Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

Ci a ion: De la Sen, M. Asymp o ic
Hype s abili y and Inpu –Ou pu
Ene gy Posi i i y o a Single-Inpu
Single-Ou pu Sys em Which
Inco po a es a Memo yless
Non-Linea De ice in he
Feed-Fo wa d Loop. Ma hema ics
2022,10, 2051. h ps://doi.o g/
10.3390/ma h10122051
Academic Edi o s: Lijun Pei and
Youming Lei
Recei ed: 24 Ap il 2022
Accep ed: 12 June 2022
Published: 13 June 2022
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ma hema ics
A icle
Asymp o ic Hype s abili y and Inpu –Ou pu Ene gy Posi i i y
o a Single-Inpu Single-Ou pu Sys em Which Inco po a es a
Memo yless Non-Linea De ice in he Feed-Fo wa d Loop
Manuel De la Sen
Ins i u e o Resea ch and De elopmen o P ocesses, Depa men o Elec ici y and Elec onics, Facul y o Science
and Technology, Uni e si y o he Basque Coun y (UPV/EHU), 48940 Leioa, Bizkaia, Spain;
[email p o ec ed]
Abs ac :
This pape isualizes he ole o hype s able con olle s in he closed-loop asymp o ic
s abili y o a single-inpu single-ou pu sys em subjec o any nonlinea and e en ually ime- a ying
con olle wi hin he hype s able class. The eed- o wa d con olled loop (o con olled plan ) con ains
a s ongly s ic ly posi i e eal ans e unc ion in pa allel wi h a non-linea and memo y- ee de ice.
The p ope ies o posi i i y and boundedness o he inpu –ou pu ene gy a e examined based on he
“ad hoc” use o he Rayleigh ene gy heo em on he unca ed ele an signals o ini e ime in e als.
The cases o minimal and non-minimal s a e-space ealiza ions o he linea pa a e cha ac e ized
om a global asymp o ic s abili y (asymp o ic hype s abili y) poin o iew. Some ela ed ex ended
esul s a e ob ained o he case when he linea pa is bo h posi i e eal and ex e nally posi i e
and o he case o inco po a ion o o he linea componen s which a e s able bu no necessa ily
posi i e eal.
Keywo ds:
posi i e ealness; hype s abili y; asymp o ic hype s abili y; passi i y; sample and
hold de ices
MSC: 93D10; 93D15; 93D20; 34D05
1. In oduc ion
Posi i e ealness is a e y ele an p ope y o linea sys ems. A posi i e eal ans e
unc ion has non-nega i e eal pa on he closed complex igh hal -plane. I has a ela i e
deg ee ( ha is, a pole-ze o excess) o 0, +1, o
−
1 [
1
–
8
]. Se e al ways and me hods o
designing such ans e unc ions in ci cui y syn hesis p oblems a e gi en in [
3
–
7
]. Thei
design in he con ex o ecu si e pa ame e adap a ion is ocused on in [
8
]. In [
9
], he global
asymp o ic s abili y p ope y is s udied o a composi e sys em wi h an asymp o ically
hype s able subsys em. A consequence o posi i e ealness o ans e unc ions is ha he
equency esponse hodog aph is con ined wi hin he i s and ou complex quad an s so
ha he maximum absolu e phase o he equency esponse is no la ge han
π/
2. On he
o he hand, i he sys em is s a e-space ealizable, hen i s ans e unc ion is p ope , ha is,
wi h no less han ze o poles, so ha i s ela i e deg ee is ei he 0 (i.e., he ans e unc ion
is bip ope , ha is, i is p ope wi h a p ope in e se) o 1. Ano he p ope y o such ans e
unc ions is ha hey a e s able, including he c i ical case, and so hey a e non-necessa ily
s ic ly s able, bu e en ual c i ical poles, i any, ha e o be single and wi h non-nega i e
associa ed esiduals. Fu he mo e, he in e ses o posi i e eal ans e unc ions a e also
posi i e eal. On he o he hand, he inpu –ou pu ene gy o he sys ems desc ibed by
posi i e eal ans e unc ions is non-nega i e o all imes. In his way, he posi i e
ealness o a ans e is associa ed wi h inpu –ou pu ene gy dissipa ion o all imes o he
co esponding dynamic sys em. I has o be poin ed ou ha posi i e ealness does no
di ec ly imply he join non-nega i i y o he inpu and he ou pu h ough ime, which
Ma hema ics 2022,10, 2051. h ps://doi.o g/10.3390/ma h10122051 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 2051 2 o 20
is he so-called ex e nal posi i i y p ope y, which also implies ha he non-nega i i y
o all imes o bo h he inpu –ou pu powe and he inpu –ou pu ene gy. A pa icula
subclass o ha se o posi i e eal ans e unc ions is ha o he so-called s ic ly posi i e
eal ans e unc ions which a e s ic ly s able, ha is, wi hou poles a he imagina y and
whose eal pa s a e s ic ly posi i e a he open igh hal -plane. Posi i e eal ans e
unc ions a e e y common in he desc ip ion o classical ci cui y in ol ing andems o
esis o s, capaci o s and induc ances.
On he o he hand, he class o non-linea and e en ually ime- a ying hype s able
con olle s is de ined by he se o con olle s which sa is y a so-called Popo ’s ype inpu –
ou pu in eg al inequali y ( e e ed o as Popo ’s hype s abili y condi ion o he whole
class o con olle s) [
9
–
20
]. In pa icula , he use o heo y in di e en adap i e con ol
p oblems is widely de eloped in [
14
,
15
] and some o he e e ences he ein. I s use ulness
