APPROXIMATE OPTIMAL CONTROL OF LINEAR TIME-DELAY SYSTEMS VIA HAAR WAVELETS

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1 Journal o Engneerng Sene and ehnology Vol., No. (6) Shool o Engneerng, aylor s Unversty APPROIAE OPIAL CONROL OF LINEAR IE-DELAY SYSES VIA HAAR WAVELES AKBAR H. BORZABADI*, SOLAYAN ASADI Shool o athemats and Computer Sene, Damghan Unversty, Damghan, Iran *Correspondng Author: borzabad@du.a.r Abstrat In ths paper, Haar wavelet benets are appled to the optmal ontrol o lnear tme-delay systems. A dsretzed orm o optmal ontrol problem at olloaton ponts based on some useul propertes o Haar wavelets transorms orgnal problem nto a nonlnear programmng (NLP). he gven numeral examples show the auray o the presented sheme n omparson wth some other methods. Keywords: Haar wavelet, Optmal ontrol problem, Dsretzaton, Lnear tmedelay system, Nonlnear programmng.. Introduton Over the last ew years we have wtnessed an ever nreasng nterest n the study o ontrol proesses governed by derent systems. One o the most mportant o these systems are delay systems. me-delay oten appears n many ontrol systems (suh as arrat, hemal or proess ontrol systems) ether n the state, the ontrol nput, or the measurements. Due to presentng delay and ts mportant onsderaton, n many pratal systems [, ], ontrol o tme-delay systems has been nterested by many engneers and sentst. Sne the analytal methods, espeally n optmal ontrol o tme-delay systems, have less ablty to mplement, the derent numeral methods to overome the problems o exat methods have been devsed. Some o these tehnques nlude, teratve dynam programmng [3], steepest desent based algorthm [4], Chebyshev seres [5], Laguerre polynomals [6], Blo-pulse untons [7], Hybrd o blo-pulse and Legendre polynomals [8], Legendre multwavelets [9], Walsh untons []. Reently, Haar wavelets have been appled extensvely or sgnal proessng n ommunatons and physs researh, and have proved to be a wonderul 486

2 Approxmate Optmal Control o Lnear me-delay Systems va Haar Nomenlatures a Haar oeent H (t) Haar matrx P Operatonal ntegraton matrx D() Delay operatonal matrx (/) /) Null matrx o order ( / ) ( / ) Gree Symbols Integral square error (t) A group o square waves (t-) Delay unton o () t Abbrevatons NLP Nonlnear programmng mathematal tool. Haar wavelets have the smplest orthogonal seres wth ompat support. In haratersts maes Haar wavelets good anddate or applaton to optmal ontrol problems []. he olloaton methods developed to solve optmal ontrol problems generally all nto two ategores, loal olloaton [] and global orthogonal olloaton [3]. In loal olloaton methods, the tme nterval onsdered s dvded nto a seres o subntervals wthn whh the ntegraton rule must be satsed. In reent years, more attenton has been oused on global orthogonal olloaton methods suh as Chebyshev, Legendre and some other. By expandng the state and ontrol varables nto peewse-ontnuous ombnaton o these nterpolatng polynomals and dervatves, then, the obetve unton and system onstrants are all onverted nto algebra equatons wth unnown oeents. In ths paper, we ntrodue an alternatve method to solve the lnear optmal ontrol wth delay systems. We ntrodue the Haar wavelets theory and propertes nludng the Haar wavelets bass and ts ntegral operatonal matrx []. he delay and produt are gven. hen we wll assume that the ontrol varables and dervatve o the state varables n the optmal ontrol problems may be expressed n the orm o Haar wavelets and unnown oeents. By usng the Haar operatonal ntegraton matrx we nd (t). he delay vetor ( t ) and U ( t ) an be alulated by usng the delay operatonal matrx and Haar operatonal ntegraton matrx. hereore, all varables n the tme-delay system are expressed as seres o the Haar amly and operatonal matrx and delay operatonal matrx. Fnally, the tas o ndng the unnown parameters that optmze the desgnate perormane whle satsyng all onstrants s perormed by a nonlnear programmng solver. In ths paper, rst Haar wavelets and ts propertes s ntrodued. hen the approxmaton o a unton by Haar wavelets s dsussed. By ntrodung operatonal ntegraton matrx and delay operatonal matrx, Haar dsretzaton method s establshed. Ater, desrbng the ormulaton o the optmal ontrol problem wth delays, the proposed method s used n the analyss o lnear tmedelay systems. Fnally, by some numeral examples the proeny o the gven approah s examned and ts results ompared wth other methods. Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

