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Chapter2.Quadraticfunctionalsforlinearretardedtypetimedelaysystem
4.5
3.5
2.5
1.5
5
4
3
2
0
0.5
1
k0=1,q=1,T=5,r=2
1.5
2
2.5
3
3.5
4
J(p)
Figure2.2.ValueoftheindexI(p)forplessthanpcrit
Wesearchforanoptimalgainwhichminimizestheindex
(2.67)
.Optimization
results,obtainedbymeansoftheMatlabfunctionfminsearch,aregiveninTable2.1.
Theseresultsareobtainedforxo=1,ko=1,!=1,andT=5.
Table2.1.Optimizationresults
Delayr
Optimalgain
Indexvalue
Criticalgain
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
3.4329
2.2020
1.5871
1.2188
0.9738
0.7993
0.6689
0.5681
0.4878
1.1014
1.4331
1.6778
1.8610
1.9997
2.1060
2.1880
2.2518
2.3019
8.50
5.89
4.58
3.80
3.28
2.91
2.64
2.43
2.26
Figure2.3showsthevalueof
x(t)
for
xo=1
,
ko=1
,
!=1
,
T=5
,
r=1
andfor
optimalvalueofP-controllergainpopt=3.4329.
