rekasius ans o -
ma ion (29) and he esul an heo y o in es iga e DIS in he
delay space and he con olle pa ame e s space o ac i e
ib a ion con ol sys ems. In addi ion, S u m sequences we e
applied o es ablish he necessa y and su icien condi ions in
iden i ying he numbe o dis inc posi i e eal oo s o D(ω).
A ma ix pencil me hodology along wi h an algeb aic
me hod we e u ilized by Ma e al. [180] o in es iga e he DIS
p oblem ia 1ass,2(s)as in (53).
E genc [196] p esen ed a me hod o de e mining he DIS
zones o a gene al RTDS wi h mul iple delays agains pa a-
me ic unce ain ies. This me hod adop ed he K onecke
summa ion scheme 1ass,1(z)as in (24) exp essed by means
o he K onecke mul iplica ion ope a o s. The sys em is DIS
i
Re(s:1(s,p)=0)<0 (59)
and all ze os zko 1ass,1(z,p)sa is y zk/∈Twhe e p
ep esen s a ec o o unknown pa ame e s. In ac ,
1ass,1z1,z2,...,znτ,p
=Xm
j=1bjz1,z2,...,zi−1,zi+1,...,znτ,pzj
i
is a sel -in e si e (symme ic) mul i a iable polynomial sa -
is ying 1ass,1(zi)=znτ
i1ass,1(1/zi) o i=1,2,...,nτ.
The ollowing unique p ope y o sel -in e si e polynomials
was u ilized: β=(2µ+1)−mwhe e βis he num-
be o i s ze os lying on T, and µis he numbe o ze os
inside D(including mul iplici y). Wi h he combina ion o
his p ope y and ano he gene al polynomial p ope y (Pel-
le ’s heo em), he ollowing su icien condi ion o DIS was
p esen ed:
Theo em 7: The sys em is DIS i (59) holds and
bµz1,z2,...,zi−1,zi+1,...,znτ,p
>Xm
j=1,j6=µbjz1,z2,...,zi−1,zi+1,...,znτ,p(60)
o µ≤m/2−1, i=1,2,...,nτ.
An expe imen al s udy e i ying his esul was p e-
sen ed in [197]. The me hodology was u he imp o ed
by Alikoç and E genc [198] whe e he Bis i z abula ion
me hod [199] was used o de e mine he loca ion o ze os wi h
espec o he uni ci cle o a single delay RTDS. The me hod
is based on a h ee- e m ecu sion o symme ic polynomials
and he numbe o sign a ia ions o hese polynomials a
z=1; namely, he sign a ia ion in a sequence o numbe s
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L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
ob ained by he solu ion o ecu si e equa ions calcula ed
om he polynomial D(z)=d1ass,1(z)/dzis e alua ed
ins ead o he condi ion (60), which enables he use o eal
a i hme ic ope a ions. Alikoç and E genc [200] ex ended his
echnique o mul iple incommensu a e delays. These esul s
can be u ilized, o ins ance, when de e mining he con olle
pa ame e s’ se obus ness wi h espec o delay alues.
Recall ha he DIS p oblem was in es iga ed also by he
semi-disc e e app oxima ion o by means o linea spline
unc ions p esen ed by Fabiano [89].
The a gumen p inciple (o , con ou in eg al) me -
hod [107], [178], was used o deal wi h he DIS as well.
Conside Theo em 4, in which ωRis he maximal posi i e oo
o RL(ω)ins ead o R(ω). Polynomial RL(ω)is cons uc ed
om R(ω), he coe icien s o which a e subs i u ed by hei
in ima ha a e independen o delay alues.
Assuming he eign o spec al me hods, ma ginal ye
in e es ing esul s we e p esen ed by Li e al. [201] whe e
he s ong DIS condi ion ia LMIs was analyzed using e-
quency domain disc e iza ion in o se e al sub-in e als and
he piecewise cons an Lyapuno ma ices. A se ies o p o-
posed s abili y c i e ia yield necessa y and su icien s ong
DIS condi ions o RTDSs wi h a single delay which is less
conse a i e han some ypical su icien LMI condi ions. I is
wo h no ing ha he no ion o s ong DIS in oduced he e is
a he di e en han ha in [165]. Namely, conside a RTDS
wi h commensu a e delays and he base delay h, he sys em is
s ongly DIS i 1(s,z)6= 0 o all s∈C+and z∈Dwhe e
z=exp(−sh). This p ope y is obus agains pe u ba ions
o pa ame e s in he s a e ma ices in (1), see [202] o de ails.