3 488 A. H. Borzabad and S. Asad. Haar Wavelets and Its Propertes he orthogonal set o Haar wavelets (t) s a group o square waves wth magntude + or - n some ntervals and zeros elsewhere. ( t) =, <, () t t <, ( t) = t <, () t <, (t) = ( t ) = t <, otherwse. (3) or =, =,,, =,,, nteger m =,( =,,, J ), ndates the level o the wavelet and =,,, s the translaton parameter. axmal level o resoluton s J. he maxmal value o s =. A smple alulaton shows that = l =, ( ) ( ) = t l t l. (4) Consequently, the untons (t) are orthogonal. hs allows us to transorm any unton square ntegrable on the nterval tme [,] nto Haar wavelets seres. 3. Funton Approxmaton by Haar Wavelets We ust ponted out that a square ntegrable unton an be expressed n terms o Haar orthogonal bass on nterval [,]. However, beore the proesson to ths transer, t s neessary to uny the tme nterval. By usng a lnear transormaton, the atual tme t an be expressed as a unton o va t = [( t t ) t ] where t s the ntal tme and t s the nal tme n a square ntegrable unton (t). Any unton () t whh s square ntegrable n the nterval [,] an be expanded n a Haar seres wth an nnte number o terms. ( t ) = a ( t ), =,, <, t [,), (5) = where the Haar oeents a = ( t ) ( ), t dt (6) are determned n suh a way that the ntegral square error Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

4 Approxmate Optmal Control o Lnear me-delay Systems va Haar = ( ( t ) a ( )), t dt = (7) s mnmum. Here s vanshed when tends to nnty. Usually, the seres expanson o (5) ontans an nnte number o terms or smooth () t. I () t s a pee wse onstant or may be approxmated as a peewse onstant, then the summaton (5) wll be termnated ater terms, that s, F ( t ) a ( t ) = A ( t ), (8) = where the oeent vetor A = [ a, a,..., a ] and ( ) = [,,..., ] t. Let us dene the olloaton ponts ts = ( s.5) /, ( s =,... ). Wth these hosen olloaton ponts, the unton s dsretzed nto a seres o nodes wth equvalent dstanes. Let the Haar matrx H be the ombnaton o () t at all the olloaton ponts. hus, H t) = [ h,..., h ] ( ( t ) ( t) ( t ) ( ) ( ) ( ) t t t = [ ( ),..., ( )] = t t. (9) ( t ) ( t) ( t ) For example, H = ( t ) ( t) =. hereore, the unton (t) may be approxmated as ( ts ) = H. () 4. Integraton o Haar Wavelets In the wavelet analyss or a dynam system, all untons need to be transormed nto Haar seres. Sne the derentaton o Haar wavelets always results n mpulse untons whh should be avoded, the ntegraton o Haar wavelets s preerred, whh should be expandable nto Haar seres wth Haar oeent matrx P ' ' t ( t ) dt = P( t ), () where P s the operatonal ntegraton matrx whh satses the ollowng reursve ormula, P / H / P =, P = [ ], () H / whh s gven n [4] and ( /) ( /) s a null matrx o order ( / ) ( / ). 5. Delay Operatonal atrx o Haar Wavelets Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

5 49 A. H. Borzabad and S. Asad he delay unton ( t ) s the sht o () t dened n (3) along the tme axs by. he delay operatonal matrx D () s gven by ( t ) D( ) ( t ), t >, t <, (3) [,] s the delay parameter. Frst, we nd the matrx D( ) = [ d ] or. he our bass untons are gven by,,, 3. By (8) and (3) we have ( t ) ( t) ( t ) ( t) = [ d ( )],, =,,3,4, 3( t ) 3( t) ( t ) ( t) 4 4 where ( t ) =, t <, and t <, ( t ) = t <, t <, 4 ( t ) = t <, 4 and 3 t <, 4 3( t ) = 3 t <. 4 o nd the entres d (),, =,,3,4, we use the nner produt. For example =., we have d =< ( t ), ( t ) >= ( t ) ( t ) dt =.9, d3 =< 3( t ), ( t ) >= 3( t ) ( t ) dt =. I we alulate all d () as d and d 3 the 4 4 operatonal matrx D () s obtaned. In partular, we have D4 4(.) = n In a smlar manner, we use the vetor unton () t wth dmenson, n n then delay matrx D () wth n an be obtaned as ollows, = ( ) ( ) = = l =, d l t l t dt l. Note that or any dmenson = then matrx s dagonal. 6. Problem Statement Consder a lnear system wth delays n both the state and ontrol desrbed by Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