in quali a i e beha iou s o dynamic sys ems and in neu al ne wo ks a e ocused on
in [
16
,
17
], while he hype s abili y in he disc e e- ime con ex is add essed in [
18
] o
linea ime- a ying sys ems. The case o impulsi e con ols in hype s abili y p oblems
is ocused on in [
19
]. On he o he hand, an impo an s abili y p ope y o he ob ained
closed-loop sys em is ha a posi i e eal ans e unc ion unde any con olle belonging
o he hype s able class o con olle s is “hype s able”. Wha his means is ha i is globally
s able in he la ge ( ha is, o any gi en ini e ini ial condi ion) in he Lyapuno ’s sense.
I he eed- o wa d ans e unc ion is s ic ly posi i e eal, hen he closed-loop sys em
is “globally hype s able”, ha is, globally asymp o ically s able in he la ge. I can be
poin ed ou ha Popo ’s hype s abili y condi ion on he con olle is also sa is ied o mo e
elemen a y s a ic non-linea con olle s in oked in he con ex o absolu e s abili y (like
he well-known Lu ’e absolu e s abili y p oblem wi hin a Lu ’e’s sec o , Popo ’s absolu e
s abili y c i e ion wi hin a Popo ’s sec o , e c.) [
20
–
22
]. The concep o hype s abili y is
closely ela ed o he mo e gene al one o dissipa i i y, o i s pa icula e sion o passi i y,
h ough he abo e-men ioned posi i i y/boundedness p ope ies o he inpu –ou pu
ene gy [
23
–
27
]. On he o he hand, a a ie y o applica ions in di e en designs in he ields
o mechanics, elec ic machine y, ci cui syn hesis, model e e ence adap i e con ol, and
delayed sys ems has been pe o med. See, o ins ance, e s. [
28
–
35
] and some e e ences
he ein o mo e de ails.
Posi i e ealness has been widely applied and linked o he hype s abili y concep in
adap i e con ol designs by aking ad an age o he la ge uni e se o use ul con olle s,
which allows a la ge lexibili y in he design o he adap i e laws, he inpu /ou pu il e s
o be used, and he amily o ee-design pa ame e s o he es ima ion algo i hm being
compa ible wi h he closed-loop s abiliza ion. Fo simila easons, hey ha e been e y
popula o he syn hesis o a wide se o egula o s in elec ical machine y p oblems.
Basically, he hype s abili y condi ion o he eedback pa ob ained unde app op ia e
ans o ma ions and equi alence manipula ions o he in ol ed equa ions is used o ge
he adap i e law, which ensu es he global s abili y o he whole scheme [
14
,
15
,
33
,
34
].
In [
36
], a double-con ec ion sys em exhibi ing chao ic beha iou wi h h ee nonlinea i ies is
discussed, and i s s abili y and dissipa i i y p ope ies and hei equilib ia a e in es iga ed.
On he o he hand, in [
37
], a chao ic dissipa i e a ac o wi h wo quad a ic nonlinea i ies,
which possesses h ee uns able equilib ium poin s, is in es iga ed. I s ealiza ion h ough
an elec onic ci cui is also desc ibed. On he o he hand, i is well-known ha disc e e-
ime models a e widely in oked in p ac ical applica ions because o hei lexibili y o
he design o app op ia e con olle s which do no need o pick up in o ma ion o all
imes, bu only a ce ain sampling ins an s, e en i he con olled sys em is o ally o ,
pa o , a con inuous- ime na u e. This ac allows o simpli ica ion o he whole design,
and, in gene al, closed-loop s abiliza ion is achie able anyway, as a e he basic needed
design pe o mances [
38
–
42
]. In his con ex , his pape also gi es some u he ideas abou
hype s abili y designs when he con inuous- ime con ol inpu o he con inuous- ime
con olled plan is gene a ed by sample and hold de ices, which pick up inpu - egis e ed
alues a p e ious sampling ins an s, which a e used o gene a e he con inuous- ime inpu .
Ma hema ics 2022,10, 2051 3 o 20
The main objec i e o his pape is o isualize he ole o he class o hype s able
con olle s in he closed-loop asymp o ic s abili y o a single-inpu single-ou pu sys em
subjec o nega i e eedback gene a ed by, in gene al, a nonlinea and e en ually ime-
a ying con olle . The con olle is any elemen wi hin a class which sa is ies a Popo ’s
ype ime-in eg al inequali y. The eed- o wa d loop consis s o a s ongly s ic ly posi i e
eal ans e unc ion ope a ing in pa allel wi h, in gene al, a non-linea , memo y- ee
de ice. The inco po a ion o such a de ice in he whole con igu a ion, while gua an eeing
he asymp o ic hype s abili y o he closed-loop sys em, is he main con ibu ion o his
wo k. Because o he in insic na u e o he hype s abili y concep , he global asymp o ic
s abili y in he la ge o he ob ained closed-loop sys em is cha ac e ized o he whole class
o con olle s sa is ying a Popo ’s ype in eg al inequali y. Special a en ion is also paid o
he p ope ies o posi i i y and he uni o m boundedness o he inpu –ou pu ene gy o
he eed- o wa d-loop o all imes so ha he con olled sys em has a dissipa i e na u e.
In pa icula , he minimum uppe -bound o such an inpu –ou pu ene gy is gi en o
