Ci a ion: Yin, C.-W.; Riaz, S.;
Zaman, H.; Ullah, N.; Blazek, V.;
P okop, L.; Misak, S. A No el
P ede ined Time PD-Type ILC
Pa adigm o Nonlinea Sys ems.
Ma hema ics 2023,11, 56. h ps://
doi.o g/10.3390/ma h11010056
Academic Edi o : An ónio Lopes
Recei ed: 7 No embe 2022
Re ised: 16 Decembe 2022
Accep ed: 18 Decembe 2022
Published: 23 Decembe 2022
Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ma hema ics
A icle
A No el P ede ined Time PD-Type ILC Pa adigm o
Nonlinea Sys ems
Chun-Wu Yin 1, Saleem Riaz 2, Haide Zaman 3, Nasim Ullah 4,* , Voj ech Blazek 5,* , Lukas P okop 5
and S anisla Misak 5
1School o In o ma ion and Con ol Enginee ing, Xi’an Uni e si y o A chi ec u e and Technology,
Xi’an 710055, China
2School o Au oma ion, No hwes e n Poly echnical Uni e si y, Xi’an 710129, China
3Elec onics Enginee ing Depa men , Uni e si y o Enginee ing and Technology Peshawa ,
Peshawa 25000, Pakis an
4Depa men o Elec ical Enginee ing, College o Enginee ing, Tai Uni e si y, Tai 11099, Saudi A abia
5ENET Cen e, VSB—Technical Uni e si y o Os a a, 708 00 Os a a, Czech Republic
*Co espondence: [email p o ec ed] (N.U.); [email p o ec ed] (V.B.)
Abs ac :
In elligen obo ics has d awn a g ea deal o a en ion due o i s high p ecision, s abili y,
and eliabili y, which a e he basic key ac o s o indus ial au oma ion. This pape p oposes an
i e a i e lea ning con ol (ILC) echnique wi h p ede ined- ime con e gence as a solu ion o an
applied enginee ing p oblem, namely, ha local ime canno be p ese when a second-o de nonlinea
sys em unde akes con ol o he accu a e acking o local ime unde any ini ial i e a i e alue. A
ime- a ying sliding su ace wi h an ini ial alue o ze o was designed, and i was heo e ically p o en
ha he ajec o y acking e o in he sliding su ace could con e ge o ze o wi hin a p ede ined ime.
The i e a i e con ol p oblem o ajec o y acking was hus changed o an i e a i e con ol p oblem
o ime- a ying sliding-mode su ace acing wi h a s a ing alue o ze o. A PD- ype closed-loop
ILC wi h a ime- a ying sliding mode su ace was designed such ha he ajec o y acking e o
con e ged and s abilized on he sliding mode su ace a e a ini e numbe o lea ning i e a ions. The
con ol goal o he sys em’s ou pu was he abili y o ack he desi ed ajec o y accu a ely wi hin a
p ede ined ime in e al, and i was achie ed by combining his wi h he p ede ined ime con e gence
cha ac e is ics o he ime- a ying sliding mode su ace. Nume ical simula ion o ajec o y acking
con ol o a epe i i e mo ion manipula o was used o e i y he e ec i eness o he p oposed
con olle and i s obus ness in he ace o ex e nal dis u bances.
Keywo ds:
i e a i e lea ning con ol; sliding mode con ol; p ede ined- ime con e gence; ime-
a ying sliding mode su ace; obo ic a m
MSC: 393D05; 37N35
1. In oduc ion
In enginee ing applica ions such as indus ial ecu en p oduc ion, ha d disk d i e
con ol, and cons uc ion obo wall building, he ou pu o he epe i i e mo ion con ol
sys em is equi ed o mo e a mechanical a m s ic ly acco ding o he desi ed ajec o y
wi hin a ini e ime in e al
[
0,
T]
. When he ini ial alue o ajec o y acking e o is ze o,
i e a i e lea ning con ol (ILC) is implemen ed. In sho , ILC can ensu e ha he sys em
ou pu ully acks he desi ed ajec o y [
1
,
2
], bu in p ac ical enginee ing applica ions i is
challenging o s ic ly loca e he ini ial s a e o he con olled sys em a he ini ial posi ion
o he desi ed ajec o y. The a bi a y ini ial alue o i e a i e lea ning con ol can only
ensu e ha he sys em ou pu accu a ely acks he desi ed ajec o y in local ime
[Ts
,
T]
,
bu he exis ing ILC con ol s a egy canno be p ese o es ima ed in he bound o ime
Ts
.
Ma hema ics 2023,11, 56. h ps://doi.o g/10.3390/ma h11010056 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 56 2 o 19
This es ic s he applica ion o ILC in p ac ical enginee ing. The e o e, he e is a s ong
need o design an i e a i e lea ning con ol algo i hm wi h a p ede ined ime o Ts.
T adi ional ILC con ol heo y is p edica ed on he assump ion ha he con olled
sys em’s i e a i e s a ing alue de ia ion is ze o. When he con olled sys em is sa is ied
wi h he i e a i e ini ial alue cons ain condi ions, he ou pu o he con olled sys em
can be in a gi en ime in e al, in s ic acco dance wi h he desi ed ajec o y [
3
–
5
] (which
is pe ec acking), bu in p ac ical enginee ing applica ions, i is di icul o mee e e y
ime cons ain wi h a ze o ini ial i e a ion alue de ia ion [
6
]. Thus, he enginee ing
applica ion o ILC heo y is limi ed. Schola s ha e con i med, h ough heo e ical analysis
and expe imen al e i ica ion, ha when he de ia ion be ween he ini ial alue o he
con olled sys em and he desi ed ajec o y is a ixed alue, o when he ini ial alue o
he sys em and he desi ed ajec o y sa is y a ce ain law, his alue o he sys em can
con e ge o he ini ial alue o he desi ed ajec o y, o which he ILC algo i hm can
also ensu e ha he ou pu o he con olled sys em ollows he desi ed ajec o y [
7
,
8
]
( o ins ance, he acking accu acy in ac ional o de con ol [
9
,
10
] and op imal lea ning
con ol [
11
]). This esul elaxes he s ic equi emen ha he i e a i e ini ial alue
a ia ion o he con en ional ILC mus be ze o [
12
,
13
], bu i is s ill compa ible wi h eal-
wo ld enginee ing applica ions. When a di e ence a ises, only i e a ions o he ILC con ol
algo i hm ha sa is y he ini ial alue c i e ia will be able o ul il he needs o he p ac ical
enginee ing applica ion.