6 Approxmate Optmal Control o Lnear me-delay Systems va Haar ( t ) = A( t ) ( t ) B ( t ) ( t ) E ( t ) U ( t ) S ( t ) U ( t ), (4) wth ntal data () =, (5) ( t ) = ( t ), t [,], (6) U ( t ) = ( t ), t [,], (7) where s an m -vetor o state; U s an q -vetor o nput; At (), Bt (), Et () and St () are ontnuous matrx untons o the tme o approprate dmensons, s a onstant speed vetor. and are delays n state and ontrol, respetvely, and the ntal unton () t and () t are ontnuous n ther respetve ntervals. he problem s to mnmze t J = H ( t, ) L( t,, U ) dt, (8) where H s a salar unton o the nal tme t and nal state varables and L( t,, U ) s a salar unton o the tme, state and ontrol U. 7. Haar Dsretzaton and me-delay Systems Analyss We dsretze the untons () t by dvdng the nterval [,], to parts o equal length t =/ and ntrodue the olloaton ponts = (.5) /, =,...,, where s the number o nodes used n the dsretzaton and also s the maxmum wavelet ndex number. We approxmate state varables x () and ontrol varables u () by Haar wavelets wth olloaton ponts,.e., x( ) x ( ), (9) u( ) u ( ), () where = [,..., ] and = [,..., ]. Usng the operatonal ntegraton x x x matrx P dened n () u u u ' ' ' ' x x () x ( ) = x ( ) d x = ( ) d x = P ( ) x. As stated n (), the expanson o the matrx () at the olloaton ponts wll the yeld the Haar matrx H = [ h,..., h ] t ellows that x ( ) = h, u( ) = h, x ( ) = C ph x, =,...,. () x u x Now we ous on the analyss o tme-delay systems. Frst hoose N as ollowng manner, N =.5, (3) =, and let N N. Let ( ) = [ x ( ), x ( ),..., x m ( )], (4) U ( ) = [ u( ), u( ),..., u q ( )], (5) ˆ ( ) = I m ( ), (6) ( ) = I q ( ), (7) Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

7 49 A. H. Borzabad and S. Asad where I m and I q dmensonal dentty matres and denotes roneer produt [5]. By (9)-() eah o x ( t ), =,,..., m and u ( t ), =,,..., q an be wrtten as x ( ) = P ( ) x (), (8) x u ( ) = ( ), (9) u where = [,,..., ] and = [,,..., ]. Usng (3)-(9) x x x x u u u u ˆ ( ) = C ( ), ( ) = ˆ ˆ C P ( ) (), (3) and also U( ) = C U ( ), (3) where C = [,,..., ], C = [,,..., ] x and Pˆ = I m P dsusson n Se.3 and a smlar notaton we an wrte ( ) = ˆ ( ), (3) C ( ) = C ( ). (33) Usng (3), (3) and (3), t an be onluded that ( ) <, ( ) = (34) C ˆ ˆ ( ˆ PD ) ( ) () < t, ( ) <, U ( ) = (35) C U D ( ) ( ) < t, where Dˆ = I m D and D = I q D. Smlarly eah entres o At (), Bt (), Et () and St () may be expanded by (8) and thus A( t ) = A ˆ ( t ), (36) B ( t ) = B ˆ ( t ), (37) E ( t ) = E ( t ), (38) S ( t ) = S ( t ), (39) where a a a b b b m m a a a m b b b m A =, B = a n a n a nm b n b n b nm nm e e e s s s q q e e e q s s s q E =, S =. e n e n e nq s n s n s nq nq nq When the Haar olloaton method s appled n the optmal ontrol problem wth tme-delay system, the varables an be set as the unnown oeents vetor o nm Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

8 Approxmate Optmal Control o Lnear me-delay Systems va Haar the dervatve o the state varables and ontrol varables together wth ntal and nal tmes, that s [ C, C,..., C, C,..., C, t, t ]. U U Consder the ost untonal (8) by = (, ( )) ( ) (, ˆˆ J H t t t L C P ( ), C ( )) dt. U Sne the Haar wavelets are expeted to be onstant steps at eah tme nterval, the above equaton an be smpled as ˆˆ U = J = H ( t, ( )) ( t t ) L(, C P ( ), C ( )), By substtutng, U, and n (4) and usng (3)-(39), or =,..., N, we have ˆ ˆ C ( ) = ( ˆ ˆ t t )(( A ( ))( C P( ) ( )) ()) ˆ ( ) ( ) ( ( ))( B ( )) ( ) ( )), E CU S and also or = N,..., N, ˆ ˆ C ( ) = ( ˆ ˆ t t )(( A ( ))( C P( ) ( )) ()) ˆ ˆ ˆ ˆ B ( )( C PD ( ) ( ) ()) ( E ( ))( CU ( )) S ( ) ( )). Also or = N,...,, ˆ ˆ C ( ) = ( ˆ ˆ t t )(( A ( ))( C P( ) ( )) ()) ˆ ( )( ˆ ˆ ( ) ˆ B C PD ( ) ()) ( E ( ))( C ( )) U S ( )( CU D( ) ( ))). Note that n (9) we ponted that or =,...,, ( ) = h. Sne the rst and last olloaton ponts are not set as the ntal and nal tme, the ntal and nal state varables are alulated aordng to = () /, = ( ) /. In ths way, the optmal ontrol o tme-delay systems transormed nto NLP or LP problem. 8. Numeral Results In ths seton, he results o applyng the method n three numeral examples are presented. Example 8. Consder the ollowng optmal ontrol problem o lnear tmedelay system x ( t ) = 4 tx ( t ) x ( t / ) u( t ), x ( t ) =, / t, wth assoated quadrat ost untonal to be mnmzed J = (4 ( ) 4 ( )). x t u t dt Usng the Haar wavelets olloaton method wth =6 olloaton pont and by (7) and (3)-(39) we have =, x x u h = 4 ( Ph x ()) h, =,,..., N, h = 4 ( Ph x ()) ( PD ( ) h x ()) h, = N,...,, x x x u Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