F. PARAMETER-DEPENDENT STABILITY
By pa ame e -dependen s abili y we mean he s abili y in es-
iga ion wi h espec o sys em pa ame e s excep o delays,
i.e., in he non-delay pa ame e space.
Recalling esea ch esul s al eady in oduced abo e again,
Dong e al. [103] e alua ed he op imal pa ame e s o he
con olle design by sea ching he global minimum o he
spec al adius o he ansi ion ma ix ha was ob ained by
means o he DQ me hod. In o de o sol e such op imiza ion
p oblems using g adien descen algo i hms, he g adien o
he spec al adius o ansi ion ma ix wi h espec o he
conce ned pa ame e s was analy ically o mula ed.
The Vande monde/Bi kho ma ix DDS app oach o mul-
iple pu ely imagina y poles by Boussaada and Niculescu
has also included non-delay pa ame e s while s udying
pa ame e -dependen exponen ial s abili y [148], [149].
O en and Mönnigmann [164] p oposed an op imiza ion
scheme ha was based on he solu ion o he H2minimiza ion
p oblem in he pa ame e space subjec o he mani old o he
c i ical pa ame e alues. In addi ion, he no mal ec o has o
be sol ed o en o ce a obus dis ance be ween any candida e
op imal s eady s a e and he c i ical mani old.
A gumen p inciple based DDS me hodology by
Xu e al. [178] can be applied o s abili y analysis wi h
espec o non-delay pa ame e alues as well.
FIGURE 3. Func ion φ7→ LmaxωLmax [204].
A simple sys ema ic equency sweeping p ocedu e o
sol ing he exac s abili y bounda ies in he pa ame e plane
p=(p1,p2) o RTDSs was p oposed by Pe ng [203]. No e
ha he me hodology has also been used o sol e he DDS
p oblem o sys ems wi h commensu a e delays as ollows:
Le exp(−sh)=exp (−jωh)=cos (ωh)−j sin (ωh)=
p1−jp2, hen a e some algeb aic ope a ions on gonio-
me ic unc ions, he po en ial s abili y bounda y plo s in
p1−p2space can be ob ained again. Since i mus hold ha
|exp(−sh)|=1, he bounda y mus in e sec he uni ci cle
in he pa ame e plane o admissible solu ions. The exac
maximum delay alue o asymp o ic s abili y hen eads
h=ω−1cos−1(p1)=ω−1sin−1(p2). I no in e sec ion is
ound, he sys em is DIS (o uns able).
The design o pa ame e s p=(p1,p2)such ha he sys em
ep esen ed by (23) is asymp o ically s able o L∈(0,Lmax]
wi h a p ede e mined (known and ixed) alue Lmax was
p esen ed by Sipahi [204]. The au ho used he Rekasius sub-
s i u ion and in oduced he sweeping pa ame e φ=ωTin
an in e al φ∈[φmin, φmax]. Then, i can be compu ed om
(48) ha exp−jωLmaxLmax=(1−jφ)/(1+jφ) o ge
a polynomial pωLmax, φ, pins ead o he quasipolynomial
1jωLmax,pwhe e he co esponding equency eads
ωLmax =2
Lmax an−1(φ)−(sgn(φ)−1)π
2,
φ6= 0,0< ωLmax ≤2π/Lmax,(61)
see Fig. 3. Then, o hese ixed alues, one should simul ane-
ously sol e he se o equa ions Re(p(p)) =Im(p(p)) =0.
Howe e , he easibili y o his solu ion mus be e i ied by
he compu a ion o he numbe o imagina y c ossing, M,
by means o Theo em 1.
Hence, i is necessa y o compu e he eigen alues o 5as
in Theo em 1, ollowed by he e i ica ion whe e hese alues
a e included in he sys em spec um . No e ha M≤n2( o
> 0) and in he e e ed esea ch s udy, he au ho en o ced
M=1 ini ially.
Sch ödel e al. [205] p esen ed a compa a i e o e iew
o ou exis ing equency-based me hods o he s a-
bili y bounda y calcula ion p oblem in he pa ame e
space, namely, he Rekasius subs i u ion me hod, he di ec
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L. Pekař, Q. Gao: Spec um Analysis o LTI Con inuous-Time Sys ems Wi h Cons an Delays
me hod [72], he K onecke mul iplica ion me hod [77]
(see also Ma e al. [180]) and he so-called ma ix sum
me hod [74]. The las one o he me hods is based on he
elimina ion o ω om he cha ac e is ic equa ion and he solu-
ion o he associa ed equa ion o z∈T. The cha ac e is ic
equa ion can be ew i en as
1ass,4(s,z):= de ((sI−M(z))) =0 (62)
whe e z=exp(−sh)and M(z)is appa en om (2) o
commensu a e delays. C ossing poles sa is y zc∈Tand
sc= ±jωc.
