On the Stabilization of a Network of a Class of SISO Coupled Hybrid Linear Subsystems via Static Linear Output Feedback


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Ci a ion: De la Sen, M. On he
S abiliza ion o a Ne wo k o a Class
o SISO Coupled Hyb id Linea
Subsys ems ia S a ic Linea Ou pu
Feedback. Ma hema ics 2022,10, 1066.
h ps://doi.o g/10.3390/
ma h10071066
Academic Edi o s: Mihail
Ioan Ab udean and Vlad Mu esan
Recei ed: 28 Feb ua y 2022
Accep ed: 22 Ma ch 2022
Published: 25 Ma ch 2022
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ma hema ics
A icle
On he S abiliza ion o a Ne wo k o a Class o SISO Coupled
Hyb id Linea Subsys ems ia S a ic Linea Ou pu Feedback
Manuel De la Sen
Depa men o Elec ici y and Elec onics, Facul y o Science and Technology, Ins i u e o Resea ch and
De elopmen o P ocesses, Uni e si y o he Basque Coun y (UPV/EHU), 48940 Leioa, Bizkaia, Spain;
[email p o ec ed]
Abs ac :
This pape deals wi h he closed-loop s abiliza ion o a ne wo k which consis s o a se o
coupled hyb id single-inpu single-ou pu (SISO) subsys ems. Each hyb id subsys em in ol es a
con inuous- ime subsys em oge he wi h a digi al (o , e en ually, disc e e- ime) one being subjec
o e en ual mu ual couplings o dynamics and also o disc e e delayed dynamics. The s abilizing
con olle is s a ic and based on linea ou pu eedback. The con olle syn hesis me hod is o algeb aic
ype and based on he use o a linea algeb aic sys em, whose unknown is a ec o equi alen o m
o he con olle gain ma ix, which is ob ained om a p e ious algeb aic p oblem e sion which is
based on he ad hoc use o he ma ix K onecke p oduc o ma ices. As a i s s ep o he s abiliza ion,
an ex ended disc e e- ime sys em is buil by disc e izing he con inuous pa s o he hyb id sys em
and o uni y hem oge he wi h i s digi al/disc e e- ime ones. The s abiliza ion s udy ia s a ic linea
ou pu eedback con ains se e al pa s as ollows: (a) s abilizing con olle exis ence and con olle
syn hesis o a p ede ined a ge ed closed-loop dynamics, (b) s abilizing con olle exis ence and i s
syn hesis unde necessa y and su icien condi ions based on he s a emen o an ad hoc algeb aic
ma ix equa ion o his p oblem, (c) achie emen o he s abiliza ion objec i e unde ei he pa ial o
o al decen alized con ol so ha he whole con olle has only a pa ial o null in o ma ion abou
couplings be ween he a ious subsys ems and (d) achie emen o he objec i e unde small coupling
dynamics be ween subsys ems.
Keywo ds:
hyb id dynamic sys ems; decen alized con ol; s abiliza ion; ou pu eedback; s a ic
ou pu eedback con olle
MSC: 93C05; 93D20; 93C55
1. In oduc ion
The s abiliza ion o dynamic sys ems ia eedback is a e y impo an opic in Con ol
Theo y since a necessa y minimum equi emen o any con olled sys em is ha i ope a es
in a s able way. The e o e, he s abiliza ion heo y is ele an in con inuous- ime sys ems,
disc e e- ime sys ems and he hyb id ones which ha e mixed con inuous- ime and disc e e-
ime pa s. See, o ins ance, [
1
–
13
] and some ela ed e e ences he ein. The disc e iza ion
o con inuous- ime sys ems can be pe o med o cons an sampling a es o o non-uni o m
ones [
3
,
6
] so as o ake he sampling a e as an ex a design unc ion which can be accommo-
da ed o he a e o a ia ion o he signals o in e es in he sys em unde s udy. The wo ks
in [
1
,
2
] ocus on he s abiliza ion o sa u a ed disc e e- ime swi ching sys ems. On he o he
hand, he wo ks in [
1
,
4
] a e ocused on he s abiliza ion o mul i a e con ol sys ems, so on
hose which ha e signals being sampled a di e en sampling a es, also wi h he objec i e
o acili a ing he accommoda ion o signals in he sys em ha , because o hei di e en
na u e, e ol e a di e en a ia ion a es o which a e needed o be sampled a di e en
a es. A use ul design echnique o s abiliza ion pu poses is he use o Lyapuno unc ions,
which can in ol e he s uc u e o he closed-loop sys em pa ame e iza ion ( ha is, he
Ma hema ics 2022,10, 1066. h ps://doi.o g/10.3390/ma h10071066 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 1066 2 o 29
one in ol ed he inco po a ion o he eedback con ol law), so as o allow he app op ia e
syn hesis o he eedback con olle , [
7
]. Some s abiliza ion p oblems also inco po a e he
ex a e o o needing o ollow he beha io o a ce ain p esc ibed model which is known
as he “model-ma ching” o “model- ollowing” objec i e. In his case, i is no only needed
o s abilize he closed-loop modes (s abiliza ion p oblem) bu also o p esc ibe he alues o
bo h he ze os and poles o he closed-loop ans e unc ion o p esc ibed alues de ined
by he e e ence model [
8
,
9
]. Di e en de ices and design echniques which should be
examined o decide on combining disc e iza ion ools wi h con inuous- ime analysis in
complex dynamic sys ems a e he use o app op ia e sampling and hold de ices [
8
–
10
]
o upda e disc e ized signal in o ma ion o con ol pu poses, he e en ual in luence o
delays ei he in he inpu o ou pu , o in he sa es, and also he possible s abiliza ion ia
s a e o ou pu eedback in ol ing ei he cen alized con ol, i.e. in ol ing all he a ailable
ou pu in o ma ion, o decen alized con ol, i.e. in ol ing only local in o ma ion o a
pa ial in o ma ion on he whole sys em. See [9,11–38] and e e ences he ein.
The main objec i e o his wo k is o deal wi h hyb id dynamic sys ems. Such sys ems
ha combine he in ol emen o bo h con inuous- ime signals and digi al signals in an
in eg a ed way ha e ecei ed impo an a en ion [
4
,
39
–
46
]. They in ol e modeling ools
which a e e y e sa ile allowing o desc ibe he whole sys em in a disc e e- ime way o a
ce ain sampling pe iod as a i s modeling s age due o he combina ion o he disc e iza ion
o he con inuous- ime subsys em wi h he ei he disc e e o digi al subsys em. In pa icula ,
he op imiza ion o inpu s and he undamen al p ope ies o such sys ems ha e ecei ed
a en ion in [
39
] and hei mul i a e sampling modeling ools o accommoda e he a ious
signals in he sys em and i s con ol pe o mance conce ns ha e been s udied in [
4
,
40
,
45
].
The main impo ance o hyb id dynamic sys ems a ises om he ac ha con inuous- ime
and digi al subsys ems usually ope a e in a combined and in eg a ed ashion in many
eal wo ld si ua ions. A second eason o es ablish such hyb id models is hei sui abili y,
o echnological implemen a ion easons, o desc ibing he use o ei he disc e e- ime o
digi al con olle s o ei he s abilize o con ol con inuous- ime plan s. Fo ha pu pose, a
wide class o linea hyb id sys ems p oposed in [
39
], and also deal wi h in [
40
–
44
], ha e
been conside ed o model- ollowing pu poses. The whole s a e o he hyb id dynamic
sys ems s udied in he abo e app oaches is desc ibed by i s con inuous- ime subs a e
being o ced by bo h he cu en inpu in con inuous ime and i s sampled alue a he las
p eceding sampling ins an as well, while he disc e e- ime o , e en ually, digi al subsys em
is d i en by he sampled con ol a sampling ins an s. In gene al, he e a e dynamical
couplings be ween bo h subs a es.
In his pape , we ocus on he closed-loop s abiliza ion o a hyb id dynamic sys em
which is a ne wo k consis ing o a andem o
q
subsys ems, each o hem being desc ibed by
a con inuous- ime subs a e oge he wi h a disc e e- ime one. In he mos gene al case, he e
a e mu ual couplings be ween bo h con inuous and disc e e subs a es o each subsys em,
couplings be ween he dynamics o he a ious subsys ems and delayed poin dynamics in
he whole sys em also wi h couplings be ween subsys ems. The closed loop s abiliza ion
o such a ne wo k is in es iga ed hough linea ou pu eedback by syn hesizing a s a ic
con olle . The possibili y o pa ial o o al lack o in o ma ion o couplings be ween subsys-
ems a ailable o he syn hesis o he con olle is also s udied. This leads o designs based
on pa ial o o al decen alized linea ou pu eedback s abilizing con ol, [
14
–
26
], which is
o in e es as a design echnique o educe he amoun o online in o ma ion o be p ocessed
o con ol he o al sys em, especially, in he case o complex high-dimensional sys ems.
The pape is o ganized as ollows. Sec ion 2p esen s he p oposed hyb id sys em
which consis s o a se o con inuous- ime sys ems wi h mu ual dynamic couplings on a
se o digi al subsys ems, bo h se s being in eg a ed in a sys em ne wo k. Each subsys em
is assumed o single-inpu single-ou pu (SISO) ype. The whole dynamics can be also
e en ually a ec ed by disc e e- ime delayed dynamics o a ini e numbe o poin delays,
and i is d i en, in gene al, by a combined ac ion o he con inuous- ime inpu along he
in e sample ime in e al oge he wi h i s sampled alues a he sampling ins an s. This
Ma hema ics 2022,10, 1066 3 o 29
sec ion also con ains wo desc ip ions o an ex ended disc e ized sys em, buil wi h he
disc e iza ion o he con inuous pa s o he whole hyb id sys em being e en ually coupled
wi h he digi al ones, whose s abiliza ion objec i e is he i s and main in ended s ep
o he s abiliza ion o he whole hyb id sys em. Sec ion 3deals wi h he s abiliza ion
h ough linea s a ic ou pu eedback o he modi ied ex ended disc e e sys em wi h ze o
inpu –ou pu di ec in e connec ion gains, wha implies basically ha he ela i e deg ee, o
pole-ze o excess in he ans e unc ion, is g ea e han one. The mechanism o designing
he con olle gain is o algeb aic ype and based on con e ing he se o equa ions o sol e
in a linea algeb aic sys em o equa ions based on a ec o o m e sion o hem being
ob ained om he use o ad hoc K onecke p oduc s o ma ices [
31
,
32
] in he o iginal
syn hesis p oblem. In gene al, he algeb aic p oblem can be: (a) non-compa ible, so ha
i has no solu ion o a p e-de ined sui ed s able closed-loop dynamics o he ex ended
disc e e sys em being de ined by a con e gen ma ix o closed-loop dynamics, o (b) i can
be algeb aically compa ible wi h ei he one (compa ible de e mina e) o in ini ely many
(compa ible inde e mina e) solu ions o he con olle o be syn hesized.
In sho , emembe ha a simple linea algeb aic sys em o equa ions
y=Ax
is
sol able in
x
, o compa ible i and only i
y∈Im(A)
. This holds i and only i
ank(A)=
ank(A,b)
(Rouché–Capelli heo em). The solu ion
x
is unique i and only i
A
is non-
singula so ha he algeb aic sys em is compa ible de e mina e. O he wise, i he Rouché–
Capelli heo em s ill holds, he e a e in ini ely many solu ions, and he algeb aic sys em
is compa ible inde e mina e. I
ank(A)< ank(A,b)
, hen
y/∈Im(A)
and he algeb aic
sys em has no solu ion so ha i is incompa ible. In he case o incompa ible sys ems, i
can be ound he bes app oxima e solu ion
x
, which minimizes
ky−Axk
by in ol ing he
pseudoin e se ma ix echniques on he singula ma ix
A
. The whole se s o compa ible
(ei he de e mina e o inde e mina e) solu ions, i hey exis , o he bes app oxima e
solu ion (i no exac solu ion exis s) can be calcula ed by pseudoin e se ma ix echniques
applied on he algeb aic sys em. In ou case, he solu ions consis o inding a linea ou pu
eedback s abilizing con olle gain, i does exis , so ha he closed-loop dynamics equalizes
some p esc ibed s abili y ma ix.
A echnical conce n is ha he algeb aic es o linea ou pu eedback s abilizabili y
canno be pe o med gene ically o some con e gen closed-loop ma ix bu only o gi en
a ge ed con e gen ma ices o closed-loop dynamics. On he o he hand, Sec ion 4 elies
on linking he exis ence o some s a ic linea ou pu eedback s abiliza ion con ol law
o he modi ied ex ended disc e e sys em wi h special Ricca i ma ix algeb aic equali ies.
Sec ion 5is de o ed o he cha ac e iza ion o keeping he s abiliza ion unde a o al o
pa ial deg ee o decen alized con ol. Such a decen aliza ion consis s o he achie emen
o he closed-loop s abiliza ion unde ei he a o al o a pa ial lack o in o ma ion abou
he couplings o mu ual dynamics be ween couples o subsys ems being ansmi ed o
he o e all con olle . In his way, each subsys em con olle ope a es jus wi h local
in o ma ion abou i s own subsys em wi h e en ually a minimum o a ailable in o ma ion
aken abou he mu ual dynamical couplings be ween he a ious subsys ems able o
achie e he closed-loop s abiliza ion. The inal pa o he a icle add esses, in Sec ion 6,
he pa icula cases o small in luences o he delayed disc e e dynamics and ha o he
couplings be ween he pai s o subsys ems in he whole dynamics o he hyb id sys em. In
hose cases, he main con olle syn hesis p ocess is pe o med on he nominal pa o he
sys em ( ha is, he one being ee o unce ain ies) wi h a su icien s abili y deg ee so as o
igh agains he in luence o he unce ain ies while keeping he closed-loop s abili y o he
whole sys em. Finally, conclusions end he pape .
No a ion
n={1, 2, ··· ,n}
Z
,
Z0+
and
Z+
a e, espec i ely, he se s o in ege numbe s, non-nega i e in ege
numbe s and posi i e in ege numbe s.
Ma hema ics 2022,10, 1066 4 o 29
R
,
R0+
and
R+
a e, espec i ely, he se s o eal, non-nega i e eal numbe s and
posi i e eal numbe s.
C
is he se o complex numbe s,
Cα={z∈C:|z|≥α}
and
Cα+={z∈C:|z|>α}
o any eal cons an α∈R0+.
Inis he n- h iden i y ma ix and 0n×mis a ze o n×m-ma ix.
Fo any squa e eal ma ix
M
,
sp(M)
is i s spec um, ha is, he se o i s eigen alues,
de (M)is i s de e minan , and adj(M)is i s adjoin ma ix and MTis he anspose o M.
Le us deno e
M=MijN=Nij
o any wo
n×m
eal ma ices
N
,
M
i
Mij ≤Nij
;
∀(i,j)∈n×m
, and deno e as
ρ(M)
he spec al adius o any gi en squa e
ma ix
M
. In he same way,
M≺N
i
N
,
M6=N
i
Mij ≤Nij
;
∀(i,j)∈n×m
. No e ha ,
a leas one pai o co esponding ma ix en ies, he associa ed inequali y is s ic , and
M≺≺ N
i
Mij <Nij
;
∀(i,j)∈n×m
. Pa icula cases a e ela ed o compa isons wi h he
ze o ma ix so ha
M∈Rn×n
0+
, o
M
0, deno es a non-nega i e ma ix, ha is,
Mij ≥
0;
∀(i,j)∈n×m
;
M(6=0)∈Rn×n
0+
, o
M
0, deno es a posi i e ma ix, ha is,
Mij ≥
0;
∀(i,j)∈n×m
wi h
M6=
0;
M∈Rn×n
+
, o
M
0, deno es a s ic ly posi i e ma ix, ha is,
Mij >0; ∀(i,j)∈n×m.
I
M
is a squa e eal ma ix, hen
M≥
0 and
M>
0 deno e ha i is, espec i ely,
posi i e semide ini e and posi i e de ini e.
M≤
0 and
M<
0 deno e ha he ma ix is
nega i e semide ini e and nega i e de ini e, espec i ely.
A squa e ma ix
M
is a s abili y ma ix i i s spec al abscissa is nega i e, i.e., i
max Re λi<
0 o
λi∈sp(M)
. whe e
sp(M)
is he se o eigen alues, o spec um, o
M
. A eal o complex squa e ma ix
M
is con e gen i and only i i s spec al adius
ρ(M)={max|λi|:λi∈sp(M)}<1.
i=√−1 is he imagina y complex uni .
σ[M(iω)]
, o
ω∈R
, s ands o he singula alues o he complex- alued a ional
ma ix M:C→Cn×m.
The
H∞
-no m o such a ma ix, which is he sup emum o i s singula alues on he
bounda y o he uni ci cle cen e ed a he o igin o he complex plane, p o ided ha i
exis s and is ini e, is deno ed by
kMkH∞
. I
M∈Rn×m
, hen he symbols
kMk∞
,
kMk1
and
kMk2
s and, espec i ely, o he
∞
, 1 and 2 ma ix no ms, ha is, o he maximum
absolu e sum o i s ows, ha o i s columns and o i s maximum singula alue.
M⊗N=Mij Nkl
is he K onecke p oduc o he eal ma ices o any o de s
M=Mij
and
N=Nij
. In pa icula , i
M
and
N
a e
m×n
and
p×q
, hen i s K onecke
p oduc is mp ×nq de ined by
M⊗N=


