A Nuercal Algorth for Reconstructng Contnuous-Te Lnear Models wth Pure Integrators fro Ther Dscrete-Te Equvalents Zaher M Kassas and Greg Morrow Abstract A nuercal algorth for reconstructng contnuous-te lnear odels wth pure ntegrators fro ther dscrete-te equvalents s derved n ths paper The algorth treats the reconstructon proble n the frequency-doan and establshes a nuercally tractable -doan to s-doan appng, whch avods the need to software packages wth sybolc atheatcal solvng capabltes The proposed algorth can be convenently pleented usng standard buldng block nuercal routnes and has been recently eployed n a coputer-aded control systes desgn software package Unlke other approaches, ths algorth s shown to avod unnecessary coputatons, whch typcally result n nuercal burden, abguty, and sngularty A nuercal exaple s llustrated deonstratng the applcaton of the proposed algorth I INTRODUCTION Identfcaton of odels s a proble of consderable portance n varous dscplnes, such as control, econocs, and sgnal processng The dentfcaton process s often done usng a dgtal coputer, fro saples of nput-output data However, the odel s usually of contnuous-te (CT) nature, and ts dynacal propertes are ost aptly descrbed n ters of dfferental equatons or transfer functons (TFs) The two basc approaches for dentfcaton of CT systes can be categored nto two ethods: () the ndrect ethod, where frst a dscrete-te (DT) odel s estated fro the sapled data and then an equvalent CT odel s calculated; and () the drect ethod, where a CT odel s obtaned drectly wthout gong through the nteredate step of deternng a DT odel [, [2, [3 Identfcaton of CT lnear te nvarant (LTI) systes can be done ether n the te-doan or n the frequencydoan Te-doan ethods usually deterne a DT odel of the syste, whle frequency-doan ethods can dentfy ether a DT (-doan) or a CT (s-doan) odel In frequency-doan based ndrect dentfcaton, where a -doan odel has to be deterned frst, the usual underlyng assuptons are: () the exctaton sgnal s constant between the saplng nstants, also known as the ero-order hold (ZOH) assupton, and () the output of the sgnal s sapled wthout bandltng t by an ant-alasng flter [4, [5 If theses assuptons are fulflled, the s-doan ratonal TF unquely deternes a correspondng -doan odel through the so-called step-nvarant (ZOH) transfor However, whle the ZOH transforaton fro the s- doan to the -doan s unque, the other drecton s Ths work has been supported by Natonal Instruents Zaher M Kassas and Greg Morrow are wth Natonal Instruents, Control Desgn & Sulaton R&D, 5 N Mopac Expwy, Austn, TX 78759, USA kassas@eeeorg and gregorrow@nco soetes abguous In [6, t was shown that t s possble to reconstruct a CT ratonal TF based on the dscreted odel at several, sutably chosen saplng rates However, the approach assued that the TF does not possess ultple poles at the sae locaton Snce ratonal TFs can be expanded nto partal fractons wth poles n the denonators, and snce the ZOH transfor s lnear, the reconstructon of the CT TF fro ts DT equvalent can be acheved by expandng the DT TF nto ts partal fractons and applyng the nverse ZOH transforaton on each Consequently, all the DT poles can be apped nto ther CT counterpart The eros cannot be related to each other by a slarly sple appng, however [7 Gven a ratonal TF, H(), correspondng to a sapled representaton of a CT TF, H(s), the partal fracton expanson (PFE) of H() wll be of the for H() n j d,j ( p ) j, where n s the nuber of poles that H() possesses, s the ultplcty of the th pole p, and {d,j ;,, ; j,, n are constants The partal fractons of H() wll be classfed to belong to one of four dfferent classes: fractons wth poles at, fractons wth poles