all imes by he la ges nega i e pa ame e , which bounds om below he ime–in eg al
Popo ’s cons ain ha de ines he class o hype s able con olle s. The main esul s a e
de i ed o he case ha he s a e-space ealiza ion o he ans e unc ion is minimal,
ha is, con ollable and obse able. The e a e also some u he ex ensions o he main
abo e esul s ha deal wi h he case o non-minimal ealiza ions, which a e s able, and
o he case when he ans e unc ion is weakly posi i e eal, o , simply, (non-s ic ly)
posi i e eal. In his las case, he inpu –ou pu ene gy is gua an eed o be non-nega i e
and uni o mly bounded o all imes, and he closed-loop hype s abili y is no asymp o ic.
Fu he ela ed esul s a e also ob ained o he case when he linea pa is bo h posi i e
eal and ex e nally posi i e. In his case, he non-nega i e and boundedness p ope ies o
he inpu –ou pu ene gy o all imes a e also ul illed by he ins an aneous inpu –ou pu
powe , o passi i y supply a e.
The esul s a e ob ained o he men ioned de ices being sa u a ed and linea , while
non-necessa ily being p opo ional o he inpu , and nonlinea , including he cons an ,
linea , and quad a ic e ms o he inpu . Some u he esul s a e also ob ained o when he
linea pa o he sys em is a pa allel connec ion o a s ic ly posi i e eal ans e unc ion
wi h a s ic ly s able one, which has a su icien ly small esonance peak compa ed o he
minimum (posi i e) alue o he eal pa o i s coun e pa in eg a ed wi h he men ioned
linea andem and connec ed in pa allel. Fu he , some applica ions a e de eloped o he
case when he con inuous- ime inpu is gene a ed om a e y gene al sampling and hold
de ice, which gene a es he cu en in e -sample inpu alue, in gene al, om i s wo las
p e ious sampled alues.
The pape is o ganized as ollows: Sec ion 2s a es he main esul s o a closed-loop
sys em whose eed- o wa d loop is a linea sys em desc ibed by a s ic ly posi i e eal
ans e unc ion ope a ing in se ies wi h a bounded nonlinea ope a o on he inpu ,
and he eedback loop is any con olle belonging o an hype s able class de ined by an
in eg al- ype, Popo ’s- ype hype s abili y cons ain . The closed-loop sys em is p o en o
be asymp o ically hype s able i he ans e unc ion o he eed- o wa d loop is s ongly
s ic ly posi i e eal. O he p o en esul s a e he in eg abili y o he squa ed inpu and he
squa ed ou pu on he whole in e al o ime and he non-nega i i y and boundedness o
he inpu –ou pu in eg al ene gy. Some ex ensions a e gi en in Sec ion 3 o : (a) weake
cons ain s ela ed o weak s ic posi i e ealness on he ans e unc ion; (b) he andems
o he s ic ly posi i e eal ans e unc ion wi h ano he s ic ly s able one which does no
ha e, in gene al, posi i e ealness p ope ies; and (c) o he al e na i e cons ain s on he
cascaded nonlinea ope a o on he inpu combined wi h he abo e a ian s. In addi ion,
in he case whe e he ans e unc ion is only posi i e eal bu no s ic ly posi i e eal,
some u he pa allel condi ions a e ob ained o he inpu and ou pu o hose in Sec ion 2.
In pa icula , some hype s abili y condi ions a e p o en i he ans e unc ion is bo h
ex e nally posi i e and posi i e eal. Howe e , he asymp o ic hype s abili y p ope y is
no concluded, in gene al. Sec ion 4de elops some applica ions o he o me heo e ical
Ma hema ics 2022,10, 2051 4 o 20
esul s o he case whe e he inpu is gene a ed om a gene al sampling and hold de ice
o speed co ec ion, which gene a es he cu en in e -sample inpu alue om he wo
las sampled alues acco ding o a co ec ing design coe icien , and, a he same ime, i
sa is ies an “ad hoc” Popo ’s- ype hype s abili y in eg al cons ain . Finally, conclusions
end he pape .
No a ion
The ollowing no a ion will be used h ough he manusc ip :
R0+=R+∪{0};R+={ ∈R: >0},
Z0+=Z+∪{0};Z+={z∈Z:z>0}, and
C0+=C+∪{iR};C+={w∈C:Rew>0},
whe e
R
,
Z
, and
C
a e he se s o eal, in ege , and complex numbe s, espec i ely, he
eal se
R
can be ex ended, including he in ini y poin s, o
¯
R=R∪{±∞}
. In he same
way, he ex ended
¯
R0+=R0+∪{+∞}
,
¯
R+=R+∪{+∞}
, and
iR ={iω:ω∈R}
a e
de ined as he se o pu e imagina y complex numbe s,
i=√−1
is he complex uni ,
u
is he unca ion in he
[0, ]
o
u:R→R
, ha is,
u (τ) = u(τ)
i
τ∈[0, ]
and
u (τ) =
0
i
τ∈(−∞, 0)∪( ,+∞)
and
( ∗h)( ) = R∞
−∞ (τ)g( −τ)dτ
, and
∀ ∈R0+
is he
con olu ion o ,h:R→R. I ,h:R→R, hen
( ∗h)( ) = Z
0 (τ)g( −τ)dτ=Z∞
−∞ (τ)g( −τ)dτ=Z∞
−∞ (τ)g ( −τ)dτ=Z∞
−∞ (τ)g ( −τ)dτ
=( ∗h)( ) = ( ∗h )( ) = ( ∗h )( )
=( ∗h) ( ) = ( ∗h ) ( ) = ( ∗h ) ( ) = ( ∗h) ( );∀ ∈R0+,
whe e
ˆ
g(s)
and
ˆ
g(iω)
a e he Laplace and Fou ie ans o ms o
g:R0+→R
, i hey exis .
The s ic ly posi i e eal ans e unc ions (in he se
SPR
), and, espec i ely, posi i e
eal ans e unc ions (in he se
PR
)
ˆ
g(s)
and
s∈C
a e analy ic in
Re s ≥
0 ( espec i ely,
in
Re s >
0) and, i hey a e s a e-space ealizable, hen hey ha e a ela i e deg ee (i.e., a
pole-ze o excess) o ei he uni y o ze o. The se