Schola s ha e p oposed con ol s a egies such as model p edic i e con ol, he ini ial
alue co ec ion me hod, he bounda y laye me hod, and he a ac o me hod
[14–16]
, bu
he ini ial alue co ec ion me hod in ol es de e mining he delay ac o in ad ance
[17,18]
.
The bounda y laye in he bounda y laye me hod is asymp o ically con e gen , which
means ha he ajec o y acking e o in he bounda y laye can only con e ge o ze o
when ime ends o in ini y, esul ing in low ajec o y acking accu acy [
19
]. The a ac o
design in he a ac o me hod has ce ain limi a ions, and some a ac o con ol s a egies
in ol e edesigning he desi ed ajec o y [20].When he ini ial alue o he acking e o
be ween he sys em ou pu and he desi ed ajec o y is
ek(
0
)≡
0, a e a ini e numbe o
i e a ions he sys em ou pu has he ull abili y o ollow he desi ed ajec o y wi hin he
ini e ime in e al,
[
0,
T]
. Howe e , when he ini ial alue o he acking e o be ween
he sys em ou pu and he desi ed ajec o y is
ek(
0
)6=
0, i means ha pa icula sys em
is able only o ack he desi ed signal wi hin he local ime in e al
[Ts
,
T]
, which is
ek( )≡
0,
∈[Ts
,
T]
. Al hough he exis ing con ol s a egies o supp ess he ini ial alue o
any i e a ion can sol e he i e a i e lea ning con e gence p oblem unde he ini ial alue o
any i e a ion, hey canno es ima e o e en se he ime,
Ts
, o achie e local con e gence in
ad ance. In some p ac ical enginee ing applica ions, i is equi ed ha he sys em ou pu
accu a ely acks he desi ed ajec o y be o e he gi en ime,
Ts
. Fo example, when a
cons uc ion obo pe o ms cons uc ion p ocesses such as conc e e oweling o wall
laying, he mechanical a m mus each he desi ed ajec o y be o e he gi en ime,
Ts
, and
epea he mo emen s ic ly acco ding o he desi ed ajec o y o ensu e he smoo hness
o conc e e oweling o he uni o mi y o wall iles. Failu e o achie e his can lead o
majo economic losses o he cons uc ion indus y, as well as aise he isk o building
collapse. Despi e he impo ance o de e mining and p ese ing he local con e gence ime
in many enginee ing applica ions, e y li le wo k has been done on he i e a i e lea ning
con ol heo y in ega d o p ede ined- ime con e gence. A he same ime, while many
cu en i e a i e lea ning con ol s a egies unde a bi a y i e a i e ini ial alues ha e been
p ojec ed mainly o i s -o de sys ems, he e a e ela i ely ew publica ions on i e a i e
lea ning con ol s a egies o second-o de nonlinea sys ems unde a bi a y i e a i e
ini ial alues.
This pape will ocus on second-o de nonlinea sys ems wi h epe i i e mo ion, and
p opose a PD- ype closed-loop i e a i e lea ning con ol s a egy based on he p ede ined-
ime con e gence sliding mode su ace, aiming o show ha he con olled sys em unde
any ini ial alue can no only ollow a local ajec o y o accu a e acking, bu also
Ma hema ics 2023,11, 56 3 o 19
p ede e mine he local con e gence ime,
Ts
, in ad ance. The main inno a ions and
con ibu ion o his s udy can be summa ized as ollows:
1.
P o ides Lyapuno s abili y c i e ion o he s abili y o nonlinea sys ems wi hin a
p ede ined ime and desc ibes he heo e ical p oo unde he gi en condi ions.
2.
P esen s a design o a ime- a ying sliding mode su ace wi h p ede ined ime
con e gence cha ac e is ics in which he con e gence ime o he ajec o y acking
e o loca ed in he sliding mode su ace can be p ese , b inging he ad an age ha
he con e gence ime is no a ec ed by he con olling cons ain s o he ini ial alue
o he i e a ion.
3.
Con e s he ajec o y acking con ol p oblem, whe e he ini ial alue o he ajec-
o y acking e o is no ze o, in o a sliding mode su ace acking con ol p oblem in
which he ini ial alue o he sliding mode su ace being ze o. Es ablishes a b idge
be ween he i e a i e lea ning con ol heo y wi h an a bi a y i e a i e ini ial alue
and he same i e a i e ini ial alue.
4.
The i e a i e lea ning con ol s a egy no only sol es he p oblem o a bi a y i e a i e
ini ial alue supp ession and simpli ies he heo e ical p oo o he con e gence o
i e a i e lea ning, i also achie es he enginee ing applica ion o he sys em ou pu ,
accu a ely acking he desi ed ajec o y wi hin a p ese local ime.
The emainde o his pape is as ollows. Sec ion 2p esen s he con ol p oblem
o mula ion and also desc ibes se e al lemmas o i e a i e lea ning con e gence p oo .
Sec ion 3p oposes an a bi a y ini ial alue supp ession s a egy based on he p ede ined
ime con e gence sliding mode con ol p inciple, men ioning i s p inciples. The Lyapuno
s abili y c i e ion o p ede ined ime con e gence o nonlinea sys ems is gi en, and a
design o a sliding mode su ace wi h he cha ac e o p ede ined ime con e gence and
ini ial alue o ze o is p esen ed. The main esul s o i e a i e con e gence a e discussed
in Sec ion 4, which also demons a es he p ede ined ime con e gence condi ion o a
PD- ype ILC. In Sec ion 5, he e ec i eness o he p oposed nonlinea con ol s a egy is
illus a ed by simula ions o a obo ic sys em, he esul s o which a e b ie ly explained.