at, fractons wth negatve real poles, and fractons wth poles elsewhere For the well-known proble of reconstructng TF odels wth negatve real poles, snce the -doan to s-doan appng nvolves takng the logarth of the pole p, and snce ln( p ) ln(p ) + j(π + 2kπ), k, ±, ±2,, the transforaton s not unque Even f we select a certan leaf of the coplex logarth functon, we end up wth a coplex pole wthout a coplex conjugate par Kollar et al n [8 suggested an extra anpulaton to overcoe ths ssue By addng a cancelng ero/pole par to the - doan odel, the transforaton becoes possble, but then abguty appears The undeterned part n the TF of the resultng s-doan syste, as t was shown, s a sgnal wth ero crossngs exactly at the saplng nstants These hdden oscllatons can be elnated by applyng a parsony prncple For the case where the -doan syste contans poles at the orgn, whch translates to a delay n the s- doan, several algorths were suggested [9, [ For the case where there are no negatve real poles n the -doan TF nor poles at the orgn, and assung that the saplng radan frequency, Ω, s larger than double the largest agnary part of the s-doan poles, the -doan
to s-doan transforaton s unque and straghtforward, analytcally Nevertheless, f a coputer-aded control syste desgn (CACSD) software tool s to be eployed to perfor autoatcally such appng, then such a tool should possess sybolc atheatcal solver capabltes Ths stes fro the fact that the -transfor of a ratonal TF n PFE for nvolves takng partal dervatves Ths paper wll focus on the case when the poles are at, whch translate to pure ntegrators n the s-doan In ths respect, a nuercal algorth for establshng -doan to s-doan appng s derved Consequently, ths algorth s eployed nto ZOH reconstructon of TF odels wth pure ntegrators It s argued that our algorth can be extended nto reconstructng TFs belongng to the fourth class dscussed above, e TFs wth poles elsewhere Ths paper s organed as follows Secton II revews relevant alternatve algorths and establshes how our approach s dfferent Secton III derves a nuercal algorth to acheve the -doan to s-doan appng Secton IV presents the algorth for reconstructng CT TF odels wth pure ntegrators, gven ther ZOH DT equvalents Secton V llustrates the applcaton of the proposed algorth through a nuercal exaple Concludng rearks and future work are presented n Secton VI II BACKGROUND AND MOTIVATION Most CACSD software packages pleent ther natve algorths n state-space [ As such, the process of reconstructng a CT TF fro ts DT equvalent nvolves realng the TF n SS, reconstructng the CT SS odel fro the DT representaton, and fnally convertng the reconstructed CT SS odel nto a TF In partcular, gve a CT LTI syste defned by the CT syste atrces (A c, B c, C c, D c ) as ẋ(t) A c x(t) + B c u(t) y(t) C c x(t) + D c u(t), where A R n n, B R n r, C R n, and D R r, the ZOH DT equvalent can be readly shown to be gven by A d e A ct, C d C c, D d D c, B d T e Acτ dτ B c [A d I A c B c, where T s the saplng te [2 Consequently, the reconstructon of the CT syste atrces, gven ther DT counterpart nvolves perforng the atrx logarth and the atrx nverse operatons, each of whch ay result n an abguty The frst abguty arses f A d has negatve real egenvalues, n whch case A c would contan coplexvalued eleents The second abguty arses whenever A d has egenvalues at wth any ultplcty, whch corresponds to A c possessng egenvalues at Ths s due to the fact that coputng B c would nvolve coputng the atrx [A d I, whch would be rank-defcent by as any egenvalues at the atrx A d would have To crcuvent ths sngularty, a coon approach s to copute the atrx equvalents through the relatonshp [ ([ Ad B d Ac B exp c I ) T, () whch avods