M11N··· M1nN
.
.
.....
.
.
Mm1N··· MmnN



ec(M)
is a eal ec o o med by he en ies o he eal ma ix
M
o de ed in he o de
o i s ows.
A†∈R ×m
is he Moo e–Pen ose gene alized in e se, o Moo e-Pen ose pseudoin-
e se, o
A∈Rm×
which sa is ies
A=AA†A
and
A†=A†AA†
. I
A∈Rm×
is o
ank
p
is, in gene al non-uniquely, ac o ized as
A=FG
wi h
ankP = ankG =p
and
F∈Rm×p
and
G∈Rp×
( hus being, espec i ely, ull column ank and ull ow ank),
hen
A†=GTGGT−1FTF−1FT
. No e ha such
F
and
G
always exis o a gi en
A
o
ank p.
The con inuous ime s a es o signals a e deno ed unde he a gumen “
”in pa en-
hesis, say
x( )
, ( unning on he non-nega i e eal se ) while he disc e e- ime ones o he
digi al ones a e deno ed wi h he a gumen “
k
” in b acke s, say
x[k]
, ( unning on he se o
non-nega i e in ege numbe s).
Ma hema ics 2022,10, 1066 5 o 29
2. Hyb id Con inuous-Time and Digi al Sys em
2.1. Simple Mo i a ion Example
To ix some ideas, we discuss a simple mo i a ing example o pu ely con inuous- ime
o disc e e- ime sys ems so ha a amily o s a ic linea ou pu eedback con olle s exis
de ined by an open ball a ound a linea ou pu eedback s abilizing con olle :
Example 1.
Conside he ei he uns able o c i ically s able disc e e cha ac e is ic polynomial
p(z)=de (zIn−A)
which desc ibes he open-loop (i.e., uncon olled) dynamics o he disc e e
con inuous- ime uns able
n
- h o de sys em o s a e
x[k]∈Rn
, con ol
u[k]∈R
and ou pu
y[k]∈Rm
o
k∈Z0+
, gi en by
x[k+1]=Ax[k]+Bu[k]
,
y[k]=Cx[k]
,
x(0)=x0
;
k∈Z0+
,
wi h
B∈Rn×
and
C∈Rm×n
wi h
max( ,m)≤n
. No e ha he sys em is s abilizable by s a ic
linea eedback s a e con ol i all he uns able open-loop modes a e con ollable, ha is, i and only
i
ank[zIn−A,B]s∈C1∩sp(A)=n
. Assume ha i is in ended o s abilize i by a s a ic ou pu
eedback con ol law
u[k]=Ky[k]=kCx[k]
;
k∈Z0+
, o some cons an con ol gain
K∈R ×m
,
so ha he closed-loop sys em becomes
x[k+1]=(A+BKC)x[k]
;
k∈Z0+
. Taking z- ans o ms
in he closed-loop equa ion unde ze o ini ial condi ions yields ha he closed-loop cha ac e is ic
polynomial is:
zIn−A−BKC =zIn−A−Am+(Am−BKC)
o any gi en
Am∈Rn×n
supposed o be con e gen ( ha is, a con e gen ma ix in he disc e e con ex ),
i.e., wi h eigen alues o modulus less han uni y so ha i exis s (zIn−Am)−1,∀z∈C1. Then,
(zIn−A−BKC)−1=hIn−(zIn−Am)−1(A+BKC −Am)i(zIn−Am)−1;∀z∈C1
exis s o all
z∈C1
i he
H∞
- no m o
(zIn−Am)−1(Am−A−BKC)
is less han one which
occu s i kAm−A−BKCkis su icien ly small o gua an ee ha




