A Simple Approach to Robust Optimal Pole Assignment of Decentralized Stochastic Singularly-Perturbed Computer Controlled Systems

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1 A Sple Appoach to Robst Optal Pole Assgnent of Decentaled Stochastc Snglaly-Petbed Copte Contolled Systes Ka-chao Yao Depatent of Indstal Edcaton and echnology Natonal Chang-ha Unvesty of Edcaton No. Sh-Da Road, Changha Cty, awan Abstact. hs pape develops a sple algoth fo havng obst optal copte contol n decentaled stochastc snglaly-petbed systes by poles assgnent. hs type of nose-dstbed syste can be often seen n copte contolled lage-scale systes sch as electc powe systes, concaton netwoks, and aeospace systes. De to that ths copte contolled syste possesses the fast esponse chaactestcs of the sbsystes, the syste analyss can be splfed by snglaly petbaton ethodology and the aggegaton atx s also appled to obtan faste calclaton. Fnally, the aggegaton atx s fond ot that wll be an potant nteeday to easly acheve the obst sb-optal poles assgnent. In the end, thee steps ae poposed to coplete the obst sb-optal pole assgnent. Key wods: obst, copte, pole, decentaled, stochastc, snglalypetbed, aggegaton atx. Intodcton Pole placeent of lage-scale systes has been a dffclt task de to the hgh denson of the systes. How to splfy the pocess of placng optal poles s the goal of ths eseach. In ths pape, the syste s concened wth decentaled stochastc snglaly-petbed copte contolled systes. Sch systes ae twote scale systes. Pactcally, copte contolled systes ae ths type of systes. hee ae soe sla eseaches elated to ths feld. G. Enea, J. Dplax, and M. Fancesch [] se a ecsve ethod to acheve optal contol wth aggegatve pole assgnent n the dscete MIMO systes. A. R. Aa and M.E. Sawan [] popose a desgn ethod fo optal contol wth egenvale placeent n a specfed egon; n 997, they pesent the wok abot the elaton between pole-placeent and lnea qadatc eglato fo dscete te systes [3]. In [4], Yao stded copte contol of decentaled snglaly-petbed systes, bt the nose dstbng factos, fast algoths and obstness ae not concened. Aong all syste pefoance eqeents, obst stablty s a paaont condton fo desgns of syste contol. Especally n [5]-[7], neos appoaches

2 have been poposed and these systes concened ae snglaly-petbed systes. Yahl Naks [8] developed a elaton fo dect calclaton of the cost fncton fo an optally contolle lnea syste wth qadatc ctea, dstbed by a coloed nose of any gven spectal densty dstbton. Jango Wang; Gangy Cao; Jn Zho [9] stdy how the optaton ethods can be sed to deal wth plant ncetanty. A weghed senstvty eo fncton s pesented fo an optal obst contolle desgn n a class of stochastc odel eos. As obseved by the pevos wok that has been done fo the stablty, enhancng pefoance, and cost naton of decentaled stochastc systes, none had focs on the obst optal pole assgnent of decentaled stochastc snglaly-petbed copte contolled systes. In ths pape, the optal poles fond ae based on a edced-ode syste odel. he optal pole egon of the close-loop syste can be ealed by adjstng the state weghtng atx and the npt weghtng atx. Afte collectng and savng all the nfoaton of the elatonshp aong the weghtng atces and the aggegaton atx, the optal feedback gan of the syste can be ndestood. Syste Pescpton he atheatcal odel of the n-ode decentaled stochastc snglaly- petbed syste s shown as: o x A x A A A A A x A B G w A x A B G w A x C C y v y v : y C v x A x A A x A B G w y C v : A B G w whee =~. he syste s a lnea te-nvaant decentaled stochastc snglaly-petbed copte contolled syste whch has n-ode and S F ndependent npts o sb-systes. x R and R ae the slow and the fast n state vaables espectvely; each sb-syste has ts own ode. R and n y R ae the npt vecto of the -th sbsyste and the otpt vecto of the -th sbsyste espectvely. A, A, A, A, C, and G ae constant atces wth (a) (b) (a) (b) (c)

