Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 The Effect of Dstace betwee Ope-Loop Poles ad Closed-Loop Poles o the Numercal Accuracy of Pole Assgmet E. Murat Bozkurt *, M. Tura Söylemez **, Member IEEE Istabul Techcal Uversty/Electrcal Egeerg Departmet, Istabul, Turkey * bozkurte@tu.edu.tr ** soylemez@elk.tu.edu.tr Abstract- The early woks o the umercal performace of the pole assgmet algorthms show that there are several parameters that affect the error made calculatos of the resultg compesators whe fxed precso arthmetc s used. It has bee already show that creasg the order of the system ad the rato of real poles to all poles of the opeloop (ad closed-loop) system (RPR) cause a expoetal crease the umercal errors made pole assgmet. I ths paper, the effect of maxmum dstace betwee opeloop system poles ad target closed-loop system poles (MD) o the performace of pole assgmet algorthms s vestgated for the frst tme. Keywords: pole assgmet, umercal accuracy, state feedback L I. INTRODUCTION ear cotrol system theory s well advaced ad several methods exst order to acheve desred desg objectves. Pole assgmet s oe of the most commoly used desg techques cotrol theory ad a umber of costructve methods ths area are descrbed the lterature []. It has bee log kow that uder certa codtos t s possble to assg the closed-loop system poles arbtrarly by state-feedback provded that the system s cotrollable []. From ths pot of vew, the process of determg closed-loop system poles s therefore coveetly called pole assgmet or pole placemet. The pole assgmet problem usg statefeedback ca be formally stated as follows: Gve a system x = Ax + Bu () where A ad B, the problem of fdg a feedback matrx K such that the matrx Ac A BK has a desred set of egevalues, s called pole assgmet problem. The egevalues of matrx A are called ope-loop poles ad those of the closed-loop matrx A BK are called closed-loop poles. The poles of closed-loop system foud uder the state feedback rule u = r Kx are determed as requred. The poles of the resultg closed-loop system, are gve by the roots of the characterstc polyomal si A + BK = 0 () or, other words, the egevalues of the closed-loop system A matrx Ac A BK. It s possble to show that egevalues of A BK ca be assged to arbtrary, real or complex cojugate, locatos f, ad oly f, (A,B) s cotrollable []. The process of determg the requred state feedback cotroller K usually volves umercal errors whe the calculatos are carred out a umercal programmg evromet where a fxed precso arthmetc s used [3]. Actually, a exact pole assgmet s possble whe symbolc algebra s used. However, symbolc algebra s ot always avalable ad also t ca be tme cosumg carryg out calculatos exactly whe t s avalable. Therefore error proe umercal calculatos are sometmes uavodable. Because of these umercal calculato errors, the resultg state feedback matrx of the pole assgmet algorthm may be calculated defectvely. Obvously, the use of a defectvely calculated state feedback matrx may result sgfcat perturbatos from the desred locatos of closed-loop poles o the resultg closed-loop system especally for hgh order systems. It s also a well-kow fact that the accuracy of pole assgmet plays a crucal role o both the robust stablty ad the performace of the resultg system. From ths pot of vew, the determato of the parameters, whch ca be effectve o the quattes of these perturbatos, s ecessary ad helpful for robustess ad performace of the resultg closed-loop system. To ths ed, the effect of the algorthm used ad the degree of the system o the performace of pole assgmet has bee vestgated [3], ad that of the rato of real poles to all poles of the ope-loop (ad closed-loop) system (RPR) has bee vestgated [4]. I ths study, aother mportat parameter, whch s amed maxmum dstace (MD) betwee ope-loop system poles ad target closed-loop system poles, effectve o the performace of the pole assgmet algorthms, has bee vestgated. It s well kow that there s o algebrac relato for the roots of polyomals of order greater tha four terms of ther coeffcets. Therefore, aalyzg the umercal errors o the pole assgmet algorthms by meas of a algebrac method s dffcult. For ths reaso, we shall follow a Mote Carlo approach order to aalyze the mportat parameters o the accuracy of the resultg compesator of the pole assgmet algorthms. O the other had, the effect of both RPR of the ope loop system poles (ORPR) ad also that of the closedloop system poles (CRPR) o the performace of the pole assgmet algorthms must be solated whe the effect of MD o the performace of pole assgmet algorthms s vestgated. Hece, the followg sectos, frst of