SPR
o he s ic ly posi i e eal ans e
unc ions is included in he se
PR
o (non-s ic ) posi i e eal ans e unc ions, he i s
ones being s ic ly s able while hose ones in he second se a e equi ed o be only s able.
The se
SSPR
, a subse o
SPR
, is a se o s ongly s ic ly posi i e eal ans e unc ions
o in e es hough he manusc ip such ha
ˆ
g∈SSPR
i , and only i ,
Re ˆ
g(s)>
0 o all
Re s ≥
0 and also o
Re s →+∞
. In addi ion, he se o he so-called weakly s ic ly
posi i e eal ans e unc ions,
WSPR
[
1
], does no necessa ily main ain he s ic posi i e
ealness o
Re s →+∞
and can be p ope ( ha is, wi h he numbe o ze os no exceeding
he numbe o poles), while no necessa ily bi-p ope (i.e., hose being p ope wi h a p ope
in e se, so wi h an iden ical numbe o poles and ze os). We no e ha he abo e se s
possess he se inclusion p ope ies
SSPR ⊂SPR ⊂PR
and
WSPR ⊂SPR ⊂PR
om
mo e es ic i e o less es ic i e condi ions. On he o he hand, s ic ly posi i e eal
ans e unc ions a e s ic ly s able, while posi i e eal ans e unc ions can ha e single
poles a he imagina y complex axis.
The main o he abo e-men ioned se s o ans e unc ions o ou cen al pu poses
in his pape is ha o he s ongly s ic ly posi i e eal ans e unc ions
SSPR,
whose
membe s ha e a s ic ly posi i e eal pa o he ans e unc ion on he open igh -hal -
plane. Such ans e unc ions a e also bi-p ope ( ha is, hey ha e he same numbe o
poles and ze os) and s ic ly s able (all poles a e in Res < 0).
We will e e o a s ic ly s able linea sys em as being one wi h all he poles o i s
ans e unc ion in he open le -hand-side complex plane
C0−={s∈C:Re s <0}
,
and we e e o a s able linea sys em as being one wi h poles in he closed le -hand-
side complex plane. In he i s case, he ma ix o dynamics is a s abili y ma ix whose
Ma hema ics 2022,10, 2051 5 o 20
eigen alues a e such poles. In he second case, some eigen alues can be alloca ed a he
imagina y complex axis. We e e indis inc ly o bo h o he abo e sys em s abili y concep s,
as well o he espec i e ans e unc ions, as a s ic ly s able, o espec i ely s able, sys ems
o ans e unc ions.
2. P oblem S a emen and Main Resul s
I is well known ha he Fou ie ans o m
ˆ
g(iω)=F(g( ))=R∞
−∞g( )e−iω d
wi h
ˆ
g:iR→C
o
g:R0+→R
exis s i
g
is absolu ely in eg able on
R
and he Laplace ans o m
ˆ
g(s) = L(g( ))=R∞
0g( )e−(σ+iω) d is de ined o a eal σ≥σ0and some σ0∈R.
Fo a gi en s able linea dynamic single-inpu
(u( ))
single-ou pu
(y( ))
sys em o
impulse esponse
g( )
:
ˆ
g(s) = ˆ
y(s)/ˆ
u(s)
is he so-called ans e unc ion, which is he
Laplace ans o m o
g( )
, which equalizes he quo ien o he Laplace ans o m o he
ou pu o he Laplace ans o m o he inpu unde null ini ial condi ions, and
ˆ
g(iω)
is i s
so-called equency esponse, which is he Fou ie ans o m o g( ).
No e ha uns able linea sys ems can s ill be analysed hough a Laplace ans o ms
con ex , bu no unde a Fou ie ans o m one.
Remembe also ha
ˆ
g(s)∈SSPR
i , and only i ,
Re ˆ
g(s)>
0 o
Re s ≥
0. This
p ope y also implies ha
Re ˆ
g(iω)>
0 and
∀ω∈
¯
R
( hus,
Re ˆ
g(iω)>
0,
∀ω∈R
, and
lim
ω→±∞Re ˆ
g(iω)>
0), and ha
ˆ
g(s)
is bi-p ope (i.e., i has he same numbe o ze os and
poles) and s ic ly s able, i.e., all i s poles a e in Re s <0.
L∞
is he se o essen ially bounded eal unc ions on
R
and
L2
is he se o squa e-
in eg able unc ions on
R
. The unc ions conside ed in his pape a e iden ically ze o on
he nega i e eal semi-axis. The e o e, i essen ial boundedness and squa e-in eg abili y,
espec i ely, a e p o en o hold on R0+, hen hey a e in L∞, espec i ely, in L2.
Fo any gi en con ol
u:R0+→R
, he ou pu o he con olled dynamic sys em
P
(o plan ), unde ze o ini ial condi ions, is:
y( ) = (g∗u)( ) + W(u );∀ ∈R0+(1)
whe e
g:R0+→R
is he impulse esponse o he linea pa , which is he Laplace in e se
ans o m o he ans e unc ion ˆ
g(s),W:R→Ris, in gene al, nonlinea , and
u( ) = − ( );∀ ∈R0+(2)
gi es he con ol ac ion unde nega i e eedback o he hype s able eedback con olle
K∈K
( he class o hype s able con olle s) o inpu
y( )
( ha is, is, he ou pu o he
eed- o wa d con olled sys em) and ou pu
:R0+→R
, which is assumed o sa is y he
subsequen Popo ’s hype s abili y inpu –ou pu in eg al condi ion o some nonze o ini e
eal cons an γ0:
Z
0 (τ)y(τ)dτ≥ −γ2
0>−∞;∀ ∈R0+(3)
The de ini ions o hype s abili y and asymp o ic hype s abili y in Popo ’s sense ollow
below. See, o ins ance, e s. [9–15].
De ini ion 1.
The con olled dynamic sys em
P
o an inpu –ou pu ela ion de ined by (1) is
hype s able i , o any con ol inpu sa is ying he in eg al inequali y (3), he ze o-s a e solu ion
x( )
o all
∈R0+
o any minimal s a e-space ealiza ion o
n
- h o de o he linea pa o (1) is
globally s able in he la ge ( ha is, o any gi en ini e ini ial condi ion
x0∈Rn
) in he sense ha
he subsequen ela ion holds o some posi i e eal cons an s δand K:
kx( )k ≤ K(kx(0)k+δ);∀ ∈R0+

