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2.2.Thequadraticfunctionalforalinearsystemwithonedelay
21
Thevalueoftheperformanceindex
(2.20)
isgivenbytheterm
(2.11)
,whichfor
system(2.18)takesaform
I=
to
/
Ż
xT(t)x(t)dt=v(I).
(2.21)
Tocalculatethevalueoftheperformanceindex
(2.21)
,whichisequaltothevalueof
theLyapunovfunctionalattheinitialstateofsystem
(2.18)
,oneneedsamathematical
formulaofthatfunctional.
2.2.2.Determinationofthequadraticfunctional
Onthespace
PC([−r,o],Rn)
wedefneaquadraticfunctional
v
positivedefnite,
diferentiable,givenbytheformula[98]
o
o
o
v(xt)=x
t(o)Ixt(o)+
T
/
xT
t(o);(9)xt(9)d9+
/
/
xT
t(9)δ(9,σ)xt(σ)dσd9
(2.22)
−r
−r
9
for
t≥to
,where
I∈Rn×n
,
;∈C1([−r,o],Rn×n)
,
δ∈C1(Ω,Rn×n)
for
Ω={(9,σ):
9∈[−r,o],σ∈[9,o]}
,
C1
isaspaceofallcontinuousfunctionswithcontinuous
derivative.
Accordingtotheformula
(2.3)xt(9)=x(t+9)
for
9∈[−r,o]
soweobtained
xt(o)=x(t).
Inthisparagraphwillbegivenaprocedureofdeterminationofthefunctional
(2.22)
coefcients.
Incalculationofthetimederivativeofthefunctional
(2.22)
willbeusedthe
followingLemma.
Thereholdstherelationship
Lemma2.11
∂xt(9)
∂t
=
∂xt(9)
∂9
.
(2.23)
Proof.
xt(9)=x(t+9)
fort≥toand9∈[−r,o]
∂xt(9)
∂t
=
∂x(t+9)
∂t
=
∂x(ξ)
∂ξ
∂ξ
∂t
=
∂x(ξ)
∂ξ
,
∂xt(9)
∂9
=
∂x(t+9)
∂9
=
∂x(ξ)
∂ξ
∂ξ
∂9
=
∂x(ξ)
∂ξ