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 all, we shall expla brefly the oto of RPR secto II. We shall the troduce some specal methods ad tools order to perform the requred aalyss about the effect of MD o the performace of the pole assgmet algorthms secto III ad IV respectvely. The method used s explaed secto V. Ths s followed by some cosderatos about the results of aalyss secto VI. I the fal secto, coclusos ad possble future research areas are gve. II. THE NOTIONS OF REAL POLE RATIO (RPR) ad EXPECTED REAL POLE RATIO (ERPR) Each possble parameter affectg the performace of the pole assgmet algorthms must be solated from others whe oe of them s to be aalyzed. Therefore, the fudametal cocepts of both RPR ad ERPR ad some smple methods order to geerate a radom system wth prescrbed RPR wll be explaed brefly before provdg the method of aalyss. Gve a set of complex umbers (closed uder complex cojugacy) Γ= { γ γ,( =,,.. ) } the Real Pole Rato (RPR) s a real umber that takes values betwee 0 ad ad s defed as the rato of the umber of real elemets to the total umber of elemets ( ) the set. That s, The umber of real elemets of Γ RPR( Γ) (3) Usg ths defto, t s possble to talk about the RPR of the ope loop system poles (ORPR) ad also that of the closed-loop system poles (CRPR). I ay case, RPR takes dscrete values for a gve order of the system ( ). If = 5, for example, RPR ca be 5, 3 or (ote 5 that complex roots of a polyomal wth real coeffcets appear cojugate pars). Smlarly, for = 6, RPR takes values from the set {0,,,}. Although t could be 3 3 meagful to exame the effect of RPR to the performace of pole assgmet at such dscrete values for a gve system order, t s dffcult to make comparsos of the results of such a aalyss for systems wth dfferet orders. Therefore the cocept of expected value of RPR (ERPR) s used the followg aalyss. ERPR s defed as the expected value of RPR for a radomly geerated set of complex umbers. That s, Expected value of umber of real elemets of Γ ERPR( Γ) (4) Lke RPR, t s possble to talk about ERPR of a ope-loop system (EORPR), ad that of a closed-loop system (ECRPR) for radomly geerated ope-loop systems ad target sets of closed-loop system poles. It should be oted that f ope-loop systems are geerated from state matrces whose elemets are depedet radom varables wth stadard dstrbutos, EORPR becomes a fucto of ad decreases asymptotcally as [5]. Therefore geeratg state-matrces drectly from radom elemets s ot eough to exame the effect of EORPR. Istead, a lear trasformato from block dagoal to a prescrbed radom state-matrx s used for geeratg state-matrces the followg. I ths approach, frst a set of complex cojugate umbers Γ o = { λ, λ,, λ} wth a gve ERPR s radomly geerated as explaed [4]. The, a ope-loop system ( AB, ) s geerated such that A= T Λ T (5) where the matrces B ad T are geerated radomly, ad Λ s a block dagoal matrx as Re Im [ λ] Im[ λ] [ λ ] Re[ λ ] Λ= Re[ λ3 ] where λ ad λ deote complex cojugate pars. III. POLE COLORING as a ERROR MEASURE for the ACCURACY of POLE ASSIGNMENT It s a well kow fact that the lack of accuracy o the resultg compesator of the pole assgmet algorthms usually causes perturbatos from the desred locatos of the closed-loop poles. Let us suppose that the error measure s the maxmum dstace betwee the desred locatos of closed-loop poles ad the correspodg perturbed closed-loop poles. However, fdg the error poses a ew problem sce t s ot always easy to determe whch