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7/21/2019 PPT Jurnal ROP
1/22

7/21/2019 PPT Jurnal ROP
2/22
Presented By
Nurul Azizah
Haqitotul Aulia
Falta U. Rosyidah
Alifa Harwitasari

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The solution of nonlinear optimal control problems is a
challengingproblem.
The recommended practice is to use different methods,
then to implement the best solution found.
To support this objective, the performance of a semi
exhaustive search methodthat uses a different
approach
The main objective is to demonstrate the performanceof the above strategy.
About This
Presentation

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A number of methods have been proposed for the solution of
nonlinear optimal control problems
Since one cannot be sure if a global optimum has been reached for
such a problem, one needs to crosscheck the results using
different optimization methods as recommended by Bojkovand uus.
Although a global optimum still cannot be guaranteed, one can
implement the best solution found by using the different methods
The semi!e"haustive search provided a smooth convergence to the
optimal solution and re#uired a significantly reduced computational
time as compared $ith the %&P algorithm. This re#uires the availability of different strategies and demonstration of
heir performance on the type of problems being solved. A semi!
e"haustive search strategy $as proposed for the solution of time!
optimal control problems
Why Use The Semiexhaustive Search

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The system is assumed to be described by the follo$ing setcof nonlinear
differential e#uations'
$here " is an (n ) *+ state vector and u is an (m ) *+ control vector bounded by
The initial state "(+ is kno$n. There may be ine#uality constraints on the
states of the type,
The final time performance inde" is given by
!x"#$, t f % & "x"t f $$, "'$
$here the final time (t f + is specified.
(ptimal )ontrol *roblem

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The proposed solution is an iterative procedure.
a specific search methodology is used to locate theoptimal control trajectory in the starting grid.
At each iteration, a narro$er grid $ith a reduced span is
formed around the optimal trajectory is sought $ithin the
narro$er grid. This approach of finding a solution to the optimiation
problem by narro$ing the grid around the optimal
control trajectory found at the previous iteration is the
same as that used by upta. /o$ever, the mainstrategy used in locating the optimal trajectory is
different.
+ethodology

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There only one control variable (m 0*+
T$o state variables (n 0 1+,the optimal control trajectory is to be found over ten time stages (P 0 *+.
To explain the algorithm
2e place only t$o values of uon either side of the initial control trajectory.There are five values (* current value 3 1 values above 3 1 values belo$+ of
u (4u 0 5+ to be considered over any stage.
et t f 0 *, and the performance inde" to be minimied be "1(t f +, that is, ("(t
f ++ 0 "1(t f + 0 "1(*+.
asic Search +ethod

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Starting from each of the five "(*+ vectors obtained over stage *, one
$ould integrate 6#. (*+ five times over stage 1 using the five values ofu.
%t can be seen that the total number of such integrations for the ten stages
is given by P* i0* 5i 0 *1,17,8
The number nine results from the follo$ing e"pression'
+aximum number of x vectors or points "p$ & "x $n & / & 0.
/o$ever, over stage , the starting number of " vectors $ill be
5(89*+ 0 15, $hich is more than nine.
)onducting 1n 2xhaustive Search

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At the first iteration, the point that had the minimum value of "1
$ithin a group $as selected. At subse#uent iterations, the
integrations $ere first carried out from each of the points in a group
over the rest of the stages using the optimal trajectory found at the
previous iteration. The point that resulted in the minimum value of
"1(t f + $ithin the group $as then selected for carrying out the
integrations using the five values of u
32SU4T

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The starting control trajectory is taken at the midpoint of :ma"and :min unless a better guess is available. The starting span is
usually (uma" 9 umin+;1 unless the search is started from a
suboptimal control trajectory $here a reduced span may be used.
The span at subse#uent iteration is reduced by multiplying the
previous span by a parameter gamma. The constraints on control variables are satisfied by keeping
the control grids $ithin the allo$able bounds.
2hen there are more than one control variable (m < *+,
integrations are done such that only one of the control variable is
varied at a time. After a certain number of iterations have been made, the search
can be restarted by rela"ing the span of the control grid
(ther 5eatures of The Search +ethod