3 appopate densons wth =~. npts and otpts. 3 Man Reslts S w R ; w and v ae dstbng noses of Fndng a easy algoth of obst sb-optal contol s the goal of ths stdy. A syste pefoances based on syste ncetantes s necessaly nvestgated and tested. Uncetantes of systes ae cased by the nevtable eos n syste odelng de to nexact and ncoplete data, splfyng appoxatons, neglected hgh feqency dynacs, and npedcted dstbances fo the envonent. In ths eseach, obst contol s defned that f the desed pefoance stll exsts afte sng the edced-ode contolles n the fll-ode systes. In ths type of patcla syste the ajo ncetanty wold be the fast state vaables of the sbsystes. Becase the oveall syste s a decentaled copte contolled syste, the esponses of copte-based sbsystes ae a lot faste than the an plant. he esponses of the fast state vaables wll de ot petty fast n the vey ntal te peod. heefoe, the oveall stcte s potentally a snglaly petbed syste. When the syste eaches Qas-steady state, the paaete can be assed as eo. De to ths phenoenon, the fast state vaables can be gnoed and the ode of the syste can be edced. hs also ses the dea that the state odel of the syste can be appoxated. In Eq. (b), the sb-staton staton vaables,,, 3 have eached qas-steady state. Hence, the syste ode s edced to the ode of the an staton whch s eqal to the denson of the slow state vaable x. Eq. (b) can be shown as: A B A x G w ) hen, ( A B A A x A G w A B A x G w ) A whee =~; ( A B A A x A G w ( B A ~ A A : x G w ) A B B A A x A A G w A x A A ae nonsngla atces. Next, ecall the state eqaton of the slow state vaable n (a). We can obtan new epesentatons fo the eqaton of slow state vaables by sng Eq. (3b): G w (3a) (3b) x [ A [ A A A A A G w ] ] x A A B A A B... A A B + : (4)

4 Now, defne G [ A A Gw ] and Let G Hw whee H s a non- w sqae atx and w w. hen, R H [ A A Gw ] w [ A A Gw ] : w R [( w w) w ] whee w [( w w) w ] s psedo ght nvese of w. An n-ode lt-npt decentaled stochastc snglaly-petbed copte contolled syste s edced nto an S=(n-F)-ode lt-npt te-nvaant syste. he state odel can be evsed as whee A B A A x A x B G A x B Hw y C v C x D v (5a) (5b) x ; C C A A ; D C A B ; A [ A A A A ] ; B A A B A A B A A B ;... w ; G [ A A G w ] ; w w ;.... heefoe, when the syste : : w data s pocessed by copte, the above eqatons wll be tansfoed nto dscetete odel as [] x (( k ) h) x ( kh) ( kh) (kh) G (6a) whee (h ) h ( ) db. h denotes the saplng ate. A h e ; y ( kh) C ( kh) C x ( kh) D ( kh) v ( kh) (6b) Now we defne anothe non-sqae atx, whch x x f (7) whee the fll-ode state vecto x = x f ; S F x R S F wth x R and R. f By the state tansfoaton, the non-sqae atx that s called the fast aggegaton atx hee that s sed as an nteeday to have tansfoaton between the fll ode odel () and the edced ode odel (5). hs non-sqae atx wll help to shoten the devaton pocess. A A A A / A / A A A / A / R = ( ) (8)

5 B (9) B R C C C ( ) () whee R denotes the psedo ght nvese and atx has all the eleents eqal to eo wth appopate se. s of Eq. (8) to Eq. () ndcate the contollng sbsyste. By applyng Eq. (), Eq. (8) can be evsed as heoetcally, L C A A A / A L C C () A / s a sngla atx and ths atx wll be ccal te to fnd the fast aggegaton atx,. Next, based on the edced-ode state odel (6), Eq. (6a) can be shown as x ( k ) x (... G ( kh) () whee s the fst coln of ; s the second coln of and so on. ( ~ ae the npts of the sbsyste one to the sbsyste. In the closeloop contol systes as Fg., we know d( K x G (3) d K x G and so on. whee d ( k ) and d ( k ) ae addtonal npts; G and G ae dstbng sgnals. d + Sbsyste () one Z y v + Man Plan A/D K A/D x X () () Othe Sbsystes Z ~ : Z : Fg.. he decentaled stochastc snglaly-petbed syste. All the sbsystes ae copte pocessng nts and assed to be eo-ode.

6 Now, f the an staton s contolled fo the sbsyste one, we can asse the d ~ d and G and G ae all dstbng nose to the contolle one. hey only affect the apltde of syste esponses. he pole locatons ae nchanged. heefoe; we evse (6) as x k ) [ K ] x ( ) JN( (kh) ( N k G (4a) whee N ( K 3K 3... K ) and K ~ K ae exstng feedback d ( gans. J... ; N ( d( : d ( y C x D ( (4b) Eq. (4) s a closed-loop state odel. Accodng to stochastc contol theoy [] and sngla petbaton ethodology [], the LQ pefoance ndex of each sbsyste n the fll ode syste: J N ( w k v Q w( R ) whee Q s the weghtng atx wth p. s. d. fo each sb-syste and Q Q. R s the weghtng atx wth p. d. fo each sb-syste. Q x w ; x s a slow state vecto and s a fast state vecto. : It can be pesented as Eq. (6) that has the edced ode state vecto, x. J N ( x k Q x R ) Wth ths constan (6), f we have contol fo the sbsyste one, we can have the optal contol whee optal k K (5) (6) K x( ) (7) ( B PB R B P (8) ) and, the P s the solton of the Rccat eqaton P N { P PB [ B PB R ] B P} N Q N (9) optal whee P s a constant atx. he not only nes the enegy se bt also stables the syste. hs stablng feedback gan stables the slow state vaables