of the desred poles correspod to whch of actual closed-loop poles. Thus, we eed to make a bjectve (oe to oe) assgmet betwee the desred locatos of poles ad the perturbed poles. Ths problem s referred as pole colorg [6]. The smple case of the desred locatos of poles ad correspodg perturbed poles for a thrd-order closedloop system s gve as Fgure..a. For the system show Fg..a there are sx possble permutatos. Three of them are show Fgure..b, Fgure..c, ad Fgure..d respectvely. Let us defe the maxmum dstace betwee pared desred ad perturbed poles for p th permutato as cost for p ad show t as J p ad the error for umercal pole assgmet problem wth J e. Here, t would be reasoable to choose the permutato whch has the smallest cost. Thus, fg..d s the proper choce from all possble permutatos. Hece, the error for pole assgmet ca be wrtte as J e m ( J ) p p! (6) = (7) ( ( d )) = m max (8) p! where d shows the dstace betwee the th par of assged poles gve by permutato p. Solvg ths problem as defed by equato (4) could be very slow,
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 sce there are! permutato to test. Fortuately there are good algorthms to address ths problem, whch fact s called the lear bottleeck assgmet problem (BAP) computer scece termology [7]. a) ρ s complex cojugate of ρ f λ s complex cojugate par of λ b) ρ λ m ( ρ λ, β ) <, otherwse (see Fgure ) jw ρ λ β ρ λ σ Ope-Loop System Poles Target Closed-Loop Poles Fg. Desred closed-loop poles ad perturbed closed-loop poles of a thrd order system s show (a) ad possble permutatos of assgmet problem of ths system are show (b), (c) ad (d) respectvely. Perturbed Poles Desred Poles IV. DISTANCE BETWEEN OPEN-LOOP POLES ad DESIRED CLOSED-LOOP POLES (MD) I le wth the error defto made above the maxmum dstace betwee ope-loop system poles ad target closed-loop system poles s called as maxmum dstace, ad shortly deoted as MD. It s possble to geerate a target closed-loop characterstc polyomal where the maxmum dstace betwee ts roots ad ope-loop pole locatos s a gve value. Aother way of thkg, f the pole colorg procedure s appled betwee ope loop poles of the gve system ad the ewly geerated set of complex umbers, the result s requred to be a desred value of MD. Let us cosder that a radom test system wth a gve ERPR s produced radomly by usg the approaches provded the prevous sectos. The, order to produce a set of complex umbers, closed uder cojugacy wth a gve MD, a smple procedure wth the followg basc steps ca the be used. Let us suppose that the desred value of MD s β. The,. Select a pole ( λ ) radomly from the pole set of the ope-loop system, for whch ERPR s kow.. If λ s real, the select a ew pot o the real axs ρ such that ρ λ = β. If λ s complex cojugate par of λ, the select a ew pot ρ o the crcle cetered λ wth radus β ad ts cojugate as ρ. 3. For all of remag ope-loop system poles λ (,, ) =, determe a target value ρ such that Fg.: The smple demostrato of the procedure whch ca be used order to geerate a target closed-loop characterstc polyomal wth a certa MD. V. THE METHOD I ths study, a Mote Carlo approach s pursued order to exame the effect of MD o the performace of pole assgmet algorthms. Here, a large umber of (usually 0000) ope-loop systems wth a certa OERPR ad the correspodg target closed-loop system poles whch has a certa MD form the ope-loop system poles are produced radomly usg the approaches provded the prevous sectos. (Note that CERPR wll be automatcally same as OERPR whe the approach gve the prevous secto s used.) The, for each par of ope-loop system ( AB, ) ad target set of closed-loop system poles γ = { γ, γ,, γ } a pole assgmet algorthm s used to calculate the requred state feedback matrx ( K ). Geerally, the spectral approach s used as the pole assgmet method, sce t s kow to have better umercal propertes comparso to some of the other well kow state feedback pole assgmet methods for sgle put systems [3]. Due to spectral approach [,8] the desred state-feedback matrx s gve by K = ζ v (9) = T T where v are recprocal egevectors (rows of the verse egevector matrx), ad ζ are scalar weghtg factors calculated as ζ q ( λ γ ) j j= (0) q ρ ( λ λ ) j j= j 3