1
de (zIn−Am)[adj(zIn−Am)(A+BKC −Am)]


H∞
=sup
ω∈R0+




adjeiωIn−Am(A+BKC −Am)
de (eiωIn−Am)



H∞
<1
No e ha he es is pe o med on he bounda y o he complex uni y ci cle cen e ed a ze o. In
addi ion, in his case, he closed-loop eigen alues a e s able, which is, in pa icula , gua an eed i
kA+BKC −Amk2≤ε<1/

(zIn−Am)−1

H∞
Now, he p oblem educes o ind i i exis s, a iple
(Am,K,∆)
, such ha
A+BKC −Am=∆
wi h
K∈R ×m
and
Am
,
∆∈Rn×n
such ha
Am
is a con e gen ma ix wi h
k∆k2=ε
o any
eal cons an
ε∈[0, ε)≡0, 1/

(zIn−Am)−1

H∞
. Since i also has o be ul illed o
ε=
0,
one concludes ha a necessa y condi ion is ha
A+BK0C=Am
o some con e gen ma ix
Am
so ha
(A,B)
has o be s abilizable and
(C,A)
de ec able, ha is,
ank[zIn−A,B]z∈C1∩sp(A)=
ankzIn−AT,CTz∈C1∩sp(A)=n
. Then, any o he s a ic linea ou pu eedback con olle o
gain
K
s abilizes he closed-loop sys em i
∆=A+BKC −Am
has a no m
ε∈[0, ε)
. Thus,
A+BKC is a con e gen ma ix o any K ∈Kd=(K:kK−K0k<1
kBkkCk

(zIn−Am)−1

H∞).
The abo e conclusion will be iden ical i
p(s)=de (sIn−A)
is he ei he uns able o
c i ically s able cha ac e is ic polynomial which desc ibes he open-loop dynamics o he linea
con inuous- ime uns able
n
- h o de sys em o s a e
x( )∈Rn
, con ol
u( )∈R
and ou pu
y( )∈Rmgi en by
.
x( )=Ax( )+Bu( );y( )=Cx( ),x(0)=x0

Ma hema ics 2022,10, 1066 6 o 29
To s abilize i unde a s a ic ou pu con ol law
u( )=Ky( )
, he open-loop sys em has o be s abi-
lizable and de ec able, ha is,
ank[sIn−A,B]s∈C0∩sp(A)= ankzIn−AT,CTs∈C0∩sp(A)=n
so ha a closed-loop equa ion
A+BK0C=Am
is achie ed o some con e gen ma ix
Am
and some
s abilizing s a ic ou pu linea eedback con olle o gain
K0
and any o he s a ic linea ou pu eedback
con olle o gain K s abilizes also he closed-loop sys em i
∆=A+BKC −Am=A+B(K−K0)C+BK0C−Am=B(K−K0)C
has a no m
ε∈[0, ε)
wi h
ε=h0, 1/

(sIn−Am)−1

∞
so ha
A+BKC
is a con e gen ma ix
o any K ∈Kc=(K:kK−K0k<1
kBkkCk

(sIn−Am)−1

∞).
2.2. Sys em S uc u e
Conside he subsequen single-inpu single-ou pu hyb id linea sys em which con-
sis s o qcoupled subsys ems:
xc( )=hxT
c1( ),xT
c2( ),··· ,xT
cq( )iT;xd[k]=hxT
d1[k],xT
d2[k],··· ,xT
dq[k]iT(1)
u( )=u1( ),u2( ),··· ,uq( )Ty( )=y1( ),y2( ),··· ,yq( )T(2)
.
xci ( )=∑q
j=1∑p
`=0Acij xcj( )+Ac s`ij xcj [k−`]+Acd`ij xdj [k−`]+bci ui( )+bc si ui[k](3)
xdi [k+1]=∑q
j=1∑p
`=0Ad`ij xdj[k−`]+Adc`ij xcj [k−`]+bdi ui[k](4)
yi( )=cT
ci xci( )+cT
csi xci[k]+cT
di xdi [k]+dci ui( )+ddi ui[k](5)
o all
∈[kT,(k+1)T)
o any in ege
k≥
0 wi h
T
being he sampling pe iod, whe e
xci( )∈Rnci
and
xdi[k]∈Rndi
;
∀i∈q
a e, espec i ely, he dimensions o he
i
- h con-
inuous and digi al subsys ems, espec i ely, whose scala inpu and ou pu a e
ui(.)
and
yi(.)
, espec i ely, o
i∈q
, and
p
is he numbe o disc e e in e nal delays. Thus,
nc=∑q
i=1nci
and
nd=∑q
i=1ndi
a e he con inuous and digi al dimensions o he whole
sys em in eg a ed by he a ious subsys em. The pa ame e iza ion o (1)–(5) is gi en by ma-
ices
Acij
,
Acs`ij ∈Rnci×ncj
,
Ad`ij ∈Rndi×ndj
,
Acd`ij ∈Rnci×ndj
,
Adc`ij ∈Rndi×ncj
;
∀i
,
j∈q
,
∀l∈p∪{0}
, which a e ma ices o con inuous and digi al dynamics;
bci
,
bcsi
,
cci
,
ccsi ∈Rnci
;
∀i∈q
, which a e con ol and ou pu ec o s o he
i
- h subsys em;
bdi
,
cdi ∈Rndi
,
∀i∈q
,
which a e con ol and ou pu ec o s o he i- h digi al subsys em; and
dci
,
ddi ∈R
, which
a e he con inuous and digi al di ec inpu –ou pu in e connec ion gains o he
i
- h con-
inuous and digi al subsys em, espec i ely;
∀i∈q
. The con inuous- ime a gumen is
deno ed by ‘( )’ while he disc e e- ime a gumen is deno ed by ‘
[k]
’ and he associa ed
con inuous and digi al a iables a e deno ed co espondingly. Thus, a con inuous a iable
a sampling ins an s is deno ed in he same way as a digi al a iable so ha
xc[k]=xc(kT)
,
u[k]=u(kT)
and
y[k]=y(kT)
o any in ege
k≥
0. Simila no a ions wi h b acke s and
pa en hesis a e o he ime a gumen s o he disc e e and con inuous a iables o he sub-
sys ems. In his way, he e is no dis inc ion in he ea men o digi al and ime-disc e ized
a iables a sampling ins an s. The o de s o all he eal cons an ma ices in (1) ag ee wi h
he dimensions o he subs a es and scala inpu and ou pu . I can be poin ed ou ha
a digi al sys em wi hin he whole hyb id s uc u e could ins ead be a dynamic sys em
being disc e ized om a con inuous one a a ce ain sampling pe iod in a si ua ion such
ha he o iginal con inuous- ime s uc u e has no speci ic in e es in he analysis since
he associa ed signals a e only ele an a he sampling ins an s. This kind o sys em can
be ea ed in he same way wi hin he p oposed hyb id con inuous/disc e e s uc u e. I
can be also poin ed ou ha a ypical s uc u e o a hyb id dynamic sys em can be ound
in cases when a con inuous sys em is in ope a ion unde a disc e ized con olle so ha
Ma hema ics 2022,10, 1066 7 o 29
he whole s uc u e has a hyb id con inuous- ime/disc e e- ime na u e consis ing o a
minimum o wo subsys ems.
The abo e sys em can be desc ibed in a compac o m as ollows:
.
xc( )=Acxc( )+∑p
`=0(Acs`xc[k−`]+Acd`xd[k−`]) +Bcu( )+Bcsu[k](6)
xd[k+1]=∑p
`=0Adc`xc[k−`]+Ad`xdj[k−`]+Bdu[k](7)
y( )=Ccxc( )+Ccsxcs[k]+Cdxd[k]+Dcu( )+Ddu[k](8)
whe e
Ac∈Rnc×nc
,
Acs`
,
Acd`∈Rnc×nd
,
Bc
,
Bcs ∈Rnc×q
,
Adc ∈Rnd×nc
,
Ad∈Rnd×nd
,
Bd∈
Rnd×q
,
Cc
,
Ccs ∈Rq×nc
,
Cd∈Rq×nd
,
Dc
,
Dd∈Rq×q
;
`=
0, 1,
. . .
,
p
, wi h
nc=∑p
i=1nci
and nd=∑p
i=1ndi a e de ined by:
Ac=