Ma hema ics 2022,10, 2051 6 o 20
De ini ion 2.
The con olled dynamic sys em
P
o an inpu –ou pu ela ion de ined by (1) is
asymp o ically hype s able i i is hype s able in he sense o De ini ion 1 and, in addi ion,
x( )→0
as →∞.
The ollowing ea u es can be emphasized conce ning he abo e
hype s abili y concep s
:
1.
The hype s abili y (asymp o ic hype s abili y) p ope y is a global Lyapuno s abili y
(global Lyapuno asymp o ic s abili y) p ope y in he la ge (i.e., on he whole s a e
space) o any con ol which sa is ies (3), which de ines a whole class o admissible
con olle s. Thus, i is no a global s abili y p ope y o a pa icula con ol law, bu i
holds inhe en ly o a whole class o con olle s. The whole class o con olle s can
include linea and nonlinea s a ic membe s, as well as ime- a ying ones, subjec o
he cons ain s o (3). I he eed- o wa d con olled plan is linea and ime-in a ian ,
i is well known ha i has o be de ined by a posi i e eal (s ic ly posi i e eal)
ans e unc ion in o de o achie e he hypes abili y (asymp o ic hype s abili y)
o he closed-loop sys em o he whole class o con olle s sa is ying he in eg al
inequali y o (3).
2.
I has been common in he classical backg ound li e a u e o use he e minology ha
a closed-loop con igu a ion (1)–(3) is hype s able i bo h he eed- o wa d block (o
con olled plan (1)) is hype s able and he eedback loop (o he class o s abilizing
con olle (2) and (3)) is hype s able as well. See, o ins ance [
13
–
15
] and some
e e ences he ein. Howe e , i can be poin ed ou ha he class o s abilizing eedback
con olle s (in he hype s abili y con ex ) can include s a ic membe s so ha i can be
p e e able o e e o he hype s abili y as a p ope y o he plan unde all con olle
membe s belonging o he hype s able class o con olle s.
3.
Since hype s abili y and asymp o ic hype s abili y a e e y wide classes o global
Lyapuno ’s s abili y, hose p ope ies can be cha ac e ized ia Lyapuno unc ion
candida es. Exhaus i e discussion on hei associa ed Lyapuno unc ion can be
ound in [
13
–
15
]. I can be poin ed ou as well ha he hype s abili y app oach is no
mo e gene al o s abili y cha ac e iza ion han he s anda d Lyapuno heo y, bu i
allows o cha ac e iza ion o he s abili y o a whole class o con olle s which sa is y
and inpu –ou pu in eg al cons ain . This class con ains e en ually linea con olle s,
classes o s a ic non-linea ones unde a sec o - ype (Lu ’e o Popo ype) cons ain ,
o e en ually ime- a ying con olle s.
4.
In De ini ion 1, i is assumed ha he s a e-space ealiza ion is minimal, ha is,
o minimum o de o he s a e o he gi en ans e unc ion, which implies ha
he ans e unc ion has no ze o-pole cancella ion and he s a e-space ealiza ion is
join ly con ollable and obse able. This cons ain is no s ic ly necessa y and some
ex ensions unde i s emo al will be gi en in Sec ion 3. Howe e , ha minimali y
cons ain helps o c ea e an easy unde s anding o he p ope y a a i s glance since
i becomes ob ious ha non-minimal ealiza ions wi h e en ual ze o-pole uns able
cancella ions in he ans e unc ion a e no s able, and so hey could ne e be
hype s able, ei he .
5.
Some basic p ope ies associa ed wi h hype s abili y o a closed-loop con igu a ion
ely on he ac ha he inpu –ou pu ene gy o he eed- o wa d block is bo h non-
nega i e and uni o mly bounded o all imes. This is he main ma hema ical ool
add essed in his esea ch o ob ain he gi en esul s.
6.
The main objec i e o his s udy is o ex end he asymp o ic hype s abili y p ope y
o he p esence o ce ain nonlinea de ices in he eed- o wa d loop, which a e
alloca ed in a se ies andem wi h he linea ime-in a ian pa , and o cha ac e ize
he s ong ype o s ic posi i e ealness o he linea ime-in a ian pa , leading o
he asymp o ic hype s abili y o he closed-loop con igu a ion.
Assump ion 1.
The con ol
u:R0+→R
is admi ed o ha e “a p io i”, any numbe o ini e
bounded discon inui ies, and a ini e numbe o impulsi e discon inui ies.
Ma hema ics 2022,10, 2051 7 o 20
The ollowing esul holds o a eed- o wa d con olled dynamic sys em unde he
class o hype s able con olle s
K
. I is conce ned wi h su iciency- ype condi ions o
he posi i i y and boundedness o he inpu –ou pu ene gy and asymp o ic anishing
condi ions o he inpu and ou pu o Punde ce ain s ipula ions on g:R0+→R.
Theo em 1. The ollowing p ope ies hold o any con olle K ∈K:
(i) The inpu –ou pu ene gy is bounded o all ime, ha is,
E( ) = Z
0y(τ)u(τ)dτ≤γ2
0<∞;∀ ∈R0+
(ii)
Assume ha
ˆ
g(s)∈SSPR
and ha
W(u)
is bounded. Then,
ess sup
∈R0+|u( )|<∞
and
lim in
→∞|u( )|≤
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω).
(iii)
I , in addi ion o he condi ions o P ope y (ii),
W(u (τ))≥λ( ,u( ))u(τ)
,
∀τ∈[0 , ]
,
and
∀ ∈R0+
o some
λ:R0+×R→R
, subjec o
in
∈R0+
λ( ,u( ))>−in
ω∈
¯
R0+
Re ˆ
g(iω)
, hen
u∈L∞∩L2
and
|u| ∈ L1
as well, so ha
u( )→0
as
→∞
, excep , e en ually, on an in e al
o ze o measu e, and
E( )∈0 , γ2
0
and
∀ ∈R0+
, ha is, he inpu –ou pu ene gy is non-nega i e
bounded o all ime. I , u he mo e,
u( )
has suppo on some eal in e al o nonze o measu e
S
,
hen
E( )∈0, γ2
0
and
∀ ≥ 1>
0, whe e
(0, 1)
is he i s connec ed componen o
S
, ha is,
he inpu –ou pu ene gy is join ly posi i e and bounded on [ 1,∞).
(i )
I , in addi ion o he condi ions o P ope ies (ii)–(iii),
W(
0
) =
0, hen
y( )
is bounded,
∀ ∈R0+
o any gi en ini e ini ial condi ions, and
y( )→0
as
→∞
. Fu he mo e,
y∈L2
so
ha bo h |u|,|y| ∈ L∞∩L1∩L2.
P oo . No e ha (3) combined wi h (2) p o es P ope y (i).
Now, no e ha by using Rayleigh (o Pa se al’s) ene gy heo em [
13
,
19
,
43
], i ollows
ha
R∞
0y(τ)u (τ)dτ=1
2πR∞
−∞ˆ
y(iω)ˆ
u (−iω)dω
, so ha , om he symme y p ope y o
he Fou ie ans o m, one ge s:
+∞>γ2
0≥E( ) = R∞
0y(τ)u (τ)dτ=R∞
0[(g∗u)(τ) + W(uτ)]u (τ)dτ
=1
2πR∞
−∞ˆ
y(iω)ˆ
u (−iω)dω=1
2πR∞
−∞ˆ
g(iω)ˆ
u (iω)ˆ
u (−iω)dω
=1
2πR∞
−∞ˆ
g(iω)|ˆ
u (iω)|2dω+R∞
0W(u (τ))(u (τ))dτ
≥1
2πin
ω∈
¯
R0+
Re ˆ
g(iω)R∞
−∞|ˆ
u (iω)|2dω+R∞
0W(u (τ))(u (τ))dτ;∀ ∈R0+,
(4)
since
ˆ
g∈SPR
implies ha
Re ˆ
g(iω)≥d=in
ω∈
¯
R
Re ˆ
g(iω)>
0 and since he hodog aph
ˆ
g(iω)
again has he symme y p ope y
Re ˆ
g(iω)=Re ˆ
g(−iω)
, and
Im ˆ
g(iω)=−Im ˆ
g(−iω)
and
∀ω∈
¯
R
(due o he symme y o he Fou ie ans o m),
one has o in e om he abo e inequali y and he Rayleigh ene gy heo em ha :
Ma hema ics 2022,10, 2051 8 o 20
∞>γ2
0≥E( )≥in
ω∈
¯
R0+
Re ˆ
g(iω)R∞
0u2
(τ)dτ+R∞
0W(u (τ))(u (τ))dτ
=in
ω∈
¯
R0+
Re ˆ
g(iω)R
0u2(τ)dτ+R
0W(u (τ))(u(τ))dτ
≥in
ω∈
¯
R0+
Re ˆ
g(iω)R
0u2(τ)dτ−sup
u∈R|W(u)|R
0|u(τ)|dτ;∀ ∈R0+
(5)
I is p o en by con adic ion ha
u( )
is essen ially bounded. I we assume ha i
is no essen ially bounded, hen he e is a s ic ly inc easing sequence
{ i}∞
i=0(⊂R0+)
such ha Z i
0u2(τ)dτ/Z i
0|u(τ)|dτ≥Mi
o some s ic ly inc easing sequence {Mi}∞
0(⊂R0+)→∞as i→∞. Then,
γ2
0
R i
0|u(τ)|dτ≥
in
ω∈R0+
Re ˆ
g(iω)R
0u2(τ)dτ
R i
0|u(τ)|dτ−sup
u∈R|W(u)|≥in
ω∈
¯
R0+
Re ˆ
g(iω)Mi−sup
u∈R|W(u)|
so ha , since W(u)is bounded,
∞>lim in
→∞