perforng the atrx nverson step [3 Nevertheless, the process of realng a gven TF n SS, perforng the coputatons, and fnally convertng back the resultng SS odel nto a TF nvolves unnecessary nteredate steps along wth coputatonally ntensve lnear algebra anpulatons Even though several algorths have been proposed to reduce such nuercal burden when coputng A c and B c, such as [4, [5, t s desrable f the reconstructon algorth does not nvolve the nteredate steps of realng the syste n SS and back to TF nor t nvolves ntensve anpulatons An attept to forulate a frequency-doan based algorth that perfors the -doan to s-doan appng was presented n [6 However, the approach n ths paper s dfferent n several aspects Frst, all the nuercal buldng blocks of our algorth are basc loopng and sple polynoal anpulatons and can be readly found n any standard nuercal software, such as LabVIEW In partcular, our algorth elnates the need to perfor PFE of polynoals of the for ( ), whch s a step that requres non-standard polynoal anpulaton buldng blocks Moreover, the proposed algorth s attractve for n-the-loop dentfcaton, where the dentfcaton algorth could resde on ebedded devces, such as processors and feld-prograable gate arrays (FPGAs), wth lted coputatonal capabltes, and where deployng ntensve lnear algebra algorths ay not be possble Second, we derve our algorth startng fro standard Laplace and -transfor tables and show how we can perfor the process of takng partal dervatves through sple nuercal polynoal anpulatons Consequently, we eploy ths -doan to s-doan appng nto reconstructng CT TFs wth pure ntegrators fro ther ZOH DT counterpart Ths algorth has been thoroughly tested and s now ade avalable through the LabVIEW Control Desgn and Sulaton Module [ and the LabVIEW Syste Identfcaton Toolkt [7 III NUMERICAL ALGORITHM FOR COMPUTING THE Z-TRANSFORM OF PURE INTEGRATOR MODELS The -transfor of an th -order ntegrator can be found n standard Laplace and -transfor table to be gven by ( ) l s ( )! { e αt, (2) where T s the saplng te n seconds [2 Equaton (2) suggests that the s-doan to -doan appng cannot be acheved through a sple nuercal algorth, and rather we need to have a sybolc atheatcal solver to acheve the transforaton fro one doan to another Ths secton wll outlne how to copute nuercally the -transfor of an th -order ntegrator
Proposton : The -transfor of an th -order ntegrator can be coputed nuercally fro H (s) s b, ( ) H (), (3) where the coeffcents {b, are to be coputed through the algorth outlne n the proof below Proof: The proof wll proceed by provng the structure of (3) and then dervng a nuercal recursve algorth for coputng the coeffcents {b, Frst, we wll rewrte (3) as H () ( ) ( )! l G (, α), (4) where we defne the auxlary functon G (, α) as { G e αt, (5) wth the ntal condton and recursve relatonshps defned respectvely as G e αt (6) G G,, 2,, (7) where the dependency of G (, α) on and α n the precedng equatons was dropped for splcty of notaton Fro the defntons of (5)-(7) ts easy to see that G C G, (8) where T e αt C (9) e αt Fro (7)-(9) we can readly copute G 2 (, α) as where P (, α) s defned as G 2 P G, () P 2 C T () Next, we wll defne the recurson P P, 2, 3, (2) Fro (7), (), and (2) we can note that G and P satsfy the condtons of Lea, except that the ndces of the ntal condtons dffer, e we have the correspondence G 2 P G g f g Fro Lea, we know that for any N we have ( ) g f g (3) Replacng g wth G + and f wth P + n (3) we get ( ) G + P G + (4) ( ) P k+ G k, (5) k k where we have used the change of suaton ndex k + to go fro (4) to (5) Fnally, we can rewrte (5) as ( ) 2 G P G (6) Now, we wll derve the relatonshp between P and P + In ths respect, we wll