Ac11 ··· Ac1q
.
.
..
.
..
.
.
Acq1··· Acqq


(9)
Acs`=


Acs`11 ··· Ac`s1q
.
.
..
.
..
.
.
Acs`q1··· Acs`qq


;Acd`=


Acd`11 ··· Acd`1q
.
.
..
.
..
.
.
Acd`q1··· Acd`qq


;`=0, 1, . . . , p(10)
Adc`=


Adc`11 ··· Adc`1q
.
.
..
.
..
.
.
Adc`q1··· Adc`qq


;Ad`=


Ad`11 ··· Ad`1q
.
.
..
.
..
.
.
Ad`q1··· Ad`qq


;`=0, 1, . . . , p(11)
Bc=block diagbc1··· bcq;
Bcs =block diagbcs1··· bcsq;Bd=block diaghbd1··· bdqi(12)
Cc=block diaghcT
c1··· cT
cqi;
Ccs =block diaghcT
cs1··· cT
csqi;Cd=block diaghcT
d1··· cT
dqi(13)
Dc=diagdc1···dcqDd=diaghdd1···ddqi(14)
The con inuous- ime subs a e e ol es h ough ime acco ding o he ollowing solu ion
equa ion ob ained om (6):
xc(kT +σ)=eAcσInc+Rσ
0e−AcτdτAc s0xc[k]
+Rσ
0eAc(σ−τ)dτ∑p
`=1Acs`xc[k−`]+∑p
`=0Acd`xd[k−`]
+Rσ
0eAc(σ−τ)dτBc s u[k]
+Rσ
0eAc(σ−τ)Bcu(kT +τ)dτ≥0 ; ∀k∈Z0+,∀σ∈(0 , T]
(15)
which can be compac ed as ollows a he sampling imes:
xc[k+1]=Ψc[k]x[k]+Γcs(T)u[k]+RT
0eAc(T−τ)Bcu(kT +τ)dτ≥0 ; ∀k∈Z0+
,∀σ∈(0 , T]
(16)
whe e
x[k]=xT
c[k],xT
d[k]T∈R(p+1)n,Ψc[k]=Ψcc[k],Ψcd[k]∈Rnc×(p+1)n(17)
Ma hema ics 2022,10, 1066 8 o 29
xT
c[k]=xT
c[k],xT
c[k−1],··· ,xT
c[k−p]∈R(p+1)nc(18)
xT
d[k]=xT
d[k],xT
d[k−1],··· ,xT
d[k−p]∈R(p+1)nd(19)
Ψcc[k]=hΦc(T)+Γcs0(T),e
Ψcc[k]i=Φc(T)+Γcs0(T),Γcs1(T),··· ,Γcsp(T)∈Rnc×(p+1)nc(20)
Ψcd[k]=hΓcd0(T),e
Ψcd(T)i=hΓcd0(T),Γcd1(T),··· ,Γcdp(T)i∈Rnc×(p+1)nd(21)
Φc(T)=eAcT,Γcs`(T)=RT
0eAc(T−τ)dτAcs`,Γcd`(T)=RT
0eAc(T−τ)dτAcdl;
l=0, 1, . . . , p(22)
Γcs(T)=ZT
0eAc(T−τ)dτBc s (23)
;
∀k∈Z0+
, wi h
n=nc+nd
,
nc=∑q
i=1nci
and
nd=∑q
i=1ndi
. The dimension o he
ex ended disc e e s a e
x[k]=xT
c[k],xT
d[k]T
is
n=(2p+1)n
. The disc e e- ime subs a e
e ol es h ough ime acco ding o he ollowing solu ion equa ion, ew i en equi alen ly
om (7):
xd[k+1]=Ψdx[k]+Bdu[k];∀k∈Z0+(24)
whe e
Ψd(T)=Ψdc(T),Ψdd∈Rnd×(p+1)n(25)
Ψdc =hAdc0,e
Ψdci=hAdc0,Adc1,··· ,Adcpi∈Rnd×(p+1)nc(26)
Ψdd =hAd0,e
Ψddi=hAd0,Ad1,··· ,Adpi∈Rnd×(p+1)nd(27)
2.3. Ex ended Disc e e Sys em
Combining (16) and (23), one concludes ha he ex ended disc e e ec o , buil wi h
he sampled alues o he con inuous- ime subs a e and he disc e e subs a e, e ol es
acco ding o he ollowing ex ended disc e e Equa ion:
x[k+1]=Adx[k]+Bdu[k]+Bcτu[k](28)
whe e
Ad=



Ψcc[k]
Ipnc
Ψdc
0p×(p+1)nc
Ψcd[k]
0p×(p+1)nd
Ψdc
Ipnd



∈Rn×n(29)
Bd=



Γcs
0pnc×2q
Bd
0pnd×2q



∈Rn×q;Bcτu[k]=


RT
0eAc(T−τ)Bcu(kT +τ)dτ
0pnc×2q
0nd×2q
0pnd×2q



∈Rn×q(30)
Fo pu poses o gene a ing he in e sample inpu om a disc e e sequence de ined a
he sampling ins an s, he ollowing echnical assump ion is made which will be use ul o
some o he coming esul s:
Assump ion 1.
Assume ha
u(kT +τ)=L(kT +τ) [k]
;
∀k∈Z0+
,
∀τ∈(0, T)
o some
con ol sequence
{ [k]}∞
k=0
and some ma ix unc ion
L:Z0+×[0, T)→Rq×q
;
∀τ∈[0 , T)
which sa is ies he cons ain s:
(1) I is pe iodic wi h pe iod
T
, ha is,
L(kT +τ)=Lk(T,τ)=L(T,τ)
;
∀k∈Z0+
,
∀τ∈[0, T)
.
(2)
I has a suppo o nonze o Lebesgue measu e on [0, T).
(3)
The Lebesgue in eg al Bc=RT
0eAc(T−τ)BcL(τ)dτexis s and i is ini e.
Ma hema ics 2022,10, 1066 9 o 29
Rema k 1.
No e ha Assump ion 1 allows a la ge a ie y o de ini ions o
L:Z0+×[0, T)→Rq×q
including isola ed bounded discon inui ies o e en a ini e numbe s o Di ac impulses on each in e al
[kT,(k+1)T)
. Mo eo e , i
u[k]
and
[k]
a e designed independen ly o each o he om, in gene al,
dis inc p e-calcula ed bounded sequences
{u[k]}∞
k=0
and
{ [k]}∞
k=0
, acco ding o some bene icial
design c i e ion, namely wi hou using a cons ain
L[k] [k]=u[k]
;
k∈Z0+
, hen he con ol
unc ion
u:Z0+×[0, T)→Rq
can ha e bounded discon inui ies a he sampling ins an s since
L[k] [k]6=u[k]
. Howe e , an ad an age o his si ua ion is ha he ex ended disc e e con ol
sequence
nuT[k], T[k]To∞
k=0∈R2q
which go e ns (28)–(30) has a dimension 2
q
ins ead o
q
such ha he po en ial s abiliza ion o such a modi ied ex ended disc e e sys em migh be achie able
unde weake condi ions han he use o he equalizing con ol cons ain
L[k] [k]=u[k]
a he
sampling ins an s.
The ou pu a sampling ins an s becomes om (8):
y[k]=(Cc+Ccs,Cd)xc[k]
xd[k]+(Dc+Dd)u[k]=Cx[k]+(Dc+Dd)u[k](31)
whe e C=Cc+Ccs, 0q×nc`,Cd, 0q×nd`.
2.4. Modi ied Ex ended Disc e e Sys em wi h Two Inpu Channels
We now gene a e he con inuous- ime con ol om a disc e e sequence being, in
gene al, dis inc o he p ima y disc e e con ol sequence
{u[k]}∞
k=0
. This s a egy allows
aking ad an age o he use o a double dimensioned con ol inpu o he ex ended disc e e
sys em which will acili a e i s po en ial s abiliza ion. In ac , he second con ol channel is
ob ained om he gene a ion o he in e sample con inuous- ime inpu om he auxilia y
disc e e sequence. As a esul , he ex ended sys em is con olled by a 2q dimensional
disc e e con ol sequence. Unde Assump ion 1, Equa ions (28)–(30), oge he wi h (32),
ake he compac o m o he ollowing modi ied ex ended disc e e sys em o s a e o
dimension
n=(2p+1)n
which desc ibes a sampling imes he join disc e ized dynamics
o he con inuous- ime sys em plus ha o he digi al one o a disc e e con ol sequence
{u[k]}∞
k=0wi h u[k]=uT[k], T[k]T∈R2q:
x[k+1]=Adx[k]+Γdu[k];y[k]=Cx[k]+Du[k](32)
whe e u[k]=uT[k], T[k]T, and
Γd=Bd,Bc∈Rn×2q;Bc=



