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s» m 6mm #4;me MI! W MJR‘ak rﬁmkm __ @ pigH ﬂ "hii .1: Lad # "22E L 1‘ (”at
% " [mat'21 2 imrt‘mle dry), \ 44m 1..., 51.. MW ouwaﬁ aimixh. Fax 11““
A v» w " 1m! 1 m; AHA EWLm+ fie) (MB # Example 13.4%. We desire to change the ﬂux density in a uuciear
.1 ; If fl!" reactor from some value at. to a vaiue of such that the energy required to drive i
the control rodsis minimizes. We therefore wish to minimize the cost function ' which approximates this desire
1
J a sf: {p3 + a: m age} 52?. We wilt assume that the reactor can be adequately represented by the point
kinetics equations with one group of delayed: neutrons ﬁepgﬁgt—ta mime/ta The foitowing parameter values are assumed: ,3 =2: 0.0068, A w 10"3 sec,
.3. an {31.1 sec“, no u 5 am e... we 6%.... a: w 5 K i0"'5, to m 0, if m 1 sec.
The TPBVP is obtained as v. «m n3 ﬁn
an‘anFlcwrlAmri: emuﬁumﬁe 1" n ~
:1#ME{”“HI} 4'11”?” i “' §]* ref: rsmmﬂrlmra}
with the boun'iary conditions reﬁt...) m 5.r em.) w 6411., w 3.20 K.W., and.
F39!) 1“ reﬁt} “ '3
Titc quasilineariaed equations take theforutjr w A]! + u, where the initial
and ﬁnaicouditions areyiﬂi) m 5,y2{{]) w 320.y'3iiiw ysifi m: 0, and where “if Emmett!) .. we) a
“if“ ““" m ' ’1 [W] 0
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ii?) i3 Erwttlei’ir) “13 .
9‘ ‘1‘ [“i‘f" ] 0 “s" i“ We: ”“2 “a“
i} 0 “.1 +3. I 4...... are} e [2Fi{r)tn”(t)1=i.0.wu 4 WHEN“)? ‘7‘] We start the quasitineariaation solution by assuming thatfag} m e... and age} w Jig: : SHE m eggs for the eeroth iteration. Figure 10 4 1 indicates the quasi“
linearised ﬂue {tensity solution at each stage in the iteration. CGM?UTATIDHAL Merrroos m Orrrstosr Srsrssts Conrnor. CH. to 50 fro” ..,
tst trlot by ooosilineoriaotionwx * '
f
40 I I I j v
2nd triol lay ooosilrneorrzotronw
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it / .
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time. seconds Fig. met Power level versus time. Sotit'rtott or ssaottttc oprtstat. oott'rnot. Paoatssrs a? oussrmtaaamm Iri a nonlinear control system, it is not generally possible to express
ptimfal control law as a product of the state vector and a timevarying
In fact, a soiation of the nonlinear, partial differential eqaations rela
he optimal control to the optimum trajectory, Mr) to 5&6), is normally
sasitil’e. Furthermore, the optimal control is highly dependent, in a
teargmanner, on the initial state vector 3:0,). This means that, for most
rearicontrol systems, only opendoop control laws are available even
h(,_____.osedioop control laws are more desirable. The desirability of
l—loop control lasts has led to the development ofspeciﬁc optima! control. is speciﬁc optimal control (3.0.0.) problem is defined in the following
sr. We are given a plant with a state equation of the form “MEI—mm—rl—iPIu'—I ...
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 Winter '09
 Bamieh

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