use the arguent of nducton, whch wll eventually yeld a recursve forula for coputng {P Frst, we wll express () as P a,j T j C j, (7) where the coeffcents a,j n (7) evaluate to a, and a, 2 Generalng (7) to any N, we have the nducton hypothess P a,j T j C j (8) Next, we wll derve a recursve forula relatng P to P + that wll ensure the nducton step, e ensurng that f P s true, then P + s also true Fro (2) and (8) we can see that P + a,j T j C j a,j T j j C j+ a,j T j+ j C j (9) Usng the change of varables k j+ n the frst suaton n (9) and splttng the last ter of the frst suaton and the frst ter of the second suaton yelds P + a,k (k )T k+ C k k a,j T j+ jc j j + a, C + (2) Now, rewrtng the suaton n (8) up to + yelds + P + a +,j T + j C j (2) Splttng out the frst and last ters of the suaton n (2) yelds P + a +,j T + j C j + a +, T + C j +a +,+ T C + (22) For the nducton step to hold true, t s requred that (22) be equal to (2) Equatng the coeffcents of equal powers of C n the two expressons gves C : a +, (23) C + : a +,+ a, (24) C j : a +,j a,j (j ) a,j j, j,, (25)
We can vew (23)-(25) as not only necessary for provng the nducton step, but also as eans for coputng recursvely P for any N Ths wll facltate the evaluaton of G, snce P and G are related through (6) Ths n turn wll facltate the coputaton of H () as t s related to G (, α) through (4) The last reanng step s to take the lt as α Frst, note that l C T Hence, takng the lt as α of (8) yelds where l P ã,j ( ) j, (26) ã,j ( ) j T a,j (27) Fro (26) we can see that P s a polynoal of order n ( ) Now, takng the lt as α of G and G, respectvely, yelds l G l G T ( ) 2 (28) Usng (6) wth 2 we can see that G 2 s the product of G and P ; hence, l G 2 s a polynoal of thrd order n ( ) tes one power of More generally, t s easy to see that for any Z, l G s a polynoal of order ( + ) n ( ) tes one power of Snce H s related to G through (4), ths proves that H s a polynoal of order n ( ) tes one power of, whch n turns proves the structure of (3) Next, we wll deterne the recurson on {b, n (3), whch wll coplete the proof Fro the relatonshp between H and G n (4), t s clear that l G has an expanson lke (3), but ts coeffcents, whch we wll denote by b,, wll dffer fro those of H by the factor ( ) ( )! So, we let where l G b, Substtutng (26) nto (6) we get l G b, ( ), (29) ( )! ( ) b, (3) ( ) 2 ( ) 2 [ + b+,k ( ) k k ã,j ( ) j ã,j ( ) j l G (3) + b+, ( ), (32) where we have used (29) to wrte (3) as (32) Hence, generally, gven a desred order > 2, we can evaluate H () through the followng recursve nuercal algorth ) Startng wth a, and a, 2, calculate ã, and ã, usng (27) Also, dentfy that b 2, and b2,2 T fro (28) and (3) Set n 2 2) Evaluate l G n n ( ) n 2 ã n n,j ( ) j (33) [ + b+,k ( ) k (34) k 3) Accuulate the resultng polynoal and dentfy the coeffcents { bn+, n+ correspondng to n+ l G bn+, n ( ) (35) 4) If n, stop; otherwse, set n n +, calculate {ã n,j n usng (23)-(25), and go to step (2) 5) Fnally, calculate the coeffcents {b, fro the coeffcents { b, usng the relatonshp (3) IV ZOH RECONSTRUCTION OF MODELS WITH PURE INTEGRATORS Proposton 2: The ZOH reconstructon of a gven TF wth -poles at s gven by { ZOH ( ) s j, (36) where the coeffcents { are to be coputed as outlned n the proof below Proof: The proof wll proceed as follows Assung that the proposed structure s true, ths wll lead to a syste of lnearly ndependent equatons, whch upon solvng we can dentfy the coeffcents { Frst, we recall that the so-called step-nvarant (or ZOH) transfor of a TF H(s) s gven by where Z H() ZOH {H(s) Z { H(s) s { H(s) { s the shorthand for Z s, (37) L { H(s) s, and L {, Z {, denote the Laplace and transfors, respectvely [2 Applyng the ZOH transfor to both sdes
