SOLVING AN OPTIMAL CONTROL PROBLEM WITH MATLAB

Transcription

1 SOLVING AN OPIMAL CONROL PROBLEM WIH MALAB RGeeharamani, SSviha Assisan Proessor, Researh Sholar KG College O Ars and Siene Absra: In his paper, we presens a Ponryagin Priniple or Bolza problem he proedre or solving a Bolza problem by sing MALAB is disssed o solving an opimal onrol problem wih ree inal ime he problem is o ind an opimal onrol and opimal sae he opimal onrol and saes are ploed in igre Keywords: Opimal Conrol, Bolza Problem, Ponryagin Priniple Inrodion he heory o opimal onrol has been developed or over ory years Wih he advanes o omper ehniqe, opimal onrol is now widely sed in mli-diiplinary appliaions sh as biologial sysems, ommniaions neworks and soioehonomi sysems e As a resl, more and more people will benei grealy by learning o solve he opimal problems nmerially An opimal onrol is a se o dierenial eqaion desribing he pahs o he onrol variables ha minimize he os nion Realizing sh growing needs, books on opimal onrol p more weigh on nmerial mehods In rerospe, [] was he irs and he lassi book or sdying he heory a well as many ineresing ases ime opimal, el-opimal an d linear gadrai reglaor problems Neessary ondiions or varios sysems were derived and eplii solions were given when possible Laer, [] proved o be a onise ye eellen book wih more engineering eamples One o he disingish eares o his book is ha i inroded several ieraive algorihms or solving problems nmerially More reenly, [] ses MALAB o solve problems whih is easier and more preise However, he nmerial mehods overed in hese books are insiien or he wide range o problems emerging rom varios ields Espeially, or hose problems wih ree inal ime and nonlinear dynamis Free inal ime problem[5] were reaed as an eqivalen variaion wih one more sae or ime However ree inal ime problem general orm and we solve he problem o ind an opimal onrol and opimal sae Basi Deiniions Opimal Conrol Consider he linear ime varying sysem and he os nional, A B J F [ Q R ] d Where A, B, Q, R are oninos on [, ],he erminal ime is speiied, F is a onsan n n symmeri posiive semideinie mari, Q is an n n symmeri posiive semideinie mari, R is an m m symmeri posiive deinie mari and is no onsrained We shall show ha he opimal onrol is a linear nion o sae,his is o he orm Where, G is an m n mari valed nion G, [, ] IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 56

2 Bolza Problem Consider he opimal onrol sysem where he perormane inde is o general orm Conaining a inal erminal os nion in addiion o he inegral os nionsh a opimal onrol problem is alled he Bolza problem Given he sysem as he perormane inde as,, J S, V,, d And he bondary ondiions as and and Where, and are n-dimensional sae and onrol veors respeively are ree, Proedre For Solving he Ponryagin Priniple For Bolza Problem Sep: Form he Ponryagin nion Sep: Minimize wr,,, V,, ',, and obain h,, Sep: Using he resls o Sep in Sep,ind he opimal, h,,,,,, Sep: Solve he se o n dierenial eqaions and Wih iniial ondiions and he inal ondiions S S IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 57

3 Sep5: Sbsie he solions o, rom sep ino he epression or he opimal onrol o sep Opimal Conrol Problem wih Free Final ime Given a seond order sysem as, and he perormane inde as, J d Find he opimal onrol and opimal sae, given he bondary ondiions as, Solion: [,] ; [,] Assme ha he onrol and sae are nonsrained We ollow he sep-by-sep proedre given a algorihm Firs by omparing he presen sysem and he perormane inde wih he general ormlaion o he Bolza problem Sep : Sep : V,, v [, ],, Where =, Form Hamilonian nion as on ind rom, =,,,, V ', IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 58

4 IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 59 Sep : Using he resls o Sep in Sep,ind he opimal as,,, 5 Sep : Obain he sae and osae eqaions rom, Solving he previos eqaions,we have he opimal sae osae as, Inegrae he above eqaions we ge, 6 7 8

5 9 Sep 5: Obain he opimal onrol rom Where,,, and are onsans evalaed sing he given bondary ondiions hereore he vales are, 8 9 Solving above eqaions we ge, 8 Finally, we have he opimal saes, osaes and onrol as, IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 6

6 6 he sysem wih opimal onrol is shown in he igre Conlsion he solion or he se o dierenial eqaions wih he bondary ondiions are solved by MALAB, he proess o solve an opimal onrol problem has been ompleed I is easy o see ha he solions or,, and = are obained by sing MALAB he opimal onrol and sae are ploed Reerenes [] Ahans M and Falb P,Opimal onrol: An inrodion o he heory and is appliaions, Dover Pbliaions 7 [] Kirik DE, An inrodion o opimal onrol heory, Dover PbliaionsIn, [] Naid DS Opimal Conrol sysemscrc Press LLC, [] Shampine LF, Kierzenka J and Reihel MW, Solving Bondary vale problems or Ordinary Dierenial Eqaions, [5] Balahandran K and Daer JP,Elemens o Conrol aheory, [6] Maski J and Srass A, Inrodion o Opimal Conrol heory, Springer-Verlag, Berlin, 98 IJSDR878 Inernaional Jornal o Sienii Developmen and Researh IJSDR wwwijsdrorg 6