Finally, Sec ion 6p esen s he conclusions.
2. Con ol P oblem Desc ip ions
Conside he ollowing second-o de nonlinea sys em wi h epe i i e mo ion cha ac e is ics:





.
x1k( ) = x2k( )
.
x2k( ) = (xk( ), ) + B( )uk( )
yk( ) = x1k( )
(1)
whe e
xk( ) = [x1k( ),x2k( )]T
indica es he s a e a iable,
yk( )∈Rm
is he ou pu a iable,
uk( )∈Rl
is he con ol inpu a iable,
k
ep esen s he numbe o i e a ions, and
∈[
0,
T]
,
B( )
is he bounded unc ion ma ix o app op ia e dimension. The unc ion
(xk( )
,
)
sa is ies he Lipschi z condi ion wi h espec o he s a e a iable,
xk( )
, in he ime in e al
∈[0, T]. Tha is means he e is a cons an ,M1>0, and unc ion (xk( ), )sa is ies
|| (xk( ), )− (xd( ), )|| ≤ M1||xk( )−xd( )|| (2)
The con ol objec i e: Le he desi ed ajec o y o he second-o de nonlinea sys em
(1) be
yd( )
in an applica ion en i onmen whe e he i e a i e ini ial alue,
yk(
0
)
, o he
second-o de nonlinea sys em (1) canno be s ic ly loca ed a he ini ial alue,
yd(
0
)
, o
he desi ed ajec o y. Design an i e a i e lea ning con olle ,
uk( )
, o make he ou pu
o sys em (1) p ecisely ack he desi ed ajec o y,
yk( )
, o e a p ede ined- ime in e al,
[Ts,T](0<Ts<T).
Ma hema ics 2023,11, 56 4 o 19
The acking e o be ween he sys em ou pu ,
yk( )
, and he a ge ajec o y,
yd( )
,
may be de e mined as ollows:
ek( ) = yk( )−yd( )(3)
Below a e some lemmas o i e a i e lea ning con e gence p oo :
Lemma 1 .
Le
w( )
,
b( )
,
a( )
be a con inuous unc ion de ined on he in e al
[
0,
T]
, and
a( )>
0.
I [21]
w( )≤b( ) + Z
0a(τ)w(τ)dτ(4)
hen w( )≤b( ) + R
0a(τ)b(τ)eR
τa(λ)dλdτ.
Lemma 2 .
Suppose ha he men ioned unc ion,
O(ξ)( )
, o he ime in e al
∈[∆
,
T]
molli ies
he ollowing condi ions [21]:
(1)
||O(ξ)( )|| ≤ M(a+R
0||ξ(s)||ds)
(2)
||O(ξ)( )−O(ζ)( )|| ≤ M(R
0||ξ(s)−ζ(s)||ds)
In he abo e o mula, i
M
and
a
a e non-nega i e cons an s, hen we can d aw he
ollowing wo conclusions:
(a)
Fo ζ( )∈C [0, T], he e exis s a unique ξ( )∈C [0, T], such ha
ξ( ) + O(x)( ) = ζ( )(5)
(b)
Acco ding o he de ini ion o he unc ion de ined as
O(ζ) = O(ξ)( )
, whe e
ξ∈C [0, T]is he only solu ion de ined by (a), he e exis s an M1>0 such ha
||O(ζ)( )|| ≤ M1(a+Z
0||ζ(s)||ds)(6)
Lemma 3 .
Le he cons an se ies
{bk}k≥0
,
bk≥
0con e ge o ze o, and he unc ion
Ok(θ)( )
sa is y [21]
||Ok(θ)( )|| =K(bk+Z
0||θk(τ)||dτ)(7)
In he p e ious exp ession,
K>
1 is a cons an . I we assume ha
Ψ( )
, which can be
× , is a dimensional ma ix o con inuous unc ions, and Ψ:C [0, T]→C [0, T], hen:
Ψ(θ)( ) = Ψ( )θ( )(8)
F om he abo e equa ions, i ollows ha when he spec al adius o Ψ< 1, hen:
lim
k→∞(Ψ+Ok)(Ψ+Ok−1)···(Ψ+O0)(θ)( ) = 0 (9)
3. A bi a y Ini ial Value Supp ession S a egy Based on P ede ined-Time
Con e gence Sliding Mode Su ace
3.1. A bi a y Ini ial Value Supp ession S a egy and I s P inciple
Acco ding o he sliding mode con ol p inciple [
22
], a sys em s a e whose ini ial alue
is loca ed a any posi ion in he s a e space can each and s abilize in he sliding mode
su ace
S(x( ))
unde he sliding mode con olle and wi hin he sliding mode su ace
(equi alen o
S(x( )) ≡
0) sliding o he equilib ium poin ,
O
. The sliding mode con ol
(SMC) law is illus a ed in Figu e 1.
Ma hema ics 2023,11, 56 5 o 19
Ma hema ics 2022, 10, x FOR PEER REVIEW 5 o 20
3. A bi a y Ini ial Value Supp ession S a egy Based on P ede ined- ime Con e -
gence Sliding Mode Su ace
3.1. A bi a y Ini ial Value Supp ession S a egy and I s P inciple
Acco ding o he sliding mode con ol p inciple [22], a sys em s a e whose ini ial
alue is loca ed a any posi ion in he s a e space can each and s abilize in he sliding
mode su ace (()) Sx unde he sliding mode con olle and wi hin he sliding mode
su ace (equi alen o (()) 0 ≡Sx ) sliding o he equilib ium poin , O. The sliding mode
con ol (SMC) law is illus a ed in Figu e 1.