of (36) yelds ( ) ZOH + s j Z [ [ + ( ) { s j+ j+ b j+, b j+, ( ) (38) (39) ( ) (4) b j+,, (4) where to go fro (38) to (39) we have used (3), and to go fro (39) to (4) we have used the fact that b j,, > j, whch enabled us to wrte the second suaton up to + Fnally, we have rearranged the suatons to go fro (4) to (4) Usng the change of varable k n the ndex of the frst suaton of (4) yelds ( ) ( ) k k b j+,k+ (42) We can readly recogne that for (42) to hold true, we need that all the coeffcents n the square brackets of the second suaton correspondng to ters wth k < to be ero, whereas the coeffcent correspondng to the ter wth k ust be Consequently, we can wrte the lnear syste b, b 2, b 3, b +, b 2,2 b 3,2 b +,2 b 3,3 b +,3 b +,+ c, c, c,2 c, (43) Therefore, we can evaluate the coeffcents { usng the lnear syste of equatons n (43) We can show that the ( + ) ( + ) atrx s nonsngular by contradcton Frst, we note that n order for the ( + ) ( + ) upper trangular atrx to be nonsngular, we need that all the dagonal eleents {b + to be nonero The proof by contradcton wll proceed as follows We wll assue that one or ore coeffcent on the dagonal of ths atrx s ero and show that ths cannot be true To do so, we note that the dagonal eleents {b +, whch facltate the evaluaton of the TFs {H () + are related to the coeffcents + { b through (3), whch we used to evaluate the auxlary TFs {G () +2, through (32) Moreover, we have showed n the proof of Lea that for any Z, l G s a polynoal of order ( + ) n ( ) Hence, the coeffcent b +,+ has to be nonero, because f t was ero, then the TF G () wll be a polynoal of order less than ( + ) n ( ) Snce ths s true for + any Z, then none of the coeffcents { b can be ero, whch copletes the proof by contradcton V EXAMPLE Ths secton wll deonstrate the applcaton of the proposed algorth nto reconstructng CT TF odels of systes wth pure ntegrators fro ther ZOH DT equvalents In ths respect, the algorths that were presented n Secton III and IV were coded nuercally and were consequently used to reconstruct CT TF odels of systes wth ntegrators up to sx ultplctes Table I shows the values of the coeffcents that correspond to the reconstructed TF n (36), where s the ultplcty of the ntegrators, and j,,, TABLE I ZOH RECONSTRUCTION OF CT TF PURE INTEGRATOR MODELS UP TO SIX MULTIPLICITIES j 2 3 4 5 6 T 2 3 4 5 6 2T 3T 4T 5T 6T T 2 T 2 T 3 2T 2 5 6T 2 7 4T 3 37 8T 2 3 2T 3 T 4 5 8T 3 7 6T 4 VI CONCLUSION 2 T 4 T 5 5 2T 5 T 6 Ths paper has presented a nuercal algorth for reconstructng CT TF odels of systes wth pure ntegrators fro ther ZOH DT equvalents The algorth used basc recurson and sple polynoal anpulatons and s well suted for CACSD software The algorth perfors the reconstructon solely n the frequency-doan; hence avods nteredate unnecessary steps of realng the TF n SS and convertng back the CT SS odel nto TF As such, the algorth avods a sngularty that ay arse f the reconstructon s done n SS, and t avods eployng coputatonally costly lnear algebra anpulatons Ths work establshed a sple -doan to s-doan appng, where the process of takng partal dervatves s realed through polynoal ultplcatons Such appng was derved fro standard Laplace and -transfor tables The generalaton of ths algorth nto reconstructng CT TFs fro ther ZOH DT equvalents wll be nvestgated n future work In ths respect, the generaled algorth wll be extended to handle poles at arbtrary locatons n the - doan It s noted that the -transfor of CT TFs wth poles at arbtrary locatons has the sae structure of the - transfor of CT TFs wth poles at s, cf (2), except for takng the lt of the partal dervatves Consequently,