Bc
0pnc×2q
0nd×2q
0pnd×2q



∈Rn×q;Bc=RT
0eAc(T−τ)BcL(kT +τ)dτ
C=Cc+Ccs , 0q×ncp,Cd, 0q×ndp∈Rq×n;D=Dc+Dd, 0q×q∈Rq×2q
(33)
The ollowing p elimina y s abilizabili y esul is o in e es o subsequen esul s o
be hen ob ained:
Rema k 2.
The ex ended disc e e sys em (32) and (33) is s abilizable by s a ic linea s a e eedback
u[k]=Kx[k]
i and only i
ankzIn−Add,Γd=n
o each
z∈C1∩spAd
[
22
,
27
,
33
,
34
].
This esul ollows di ec ly om he Popo –Bele i ch–Hau us s abilizabili y es , [
40
,
41
] o disc e e
sys ems applied o (32) and (33) by using a simila i y ans o m on
Ad
o a iangula o m
ˆ
Ad="ˆ
Ad11 ˆ
Ad12
0ˆ
Ad22 #
, which makes he associa ed con ol ma ix
ˆ
Γd="ˆ
Γd1
0#
and he
ans o med s a e ec o becomes
ˆ
x( )=ˆ
x1( )
ˆ
x2( )
, such ha
ˆ
Add22
is s able desc ibing he
Ma hema ics 2022,10, 1066 16 o 29
o he modi ied ex ended disc e e sys em (32) and (33). The con olle gain sa is ies he
equi alen condi ions (65) and (66), subjec o (68), since s a ic s abilizing con olle gains
K
exis such ha (65) is an algeb aic compa ible sys em so ha
ankAT
dPΓd⊗CT= ankhAT
dPΓd⊗CT,−AT
d⊗Ad−Iˆ
n2−AT
d⊗GTΓT
d+GTΓT
d⊗GTΓT
di (74)
In addi ion, (67), subjec o (69), ollows by eplacing
G
in he las addi i e le -hand-
side e m o (64), ob ained om (53) in o (64), and hen calcula ing
K
wi h he gene al
solu ion based on pseudoin e sion ules.
To p o e P ope y (iii), no e ha one ob ains by equalizing he wo igh -hand-sides o
(66) and (67) ha
K1−ΓT
dPΓd+R−1GT†K3C†+K2Iq−CC†−K4+ΓT
dPΓd+R−1GT†GTΓT
dPΓd+R−1K4CC†=0 (75)
which leads o (71) a e eplacing (68) and (69) in i s le -hand side. Since
K2
,
K4
a e
a bi a y, one ob ains (72) by aking
K2=K4−ΓT
dPΓd+R−1GT†GTΓT
dPΓd+R−1K4CC†Iq−CC††(76)
o , simply, by ze oing
K2
and
K4
. On he o he hand, (73) ollows by eplacing
K2
,
Equa ion (72), in (68), and he ob ained esul , oge he wi h and (76), in (66). 
Rema k 7.
Theo em 3(iii) implies ha , in gene al,
G
is no unique in Theo em 3(ii). As a esul ,
P
is posi i e de ini e and unique in (64) once
G
has been ixed o each gi en symme ic posi i e
ma ix R i and only i he pai Ad,Γdis con ollable.
I u ns ou ha a gene al applica ion o Theo em 3 migh be e y in ol ed in he
cases o a ce ain dimensionali y, and gene alized in e ses no being coinciden wi h he
s anda d ones a e in ol ed in he compu a ions. Howe e , i can be use ul o discussing
in a closed o m he exis ence o a s abilizing s a ic linea ou pu eedback con olle o he
ex ended disc e e sys em.
5. Decen alized e sus Cen alized Con ol o he Ex ended Disc e e Sys em
I is now discussed i he s abilizing con ol gain can be spa se i no in i s o -diagonal
en ies and how spa se i can be. As i is admi ed o being mo e spa se in i s o -diagonal
pa , mo e in o ma ion could be dele ed o each indi idual subsys em om he emaining
ones while s ill keeping he s abiliza ion p ope y o he whole sys em. No e ha he s a ic
con olle gain is o he o m:
K=Kd+Kod =K1
K2;Kd:=K1d
K2d;Kod :=K1od
K2od (77)
whe e he abo e six column ma ix blocks a e squa e q-ma ices and
Kid
and
Kiod
a e
diagonal, espec i ely, and o diagonal ze o en ies, o
i=
1, 2. No e ha
K
has 2
q
diagonal en ies and 2
q(q−1)
o non-diagonal ones. Assume ha he whole amily o such
s abilizing con olle s ia linea ou pu eedback o he modi ied ex ended disc e e sys em
is
K
. No e ha he abo e conside a ion is only o in e es i
q>
1, i.e., i he e a e a leas
wo coupled subsys ems in he whole s uc u e. The whole decen aliza ion implies ha
each subsys em is con olled by a con ol inpu which has a ailable in o ma ion only on i s
own ou pu . The wo subsequen de ini ions ely on how s ong he decen aliza ion o he
ou pu in o ma ion is o make possible he s abiliza ion o he whole coupled sys em.

Ma hema ics 2022,10, 1066 17 o 29
De ini ion 1.
The maximum decen alized deg ee o ou pu linea eedback s abiliza ion (MDdos) o
he ex ended disc e e sys em is he maximum numbe o non-diagonal ze o en ies
i∈[0 , 2q(q−1)] ∈Z0+in Kod, be ween all he gains K ∈K.
De ini ion 2.
The minimum cen alized deg ee o ou pu linea eedback s abiliza ion (mCdos) o he
ex ended disc e e sys em is he minimum numbe o non-diagonal ze o en ies
i∈[0 , 2q(q−1)] ∈Z0+
in Kod, be ween all he gains K ∈K.
I can be obse ed ha De ini ions 1 and 2 ha e only sense o
q≥
2 since, i
q=
1,
ha is, he whole sys em consis s o a single subsys em, hen he e is no dis inc ion be ween
cen alized and decen alized con ol. No e ha , i ially, 2
q(q−1)
= (mCdos) + (MDdos).
No e also ha i MDdos = 2
q(q−1)
, hen he linea ou pu eedback s abiliza ion o he
ex ended disc e e sys em may be pe o med wi h some ully decen alized con ol o
gain
K=Kd∈K
, ha is, he whole closed-loop s abiliza ion may be pe o med unde
indi idual con olle s o each subsys em which only ake in o ma ion on he ou pu o
such a subsys em, ha is, jus o one o he componen s o he ou pu ec o which is he
ou pu o he in ol ed subsys em. Fu he mo e, no e ha i MDdos = 2
q(q−1)
, hen he
closed-loop s abiliza ion can only be pe o med unde ully cen alized con ol, i.e., each
subsys em has o acqui e a ailable in o ma ion on he ou pu s o all he subsys ems in he
whole s uc u e.
The subsequen esul add esses he closed-loop ully decen alized s abiliza ion o
he modi ied ex ended disc e e sys em ia linea ou pu eedback based on Theo em 2 and
on Theo em 3.
Theo em 4. Assume ha Dc+Dd=0. Then, he ollowing p ope ies hold:
(i)
Assume ha he e exis s some con e gen ma ix
Acld
such ha
ΓdKC =Acld −Ad
is
sol able wi h a solu ion:
K=Kdiag +Kodiag =Γ†
dAcld −AdC†∈K(78)
o , equi alen ly,
ecK=Γ†
d⊗C†T ec Acld −Ad(79)
In addi ion, assume also ha
ank hI2q2−Γ†
dΓd⊗C†TCTTi= ankI2q2−Γ†
dΓd⊗C†TCT,Γ†
d⊗C†T ec Acld −Adod(80)
Then,
MDdos =
2
q(q−1)
so ha he closed-loop modi ied ex ended disc e e sys em can be
s abilized wi h ully decen alized con ol which alloca es he closed-loop modes o he modi ied
ex ended sys em a he eigen alues o Acld.
(ii)
Assume ha he hypo heses o Theo em 3 and (72) hold. Assume also ha
ank Ω2=
ankΩ2, ec Ω1od , whe e
Ω1=ΓT
dPΓd+R−1ΓT
dPA −ΓT
dPΓd+R−1GT†P−AT
dPAd−CTC+AT
dPΓdΓT
dPΓd+R−1+GTΓT
dPAd
−ΓT
dPΓd+R−1ΓT
dPC†=Ω1d+Ω1od
(81)
Ω2=Iq⊗Iq−C C†TIq−CC†T−ΓT
dPΓd+R−1GT†GTΓT
dPΓd+R−1⊗Iq−C C†TIq−CC†TCC†T(82)
Then,
MDdos =
2
q(q−1)
so ha he closed-loop modi ied ex ended disc e e sys em can
be s abilized wi h ully decen alized con ol which alloca es he closed-loop modes a he
eigen alues o some exis ing con e gen ma ix.
Ma hema ics 2022,10, 1066 18 o 29
P oo .
P ope y (i) ollows om (53) and (54) by aking in o accoun ha
CT†=C†T
since
(78) is a pa icula solu ion wi h
X=
0, hen
ec(X)=
0, o
ΓdKC =Acld −Ad
which is
sol able i and only i (80) holds, and hen he e is a eal ma ix
X
o o de 2
q2×q
gi en by
hI2q2−Γ†
dΓd⊗CTC†Ti ec (X)= ec ΓdΓ†
dAcld −AdC†Cod
=ΓdΓ†
d⊗CTC†T ecAcld −Adod
(83)
See (58), such ha he e is some
K=Kd∈K
since
Kod =K−Kd=
0 i (80) holds,
since one has ha he gene al solu ion in Kwhich includes as a pa icula case (79) is:
ecK= ecKd+ ecKod= ecKd
=Γ†
d⊗C†T ec Acld −Ad+hI2q2−Γ†
dΓd⊗C†TCTi ec (X)
(84)
=Γ†
d⊗C†T ec Acld −Add
+Γ†
d⊗C†T ec Acld −Adod
+hI2q2−Γ†
dΓd⊗C†TCTi ec (X)(85)
=Γ†
d⊗C†T ec Acld −Add
(86)
In addi ion, (86) holds by ze oing he second addi i e e m o he igh -hand side o
(85) by he choice o a solu ion
ecX
which exis s since (80) holds. P ope y (i) has been
p o ed. To p o e P ope y (ii), no e ha i he hypo heses o Theo em 3 and (72) hold, hen
a se o s abilizing con olle gains sa is ying (73) can be calcula ed which can be ec o ized
as ollows:
ecK= ec(Ω1d)+Ω1od +Ω2 ec K4 (87)
No e ha
ec(Ω1od)+Ω2 ecK4=
0, i
ank(Ω2)= ankΩ2, ec Ω1od 
, o
ec K4=−Ω2⊗I2q2 ec Ω1od 
, hen
ecK= ecKd= ec(Ω1d)
so ha a ully
s abilizing con olle o gain
K=Kd∈K
s abilizes he closed-loop sys em unde linea
ou pu eedback ully decen alized s abiliza ion. P ope y (ii) has been p o ed. 
Rema k 8.
No e ha Theo em 4 elies on he ully decen alized ou pu eedback s abiliza ion
h ough a s a ic con olle o he modi ied ex ended disc e e sys em. I s ex ension o a pa ial
decen alized s abiliza ion is di ec unde simila ools ia al e na i e, mo e gene al decomposi ions
ecK= ecKqd+ ecKqod
in Theo em 4(i) and
Ω=Ω1qd +Ω1qod
o Theo em 4(ii) in
quasi-diagonal and o -quasi-diagonal column ma ix blocks by including he en a i e minimum
numbe o he quasi-diagonal en ies coming, dele ing hem om he o -quasi-diagonal blocks. In
his case, he decen alized s abiliza ion is no ull and can ha e di e en deg ees o decen aliza ion
depending on he o -diagonal en ies ans e ed o he quasi-diagonal column ma ix blocks.
Example 2.
Conside he ollowing hyb id delay- ee sys em consis ing o wo subsys ems gi en by:
.
xc1( )=xc2( )
xd1[k]=−xc1[k]−2xd1[k]+u[k]
(88)
.
xc2( )=−xc2( )−0.8xc1( )−2xd1[k]+u( )+0.8u[k]
y( )=y1( )=xd1[k];∀k∈Z0+,∀ ∈[kT (k+1)T)
subjec o any gi en ini e ini ial condi ions, whe e
T
is he sampling pe iod. The disc e iza ion
o (88) yields he ollowing desc ip ion o hi d o de h ough ex ended disc e e ec o
x[k]=
(xc1[k],xc2[k],xd1[k])T:
Ma hema ics 2022,10, 1066 19 o 29