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 where ρ are scalars gve by ρ v B () After fdg the state-feedback matrx, K, the closedloop system A matrx A c = A BK s calculated. The egevalues of ths matrx gves the acheved set of closed-loop system poles, whch usually are dfferet tha desred set of poles. Usg the approach provded secto III, the dstace betwee the acheved ad the desred set s the calculated as the error of pole assgmet. Ths calculato s carred out for each par of ope-loop system ( AB, ), ad target set of closed-loop system poles ad the average (ad also stadard devato) of the error s foud for gve value of MD. VI. THE RESULTS OF THE ANALYSIS I ths secto, we exame the effect of MD o the performace of the pole assgmet. Therefore, order to exame the effect of MD o the umercal accuracy of resultg compesator of the pole assgmet, the expected value of both ORPR (OERPR) ad CRPR (CERPR) are set to a fxed value (0.5), ad the value of MD s chaged betwee 0 - ad 0 7. For each value of MD, 0000 test systems ad a target set of closed-loop system poles for each system are geerated. The usg the error aalyss method gve the prevous secto, the average values of the errors ad the stadard devatos of the errors are foud. The results are depcted Fgure 3. Note that the vertcal axs Fgure 3 represets the logarthmc error (maxmum dstace) betwee the acheved ad the desred sets of closed-loop system poles ad horzotal axs represets the logarthm of the value of MD. It s possble to roughly state that t shows the accuracy of the acheved closed-loop system poles wth respect to desred oes. For example, whe MD s 0 4 the maxmum dstace betwee desred ad acheved set of closed-loop systems s expected to be aroud 0-7. It ca be observed from the fgure that the error made by pole assgmet creases expoetally as MD chages betwee 0 ad 0 6. I addto, the effect of MD o the accuracy of the resultg compesator of pole assgmet algorthm ca be aalyzed for dfferet values of OERPR (ad CERPR). The results for system orders 0 ad 5 are depcted Fgure 4 ad Fgure 5, respectvely. The relato betwee ERPR ad MD ca be examed more deeply whe we regard the Fgure 4. T 0-5 -0 N=0-5 - 0 3 4 5 Fg.3: The effect of (MD) o the accuracy of the pole assgmet algorthm. The average value of error s show by the thck le ad stadard devato bads are show by doted les. The MD creased logarthmcally, ad the graphcs are obtaed whe the order of the system s 0 (N=0) ad ERPR s 0.5. (0000 test systems are used for each value of MD). - -4-0 - OERPR=CERPR= OERPR=CERPR=0.75 OERPR=CERPR=0.5 OERPR=CERPR=0.5 OERPR=CERPR=0-4 - 0 3 4 5 Fg.4: The average values of the pole assgmet errors for dfferet values of ERPR ad MD. The vertcal axs represets the logarthmc error ad horzotal axs represets the logarthm of the value of MD. The order of the system s 0. (0000 test systems are used for each value of MD, ERPR). The error made pole assgmet creases pecewse learly as MD creases as t ca be observed from Fgure 4. The rate of the error curves, however, chages as ERPR chages betwee 0 ad. Ths result dcates that the umber of real valued poles ca be cosdered as a olear multpler for the effect of MD o the errors made pole assgmet. A smlar result s preseted Fgure 5, where the system order s take as 5. The dfferece betwee Fgure 4 ad Fgure 5 s that the error curves are shfted upward Fgure 5, as expected. 4
