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2.2.Thequadraticfunctionalforalinearsystemwithonedelay
Takingintoaccount(2.34)and(2.51)oneobtains
δ(9,σ)=BTKT(9−σ).
25
(2.52)
Inthiswayoneobtainedallcoefcientsofthefunctional
(2.22)
.Thiscoefcients
dependonthematrices
A
and
B
ofsystem
(2.12)
.Thetimederivativeofthefunc-
tional(2.22)isnegativedefnite.
2.2.3.Theexamplesoftheparametricoptimization
2.2.3.1.Theparametricoptimizationof
theinertialsystemwithdelayandaP-controller
Thesystemwithzeroequilibriumpoint
Letusconsiderafrstorderinertialsystemwithdelaydescribedbyequation
(see[12])
(
I
I
I
I
4
I
I
I
I
l
x(o)=xo,
x(9)=o,
u(t)=−px(t)
dx(t)
dt
=−
!
Tx(t)+
ko
Tu(t−r)
(2.53)
t≥o
,
x(t)∈R
,
9∈[−r,o)
,
p,ko,T,!,xo∈R
,
r≥o
.Theparameter
ko
isagainof
aplant,
p
isagainofaP-controller,
T
isasystemtimeconstant,
xo
isaninitialstate
ofasystem.Inthecase
!=1
anequation
(2.53)
describesastaticobjectandinthe
case!=oanequation(2.53)describesanastaticobject.
Onecanreshapeanequation(2.53)totheform
(
I
4
I
l
x(o)=xo,
x(9)=o
dx(t)
dt
=−
Tx(t)−
!
kop
Tx(t−r),
(2.54)
fort≥o.
Onesearchesforaparameter
p
whichminimizestheintegralquadraticperfor-
manceindex
I=
/
o
Ż
x2(t)dt.
(2.55)
Thequadraticfunctionalvisdefnedbytheformula
o
o
o
v(x(t+·))=Ix2(t)+
/
x(t);(9)x(t+9)d9+
/
/
x(t+9)δ(9,σ)x(t+σ)dσd9.
−r
−r
9
Accordingtoaterm(2.21)aperformanceindexvalueisgivenbytheformula
I=
/
o
Ż
x2(t)dt=v(x(t+·))|
|t=o.
(2.56)
