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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.

    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 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

    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

    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

    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

    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

    * 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"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 *

    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.