9 494 A. H. Borzabad and S. Asad J Ph x h = (4( x ()) 4( u ) ), = where x and u are the unnown varables o NLP and by (3) N =8. he obtaned mnmum value o the ost untonal s J = whh s muh better than J = 5.73, reported n [9]. Agan the results has been examned usng 3 olloaton ponts. In Fgs. and, one an observe the dagram o approxmate optmal ontrol and state untons, respetvely. Fg.. he approxmate optmal ontrol nput n Example 8.. Fg.. he approxmate optmal traetory n Example 8.. Example 8. Consder the problem o mnmzng J = () () ( u ( t )) dt, (4) subet to the system o the delayed derental equatons ( t) = ( ) ( ) ( ),, t t u t t (4) Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

10 Approxmate Optmal Control o Lnear me-delay Systems va Haar ( t) = x ( t) x( t), () =, ( t) =, t. (4) As prevous example, the mnmzaton o J subet to (4) and (4) has been obtaned usng the proposed method. Usng 6 olloaton ponts or Haar wavelets dsretzaton method, the optmal value s obtaned J =.6986, whh s better than J = and J = 3.399, reported n [9] and [6], respetvely. he ontrol varable ut () and the state varables x () t, x () t or two derent number o olloaton ponts, = 6 and = 3, depted n Fgs. 3 and 4, respetvely. Fg. 3. he approxmate optmal ontrol nput n Example 8.. Fg. 4. he approxmate optmal traetory n Example 8.. Example 8.3 In ths example, the delay s onsdered n ontrol and state varables. he problem s mnmzaton o the untonal Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

11 496 A. H. Borzabad and S. Asad J = ( ( ) ( )), x t u t dt subet to delayed derental equaton x ( t ) = x ( t ) x ( t ) u( t ) u ( t ), 3 3 x ( t ) =, t [,], 3 u ( t ) =, t [,]. 3 Usng 6 olloaton ponts n Haar wavelets dsretzaton method the optmal value s obtaned J =.4. hs value ompares well wth those gven n []. he near optmal ontrol and state varables whh are obtaned by the Haar wavelet dsretzaton method are shown n Fgs. 5 and 6 or =6 and = 3, respetvely. Fg. 5. he approxmate optmal ontrol nput n Example 8.3. Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

12 Approxmate Optmal Control o Lnear me-delay Systems va Haar Fg. 6. he approxmate optmal traetory n Example Conlusons In ths paper usng the propertes o Haar wavelets, a olloaton based method s presented or the resoluton o optmal ontrol governed by lnear tme delay systems. he gven manner s based on onvertng the orgnal problem to a nonlnear programmng problem. One nterestng advantage o the proposed method s ts smplty. he numeral results show that nreasng the number o ponts, t s possble to mprove the obetve unton as well as the onvergene o approxmate soluton o the problem may lead to the exat optmal soluton. Also the derved results ndate the that the proposed approah leads to nd the traetory and ontrol untons that the orrespondng obetve unton s better than some other methods. Reerenes. Jamshd,.; and Wang, C.. (984). A omputatonal algorthm or largesale nonlnear tme-delay systems. IEEE ransatons on Systems an and Cybernets, 4(), -9.. Rabah, R.; and Slyar, G. (7). On exat ontrollablty o lnear tme delay systems o natural type, Applatons o me Delay Systems, Leture Notes n Control and Inormaton Senes, 35, Dadebo, S.; and Luus, R. (99). Optmal ontrol o tme-delay systems by dynam programmng. Journal o Optmzaton heory and Applatons, 3(), Furutaa, K.; Yamataa,.; and Sato, Y. (988). Computaton o optmal ontrol or lnear systems wth delay. Internatonal Journal o Control, 48(), Journal o Engneerng Sene and ehnology Otober 6, Vol. ()

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