sup
u∈R|W(u)|+γ2
0
R i
0|u(τ)|dτ−Miin
ω∈
¯
R0+
Re ˆ
g(iω)

≥0, (6)
a con adic ion since
in
ω∈
¯
R
Re ˆ
g(iω)>
0 and
{Mi}∞
0(⊂R0+)→∞
as
i→∞
. Thus,
ess sup
∈R0+|u( )|<∞.
I is now p o en ha
lim in
→∞|u( )|≤
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω)
. Assume, on he con a y, ha he e
is some |u|>
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω)>0 such ha lim in
→∞|u( )|=|u|. Then,
γ2
0≥lim in
→∞

in
ω∈
¯
R0+
Re ˆ
g(iω)|u|−sup
u∈R|W(u)|

Z +θ
|u(τ)|dτand ∀θ∈R0+(7)
so ha one ge s he subsequen con adic ion:
∞>γ2
0≥lim
θ→∞

lim in
→∞

in
ω∈
¯
R0+
Re ˆ
g(iω)|u|−sup
u∈R|W(u)|

Z +θ
|u(τ)|dτ

=∞.
Then, ei he
lim in
→∞|u( )|≤
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω)
o
lim in
→∞|u( )|>
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω)
and
u( )→0
as
→∞
, excep e en ually on a ime in e al o ze o measu e, he second condi ion
Ma hema ics 2022,10, 2051 9 o 20
being a con adic ion i sel . The e o e,
lim in
→∞|u( )|≤
sup
u∈R|W(u)|
in
ω∈
¯
R0+
Re ˆ
g(iω)
. P ope y (ii) has
been p o en.
Now, also assume ha
W(u (τ))≥λ( ,u( ))u(τ)
,
∀τ∈[0 , ]
, and
∀ ∈R0+
, so ha one
ob ains om (5) ∞>γ2
0≥E( )≥in
ω∈
¯
R0+
Re ˆ
g(iω)R∞
0u2
(τ)dτ+R∞
0λ( ,u( ))u(τ)u (τ)dτ
≥

in
ω∈
¯
R0+
Re ˆ
g(iω)+in
∈R0+
λ( ,u( ))

Z∞
0u2
(τ)dτ
=

in
ω∈
¯
R0+
Re ˆ
g(iω)+in
∈R0+
λ( ,u( ))