t s argued that the generaled algorth wll follow the dervatons outlned n ths work APPENDIX Lea : Gven two sequences of -tes dfferentable functons, wth respect to the varable α, {f and {g, satsfyng then t follows that f f,, 2, (44) g g,, 2, (45) g f g, (46) g g ( ) f g, (47) where ( ) s the bnoal coeffcent, defned as ( )! ( )!! (48) Proof: Fro (44)-(46) t s easy to see that g 2 f g + f g (49) g 3 f 2 g + 2 f g + f g 2 (5) g 4 f 3 g + 3 f 2 g + 3f g 2 + f g 3 (5) Now, consder the frst few evaluatons of the bnoal theore (a + b) a b + a b (52) (a + b) 2 a 2 b + 2 a b + a b 2 (53) (a + b) 3 a 3 b + 3 a 2 b + 3a b 2 + a b 3 (54) Generally, for any N, the bnoal theore s stated as ( ) (a + b) a b (55) We can easly see the effect of ultplyng both sdes of (55) by the ter (a + b) as ( ) [a (a + b) + b (a + b) ( ) [a + b + a b + + ( ) + a + b (56) Now, consder takng the partal dervatve wth respect to α of the ter f g {f g f g + f g f + g + f g + (57) Consequently, we fro (49)-(57) we can establsh the followng analogy: the process of takng successve partal dervatves wth respect to α s equvalent to successve ultplcatons by a bnoal Hence, we can ake the followng assocatons whch yelds (47) f a g b ( ) (a + b) g f g (a + b), REFERENCES [ N Snha and G Lastan, Identfcaton of contnuous-te ultvarable systes fro sapled data, Int J Control, vol 35, no, p 726, 982 [2 H Garner, M Mensler, and A Rchard, Contnuous-te odel dentfcaton fro sapled data: pleentaton ssues and perforance evaluaton, Int J Control, vol 76, no 3, pp 337 357, 23 [3 M Matsubara, Y Usu, and S Sugoto, Identfcaton of contnuouste MIMO systes va sapled data, Control-Theory and Advanced Technology, vol 2, no 5, pp 9 36, October 26 [4 J Schoukens, R Pntelon, and H V Hae, Identfcaton of lnear dynac systes usng pecewse constant exctatons: use, suse and alternatves, Autoatca, vol 3, no 7, pp 53 69, 994 [5 L Ljung, Syste Identfcaton Theory for the User, 2nd ed Upper Saddle Rver, NJ: Prentce Hall, 999 [6 T Chen and D Mller, Reconstructon of contnuous-te systes fro ther dscretatons, IEEE Trans on Autoatc Control, vol 45, no, pp 94 97, 2 [7 K Astro, P Hagander, and J Sternby, Zeros of sapled systes, Autoatca, vol 2, no, pp 3 38, 984 [8 I Kollar, G Frankln, and R Pntelon, On the equvalence of - doan and s-doan odels n syste dentfcaton, n Proc IEEE Instruentaton and Measureent Technology Conference 96, Brussels, Belgu, June 996, pp 4 9 [9 J Neeth and I Kollar, Step-nvarant transfor fro - to s- doan: A general fraework, n Proc IEEE Instruentaton and Measureent Technology Conference, Baltore, MD, May 2, pp 92 97 [ A Kunetsov and D Clarke, Sple nuercal algorths for contnuous-to-dscrete and dscrete-to-contnuous converson of the systes wth te delays, n th IFAC Syposu on Syste Identfcaton Sysd 94, vol 3, Copenhagen, Denark, July 994, pp 633 638 [ NI, LabVIEW Control Desgn Toolkt User Manual, August, 27 [2 G Frankln, J Powell, and M Workan, Dgtal Control of Dynac Systes Systes, 3rd ed Menlo Park, CA: Addson Wesley, 997 [3 BWttenark, K Astro, and K Aren, Coputer control: An overvew, n IFAC Professonal Brefs, Lund Insttute of Technology, 22, pp 5 6 [4 L Sheh, J Tsa, and S Lan, Deternng contnuous-te state equatons fro dscrete-te state equatons va the prncpal qth root ethod, IEEE Trans on Autoatc Control, vol 3, no 5, pp 454 457, 986 [5 V Felu, J A Cerrada, and C Cerrada, An algorth to copute the contnuous state odel fro ts equvalent dscrete odel, Control- Theory and Advanced Technology, vol 4, no 2, pp 23 24, 988 [6 V Felu, A transforaton algorth for estatng syste laplace transfor fro sapled data, IEEE Trans on Systes, Man, and Cybernetcs, vol 6, no, pp 68 73, 986 [7 NI, LabVIEW Syste Identfcaton Toolkt User Manual, August, 26