xc1[k+1]
xc2[k+1]
xd1[k+1]
=
−0.8T+e−T+0.2 1 −e−T−2T−e−T−1
0.81−e−Te−T21−e−T
−1 0 −2


xc1[k]
xc2[k]
xd1[k]
+

0.8T+e−T−1
0.8e−T−1
1
u[k]
+1.5

RT
01−e−(T−τ)u(kT +τ)dτ
RT
0e−(T−τ)u(kT +τ)dτ
0


;y[k]=(0 , 0 , 1)

xc1[k]
xc2[k]
xd1[k]

(89)
To de ine an auxilia y inpu sequence
{ [k]}∞
k=0
gene a e he con inuous con ol inpu
u(kT +τ)=L(T,τ) [k]in he in e sample in e als wi h L(T,τ)=1
1−e−(T−τ)1
e−(T−τ)0T;
∀k∈Z0+
,
∀τ∈(0 , T)
. Fo a sampling pe iod o
T=
0.4 s, he ma ix o dynamics o
he uncon olled ex ended disc e e sys em Equa ion (89) has as eigen alues
z1=
0.21226 and
z2,3 =−
1.26942
±
3.12457
i
, he wo complex conjuga e ones being uns able. The ex ended disc e e
con ol 3
×
2ma ix associa ed wi h he wo-dimensional ex ended con ol sequence
{u[k], [k]}∞
k=0
o T =0.4 s becomes:
Γd=Bd,Bc=

0.05625
−0.263744
1
0.6
0.6
0
(90)
The s a ic con olle gain o he ex ended disc e e sys em is o he o m
K=K1,K2T∈R2
leading o he ollowing closed-loop ma ix o dynamics o he modi ied ex ended disc e e sys em
Acld =
−0.8T+e−T+0.2 1 −e−T2e−T−T−1+0.1125 +0.6K2
0.81−e−Te−T0.65936 −0.526549 +0.6K2
−1 0 K1−2
(91)
Since
T=
0.4 , i
K1=
2and
K2=
0.34686
/=
0.5781 , hen a solu ion o Equa ion (42) is
ecK=K1,K2T=(2, 0.5781)
, which is sol able acco ding o (35), o a a ge ed ma ix o
closed-loop dynamics gi en by he o de ed ow-pe - ow ec o de ined by:
ecAcld=(−0.656256, 0.32968 , 0 , 0.263744 , 0.67032 , 0.479671 , −1, 0, 0)(92)
The o de ed ow-pe - ow ec o co esponding o he ma ix ΓdKCTis gi en by:
ecΓdKCT=0, 0, 0.05625K1+0.6K2, 0, 0, −0.2632744K1+0.6K2, 0, 0, K1(93)
co esponding o he ma ix:
Acld =
−0.656256 0.32968 0
0.263744 0.67032 0.479671
−100

wi h cha ac e is ic polynomial
p(z)=z3−
0.0146
z2−
0.526853
z+
0.150995 whose ze os a e all
s able wi h alues
z1=−
0.83432;
z2,3 =
0.42419
±
0.03213
i
. As a esul , as
k→∞
,
{x[k]}→0
,
{y[k]}→0
,
{u[k]}→0
,
{ [k]}→0
,
xd1[k]→0
,
xc1[k]→0
,
xc2[k]→0
o any gi en i-
ni e ini ial condi ions. Since
{ [k]}→0
as
k→∞
,
u( )→0
as
→∞
,
xc1( )→0
,
xc2( )→0
and
y( )→0
as
→∞
. The con olle is o decen alized ype since i only picks up
in o ma ion o he i s subsys em h ough i s ou pu which is also he global ou pu o he whole sys em.
Ma hema ics 2022,10, 1066 20 o 29
6. Cases o Small In luences o he Delayed Disc e e Dynamics and o he Couplings
be ween Subsys ems
No e ha he closed-loop ex ended disc e e sys em can be e- o mula ed wi h i s
s a e e olu ion by aking in o accoun a sepa a ion o e ms wi h associa ed su icien ly
small no ms in he ele an equa ions o (17) o (27) and (32) and (33) associa ed wi h he
delayed dynamics:
x[k+1]=ˆ
Adx[k]+ˆ
Bdu[k]+Bcτu[k]+e
Adx[k]
=ˆ
Ad+ΓdKC +e
Adx[k]=Acld x[k](94)
whe e
ˆ
Ad=



Φc(T)+Γcs0(T)0nc×pnc
Ipnc
Adc00nc×pnc
0p×(p+1)nc
Γcd0(T)0nc×pnd
0p×(p+1)nd
Ad00nc×pnd
Ipnd



;
e
Ad=




0nc×nce
Ψcc(T)
0pnc×pnc
e
Ψdc
0p×(p+1)nc
0nc×nde
Ψcd(T)
0p×(p+1)nd
e
Ψdd
0pnd×pnd





(95)
ˆ
Ad+ΓdKC
=



Φc(T) + Γcs0(T) + Γcs(T)K1+BcK2)(Cc+Ccs)x0nc×pncΓcd0(T) + (Γcs(T)K1+BcK2)Cdx0nc×pnd
Ipnc0p×(p+1)nd
Adc0+BdK1(Cc+Ccs)x0nc×pncAd0+BdK1Cdx0nc×pnd
0p×(p+1)ncIpnd



(96)
Because o i s s uc u e, he eigen alues o
ˆ
Ad
a e a ze o eigen alue o mul iplici y
2pn plus he n=nc+ndaddi ional eigen alues o
ˆ
Ad0=Φc(T)+Γcs0(T)Γcd0(T)
Adc0Ad0(97)
By he same eason, he se o eigen alues o
ˆ
Ad+ΓdKC
a e a ze o eigen alue o
mul iplici y 2pn plus he n=nc+ndex a eigen alues o
ˆ
Ad∗=Φc(T)+Γcs0(T)+Γcs(T)K1+BcK2(Cc+Ccs)+Γcs(T)K1+BcK2Cd
Adc0+BdK1(Cc+Ccs)Ad0+BdK1Cd(98)
Fu he mo e, no e ha
zIn−Acld =zIn−ˆ
Ad+ΓdKC−e
Ad=zIn−ˆ
Ad−ΓdKCIn−zIn−ˆ
Ad−ΓdKC−1e
Ad;∀z∈C sp ˆ
Ad (99)
Addi ionally, ha
de zIn−Acld=de zIn−ˆ
Ad−ΓdKC×de In−zIn−ˆ
Ad−ΓdKC−1e
Ad(100)
Since
de zIn−Acld
and
de zIn−ˆ
Ad−ΓdKC
a e en i e unc ions, hey ha e he
same numbe o ze os in he open uni ci cle o he complex plane cen e ed a he o igin
{z∈C:z<1}i
de zIn−Acld−de zIn−ˆ
Ad−ΓdKC
=de zIn−ˆ
Ad−ΓdKC1−de In−zIn−ˆ
Ad−ΓdKC−1e
Ad
<de zIn−ˆ
Ad−ΓdKC
(101)
Ma hema ics 2022,10, 1066 21 o 29
a he bounda y
{z∈C:z=1}
o such a ci cle (Rouché heo em, [
47
]) p o ided ha
zIn−ˆ
Ad−ΓdKC−1exis s, equi alen ly, i
1−de In−zIn−ˆ
Ad−ΓdKC−1e
Ad<1 o |z|=1 (102)
which holds i
0<de In−zIn−ˆ
Ad−ΓdKC−1e
Ad<1 o |z|=1 (103)
Tha is, gua an eed i he
H∞
-no m