7 of the syste. hee wll be no contol to the fast state vaables; theefoe, stablty of the fast state vaables s eqed. Ftheoe, n steady state, the optal contol cost fo the sbsyste one can also be obtaned as he obst sb-optal poles ae whee Pd J d optal x N () Px () () P eg[ B K ] () s the desed sb-optal pole locatons. he optal feedback contol and the optal costs of the sbsyste two to the sbsyste can be fond by the sae pocede sed n the sbsyste one. Fo a sccessfl state feedback desgn, stablablty s a necessay condton, and contollablty s a sffcent condton. In the fogong pocess, we se the edced ode state odel (6) and exstng feedback gans K ~ K to copte the sb-optal feedback gan of the sbsyste one: K ( B PB R ) B P, f the contol s pefoed fo the N sbsyste one. Now, we wold lke to fnd the sb-optal feedback gan, K, fo the ognal fll ode syste by sng the aggegaton atx, then the npt of the sbsyste one. K x () whee x denotes edced-ode state. Accodng to the state tansfoaton technqe, Eq. () can be shown as whee x f fnd the elatonshp Also, K x f (3) K x f (4) denotes fll-ode state. By copang wth Eq. (3) and Eq. (4), we can K K (5a) K K (5b) whee K s the obst sb-optal feedback gan pleentng n the ognal fll ode syste. K s the obst sb-optal feedback gan obtaned fo the edced ode syste. s the fast aggegaton atx. Fo the sbsyste two to the sbsyste can follow the sae ethod as the sbsyste one to fnd the optal poles by the aggegaton atx. 4 Illstatons he whole syste s a ffth-ode syste wth thee fst ode sbsystes and thee npts. he state odel s shown as

8 x.5 x x..3. x y y v y G (6a) 3.6 (6b) whee x and x ae slow state vectos that ae second-ode.,, 3 ae all fast state vectos and fst-ode ndvdally. w and v ae dstbng noses of npts and otpts. heefoe, when the syste eseaches qas-steady state, and the syste can be edced to a second ode syste sch as.545 x x Next, we dgte ths edced-ode odel to dscete-te doan wth the saplng peod.. (7).985 x( k ) x (.5.89 ( ) 3 k (8) In ths exaple, the oveall state vecto w s concened wth x w. Now, f we want to have the optal contol n the sbsyste one, by assng the exstng K and K 3 5 5, we can ewte the odel as x( k ) x( ( (9) If the pefoance ndex of the slow state vecto fo the sbsyste one s N J ( x Q x( R ) (3) k whee Q ; R. he optal contol of the sbsyste one: optal x ( (3)

9 wth P he pole locatons of ths optal contol ae and.84; theefoe, the syste s stabled by the contolle, too. If the ntal condton x ( ), the optal cost can be calclated as optal J x () Px () 5 (3) Fo the obst contol test, ths optal edced-ode contol wll be placed back to the ognal fll-ode odel. If the desed pefoance stll exst, we have a obst contol syste. If., we can have the sae dscete-te odel as: w( k ) w , n the fll he pole locatons of the syste ae.9853 and.86. We can see the locatons ae vey close to the desed pole locatons; theefoe, we have a obst contol syste. x Also, by assng y, we can have the syste 3 esponses based on the sbsyste one wth h=. and =. as follows: optal Now, we se the optal feedback gan, K ode syste wth K and K ( ( ) 3 k Open-Loop Zeo-Inpt Response.5 Apltde e(. sec.) Fg.. he open-loop eo-npt esponse of the fll-ode syste wth the slow state poles at.995 and.938.

