# ppt jurnal rop

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• 7/21/2019 PPT Jurnal ROP

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• 7/21/2019 PPT Jurnal ROP

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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.

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 cross-check 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 Semi-exhaustive 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 optimi-ation

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 minimi-ed 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

*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

minimi-es 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 optimi-ation 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

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

minimi-es 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

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"imi-ed.

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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 *

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.