Equa ion(62)can also be e o mula ed as de ((zU−V)) =
0 whe e U,Va e ma ices ha include K onecke sum and
mul iplica ion ope a ions (see [205] o mo e de ail). By
sol ing his equa ion o z∈T, he c ossing equencies can
be ob ained om 1ass,4(ω)=1ass,4(jω, zc)=0 and he
co esponding delays ia τk=ω−1(a g(z)+2kπ),k∈Z.
In [205], a gene aliza ion o he p oblem o calcula ing he
s abili y egion o TDSs in he delay and non-delay pa am-
e e space (which is e y close o he CTCR pa adigm) was
also gi en o he eade . In addi ion, h ee ypes o s abili y
bounda ies we e in oduced.
I is wo h no ing ha mos o he esul s on pa ame e -
dependen s abili y we e ob ained o con ol sys ems
and ela ed asks o con olle pa ame e s uning, see
e.g. [119], [124], [125], [146], [165], [195], which goes,
howe e , beyond he objec i e o his su ey ha is aimed
a sys em analysis.
Selec ed esul s om he gene al pa o his sec ion (and
also om he p e ious one in some cases) a e summa ized
in Table 2 o p o ide he eade wi h an o e iew o he
heo e ical s abili y s udies.
VII. ENGINEERING APPLICATIONS AND CASE STUDIES
This sec ion is ocused on he commen ed lis o aca-
demic o p ac ical applica ions o he me hods o LTI-
TDSs spec um analysis. No e ha his sec ion is summa ized
in Table 3.
Me hod app oxima ing o ( ), in oduced he ein in sec-
ions om IV o VI, can be ound in he li e a u e as a o i e
ools o he analysis o milling p ocesses – un o una ely,
hese models o ype (23) usually include a ime-dependen
s a e ma ix A1.
Recall ha Tang e al. [98] p edic ed milling s abili y ia
an imp o ed FD me hod wi h Lag ange polynomial in e po-
la ion, and he au ho s p esen ed a compa ison wi h SD and
NI echniques as well. The same p oblem was sol ed using
FD and NI me hods in [95] and [99], espec i ely.
The DQ me hod u ilized o he s abili y analysis o milling
p ocesses was p esen ed by Ding e al. [102].
Quo e al. [207] u ilized he hi d o de FD me hod o
ge he exac s abili y bounds. Ozoegwu [208] p esen ed a
me hod being e y close o he FD one, ye he leas squa es
(also called he hype hi d-o de ) app oxima ion was applied
ins ead o he in e pola ion p ocedu e.
The hype hi d-o de app oxima ion ollowed by he
NI me hod was u he used by Ozoegwu e . al. [209] and
ex ended hi d and ou h o de ec o NI schemes o one-
deg ee-o - eedom and wo-deg ee-o - eedom milling p o-
cesses in [100]. The same au ho s also p esen ed he use o
he SE me hod while analyzing he cha e s abili y o a h ee-
oo h plas ic end-milling CNC machine [210].
These app oxima ion me hods, howe e , ha e been applied
in o he indus ial applica ions as well. Khasawneh [211]
u ilized he SE me hod wi h he ba ycen ic Lag ange o -
mula o analyze he s abili y o machining p ocesses which
may lose s abili y due o cha e ib a ions, i.e., sel -exci ed
ib a ions due o he su ace egene a ion e ec . A sho
(in i s o m) ye comp ehensi e (in i s con en ) o e iew o
nume ical echniques ha a e based on a ini e dimensional
app oxima ion o he in ini e dimensional sys em used o
he s abili y p edic ion o machining p ocesses was p esen ed
by Inspe ge e al. [11]. This ype o cha e occu s due o
wo kpiece o a ions o dynamic cu ing load changings.
Kisho e al. [8] discussed s abili y analysis using spec-
al disc e iza ion o ime-delayed elec ic powe sys ems,
namely, he 4-gene a o and he 14-gene a o Sou heas Aus-
alian powe sys ems.
The PsC me hod was used o ge he disc e e mapping
he e, and he au ho s compu ed he igh mos poles and he
spec al abscissa o e a wide ange o ime delays, which
cha ac e izes a pa ial solu ion o he DDS p oblem. As in o-
duced abo e, Ye e al. p oposed an i e a i e PsC me hod o
spec al analysis o la ge DCPPS o o e come compu a ional
p oblems wi h spa se ma ix app oxima ion o he in ini esi-
mal gene a o [84], [86], and he solu ion ope a o [87].