Z
0u2(τ)dτ;∀ ∈R0+, (8)
and, i , u he mo e,
in
∈R0+
λ( ,u( ))>−in
ω∈
¯
R0+
Re ˆ
g(iω)
, hen, in addi ion o he p e iously
p o ed p ope ies
ess sup
∈R0+|u( )|<∞
and
u∈L2
, hen
u∈L∞∩L2
, so ha
u( )→0
as
→∞
, excep e en ually on a ime in e al o ze o measu e, and i ollows om (8) ha
E( )∈0 , γ2
0
and
∀ ∈R0+
; ha is he, inpu –ou pu ene gy is non-nega i e bounded o
all imes. I , in addi ion,
u( )
has suppo on some eal in e al o nonze o measu e
S
, hen
E( )∈0, γ2
0
and
∀ ≥ 1
, whe e
(0, 1)
is he i s connec ed componen o
S
. P ope y
(iii) has been p o en.
P ope y (i ) ollows since ˆ
g(s)∈SSPR, hus:
(1) he bi-p ope and s ic ly s able sys em o he o m
ˆ
g(s) = ˆ
g1(s) + d
wi h
d=in
ω∈
¯
R0+
Re ˆ
g(iω)>
0, since
ˆ
g1(s)∈PR
is he p ope o uni y ela i e o de wi h
Re ˆ
g1(s)≥0 o Re s ≥0 and lim
|s|→∞Re ˆ
g1(s) = 0;
(2) W(u (τ))≥λ( ,u( ))u(τ),τ∈[0 , ];∀ ∈R0+; and
(3)
in
∈R0+
λ( ,u( ))>−d
hen
sup
∈R0+|y( )|<∞
and
y( )→0
as
→∞
o any gi en
ini e ini ial condi ions since:
(a) I has al eady been p o en ha
ess sup
∈R0+|u( )|<∞
and
u( )→0
as
→∞
(excep
e en ually on a se o ze o measu e), and u he mo e,
(b) o any e en ually non-ze o ini ial condi ions
x(
0
) = x0
, he solu ion o (1) has an
ex a addi i e unc ion
y0:R0+→R0+
, which is bounded and exponen ially anishing,
since ˆ
g(s)is s ic ly s able and g( )also asymp o ically anishes, so ha
|y( )|=|(g∗u)( )|+|W(u )|+|y0( )|≤y<∞;∀ ∈R0+(9)
lim
→∞y( ) = lim
→∞(g∗u)( ) + W(0) + lim
→∞y0( ) = 0; ∀ ∈R0+(10)
since W(0) = 0. Since |u| ∈ L∞∩L1∩L2, hen |y| ∈ L∞∩L1∩L2.
The ollowing esul is a di ec ex ension o Theo em 1 i he e a e wo linea s ic ly
s able sys ems in pa allel connec ion in he eed- o wa d loop, wi h one o hem ha ing
s ic posi i e ealness p ope ies.
Co olla y 1. Assume ha he ou pu o he sys em is
y( ) = ((g+ga)∗u)( ) + W(u );∀ ∈R0+(11)
Ma hema ics 2022,10, 2051 16 o 20
(a)
i
λc( )≡
0 and
∀ ∈R0+
hen he sampling and hold de ice is a ze o-o de hold
(Z0H) and u(kT +τ)=uk,∀τ∈[0 , T), and ∀k∈Z0+;
(b)
i
λc(kT +τ)=τ
,
∀τ∈[0 , T)
, and
∀k∈Z0+
, hen he sampling and hold de ice is a
i s -o de -hold (FOH) and
u(kT +τ)=u(kT)+τ
T(u(kT)−u[(k−1)T])
,
∀τ∈[0 , T)
,
and ∀k∈Z0+; and
(c)
i
λc:R0+→(0 , T)
, hen he sampling and hold de ice is a speed co ec ion hold
(SCH):
u(kT +τ)=u(kT)+λc(kT +τ)
T(u(kT)−u[(k−1)T]),∀τ∈[0 , T), and ∀k∈Z0+.
In he case whe e
λc(kT +τ)=kcτ
,
∀k∈Z0+
, o some
kc∈(0 , 1)
,
∀τ∈[0 , T)
, hen
he SCH is o cons an slope
kc
, and SCH(
kc
). I
kc∈(0 , 1)
, hen he de ice is named as a
pa ial speed co ec ion hold (PSCH).
We now discuss he hype s able design o he mo e gene al SCH sampling and hold
de ice. Assume ha eedback o he hype s able eedback con olle
K∈Kd
whe e
Kd
has an inpu
y( )
( ha is, i is he ou pu o he eed- o wa d con olled sys em) and ou pu
( )(=−u( )):R0+→R
, which is assumed o sa is y he ollowing con inuous/disc e e
Popo ’s hype s abili y inpu –ou pu in eg al condi ion:
Z
0y(σ) (σ)dτ=Z
0(−u(σ))y(σ)dσ
=− k−1
∑
j=0Z(j+1)T
jT u(σ)y(σ)dσ+Zτ
0u(kT +σ)y(kT +σ)dσ!
=−γ2
0+Zk−1
j=0Z(j+1)T
jT
ε2(σ)dσ+Zτ
0
ε2(kT +σ)dσ≥ −γ2
0=>−∞
=kT +τ,∀k∈Z0+,τ∈[0, T], (24)
o some a bi a y squa e-in eg able
γ,ε:R0+→R
ul illing 0
≤R
0ε2(σ)dσ<R
0γ2(σ)dσ<
γ2
0<+∞
;
∀ ∈R0+
, wi h
(y(kT)=0)⇒[(ε(kT)=0)∧u(kT)=0]
and
|ε(kT)|<|y(kT)|
i
y(kT)6=
0,
∀k∈Z0+
, and
γ2
0=lim
→∞R
0γ2(τ)dτ
o some ini e nonze o eal cons an
γ0
.
No e ha
γ( ),ε( )→0
as
→∞
. The pu pose o he auxilia y unc ions
γ,ε:R0+→R
is o gua an ee (24) unde equali ies o all
∈R0+
. Equa ion (24) holds o all
∈R0+
,
subjec o (23), i :
λc(kT +τ)
T(u(kT)−u[(k−1)T]) +u(kT)y(kT +τ)=γ2(kT +τ)−ε2(kT +τ);∀τ∈[0 , T),∀k∈Z0+, (25)
which, a he sampling ins an s, i.e., τ=0, becomes:
u(kT)y(kT)=γ2(kT)−ε2(kT);∀k∈Z0+, (26)
which, when eplaced in (25), yields:
h1+λc(kT+τ)
Tγ2(kT)−ε2(kT)
y(kT)−λc(kT+τ)
T
γ2[(k−1)T]−ε2[(k−1)T]
y[(k−1)T]iy(kT +τ)
=γ2(kT +τ)−ε2(kT +τ);∀τ∈[0 , T),∀k∈Z0+,
(27)
which ensu es he pa icula needed cons ain a he sampling ins an s:
λc(kT)=γ2(kT)−ε2(kT)y(kT)−γ2(kT)−ε2(kT)y(kT)
(γ2(kT)−ε2(kT))y[(k−1)T]+(ε2[(k−1)T]−γ2[(k−1)T])y(kT)
Ty[(k−1)T]
y(kT)=0 and ∀k∈Z0+,