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Pengujian dengan membandingkannya dengan algoritma %&P di
variabel berikut, yang mempengaruhi konvergensi terus sama di
kedua metode.
=umlah nilai terdistribusi merata yang masing!masing elemen dari
vektor u diperbolehkan untuk mengambil (4u+. &alam mulai
jaringan kontrol , nilai!nilai dari u dibagikan seragam antara :>A?dan umin sekitar titik tengah.
(b+ =umlah nilai bah$a setiap elemen dari vektor " diperbolehkan
untuk mengambil pada setiap tahap (4"+. @leh karena itu, jumlah
maksimum vektor negara atau poin (4p+, dari yang integrasidilakukan pada setiap tahap adalah sama di kedua metode.
(c+ aktor reduksi span, , dimana rentang (range+ dari jaringan
control berkurang pada setiap iterasi.
*engu6ian

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This e"ample considers a problem studied by Ceo DEF and uus DGF
$here the system is described by the follo$ing differential e#uations'
d"*;dt 0 "1 (5+
d"1;dt 0 9"8u 3 *Ht 9 E (H+
d"8;dt 0 u (7+"(+ 0 D 9* 9p5 FT and t f 0 *.
Example Problem 1

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The control is bounded by
7' u 8#, "9$
a piece$ise constant control trajectory over * time stages is to be found that
minimies the follo$ing functional'
& : 8# !x/ 8 ; x// ; #.###here x'"#$ & #. dx'?dt & x/
8 ; x// ; #.###

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The convergence
of the methods to
the optimum and
the
computational
times for *
iterations aresho$n
3esult

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or this problem, the different values of did not have much effect on the convergence.
difference e#uations represent the concentrations of the three key components atstage k.
2here x"#$ & ! 8 # # %T.
The control u*(k+ is related to the temperature T at stage k through the relation u8"k$ &
8#'e7###?T "k$ ...(*I+ and u1(k+ is the residence time in stage k. The constraints are as
follo$s'
T "k$ 0', "8

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This e"ample considers the follo$ing nondifferential system that has been studied
by Thomopoulos and Papadakis $ho sho$ed the difficulty of convergence of
several optimiation techni#ues. This problem $as also considered by uus .
dx8?dt & x/ "8A$
dx/?dt & 7x8 7 x/ ; u ; d "89$
dx?dt & ise constant control profile over 8#
time stages such that x"t f $ is minimized "t f & / s$.
6"ample problem 8

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Jonvergence to the minimum for
6"ample problem 8 (4u 0 5, 4" 0
8, 4p 0 17+.
The proposed solution
converges smoothly to
the optimum. 5or & #.0,
both methods re#uired
more than 8# iterations
to converge. Thecomputational time for
the proposed solution for
the * iterations $as
about 8?8 to 8? as
compared $ith that of the
%&P algorithm.

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Example Problem
This e"ample considers a JSTK problem that has multiple solutions
and has been studied by uus. The system is described by thefollo$ing differential e#uations.
The problem is to find a control trajectory over * time stages that
minimies follo$ing functional'

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The convergence of the
methods to the global
optimum is sho$n belo$,
$hich is not affected by thepresence of a local
optimum. Again, for & #.0,
both methods re#uired
more than 8# iterations to
converge. Thecomputational time for the
proposed solution for the
* iterations $as about 8?/
as compared $ith that of
the %&P algorithm.

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Example Problem This e"ample considers the tubular reactor problem that has been
studied by uus D8F. The mass balance is described by the follo$ingdifferential e#uations.
The final time (t f + 0 * s and the constraint on the temperature isH18.*H T E18.*H. The problem is to find a piece$ise constanttemperature profile over *5 time stages such that "8(t f + is ma"imied.

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The convergence of the
methods to the optimum is
sho$n %n this case, the %&P
algorithm took a better jumpto$ards the optimum at the
first iteration. The rest of the
convergence trajectories $ere
essentially the same.
The value of did not have
much effect on theconvergence.
The computational time for the
proposed solution for the *
iterations $as about *;1 as
compared $ith that of the %&P
algorithm.

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Conclusions
The semi!e"haustive search used in this paper provides
an alternate method for solving the nonlinear optimal
control problems. As noted in 6"ample 1, the %&P
algorithm did not converge to the optimum $hen startedfrom the midpoint of the given control bounds. The semi!
e"haustive search, on the other hand, provided a smooth
convergence to the optimum $ithin one pass on the five
e"ample problems tested. >oreover, it re#uired
significantly reduced computational times.