Figu e 1. Schema ic diag am o he p inciple, sliding mode con ol p inciple.
This s udy o e s an a bi a y i e a i e ini ial alue supp ession con ol echnique
based on a p ede ined- ime con e gence sliding mode su ace o ackle he a bi a y s a -
ing alue issue in i e a i e lea ning con ol, using a con ol goal sys em (1) and he SMC
p inciple.
When he s a ing s a e o he k- h i e a i e lea ning is a any poin in space, i is he
same as he ini ial alue o ajec o y o ou e acking e o s, (0) 0
k≠e. By applying he
SMC concep , i is possible o build a sliding mode su ace, (())
k Se ,wi h p ede ined- ime
con e gence cha ac e is ics and an i e a i e lea ning con olle , ()
k u, allowing he con-
olle , ()
k u, o d i e e o s, ()
k e, ha a i e a any s a ing posi ion and s abilize in he
sliding mode su ace (equi alen o (())0
k ≡Se ). When he acking e o ()
k e is s abi-
lized in he sliding mode su ace acco ding o he p ede ined- ime con e gence cha ac e -
is ics o he SMC su ace, and when he acking e o , ()
k e, e u ns o ze o wi hin he
p ede e mined pe iod,
s
T; ha is, when () 0, [ , ]
ks
TT≡∈e, he goals o supp essing he
issue o andom s a ing alues and achie ing p ecise acking o he in ended ajec o y
a e bo h ealized.
To achie e he ajec o y acking e o sa is ying () 0
k ≡e wi hin he p ede ined-
ime in e al [,]
s
TT∈, se e al co e p oblems p esen hemsel es. The i s is ensu ing
ha he acking e o , ()
k e, con e ges and s abilizes wi hin he sliding mode su ace
()
k
S
e a e ini e i e a i e lea ning, ha is, lim ( ( )) 0
k
k
→∞ =Se . The second is ha in he slid-
ing mode su ace (())
k Se , ajec o y acking e o ()
k e con e ges o he equilib ium
poin in a p ede ined- ime,
s
T, ha is, lim ( ) 0
s
k
T
→=e. Based on he abo e wo co e p ob-
lems, his pape designs a con olle ha supp esses he a bi a y i e a i e ini ial alue
p oblem in wo s eps. The i s s ep is o design he sliding mode su ace (())
k Se wi h
he cha ac e is ic o con e ging o he equilib ium poin wi hin he p ede ined- ime,
s
T,
o ensu e ha he acking e o ()
k e in sliding mode su ace (())
k Se con e ges o he
equilib ium poin wi hin he p ede ined- ime,
s
T. The second s ep is o design an i e a i e
O
() 0
s
=x
0
x
A
Figu e 1. Schema ic diag am o he p inciple, sliding mode con ol p inciple.
This s udy o e s an a bi a y i e a i e ini ial alue supp ession con ol echnique based
on a p ede ined- ime con e gence sliding mode su ace o ackle he a bi a y s a ing alue
issue in i e a i e lea ning con ol, using a con ol goal sys em (1) and he SMC p inciple.
When he s a ing s a e o he k- h i e a i e lea ning is a any poin in space, i is he
same as he ini ial alue o ajec o y o ou e acking e o s,
ek(
0
)6=
0. By applying
he SMC concep , i is possible o build a sliding mode su ace,
S(ek( ))
,wi h p ede ined-
ime con e gence cha ac e is ics and an i e a i e lea ning con olle ,
uk( )
, allowing he
con olle ,
uk( )
, o d i e e o s,
ek( )
, ha a i e a any s a ing posi ion and s abilize in
he sliding mode su ace (equi alen o
S(ek( )) ≡
0). When he acking e o
ek( )
is s abilized in he sliding mode su ace acco ding o he p ede ined- ime con e gence
cha ac e is ics o he SMC su ace, and when he acking e o ,
ek( )
, e u ns o ze o wi hin
he p ede e mined pe iod,
Ts
; ha is, when
ek( )≡
0,
∈[Ts
,
T]
, he goals o supp essing
he issue o andom s a ing alues and achie ing p ecise acking o he in ended ajec o y
a e bo h ealized.
To achie e he ajec o y acking e o sa is ying
ek( )≡
0 wi hin he p ede ined- ime
in e al
∈[Ts
,
T]
, se e al co e p oblems p esen hemsel es. The i s is ensu ing ha
he acking e o ,
ek( )
, con e ges and s abilizes wi hin he sliding mode su ace
S(ek)
a e ini e i e a i e lea ning, ha is,
lim
k→∞S(ek( )) =
0. The second is ha in he sliding
mode su ace
S(ek( ))
, ajec o y acking e o
ek( )
con e ges o he equilib ium poin in
a p ede ined- ime,
Ts
, ha is,
lim
→Ts
ek( ) =
0. Based on he abo e wo co e p oblems, his
pape designs a con olle ha supp esses he a bi a y i e a i e ini ial alue p oblem in
wo s eps. The i s s ep is o design he sliding mode su ace
S(ek( ))
wi h he cha ac e is ic
o con e ging o he equilib ium poin wi hin he p ede ined- ime,
Ts
, o ensu e ha he
acking e o
ek( )
in sliding mode su ace
S(ek( ))
con e ges o he equilib ium poin
wi hin he p ede ined- ime,
Ts
. The second s ep is o design an i e a i e lea ning con olle
o ensu e he con e gence o i e a i e lea ning, so ha he acking e o
ek( )
eaches and
s abilizes in he sliding su ace S(ek( )).
3.2. P ede ined-Time Con e gence Lyapuno S abili y C i e ion and Sliding Mode Su ace Design
P ede ined- ime con e gence is he key o supp essing he a bi a y ini ial alue o
i e a ion and ensu ing ha he ajec o y acking e o can achie e accu a e acking be o e
he p ede ined- ime, Ts. The de ini ion o p ede ined- ime s abili y is gi en below.