Ma hema ics 2022,10, 2051 17 o 20
which gua an ees
u( ) = u(kT)=γ2(kT)−ε2(kT)
y(kT)
i
=kT
and
∀k∈Z0+
, p o ided ha
y(kT)6=0 and u(kT)=0 i y(kT)=0.
Thus, he hype s able con olle is syn hesized as ollows:
(a)
I
=kT
(sampling ins an s), hen
u(kT)=γ2(kT)−ε2(kT)
y(kT)
wi h
|ε(kT)|<|y(kT)|
i
y(kT)6=0 and u(kT)=0, wi h |ε(kT)|=|y(kT)|i y(kT)=0 and ∀k∈Z0+.
(b)
I 6=kT (in e -sampling ins an s):
u(kT +τ)=u(kT)+λc(kT +τ)
T(u(kT)−u[(k−1)T]) and ∀k∈Z0+,
λc(kT +τ)=γ2(kT +τ)−ε2(kT +τ)y(kT)−γ2(kT)−ε2(kT)y(kT +τ)
(γ2(kT)−ε2(kT))y[(k−1)T]+(ε2[(k−1)T]−γ2[(k−1)T])y(kT)
Ty[(k−1)T]
y(kT +τ),
∀τ∈(0 , T), and ∀k∈Z0+
|ε(kT +τ)|<|y(kT +τ)|;∀τ∈(0 , T),∀k∈Z0+.
No e om (27) ha as
{u(kT)}∞
k=0→0
,
lim
k→∞(|y(kT +τ)|−|y(kT +τ)|)=
0;
∀τ∈[0 , T)and ∀k∈Z0+.
The con ol law o he o m (23), ob ained om an SCH sampling and hold de ice,
which is wi hin he class o hype s able con olle s subjec o he in eg al Popo ’s- ype
cons ain (24), sa is ies Theo em i i is he eedback loop o a ans e unc ion
ˆ
g(s)∈SSPR
in pa allel wi h a non-linea de ice W(u)unde he gi en hypo heses in he heo em.
Rema k 4.
I can be poin ed ou ha he Popo ian hype s abili y cons ain s o (24) [
10
–
12
]
a e mo e gene al because o he mo e gene al inpu s gene a ed om hei sampled alues han
he pa allel cons ain s associa ed wi h disc e e alues o he inpu s and ou pu s being o he
o m
∑j
j=0y(kT) (kT)≥ −γ2
0>−∞
;
∀k∈Z0+
. A pa allel esul o Theo em 1 and i s
co olla ies could be easily ob ained by applying he Rayleigh heo em on he uni complex ci cle
and using he Z- ans o m o he impulse esponse o he linea pa . Basically, he in eg als o (4)
would be changed o he sums
ˆ
g(iω)→ˆ
gD(z)
o
z=eiθ
and
θ∈[0 , 2π)
( hen
|z|=
1), wi h
ˆ
gD(z) = Z(1−e−Ts)g ˆ(s)
s
whe e
1−e−Ts
s
is he ans e unc ion o he ZOH. I u ns ou ha
d=in
ω∈
¯
R0+
ˆ
g(iω)=in
θ∈[0 ,2π)
ˆ
gDeiθ>
0since
ˆ
g(s)∈SSPR
, which is hen also bi-p ope and
s ic ly s able, ha is,
ˆ
ga(z)
and
ˆ
g(s)
ha e an iden ical inpu –ou pu in e connec ion gain, which
is he quo ien o he leading coe icien s o bo h hei nume a o and denomina o polynomials.
Howe e , he disc e iza ion app oach add essed in his way has only in o ma ion a he sampling
ins an s, a he han o all ime, and i is o in e es only o he use o disc e iza ion unde a ZOH
and no o mo e gene al sampling and hold de ices. E en in his case, no e ha he ou pu is no
ully add essed in he inpu –ou pu ene gy o mulas o Theo em 1 and he la e co olla ies since he
ou pu is no piece-wise cons an .
5. Conclusions
The pape has in es iga ed he asymp o ic hype s abili y o a single-inpu single-
ou pu closed-loop con ol con igu a ion whose eed- o wa d loop consis s o a pa allel
connec ion o a s ongly s ic ly posi i e eal ans e unc ion, oge he wi h (in gene al)
a non-posi i e nonlinea ope a o which has o sa is y some discussed condi ions. The
eedback loop consis s o , in gene al, a nonlinea and, pe haps, ime- a ying con olle
which sa is ies a Popo - ype in eg al inequali y. The global asymp o ic s abili y is p o en
o be “in he la ge”, ha is, i is gua an eed o any gi en ini e ini ial condi ion, and he
asymp o ic hype s abili y p ope y implies ha he closed-loop asymp o ic s abili y is
gua an eed independen ly o he pa icula con olle employed wi hin he abo e class.
Ma hema ics 2022,10, 2051 18 o 20
The p ope y is add essed by p o ing, ough Pa se al’s heo em, ha he inpu –ou pu
ene gy o he eed- o wa d loop is always posi i e and bounded o all imes. Ex a
su iciency- ype condi ions o keep he asymp o ic hype s abili y p ope y a e ob ained
unde he inco po a ion o an addi ional s ic ly s able linea and ime-in a ian sys em.
In pa icula , and in o de o keep he hype s abili y p ope ies o he whole closed-loop
con igu a ion, i s equency esponse esonance gain, which is su icien ly small, is ela ed
o he minimum alue o he eal pa o he impulse esponse associa ed wi h he s ongly
s ic ly posi i e eal ans e unc ion. A case s udy is p o ided, which is conce ned wi h
he use o a ac ional sampling and hold de ice o gene a e he con inuous- ime inpu
om hei sampled alues a a cons an sampling a e.
Funding:
This esea ch was unded by he Spanish Go e nmen and he Eu opean Commission,
g an numbe RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and by he Basque Go e nmen , g an
numbe IT1207-19. The APC was unded by g an RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE).
Acknowledgmen s:
The au ho is g a e ul o he Spanish Go e nmen and he Eu opean Commission
o i s suppo h ough g an RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and o he Basque
Go e nmen o i s suppo h ough g an IT1207-19. He is also g a e ul o he Re e ees by hei
use ul sugges ions and commen s.
Con lic s o In e es : The au ho decla es no con lic o in e es .
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