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2.2.Thequadraticfunctionalforalinearsystemwithonedelay
27
K(9)=−
kop
TA
sinhA9
−
T(Asinh
Ar
2+
kop
!+kop
T
cosh
Ar
2)(cosh
Ar
2
+
!+kop
TA
sinh
Ar
2)coshA9,
(2.65)
δ(9,σ)=
k2
T2A
op
2
sinhA(9−σ)
+
T2(Asinh
Ar
2+
k2
op
!+kop
2
T
cosh
Ar
2)(cosh
Ar
2
+
!+kop
TA
sinh
Ar
2)coshA(9−σ).
(2.66)
Nowaperformanceindexvalueiscalculated
I=
2(Asinh
Ar
2+
x2
o
!+kop
T
cosh
Ar
2)[cosh
Ar
2
+
!+kop
TA
sinh
Ar
2].
(2.67)
Figure2.1showsthevalueoftheindex
I(p)
for
xo=1
,
ko=1
,
!=1
,and
T=5
and
r=2.Youcanseethatthereexistsacriticalvalueofthegainpcrit.Thesystem(2.54)
isstableforgainslessthancriticaloneandunstableforgainsgreaterthancritical.
Figure2.1.ValueoftheindexI(p)forpgreaterthanpcrit
Figure2.2showsthevalueoftheindex
I(p)
for
p
lessthancriticalgain.Youcan
seethatthefunctionI(p)isconvexandhasaminimum.