Milano [138], and Milano and Anghel [144] used he PsC
echnique o compu e poles o a la ge DCPPS and compa ed
i wi h some o he disc e iza ion schemes o ge a ini e-
dimensional app oxima ion o he solu ion ope a o ( ). Fo
hese esul s, see also Table 1.
The pseudospec al me hod by B eda e al. [58] was u i-
lized by Coelho e al. [212] o analyze he spec um o a
single delay RTDS exp essing he eedback con ol sys em
o an islanded mic og id composed o wo o mo e ol age
sou ce in e e s wi h communica ion delays.
Sensi i i y analysis o he poles was conduc ed by
Zhao [213] in o de o e eal he dynamic s abili y ma gin
and o iden i y he p ope ange o he con ol pa ame e s, o
an islanded medium- ol age mic og id placed in he Dongao
Island. Un o una ely, he au ho s did no e e o he used
me hod.
Dong e al. [214] p oposed a s abili y analysis me hod o
he hyb id ene gy s o age sys ems wi h delays and applied
i o a lab-scale DC mic og id. The s abili y ma gin (i.e.,
he maximum s abilizable delay) was compu ed by he de e -
mina ion o pu ely imagina y poles. The leading idea o he
c i ical poles compu a ion is based on he assump ion ha all
delays a e a ional numbe s o hey can be app oxima ed by
he a ional numbe s. Then, one can ew i e he cha ac e is ic
equa ion 1(jωc)=0 so ha i s solu ion has a pe iod o 2π.
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TABLE 2. S abili y s udies ela ed o pole loci – heo y (Sec ion i).
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TABLE 2. (Con inued.) S abili y s udies ela ed o pole loci – heo y (Sec ion i).
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TABLE 3. S abili y s udies ela ed o pole loci – applica ions (Sec ion ii).
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TABLE 3. (Con inued.) S abili y s udies ela ed o pole loci – applica ions (Sec ion ii).
Hence, he in e al [0,2π)is disc e ized, and sys em poles
a e hen compu ed in e e y single disc e e s ep inside his
in e al.
Al hough he Lambe W unc ion has a limi ed u iliza ion
due o model es ic ions, some enginee ing applica ions can
be ound. Fo ins ance, Pe i e al. [215] s udied eac ion-
di usion sys ems wi h a ime delay conside ed in he complex
ne wo ks in he amewo k o Tu ing ins abili ies. Explici
analy ic condi ions o he onse o pa e ns as a unc ion
o he main in ol ed pa ame e s, he ime delay, and he
ne wo k opology we e ob ained using he scala Lambe
W unc ion. The au ho s hen p edic ed whe he o no he
sys ems would exhibi a wa e pa e n associa ed wi h a Hop
bi u ca ion, o a s a iona y Tu ing pa e n. Yi e al. [216] used
he unc ion o ob ain he igh mos poles o neu al ne wo ks
wi h ime delays and pa ame ic unce ain ies modeled by a
single delay RTDS. Howe e , no e also ha mo e pa icula
applica ions o he Lambe W unc ion ha e been made con-
ce ning con olle design, see e.g. [109], [217], and e e ences
he ein.
Niu e al. [123] used he Padé app oxima ion o es i-
ma e he spec um o a powe sys em wi h a ime delay,
see Table 1. Gölgeli and Özbay [218] u ilized he YALTA
so wa e o in es iga e he unique local s abili y by analyz-
ing he impac o he nico ine exposu e on he choles e ol
biosyn hesis. The so-called delay-dependen coupling (DDC)
was conside ed in [219] o p e en ins abili y in a mul i-agen
sys em in which agen s communica e wi h each o he unde
homogeneous delays, while a emp ing o each consensus.
The sys em model has a simpli ied o m as ollows
˙
x( )= (L)Ax ( −L)(63)
whe e (L) ep esen ed he DDC as a unc ion o he delay
alue L. The main idea while designing he s abili y o (63)
was based on he ollowing o mula o he compu a ion o
he delay ma gin Lmax.
Lmax =1
(L)min
k
ηk/2
2αksinηk/2(64)
whe e αk,ηka e ela ed o he pa icula eigen alue sko A
acco ding o Fig. 4.
T ajec o ies o poles we e ob ained ia TRACE-DDE
ool [220]. No e ha a mul i-agen consensus dynamics unde
FIGURE 4. Eigen alues sko A wi h espec o αk,ηk[219].
a communica ion delay, he delay ma gin and he ne wo k
opologies o educe he du a ion o each consensus we e
also in es iga ed by means o he igh mos poles e.g. in
wo ks [221], [222], which, howe e , can be conside ed as
con ol a he han analy ic asks.