De ini ion 1.
Fo a second-o de nonlinea sys em (1), i he e is a p ese cons an ,
Ts>
0, such
ha o any ∈[0, ∞] he condi ion is sa is ied:
when →Ts, lim
→Ts
y( ) = 0; When ≥Ts, i has y( )≡0
Then he second-o de nonlinea sys em (1) is globally p ede ined- ime s able.

Ma hema ics 2023,11, 56 6 o 19
A Lyapuno s abili y c i e ion o he p ede ined- ime con e gence is p esen ed below
and p o en heo e ically in o de o acili a e he assessmen o he nonlinea sys em’s
global p ede ined- ime con e gence.
Theo em 1.
In a nonlinea sys em (1), o any gi en p ede ined- ime,
Ts>
0, i he e exsi s a
posi i ely de ini e and adially unbounded Lyapuno unc ion, V( ), which sa is ies
.
V( )≤ − π
2λTs√ab(aV1−λ( ) + bV1+λ( )) (10)
whe e he pa ame e s sa is y 0<λ<1, a>0, b>0, hen
(1) i
V(
0
)6=
0, hen he sys em is global p ede ined- ime s able and con e ges o he equilib ium
ime: s=2Ts
πa c anqb
aVλ(0)<Tsand
(2)
V(
0
) =
0, hen
V( )≡
0, meaning ha he sys em s a e is always a he equilib ium poin .
P oo .
Acco ding o
.
V( )≤ − π
2λTs√ab (aV1−λ( ) + bV1+λ( ))
, adding he nonnega i e
cons an ∆≥0 o he igh hand means ha
.
V( ) = −π
2λTs√ab(aV1−λ( ) + bV1+λ( )) −∆
changes i o ma , gi ing:
dV( )
d =−π
2λTs√ab aV1−λ( )(1+b
aV2λ( )) −∆
=−π
2λTs√ab aV1−λ( )(1+b
aV2λ( ) + 2λTs√ab
πaV1−λ( )∆)(11)
A e ans o ming (11), we ob ain
π√a
2Ts√bd =−λVλ−1( )dV( )
1+ (qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆
=−dVλ( )
1+ (qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆
(12)
A e compiling di e en ia ion (12), in o de o make he in eg a ion p ocess simple ,
we ha e
b
a
π
2Ts a
bd =π
2Tsd =−d(qb
aVλ( ))
1+ (qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆
(13)
Assume ha
V( s) =
0 a ime
s
and in eg a e bo h sides o Equa ion (13) simul-
aneously on
(
0,
s]
. Since
V( )≥
0,
∆≥
0, hen
2λTs√ab
πaV1−λ( )∆≥
0 and, acco ding o
lim
→∞a c an( ) = π
2, we ha e:
R s
0π
2Tsd =−R s
0
d(qb
aVλ( ))
1+(qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆
=−RV( s)
V(0)
d(qb
aVλ( ))
1+(qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆
=RV(0)
0
d(qb
aVλ( ))
1+(qb
aVλ( ))2+2λTs√ab
πaV1−λ( )∆≤RV(0)
0
d(qb
aVλ( ))
1+(qb
aVλ( ))2
⇒π
2Ts| s
0≤a c anqb
aVλ( )|V(0)
0⇒π
2Ts s≤a c anqb
aVλ(0)
⇒ s≤2Ts
πa c anqb
aVλ(0)≤2Ts
ππ
2=Ts
(14)
Ma hema ics 2023,11, 56 7 o 19
When
V(
0
) =
0,
s≤2Ts
πa c anqb
aVλ(0)=
0, i means ha
V( )≡
0 is always
ue, indica ing ha he sys em s a e has always been a he equilib ium poin . 
The p ede ined- ime con e gence s abili y c i e ion o he nonlinea sys em is mainly
used o de e mine whe he he ajec o y acking e o loca ed in he sliding mode su ace
can con e ge o he o igin wi hin a p ede ined ime. We can de i e a p ede ined- ime
sliding mode su ace. The main goal o he p oposed su ace is o con e ge a ha pa -
icula p ede ined ime. This kind o ime- a ying sliding su ace wi h p ede ined ime
cha ac e is ics can be desc ibed as ollows:
S(e( )) = 










.
e( ) + π
2λTs√ab (ae1−λ( ) + be1+λ( ))−
.
e(0) + π
2λTs√ab (ae1−λ(0) + be1+λ(0))exp(−α ) ≤ ∆
.
e( ) + π
2λTs√ab (ae1−λ( ) + be1+λ( )) ∆≤ ≤T
(15)
whe e he pa ame e s sa is y 0
<λ<
1,
a>
0,
b>
0,
α>
0,
Ts
is he p ede ined- ime, and
∆
is a smalle cons an ha sa is ies
∆Ts
. Subsequen ly, a ajec o y acking e o ,
e( )
,
in he sliding mode su ace may go o ze o wi hin a p ede ined- ime
Ts
, which is gi en in
he o m o a heo em, and can be p o ed heo e ically.
Theo em 2.
Fo any p ede ined- ime
Ts>
0, when he sliding mode su ace (15) sa is ies
S(e( )) =
0, i shows ha he e o
e( )
will con e ge o ze o wi hin he p ede ined- ime,
Ts
,
and he con e gence ime is:
s=2Ts
πa c an
sb
aVλ
1(0)
<Ts(16)
whe e a =a21−0.5λ,b=b21+0.5λ,λ=0.5λ.
P oo . No e a iable as:
ω( ) = .
e(0) + π
2λTs√ab(ae1−λ(0) + be1+λ(0))exp(−α )(17)
When
S(e( )) =
0 has
.
e( ) = 


−π
2λTs√ab (ae1−λ( ) + be1+λ( )) + ω( ) ≤ ∆
−π
2λTs√ab (ae1−λ( ) + be1+λ( )) ∆≤ ≤T
,
es ablish he Lyapuno unc ion as V1( ) = 0.5eT( )e( ), and de i e i o ge
.