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 - -4-0 - OERPR=CERPR= OERPR=CERPR=0.75 OERPR=CERPR=0.5 OERPR=CERPR=0.5 OERPR=CERPR=0 values whe RPR 0.5. - -4-0 5 MD > 0 for systems wth order 0 ad Spectral Mappg Ackerma -4-0 3 4 5 Fg.5: The average values of umercal errors of the pole assgmet algorthm for dfferet values of OERPR, CERPR ad MD. The order of the system s 5. (0000 test systems are used for each value of OERPR ad CERPR). Error curves for a fxed ERPR (0.5) are depcted o Fgure 6 for dfferet values of system orders (=0, 5 ad 0). For low value of MD, error curves are shfted upwards as the order of the system creases. A terestg fact observed ths fgure, however, s that the value of Log( MD) s the domatg parameter determato of the errors for hgh values of MD ( MD > 0 ). Ths s qute surprsg as oe would expect the order of the system to be the ma parameter determato of the errors made pole assgmet. -0 - N=0 N=5 N=0-4 - 0 3 4 5 Fg.6: The average values of perturbatos from desred locatos for each value of MD are show for 0 th order, 5 th order ad 0 th order systems. (OERPR ad CRPR s 0.5). Obvously, the algorthm used pole assgmet s a mportat factor that affects the errors o the acheved closed-loop system poles. It s a well kow fact that the spectral method has better umercal propertes comparso to may of the other methods for statefeedback pole assgmet. Nevertheless, examato of Fgure 6 reveals that ths statemet s true oly for smaller 4 values of MD ( MD < 0 ). For large values of MD, the performace of the spectral approach becomes worse comparso to other methods. I partcular, the error made Ackerma s method s observed to have smaller - -4-0 3 4 5 Fg.7: The average values of pole assgmet errors wth respect to MD for dfferet pole assgmet methods. The order of the system s 0 ad both OERPR ad CERPR s 0.5. (0000 test systems are used for each value of MD). VII. CONCLUSION I the prevous studes, several parameters that affect the error made state-feedback pole assgmet have bee vestgated. These parameters clude the algorthm to be used, the order of the system, ad the rato of the umber of real poles to the order of ope ad closed-loop systems. I ths paper, the affect of the maxmum dstace betwee ope-loop system poles ad target closed-loop poles (MD) o the performace of the pole assgmet s show for the frst tme. For ths purpose, a smple procedure s provded to geerate a set of complex umbers wth a gve MD. I partcular, t s show that the umercal errors resultg from pole assgmet algorthms crease pecewse learly wth respect to MD. The results suggest that the poles of the closed-loop system should be chose close to the poles of ope-loop system whe possble to prevet umercal errors. A terestg result s that the performace of spectral algorthm becomes worse tha that of Ackema s algorthm for hgh values of MD. It should be oted that the umercal accuracy of pole assgmet algorthms are also depedet o other parameters such as the codto umber of the cotrollablty matrx of a gve system ad the umercal accuracy of umercal calculatos carred out. Therefore the results obtaed here (umercal values of errors foud through expermets) may chage. However, we observe that the treds foud ths study are cosstet ad does ot chage f all the remag parameters are fxed. Future research should clude the effects of the other possble parameters such as the codto umber of the cotrollablty matrx of a gve system to the performace of pole assgmet algorthms. Also algebrac formulatos to relate these parameters to the error should be sought. 5
Proceedgs of the 5th Medterraea Coferece o Cotrol & Automato, July 7-9, 007, Athes - Greece T9-00 REFERENCES [] M.T Söylemez, 999. Pole Assgmet for ucerta systems, UMIST Cotrol Systems Cetre Research Studes. ISBN 0 86380 46 X. [] W. M. Woham, 967. O pole assgmet mult-put cotrollable lear systems IEEE Tras. Automat. Cotr., vol., pp. 660 665. [3] M.T Söylemez, Muro, N., 998 Pole assgmet ad symbolc algebra: a ew way of thkg, UKACC Iteratoal Coferece o CONTROL 98, -4 September 998, Coferece Publcato No.455, IEE. [4] E.M Bozkurt, M.T Söylemez, The Effect of Real Pole Rato o the Performace of Pole Assgmet Algorthms ECC 07, Kos, Greece. [5] Edelma, A., Kostla E., Shup M. 993. How May Egevalues of a Radom Matrx are Real? AMS appled mathematcs [6] M.T Söylemez ad N.Muro, Robust pole assgmet ucerta systems IEE Proc-Cotrol Theory Appl., Vol. 44, No 3, May 997 [7] Burkard, R.E., Dergs, U.: Assgmet ad matchg problems (Sprger-Verlag, 980) [8] Muro, N. 969. Computer Aded Desg of Multvarable Cotrol Systems, PhD thess, UMIST, Machester. 6