zIn−ˆ
Ad−ΓdKC−1e
Ad

H∞
o
zIn−ˆ
Ad−ΓdKC−1e
Adis less han uni y, ha is, i
σeiθIn−ˆ
Ad−ΓdKC−1e
Ad=max
0≤θ<2π


eiθIn−ˆ
Ad−ΓdKC−1e
Ad


2
<1
which holds o su icien ly small


e
Ad

2
. Thus, since
ˆ
Ad+ΓdKC
is con e gen i
ˆ
Ad∗
is
con e gen , we ha e p o ed he ollowing closed-loop global asymp o ic s abili y esul by
aking in o accoun also Rema k 4:
Theo em 5.
Assume ha
Dc+Dd=
0. I
ˆ
Ad∗
is con e gen and


e
Ad

2
is su icien ly small,
acco ding o


zIn−ˆ
Ad−ΓdKC−1e
Ad

H∞
<
1, hen
Acld
is con e gen . As a esul , he
esul ing closed-loop modi ied ex ended disc e e sys em is globally asymp o ically s able in he
sense ha , o any gi en ini e ini ial condi ions, he sequences
{u[k]}∞
k=0
and
{x[k]}∞
k=0
a e
bounded, and
{u[k]}∞
k=0→0
and
{x[k]}∞
k=0→0
. Mo eo e ,
u( )→0
,
xc( )→0
,
xc( )→0
and
y( )→0
as
→∞
, so ha he comple e hyb id sys em is also globally asymp o ically s able.
In iew o (9)–(11) and (20)–(22), he delay- ee dynamics couplings o each subsys em
wi h he emaining ones wi hin he whole ne wo k a e e lec ed in
ˆ
Ad+ΓdKC
de ined
in (96) by he o -diagonal ma ix blocks o he ma ices
Φc(T)=Φcd(T)+Φcod(T)=
eAcd T+RT
0eAcd(T−τ)dτAcod
,whe e
Acd
and
A0cd
a e he diagonal (subsc ip ed wi h “d”)
and o -diagonal (subsc ip ed wi h “od”) ma ix blocks o
Ac
,
Acs0=Acsd +Acsdo
and
Adc0=Adcd +Adcdo
. To e alua e when he closed-loop s abiliza ion by ully decen alized
con ol is possible unde su icien ly weak couplings be ween he a ious subsys ems and,
a he same ime, su icien ly weak delayed dynamics, we now u he decompose he
con olle gain as K=hKT
1,KT
2iT=Kd+Kod acco ding o (77) o yield:
Acld =ˆ
Add +ΓdKdC+ˆ
Adod + +ΓdKodC+e
Ad(104)
whe e
¯
ˆ
Ada =ˆ
Add +¯
Γd¯
Kd¯
C
=



Φc(T)+Γcs0(T)+(Γcs(T)¯
K1d+Bc¯
K2d)(Cc+Ccs)0nc×pncΦc(T)+Γcs0(T)+(Γcs(T)¯
K1d+Bc¯
K2d)Cd0nc×pnd
Ipnc0p×(p+1)nd
Adc0+Bd¯
K1d(Cc+Ccs)0nc×pncAd0+Bd¯
K1dCd0nc×pnd
0p×(p+1)ncIpnd




(105)
e¯
Ada =ˆ
Adod +¯
Γd¯
Kod ¯
C+e¯
Ad





(Γcs(T)¯
K1od +Bc¯
K2od )(Cc+Ccs)+e¯
Ψcc(T)0pnc×pncΓcs(T)K1od +BcK2odCd+e
Ψcd(T)
0pnc×pnc0p×(p+1)nd
e
Ψdc(T)+BdK1od(Cc+Ccs)e
Ψdd(T)+BdK1odCd
0pnc×(p+1)nc0pndc×pnd





(106)
ˆ
Ada∗=Φc(T)+Γcs0(T)+Γcs(T)K1d+BcK2d(Cc+Ccs)+Γcs(T)K1d+BcK2dCd
Adc0+BdK1d(Cc+Ccs)Ad0+BdK1dCd(107)

Ma hema ics 2022,10, 1066 22 o 29
In addi ion,
ˆ
Ada∗
has he same numbe o s uc u al nonze o eigen alues as
ˆ
Add
in he same way as i has
ˆ
Ad∗
Equa ion (98) e sus
ˆ
Ad0
Equa ion (97). The app op ia e
modi ica ion o Theo em 5 by aking in o accoun (104)–(107) unde su icien ly small
couplings o mu ual dynamics be ween pai s o subsys ems leads o he subsequen esul :
Theo em 6.
Assume ha
Dc+Dd=
0. I
ˆ
Ada∗
is con e gen and


e
Ada


is su icien ly
small sa is ying


zIn−ˆ
Add −ΓdKdCd−1e
Ada

H∞
<
1, hen
Acld
is con e gen unde ully
decen alized con ol, ha is, MDdos = 2
q(q−1)
and he closed-loop modi ied ex ended disc e e
sys em is globally asymp o ically s able in he sense ha , o any gi en ini e ini ial condi ions,
he sequences
{u[k]}∞
k=0
and
{x[k]}∞
k=0
a e bounded, and
{u[k]}∞
k=0→0
and
{x[k]}∞
k=0→0
.
Mo eo e ,
u( )→0
,
xc( )→0
,
xc( )→0
and
y( )→0
as
→∞
, so ha he comple e
hyb id sys em is also globally asymp o ically s able.
Rema k 9.
I u ns ou ha , o he pa ial decen alized s abiliza ion p oblem wi h he maximum
deg ee o decen aliza ion and, co espondingly, wi h he minimum deg ee o cen aliza ion, Theo em
6 can be di ec ly e-add essed as a pa allel esul in he sense ha
ˆ
Ada
,
e
Ada
and
ˆ
Ada∗
can be e-
placed, espec i ely, by
ˆ
Ada(MDdos)
,
e
Ada(MDdos)
and
ˆ
Ada∗(MDdos)
de ined acco dingly o an
es ima ion o he maximum decen aliza ion deg ee
MDdos ={max i ∈[0 , 2q(q−1)] ∈Z0+}
such ha :
(a)
Kdao
= minimum numbe be ween
K1od
and
K2od
o o -diagonal en ies o be used in he
e-de ini ion o
ˆ
Ada(MDdos)
, p e iously de ined in (105), by eplacing
Kd→Kd+Kdao
in ˆ
Adbeing de ined in (95);
(b)
Kod −Kdao
o be used in he e-de ini ion o
e
Ada(MDdos)
, p e iously de ined in (106), by
eplacing Kod →Kod −Kdao in e
Adbeing de ined in (85);
(c)
Re o mula e Theo em 6 acco ding o he wo abo e eplacemen s.
The abo e modi ica ion o Theo em 6 is based on an es ima ion o he maximum decen aliza ion
deg ee, a he han on such a deg ee i sel , since Theo em 6 is a he a local obus ness s abili y esul
o su icien ly weak delayed dynamics and su icien ly weak coupling dynamics be ween he a ious
pai s linking he
q
subsys ems. In ac , he esul is based on he s abili y o a nominal closed-loop
sys em wi hou delayed dynamics and couplings be ween each pai o he a ious subsys ems and a
su icien smallness o he emaining con ibu i e e ms o he whole dynamics.
I is also possible o ew i e, equi alen ly, (94) by decomposing he con olle in o
wo pa s, one o be used o add ess he nominal closed-loop design while he o he being
used o pa ially compensa e he e ec o unce ain ies in he closed-loop dynamics. The
esul ing e sion o (94) is:
x[k+1]=ˆ
Ad+ΓdK∗C+ΓdK−K∗C+e
Adx[k]=Acld x[k]
The subsequen example isualizes he abo e ideas.
Example 3.
Conside he ollowing hyb id delay- ee sys em o sampling pe iod
T=
0.1 s which
consis s o wo subsys ems desc ibed by:
.
x1c1( )=x1c2( )−x1c2[k]
x1d1[k+1]=−x1d1[k]+x1c1[k]+α0.1x1c1[k]+∑3
i=1α12ix2di[k]+u[k]
x2d1[k]=x2d2[k];x2d2[k]=x2d3[k]
x2d3[k+1]=0.15x2d1[k]−0.1x2d2[k−1]+1.05x2d3[k]+∑2
i=1α21ix1di[k]
y( )=y1( )=x1c1[k];∀k∈Z0+,∀ ∈[kT (k+1)T)
(108)
Ma hema ics 2022,10, 1066 23 o 29
The
α(.)
akes accoun o small dynamic coupling unce ain ies o no e y p ecise knowledge.
The whole ex ended disc e e sys em o s a e
x[k]=(x1c1[k],x1d1[k],x2d1[k],x2d2[k],x2d3[k])T
,
wi h he con inuous pa disc e ized o he pe iod
T=
0.1, is desc ibed by he ollowing equa ions:
x1c1[(k+1)T]=x1c2[kT]
x1d1[k+1]=−x1d1[k]+x1c1[k]+α111x1c1[k]+∑3
i=1α12ix2di[k]+u[k]
x2d1[k]=x2d2[k];x2d2[k]=x2d3[k]
(109)
x2d3[k+1]=0.15x2d1[k]−0.1x2d2[k−1]+1.05x2d3[k]+∑2
i=1α21ix1di[k]+u[k]
y( )=y1( )=x1c1[k];∀k∈Z0+,∀ ∈[kT (k+1)T)
which can be ew i en in a compac o m, which is also in companion con ollabili y o m [
27
,
48
],
o each o he subsys ems as ollows:
Ad=Ad11 Ad12
Ad21 Ad22 =





0 1 0 0 0
1−1 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0.15 −0.1 1.05