V1( ) = eT( ).
e( )
=


−π
2λTs√ab (aeT( )e1−λ( ) + beT( )e1+λ( )) + eT( )ω( ) ≤ ∆
−π
2λTs√ab (aeT( )e1−λ( ) + beT( )e1+λ( )) ∆≤ ≤T
=


−π
2λTs√ab (a21−0.5λV11−0.5λ( ) + b21+0.5λV11+0.5λ( )) + eT( )ω( ) ≤ ∆
−π
2λTs√ab (a21−0.5λV11−0.5λ( ) + b21+0.5λV11+0.5λ( )) ∆≤ ≤T
=


−π
2λTs√ab (aV11−λ( ) + bV11+λ( )) + eT( )ω( ) ≤ ∆
−π
2λTs√ab (aV11−λ( ) + bV11+λ( )) ∆≤ ≤T
(18)
whe e
a=a
2
1−0.5λ
,
b=b
2
1+0.5λ
,
λ=
0.5
λ
. Due o he ac ha
∆
sa is ies
∆Ts
, and
ω( )
con ains he exponen pa
exp(−α )
, when
α
akes a la ge alue,
ω( )
can quickly
Ma hema ics 2023,11, 56 8 o 19
end o ze o, so he in luence o
eT( )ω( )
on
.
V1( )
is e y small and (18) can ac ually be
equi alen o: .
V1( ) = −π
2λTs√ab(aV11−λ( ) + bV11+λ( )) (19)
Acco ding o Theo em 1, when he sliding mode su ace (15) ul ils he condi ion
S(e( )) =
0, which implies an e o ,
e( )
, i may con e ge o ze o wi hin he p ede ined-
ime, wi h a con e gence ime o
s=2Ts
πa c an
sb
aVλ
1(0)
<Ts
F om he abo e analysis, i can be seen ha he p ede ined- ime con e gence sliding
mode i e a i e lea ning con ol s a egy p oposed in his pape can be desc ibed as design-
ing an i e a i e lea ning con olle o a sliding mode su ace,
S(ek( ))
, so ha he sliding
mode su ace
S(ek( ))
con e ges o 0 a e i e a i e lea ning. This means ha he ajec o y
acking e o ,
ek( )
, will con e ge o 0 wi hin he p ede ined- ime,
Ts
, achie ing he con ol
pu pose o accu a ely acking he desi ed ajec o y wi hin he p ese in e al [Ts,T].
This con ol s a egy ans o ms he ajec o y acking con ol p oblem wi h an ini ial
ajec o y acking e o alue ha is no ze o in o a sliding mode su ace acking con ol
p oblem wi h he ini ial alue o he sliding su ace a ze o. I also es ablishes a b idge
connec ing he i e a i e lea ning con ol heo y o a bi a y i e a i e ini ial alue and he
same i e a i e ini ial alue. The heo e ical connec ing b idge no only sol es he a bi a y
ini ial alue p oblem o i e a ion bu also simpli ies he heo e ical p oo o he con e gence
o i e a i e lea ning and can ake ad an age o he exis ing heo e ical achie emen s o
i e a i e lea ning con ol.
Fo con enience, in he heo e ical p oo o he con e gence o i e a i e lea ning, we
will ake S(ek( )) as Sk( ).
4. Con e gence Analysis o PD-Type ILC
To ensu e ha he ou pu o he nonlinea sys em accu a ely acks he desi ed a-
jec o y wi hin he p ede ined- ime in e al,
[Ts
,
T]
, he sliding mode su ace
Sk( )
mus
con e ge o ze o. The e o e, le he desi ed ajec o y o he sliding mode su ace
Sk( )
be
Sd( ) = 0, and .
Sd( ) = 0, hen deno e he acking e o δSk( ) = Sk( )−Sd( ) = Sk( ).
The PD- ype closed-loop i e a i e lea ning con olle o sliding mode su aces is
designed as:
uk+1( ) = uk( ) + H1δSk+1( ) + H2δ.
Sk+1( )
=uk( ) + H1Sk+1( ) + H2
.
Sk+1( )(20)
whe e
H1
is p opo ional gain ma ix,
H2
is di e en ial gain ma ix, and
H1
and
H2
a e
posi i e-de ini e ma ixes.
Theo em 3.
In ela ion o he second-o de nonlinea sys em de ined in (1), i he PD- ype i e a i e
lea ning con olle is he con olle which is desc ibed in (19), and he spec al adius sa is ies
ρ([I−BH2]−1)<1 (21)
hen, he sliding mode su ace
Sk( )
will con e ge o ze o unde he condi ion o
k→∞
, ie
lim
k→∞Sk( ) = 0.
Ma hema ics 2023,11, 56 9 o 19
P oo . In oduce a iable as:
( ) = −.
yd( ) + 




π
2λTs√ab (ae1−λ( ) + be1+λ( )) −ω( ) ≤ ∆
π
2λTs√ab (ae1−λ( ) + be1+λ( )) ∆≤ ≤T
.
( ) = −..
yd( ) + 


π
2λTs√ab (a(1−λ)e−λ( ) + b(1+λ)eλ( )).
e( ) + αω( ) ≤ ∆
π
2λTs√ab (a(1−λ)e−λ( ) + b(1+λ)eλ( )).
e( ) ∆≤ ≤T
(22)
Then S( ) = .
y( ) + ( )
.
S( ) = ..
y( ) + .
( )
Acco ding o Fo mula (1), we ha e
Sk( ) = .
x1k+ ( ) = x2k+ k( )
.
Sk( ) = .
x2k+.
k( ) = (xk, ) + Buk+.
k( ),F(Sk, ) + Buk
(23)
whe e
F(Sk
,
) = (xk
,
) + .
k( )
. Because
(xk( )
,
)
is a unc ion o
x1k
,
x2k
, and wi hin he ime
in e al
∈[
0,
T]
, i sa is ies he Lipschi z condi ion ega ding
xk( )
and
Sk( ) = x2k+ k( )
.