Sipahi e al. [223] s udied how he memo y o d i e s
modeled by dis ibu ed delays a ec s he decision-making
p ocess in a ca ollowing scena io, in which each d i e aims
a keeping a ixed ime-headway wi h espec o he p eced-
ing ehicle. When analyzing s abili y, he app oxima ion o
delays was done by using he asymp o ic (limi ) p ope ies
o dis ibu ed delay e ms and he Taylo se ies expansion.
The spec um was compu ed by means o he QPmR oolbox.
The au ho s in e alia ound ha he dynamics can exhibi
he spec um simila o NTDSs o some in e connec ion
schemes, al hough he model does no i in he s anda d
NTDSs.
Single and double Hop and he pi ch o k bi u ca ion anal-
yses we e p esen ed by Ding e al. [224] o an ac i e con-
ol sys em o he ball al e in glue dosing p ocesses o
pa icleboa d. Fo a double Hop bi u ca ion, he mul iple
ime scales me hod ins ead o he habi ual Puiseux se ies was
used, which is based on he ollowing o m o he solu ion
o (1).
x( )=X∞
i=112i−1
2xi(T0,T1, . . .)(65)
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LIBOR PEKAŘ was bo n in Zlín, Czech Republic,
in 1979. He ecei ed he B.S. deg ee in au oma ion
and in o ma ics, he M.S. deg ee in au oma ion
and con ol enginee ing in consump ion indus y,
and he Ph.D. deg ee in echnical cybe ne ics om
Tomas Ba a Uni e si y in Zlín, in 2002, 2005, and
2013, espec i ely.
F om 2006 o 2013, he was a Junio Lec u e
wi h he Facul y o Applied In o ma ics, Tomas
Ba a Uni e si y in Zlín. F om 2013 o 2018, he
was a Senio Lec u e wi h Tomas Ba a Uni e si y in Zlín, whe e he has
been an Associa e P o esso since 2018. He has au ho ed wo book chap e s,
o e 40 jou nal a icles, and o e 70 con e ence pape s. His esea ch in e es s
include analysis, modeling, iden i ica ion, and con ol o ime-delay sys ems,
algeb aic con ol me hods, and au o uning and op imiza ion echniques. He
has been he Lead Gues Edi o o jou nals Ma hema ical P oblems in
Enginee ing and Ad ances in Mechanical Enginee ing, and an Edi o o he
Ma hema ical P oblems in Enginee ing since 2018.
D . Pekař was a ecipien o he Rec o s’ Awa d o he bes Ph.D.
hesis in he Facul y o Applied In o ma ics, Tomas Ba a Uni e si y in Zlín,
in 2013, and he Lau ea e o he ASR Semina y Ins umen a ion and Con ol
in 2007 and 2009.
QINGBIN GAO ecei ed he B.S. deg ee in
mechanical enginee ing om he Ha bin Ins i u e
o Technology, China, in 2011, and he Ph.D.
deg ee in mechanical enginee ing om he Uni e -
si y o Connec icu in 2015. He was an Assis an
P o esso wi h he Depa men o Mechanical and
Ae ospace Enginee ing, Cali o nia S a e Uni e -
si y, Long Beach, om 2015 o 2018. Since 2018,
he has been an Assis an P o esso wi h he Depa -
men o Mechanical Enginee ing, The Uni e si y
o Alabama. His main esea ch ocuses on he s abili y analysis and con ol
syn hesis o ime-delay sys ems wi h applica ions o mul i-agen sys ems,
manu ac u ing, connec ed ehicles, human lea ning, and powe sys ems and
ib a ions.
He was a ecipien o he Bes Con e ence Pape Awa d o he 19 h
In e na ional Con e ence on Ne wo king, Sensing, and Con ol (ICNSC)
in 2017 and he 6 h Ame ican Socie y o Mechanical Enginee s (ASME)
Dynamic Sys ems and Con ol Con e ence (DSCC) in 2013. He has se ed
as a Session Chai o 2017 ASME DSCC, 2017 ICNSC, and 2016 IEEE
Ame ican Con ol Con e ence (ACC). He has se ed as an Associa e Edi o
o 2017 ACC, 2018 ACC, and 2017 ASME DSCC. He has also se ed
as a Re iewe o o e 100 pape s om o e 30 jou nals, including bu
no limi ed o Au oma ica, he IEEE TRANSACTIONS ON AUTOMATIC CONTROL,
and Mecha onics. He is cu en ly a Gues Edi o o he IEEE ACCESS and
Ad ances in Mechanical Enginee ing.
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