(110)
B=





0000
1000
0000
0000
0001





;C=100 0 0
001 0 0 (111)
e
Ad="e
Ad11 e
Ad12
e
Ad21 e
Ad22 #=





00000
0 0 α121 α122 α123
00000
00000
α211 α212 000





(112)
ece
Ad=(0 , 0, 0 , 0, 0 , 0 , 0 , α121 ,α122,α123 , 0, 0 , 0, 0 , 0 , 0 , 0, 0 , 0, 0 , α211 ,α212 , 0 , 0 , 0T
whe e
e
Ad
is he ma ix dynamics o he unce ain ies. The ma ix
Ad
is no con e gen since i
has wo uns able eigen alues
z=
1.08521 and
z=−1+√5
2
. The con olle is p oposed o ha e
he s uc u e:
K=K1
K2=



K111
K112
K121
K122
K211
K212
K221
K222



(113)
Leading o a closed-loop dynamics o he whole ex ended disc e e sys em gi en by he ma ix:
Acld =Ad+BKC +e
Ad=





0 1 0 0 0
1+K111 −1K112 +α121 α122 α123
0 0 0 1 0
0 0 0 0 1
K221 +α211 α212 0.15 +K222 −0.1 1.05





(114)
Ma hema ics 2022,10, 1066 24 o 29
which can be equi alen ly decomposed also as
Acld =Acld∗+e
Acld
in e ms o a closed-loop coupling
nominal and unce ain dynamics be ween bo h subsys ems being gi en by he ma ices:
Acld∗=





0 1 0 0 0
1+K111 −1 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 0.15 +K222 −0.1 1.05





(115)
e
Acld =





0 0 0 0 0
0 0 K112 +α121 α122 α123
0 0 0 1 0
0 0 0 0 1
K221 +α211 α212 0 0 0





(116)
Then,
ecK=K111,K112 ,K121,K122,K211 ,K212 ,K221 ,K222T
Because o he spa se s uc u e o he ma ix o dynamics, he whole numbe o con olle
en ies is simpli ied by ze oing di ec ly
K121
,
K122
,
K211
,
K212
. Mo eo e ,
K112
and
K221
a e used
o add ess he achie emen o he su icien no m smallness o he unce ain ies ec o , so hey a e
also ze oed in he unknowns ec o
ecK
and ans e ed o
ece
Ad
so ha he nominal linea
algeb aic Equa ion (42) is sol ed in he unknown ec o :
ecK=K111, 0 , 0, 0, 0 , 0 , 0, K222T
wi h
ecAd=(0 , 1, 0 , 0, 0 , 1 , −1 , 0 , 0, 0 , 0 , 0, 0 , 1 , 0 , 0, 0 , 0, 0 , 1 , 0 , 0 , 0.15 , −0.1 , 1.05T
and
ece
Ad=0 , 0, 0 , 0, 0 , 0 , 0 , K112 +α121 ,α122,α123 , 0, 0 , 0, 0 , 0 , 0 , 0, 0 , 0, 0 , K221 +α211 ,α212 , 0 , 0 , 0T
One checks he s a ic con olle syn hesis sol abili y o h ee in ended ma ices o he nominal
closed-loop dynamics ( ha is, excluding he con ibu ion o he unce ain ies, which a e inco po a ed
o he ma ix
e
Ad
, in his i s syn hesis s ep) which a e, espec i ely, de ined depending on he
unknown K111 by:
ecAcld∗1=0 , 1, 0 , 0, 0 , 1 +K111 ,−1 , 0 , 0, 0 , 0 , 0, 0 , 1 , 0 , 0, 0 , 0, 0 , 1 , 0 , 0 , 0.20 , −0.1 , 1.05T
ecAcld∗2=0 , 1, 0 , 0, 0 , 1 +K111 ,−1 , 0 , 0, 0 , 0 , 0, 0 , 1 , 0 , 0, 0 , 0, 0 , 1 , 0 , 0 , 0.15 , 0.30 , 1.05T
ecAcld∗3=0 , 1, 0 , 0, 0 , 1 +K111 ,−1 , 0 , 0, 0 , 0 , 0, 0 , 1 , 0 , 0, 0 , 0, 0 , 1 , 0 , 0 , 0.15 , 0.40 , 1.05T
Now, no e ha he closed-loop cha ac e is ic polynomials which de ine he espec i e closed-loop
sel -dynamics o bo h subsys ems in he ex ended disc e ized sys em, a e compensa ion ia s a ic
linea ou pu eedback, a e:
p1(z)=z2+z−1+K111;p2(z)=z3−1.05z2+0.1z−0.15 +K222(117)
The i s one depends on he s ill unde e mined
K111
. The eigen alues o
Acld∗
a e i -
ially he ze os o he p oduc o bo h cha ac e is ic polynomials
p1(z)p2(z)
since he ma ices
o a ge ed closed-loop dynamics a e in companion o ms in he sel -dynamics o bo h subsys-
ems in eg a ed in he ex ended disc e e one. No e ha
p1(z)
is s able o
K111 ∈−5
4,−1
2
while
p2(z)
is s able o
K222 =−
0.35 wi h ze os
−
0.34454 and 0.69727
±
0.30707
i
, o
K222 =−
0.45 wi h ze os
−
0.41913 and 0.73457
±
0.41973
i
o o
K222 =−
0.55 wi h ze os
−
0.47971 and
0.7649 ±0.49881i
. Those ze os a e in
Acld∗i
, espec i ely, o i = 1, 2, 3. How-
Ma hema ics 2022,10, 1066 25 o 29
e e , in he absence o closed-loop compensa ion h ough he choice
K222 =
0, he polynomial
p2(z)=z3−
1.05
z2+
0.1
z−
0.15 is no s able ha ing a ze o
z=
1.08521. In summa y, he
abo e sys em in he absence o coupling dynamics is uns able in he absence o con ol, ha is, he
open-loop sys em is uns able. Howe e , he closed-loop one can be s abilized wi h linea s a ic ou pu
eedback con ol jus wi h wo nonze o scala gains, ha is, wi h wo nonze o en ies in he con ol
gain ma ix (113). Wi h bo h sel -dynamics being s able unde he condi ions gi en o he choices
o K111 and K222, one concludes ha Acld∗is con e gen .
I u ns ou ha any no m o
e
Acld
is a bi a y small o
α=max(|α122|
,
|α123|
,
|α212|
,
K112 +α121
,
K221 +α211 )
being a bi a y small. Unde he gi en condi ions which gua an ee
ha
p1(z)
and
p2(z)
a e s able, so ha
Acld∗
is con e gen , i ollows ha
Acld
is also con e gen i
α
is su icien ly small ela ed o 1
/

(zI5−Acld∗)−1

H∞
so since, o any complex numbe
z
which
is no an eigen alue o Acld∗, one has ha
zI5−Acld =(zI5−Acld∗)I5−(zI5−Acld∗)−1e
Acld∗(118)
so ha he eigen alues o Acld a e no in C1. In pa icula , no e he ollowing ea u es:
(a)
Assume ha
α121
and
α211
a e known p ecisely. Then, he addi ional choices o he p e iously
unspeci ied gains
K112 =−α121
and
K221 =−α211
as en ies o he con olle gain gua an ee
ha
Acld
is con e gen , so ha he ex ended closed-loop sys em is
s able i
α0=min(|α122|,|α123|,|α212|)
is su icien ly small sa is ying
α0<
1
/ √5sup
0≤θ<2π

eiθI5−Acld∗−1

2!
a e using he no m inequali y


e
Acld

2≤
√5min

e
Acld

∞,

e
Acld

1[49], o he ma ix e
Acld o o de 5.
(b) Assume ha
α121
and
α211
a e no known p ecisely bu hey a e known o belong o known espec i e
eal subse s
α
−121
,α121
and
α
−211
,α211
, which is a easonable assump ion in p ac ice. Then,
choose he p e iously unspeci ied gains
K112 =−α121∗=−α
−121
+α121
2
and
K221 =−α221∗=
−α
−221
+α221
2
as en ies o he con olle gain gua an ee ha
Acld
is con e gen , so ha he ex ended
closed-loop sys em is s able i
α=max(|α122|
,
|α123|
,
|α212|
,
α
−121
,
|α121,|
,
1
2α121−α
−121
,
1
2α221−α
−221)is su icien ly small so ha α<1/ √5sup
0≤θ<2π

eiθI5−Acld∗−1

2!.
As a esul , as
k→∞
,
{x[k]}→0
,
{y[k]}→0
,
{u[k]}→0
,
x2dj[k]→0
(
j=
1, 2, 3
)
,
x1ci[k]→0
(
i=
1, 2) o any gi en ini e ini ial condi ions. Mo eo e ,
u( )→0
as
→∞
,
x1ci( )→0
(
i=
1, 2) and
y( )→0
as
→∞
. The con olle is o decen alized ype since i
only picks up in o ma ion o he i s subsys em h ough i s ou pu which is also he global ou pu o
he whole sys em.
No e ha o he spa se con ol and ou pu ma ices de ined in (111), ou o he eigh con ol
gains which a e en ies o he con ol ma ix (113) do no play a ole in he closed-loop ma ix o
dynamics and can be ze oed.
Example 4.
Assume a i h o de sys em as ha o Example 4 bu wi h, in gene al, a less spa se
pa ame e iza ion o he con ol and ou pu ma ices. In his case, he s abili y o he sel -dynamics o
bo h subsys ems and he in luence o he coupling dynamics o keep he achie ed closed-loop s abili y
migh be mo e di icul o deal wi h. The gene al idea o s abilizing he uncoupled dynamics unde a
su icien ly small in luence o he coupling one will be add essed as ollows. Assume ha he ma ix
o he closed-loop dynamics is pa i ioned in o ou block ma ices as:
Acld =Acld11 Acld12
Acld21 Acld22 (119)