10 Closed-Loop Zeo-Inpt Response wth Optal Redced-ode Contolle 8 6 Apltde e(. sec.) Fg. 3. he closed-loop eo-npt esponse wth the optal edcedode contolle shftng poles to.98 and.84. he obstness bond of the obst contol syste can be fond by changng the vale of. able shows how the poles shft when the vale of changes. able. he obst contol test. Poles 5.e , , , , , , , , ,.7673 In ths case, f we asse the syste pefoance allows.3 shft at each pole locaton, when <.5, we can have a obst contol syste. he sb-optal, edced-ode contol that pefos nsde ths bond s call obst, decentaled, sb-optal edced-ode contol. In ths case, the appoxated optal poles,.98 and.84, ae sed to copae wth the shftng poles cased by syste ncetantes. he obst sb-optal contol, the sb-optal costs, and the obst contol tests of the sbsyste two and the sbsyste thee can jst follow the sae pocede sed n the sbsyste one. Afte the edced-ode feedbacks ae affed to be obst, fo the fll ode feedback gans can be fond by Eq. (8) to Eq. (). In Eq. (9)

11 .8.8 In Eq. (), = t.3 t 6 t t 6 C t t 7 C t t3 t 4 t5 t 7 t8 t9 t C L ( ) heefoe; t t /.3 /.5.4 /.4 /.35 /.3 / t t C he aggegaton atx s solved as and, the fll-ode feedback K K = whee K fond n Eq. (3). he fll ode feedback gans of the est of the sbsystes can follow the sae pocede as above. 5 Conclsons he fll ode sb-optal feedback of the decentaled copte contol of stochastc snglaly-petbed syste can be fond easly thogh the aggegaton atx and cople steps; oeove, the fond obst sb-optal edced ode feedback gan can also acheve the desed pefoance wth deceasng cost. By sng the edced-ode state odel obtaned fo pefong the snglaly ethodology, the obst edced ode feedback gan can be calclated based on the slow LQ pefo ndex. Next, the fll-ode feedback gan can be fond by ltplyng the fast aggegaton atx as Eq. (5b). he effect by applyng the fll ode feedback and edced ode feedback wll have sla pefoance. hese two types of feedback gans povde the deand of syste to adjst the contol stats and (33) (34)

12 pefoance. he copleton of ths algoth helps s to analyss the decentaled copte contol of stochastc snglaly-petbed syste and fast to fnd the sboptal feedback gan fo the fll-ode contol and edced-ode contol. hee steps of fndng the obst sb-optal poles of the syste ae pesented as below:. Fnd the fast aggegaton atx fo Eq. (8)-().. Fnd the edced-ode sb-optal obst feedback gan of the syste fo Eq. (7). 3. Fnd the fll-ode sb-optal feedback gan of the ognal syste fo Eq. (5b). Refeences. G. Enea, J. Dplax and M.Fancesch, Dscete Optal Contol wthaggegaton Pole Placeent, IEE Poceedng-D, Vol. 4, No. 5, pp.39--pp.3 (993).. A. Aa and M. E. Sawan, Desgnof Optal Contol Systes wth Egenvale Placeent n a Specfed Regon, Optal Contol Applcatons & Methods, Vol. 6, pp.49--pp.54 (993). 3. A. Aa and M. E. Sawan, Relaton between Pole-placeent and Lnea Qadatc Reglato fo Dscete-te Systes, J. Fankln, Vol.334B, No. 4, pp.55--pp53, (997). 4. Ka-chao Yao, Copte Contol of Decentaled Snglaly-petbed Syste, Fst Intenatonal NAISO Conges on Atonoos Intelgent Systes (ICAIS ), Astala, CD #8--KY-47 (). 5. Son, J.-W.; L, J.-, Robst stablty of Nonlnea Snglaly Petbed Syste wth Uncetantes, IEE Poceedngs - Contol heoy and Applcatons, Vole 53, Isse, pp. 4-- (6). 6. Inha Hyn; Sawan, M.E.; Dong G Lee; Dongwook K, Robst stablty fo decentaled snglaly petbed nfed syste, Poceedngs of Aecan Contol Confeence, pp (6). 7.Fdan, E., Robst Sapled-data H Contol of Lnea Snglaly Petbed Systes, IEEE ansactons on Atoatc Contol, Vole 5, Isse 3, pp (6). 8. Yahl Naks, Cost Fncton Calclaton fo Statonay, Lnea-Qadatc Syste wth Coloed Nose, IEEE ansacton on Atoatc Contol, Vol.37, No., pp (99). 9. Jango Wang; Gangy Cao; Jn Zho, Optal Robst Contol fo a Class of Stochastc Model Eos, he Sxth Wold Congess on Intellgent Contol and Atoaton, WCICA 6, Vole, Page(s): (6).. Kal J. Asto and Bjon Wttenak, Copte-contolled Systes: heoy and Desgn, Pentce-Hall Inc, NJ (997).. R. Stengel, Stochastc Optal Contol, John Wley & Sons, NY (986).. DS Nad, Sngla Petbaton Methodology n Contol Systes, Pete Peegns Ltd, London (988).