The e o e, he unc ion
F(Sk
,
)
also sa is ies he Lipschi z condi ion wi h a iable
Sk
. Tha
is, he e is a cons an M2>0, wi h
||F(Sk, )−F(Sd, )||≤M2||Sk−Sd|| (24)
The unique solu ion o his di e en ial equa ion is ob ained acco ding o
.
Sk=F(Sk, ) + Buk
add Sk(0) = 0.
Sk=Z
0[F(Sk) + Buk]dτ=Z
0F(Sk)dτ+Z
0Bukdτ(25)
Then:
Sk+1( ) = R
0F(Sk+1)dτ+R
0Buk+1dτ=
R
0F(Sk+1,τ)dτ+R
0Bukdτ+R
0B[H1Sk+1(τ) + H2
.
Sk+1(τ)]dτ=
R
0[F(Sk+1)−F(Sk)]dτ+Sk+R
0BH1Sk+1(τ)dτ+BH2Sk+1−R
0Sk+1(τ)dBH2
dτdτ
(26)
Then
I−BH2]Sk+1( ) = Sk( ) + Z
0[F(Sk+1,τ)−F(Sk,τ)]dτ+Z
0BH1Sk+1dτ−Z
0Sk+1
dBH2
dτdτ(27)
is de o med o ob ain
Sk+1=[I−BH2]−1{Sk+Z
0[F(Sk+1)−F(Sk)]dτ+Z
0BH1Sk+1dτ−Z
0Sk+1
dBH2
dτdτ} (28)
No e Ψ( ) = [I−BH2]−1, and is de ined as
Kk+1(Sk+1)( ) = −[I−BH2]−1{Z
0[F(Sk+1)−F(Sk)]dτ+Z
0BH1Sk+1dτ−Z
0Sk+1
dBH2
dτdτ} (29)
Then:
Sk+1( ) + Kk+1(Sk+1)( ) = Ψ( )Sk( )(30)
Ma hema ics 2023,11, 56 16 o 19
Ma hema ics 2022, 10, x FOR PEER REVIEW 17 o 20
Time(s)
Figu e 9. Con ol inpu in he en h i e a ion.
5.2. Case 2: Robus ness (Wi h Dis u bances)
In o de o e i y he obus ness o he con olle o ex e nal dis u bances while keep-
ing he con ol pa ame e s he same as in Case 1 and he ini ial alue o he i e a ion in
each i e a i e lea ning p ocess andomly gene a ed, a nume ical simula ion was pe -
o med a e adding an ex e nal dis u bance
( ) [3sin( ),1(1 e )]
−
=−d
o he dynamic sys-
em o he wo deg ee o eedom manipula o . The ajec o y acking e ec a e 10 i e -
a ions is shown in Figu e 10. The i e a i e con e gence diag am o he mean absolu e e o
wi h espec o he numbe o i e a ions is shown in Figu e 11.
Figu e 10. T ajec o y acking e ec o he en h i e a ion.
Figu e 9. Con ol inpu in he en h i e a ion.
Ma hema ics 2022, 10, x FOR PEER REVIEW 17 o 20
Time(s)
Figu e 9. Con ol inpu in he en h i e a ion.
5.2. Case 2: Robus ness (Wi h Dis u bances)
In o de o e i y he obus ness o he con olle o ex e nal dis u bances while keep-
ing he con ol pa ame e s he same as in Case 1 and he ini ial alue o he i e a ion in
each i e a i e lea ning p ocess andomly gene a ed, a nume ical simula ion was pe -
o med a e adding an ex e nal dis u bance
( ) [3sin( ),1(1 e )]
−Τ
=−d
o he dynamic sys-
em o he wo deg ee o eedom manipula o . The ajec o y acking e ec a e 10 i e -
a ions is shown in Figu e 10. The i e a i e con e gence diag am o he mean absolu e e o
wi h espec o he numbe o i e a ions is shown in Figu e 11.
Figu e 10. T ajec o y acking e ec o he en h i e a ion.
Figu e 10. T ajec o y acking e ec o he en h i e a ion.
I can be seen om Figu es 10 and 11 ha when an ex e nal dis u bance was added
o he dynamic sys em o he manipula o , he end posi ion o he manipula o could
s ill achie e high-p ecision acking o he desi ed ajec o y a e he p ese con e gence
ime. By compa ing Figu e 7wi h Figu e 10, i can be seen clea ly ha a e he ex e nal
dis u bance was added o he sys em, he absolu e a e age e o alue and a ying end
a e he p ese con e gence ime did no change. This indica es ha he addi ion o an
ex e nal dis u bance o he mechanical a m dynamics sys em had no e ec on he i e a i e
con e gence accu acy and con e gence speed. The p oposed con olle was able o achie e
high-p ecision ajec o y acking wi hin he p ede ined ime. I can he e o e be concluded
ha he i e a i e lea ning con ol algo i hm p oposed in his pape is s ongly obus in he
ace o bounded ex e nal dis u bances.

Ma hema ics 2023,11, 56 17 o 19
Ma hema ics 2022, 10, x FOR PEER REVIEW 18 o 20
Figu e 11. I e a i e con e gence end.
I can be seen om Figu e 10 and Figu e 11 ha when an ex e nal dis u bance was
added o he dynamic sys em o he manipula o , he end posi ion o he manipula o
could s ill achie e high-p ecision acking o he desi ed ajec o y a e he p ese con e -
gence ime. By compa ing Figu e 7 wi h Figu e 10, i can be seen clea ly ha a e he
ex e nal dis u bance was added o he sys em, he absolu e a e age e o alue and a y-
ing end a e he p ese con e gence ime did no change. This indica es ha he addi ion
o an ex e nal dis u bance o he mechanical a m dynamics sys em had no e ec on he
i e a i e con e gence accu acy and con e gence speed. The p oposed con olle was able
o achie e high-p ecision ajec o y acking wi hin he p ede ined ime. I can he e o e
be concluded ha he i e a i e lea ning con ol algo i hm p oposed in his pape is
s ongly obus in he ace o bounded ex e nal dis u bances.
6. Conclusions
This pape p esen ed a s udy in o he p oblem o accu a e acking con ol o a sec-
ond-o de nonlinea sys em wi h a bi a y i e a i e ini ial alues wi hin a p ese ime in-
e al. Fi s , a ime- a ying sliding mode su ace wi h p ede ined- ime con e gence and
ze o ini ial alue cha ac e is ics was cons uc ed, and he Lyapuno s abili y c i e ion o
p ede ined- ime con e gence, which is used o e alua e he ajec o y acking e o in he
sliding mode su ace ha can con e ge o he o igin a he p ede ined- ime, was gi en.
Second, i was heo e ically p o en ha he p ede ined con e gence ime was independ-
en o he ini ial alue o i e a ion and con ol pa ame e s. Fu he mo e, in he p ocess o
designing an i e a i e lea ning con olle , he i e a i e con ol p oblem o ajec o y
acking unde an a bi a y ini ial alue o i e a ion was ans o med in o a ime- a ying
sliding mode su ace acking i e a i e con ol p oblem when he ini ial i e a ion alue is
ze o. This es ablishes a b idge o con e ing he heo y o i e a i e lea ning con ol be-
ween he a bi a y ini ial alue and he same ini ial alue. Finally, we p esen ed a design
o a PD- ype closed-loop i e a i e lea ning con olle based on a ime- a ying sliding
mode su ace. I was p o ed heo e ically ha he ajec o y acking e o o a second-
o de nonlinea sys em can con e ge and s abilize wi hin he sliding mode su ace a e
lea ning wi h a ini e numbe o i e a ions. This con i med ha he sys em ou pu was
capable o accu a ely acking he desi ed ajec o y wi hin a p ede ined ime.
This wo k expands he applica ion a ea o i e a i e lea ning con ol heo y in p ac i-
cal enginee ing applica ions.
Figu e 11. I e a i e con e gence end.
6. Conclusions
This pape p esen ed a s udy in o he p oblem o accu a e acking con ol o a second-
o de nonlinea sys em wi h a bi a y i e a i e ini ial alues wi hin a p ese ime in e al.
Fi s , a ime- a ying sliding mode su ace wi h p ede ined- ime con e gence and ze o ini ial
alue cha ac e is ics was cons uc ed, and he Lyapuno s abili y c i e ion o p ede ined- ime
con e gence, which is used o e alua e he ajec o y acking e o in he sliding mode su ace
ha can con e ge o he o igin a he p ede ined- ime, was gi en. Second, i was heo e ically
p o en ha he p ede ined con e gence ime was independen o he ini ial alue o i e a ion
and con ol pa ame e s. Fu he mo e, in he p ocess o designing an i e a i e lea ning con olle ,
he i e a i e con ol p oblem o ajec o y acking unde an a bi a y ini ial alue o i e a ion
was ans o med in o a ime- a ying sliding mode su ace acking i e a i e con ol p oblem
when he ini ial i e a ion alue is ze o. This es ablishes a b idge o con e ing he heo y o
i e a i e lea ning con ol be ween he a bi a y ini ial alue and he same ini ial alue. Finally,
we p esen ed a design o a PD- ype closed-loop i e a i e lea ning con olle based on a ime-
a ying sliding mode su ace. I was p o ed heo e ically ha he ajec o y acking e o o a
second-o de nonlinea sys em can con e ge and s abilize wi hin he sliding mode su ace a e
lea ning wi h a ini e numbe o i e a ions. This con i med ha he sys em ou pu was capable
o accu a ely acking he desi ed ajec o y wi hin a p ede ined ime.
This wo k expands he applica ion a ea o i e a i e lea ning con ol heo y in p ac ical
enginee ing applica ions.
Au ho Con ibu ions:
Concep ualiza ion and heo e ical p oo , C.-W.Y. and S.R.; o mal analysis, S.R.
unding acquisi ion, V.B., L.P. and S.M.; me hodology, H.Z.; p ojec adminis a ion, N.U.; esou ces,
H.Z.; w i ing—o iginal d a , C.-W.Y.; w i ing— e iew and edi ing, N.U., V.B. and S.M. All au ho s
ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This wo k was suppo ed in pa by he Doc o al G an Compe i ion VSB—Technical Uni e si y
o Os a a, unde G an CZ.02.2.69/0.0/0.0/19073/0016945; in pa by he Ope a ional P og amme
Resea ch, De elopmen and Educa ion, unde P ojec DGS/TEAM/2020-015; in pa by he Pa ial
Discha ge De ec ion in Insula ion Sys ems, Na ional Cen e o Ene gy, unde P ojec TN01000007. This
wo k also ecei ed pa ial suppo om Tai Uni e si y Resea che s Suppo ing P ojec numbe (TURSP-
2020/144), Tai Uni e si y, Tai , Saudi A abia.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Ma hema ics 2023,11, 56 18 o 19
Acknowledgmen s:
The au ho s acknowledge he suppo ecei ed om he Doc o al G an Compe i-
ion VSB—Technical Uni e si y o Os a a, unde G an CZ.02.2.69/0.0/0.0/19073/0016945; in pa by
he Ope a ional P og amme Resea ch, De elopmen and Educa ion, unde P ojec DGS/TEAM/2020-
015; in pa by he Pa ial Discha ge De ec ion in Insula ion Sys ems, Na ional Cen e o Ene gy,
unde P ojec TN01000007. This wo k also ecei ed pa ial suppo om Tai Uni e si y Resea che s
Suppo ing P ojec numbe (TURSP-2020/144), Tai Uni e si y, Tai , Saudi A abia.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .