I J E E E C Internatonal Journal of Electrcal, Electroncs ISSN No. (Onlne) : 77-66 and Computer Engneerng 1(): -7(01) Specal Edton for Best Papers of Mcael Faraday IET Inda Summt-01, MFIIS-1 A New Recursve Metod for Solvng State Equatons Usng Taylor Seres Sucsmta Gos*, Ans Deb** and Gautam Sarkar*** *Department of Electrcal Engneerng, MCKV Insttute of Engneerng, Llua, Howra, (WB) ** Department of Appled Pyscs, Unversty of Calcutta, Kolkata, (WB) ***Department of Appled Pyscs, Unversty of Calcutta, Kolkata, (WB) (Receved 15 October, 01 Accepted 01 December, 01) ABSTRACT: Te paper presents applcaton of Taylor seres to solve states of te control system. Te systems are analyzed by frst Taylor seres and as well as second Taylor seres. A new recursve metod for te soluton s dscussed ere. Te state equaton of a lnear tme nvarant and dfferental equaton of a nonlnear tme nvarant system are consdered as eample to clarfy te metod. Te states of two systems are ten solved by frst and second Taylor seres bot and also compared wt te eact soluton wc are reasonably close to eac oter. Ts comparson s sown n tables and graps bot to valdate te new approac. Inde Terms Lnear tme nvarant, nonlnear tme nvarant, states, Taylor seres. I. INTRODUCTION Te concept of Taylor seres epanson was formally ntroduced by te Engls matematcan Brook Taylor n 1715. Te concept, toug qute old, as not lost ts mportance and s used n many areas of matematcal analyss. Taylor s seres s an epanson of a functon nto an nfnte seres of a varable t, or nto a fnte seres plus a remander term. Te coeffcents of te epanson or of te subsequent terms of te seres nvolve te successve dervatves of te functon. In ts paper Taylor seres s utlzed to solve dynamc dfferental equatons [1] of a control system, e. g., state equatons, to determne te states of te system. Several works ave been done to form ortogonal operatonal matr usng Taylor seres [3]. Tme delay system as also been analyzed along wt dentfcaton of parameters [4]. Te presented metod eren uses Taylor appromaton to solve state as well as dfferental equatons n a recursve manner. Haar functon [6], Wals functon [7], Block pulse functon [8] are te matematcal tools used for state space problem soluton. Later, ybrd functons [9, 10], trangular functon [11] ave been used etensvely. But te Taylor seres tecnque s muc smpler and a powerful tool as well. II. TAYLOR APPROXIMATION A. Frst Taylor appromaton [] Consder a tme functon f(t) n an nterval of wdt, t (, (+1)). A frst Taylor appromaton f1(t) of te functon f(t) around a pont f1(t) f(μ) + f(μ)(t - μ) were μ (,( + 1) ) μ s represented as (1) If te pont μ concdes wt te leadng termnal pont, ten μ = and equaton (1) becomes f1(t) = f() If t=(+1) n (), ten + f()(t - ) () f1{( + 1)} = f() + f() (3) f1{( + 1)}n equaton (3) s te ntal value of f(t) for te net nterval t( +1),( ( + ). )
Gos, Deb and Sarkar 3 B. Second Taylor appromaton A second Taylor appromaton f(t) of te functon f(t) around te same pont μ s represented as 1 f(t) f(μ) + f(μ)(t - μ) + f(μ)(t - μ) (4)! Wt te assumpton μ =, (4) reduces to 1 f(t) = f() + f()(t - ) + f()(t - ) (5)! In (5), as before, we set t=(+1) ten (6) f{( + 1)} = f() + f() + f()! f{( + 1)} s te ntal value of f(t) for te net nterval t( +1),( ( + ). ) III. SOLUTION OF STATE EQUATION VIA TAYLOR APPROXIMATION A. Lnear tme nvarant (LTI) system Consder te state equaton of a lnear tme nvarant (LTI) system as (t) = A(t) + B(t) U (7) (0) and = 0 Dfferentatng (7), we ave = + U (t) A (t) B(t) (8) For t=, equatons (7) and (8) can be rewrtten as () = A() + B() U (9a) = + U () A () B() (9b) Usng (3) n equaton (9), we can wrte followng recursve equatons 1() = (0)(0) 1{( + 1)} = 1() 1() (10) were, =1,, 3,, N, N beng a large number. Tus, from (10 ), usng (9a) and knowng U(t), we can solve for te state vector (t) recursvely va frst Taylor appromaton. Smlarly, to obtan a more accurate recursve soluton, we use second Taylor appromaton as gven n (6) to get ()} = (0)(0)(0) +! {( + 1)} = () () + ()! (11) were, =1,, 3,, N, N beng a large number. Tus, from (11), usng bot (9a) and (9b), we can solve for te state vector (t) recursvely va second Taylor appromaton. Equatons (10) and (11) may be solved wt a fed step sze or any dynamc step sze, wc means can be canged durng recurson. B. Eample Consder te lnear tme nvarant system (t) 1 1.8 1.8 = + 5 1 0 (t)(t) U (1) and 0 (0) = = wt unt step nput. 0 0 Te eact soluton s 1(t) = 0.18-0.18ep(-t) cos 3t + 0.54ep(-t) sn 3t (13a) and (t) = 0.9-0.9ep(-t) cos 3t - 0.3ep(-t) sn 3t (13b) Usng equatons (9a) and (10) we can solve for te states va frst Taylor appromaton, and usng equatons (9a), (9b) and (11), we can solve te states wt second Taylor appromaton. Te results are gven n Table 1 and Table. For computaton, we ave consdered a tme nterval T = s and te number of steps m = 0, so tat = 0.1 s. Fgure 1 sows te recursve solutons obtaned va frst Taylor appromaton wle fgure sows te recursve solutons obtaned usng second appromaton. Soluton by 1 st Taylor seres s represented by Taylor1 and nd Taylor seres by Taylor n grap. To compare te credblty of te results, bot te fgures sow te eact soluton of te states 1 and. As epected, te second appromaton s way better tan te frst appromaton. In fgure 3, all te results,. e., recursve soluton usng frst Taylor appromaton as well as second Taylor appromaton, and te eact soluton, are sown togeter for better clarty. If we gradually ncrease m, keepng te tme nterval T fed (. e., s decreased), te frst Taylor appromaton mproves gradually. It s observed tat for m = 160 te recursve soluton almost overlaps te eact soluton. Ts s sown n fgure 4.
Gos, Deb and Sarkar 4 Fg. 1. Soluton of te states 1 and va frst Taylor appromaton compared wt te eact soluton for m=0. Table 1: Recursve soluton of te state (1) obtaned va frst and second Taylor appromaton compared wt te eact soluton (for T= s, m=0 and =0.1 s). Tme (Sec.) Eact soluton 1(t) X1 Pontwse recursve soluton of 1(t) wt 1st appromaton 0 0 0 0 0.1000 0.1688 0.1800 0.1710 0.000 0.3080 0.340 0.3108 0.3000 0.4105 0.4716 0.41 0.4000 0.4737 0.5591 0.473 0.5000 0.4990 0.5999 0.4956 0.6000 0.4911 0.5947 0.4846 0.7000 0.4566 0.5485 0.4475 0.8000 0.4035 0.4700 0.396 0.9000 0.3400 0.3705 0.384 1.0000 0.736 0.618 0.64 1.1000 0.108 0.1558 0.01 1.000 0.1566 0.069 0.1495 1.3000 0.1144-0.0091 0.1104 1.4000 0.0857-0.0550 0.0850 1.5000 0.0707-0.077 0.0731 1.6000 0.068-0.0634 0.0734 1.7000 0.076-0.0307 0.0835 1.8000 0.091 0.0198 0.1006 1.9000.0000 0.1131 0.136 0.081 0.1464 Pontwse recursve soluton of 1(t) wt nd appromaton 0.118 0.1445 Fg.. Soluton of te states 1 and va second Taylor appromaton compared wt te eact soluton for m=0. Table : Recursve soluton of te state () obtaned va frst and second Taylor appromaton compared wt te eact soluton (for T= s, m=0 and =0.1 s). Tme (Sec.) 0 0.1000 0.000 0.3000 0.4000 0.5000 0.8000 0.9000 1.0000 1.1000 1.000 1.3000 1.4000 1.5000 Eact soluton (t) X Pontwse recursve soluton of (t) wt 1st appromaton 0 0 0 0.0418 0 0.0450 0.153 0.0900 0.1607 0.3115 0.50 0.330 0.4940 0.466 0.5083 0.6799 0.6959 0.6951 0.8519 0.96 0.8658 0.9970 1.1309 1.0077 1.1071 1.91 1.1130 1.1787 1.3979 1.1788 1.1 1.4433 1.066 1.116 1.499 1.007 1.1831 1.3648 1.168 1.1343 1.598 1.1169 1.0733 1.19 1.055 1.0078 0.9888 0.9907 0.9444 0.8536 0.999 0.8886 0.7365 0.8778 0.8439 0.6475 0.8374 0.813 0.596 Pontwse recursve soluton of (t) wt nd appromaton 0.8105 1.6000 0.7944 0.5740 0.7968
Gos, Deb and Sarkar 5 k 3k = (16b) f f (0) and (0) can () () () () So, knowng (0), usng (16a) and (16b), be obtaned. Usng (3), for te unknown functon (t), we can wrte te followng equatons as 1(1) = (0) + (0) 1{( + 1)} = 1() + 1() (17) Fg. 3. Comparson between frst appromaton and second appromaton for m = 0. Fg. 4. Taylor appromaton upto frst appromaton and for m = 160. A. Non-lnear tme nvarant (NLTI) system Consder te followng nonlnear dfferental equaton 3 f k k 0 + + = (14) were, f, k and k are constants. Let te ntal state be (0) Dfferentatng (14), we ave f k 3k = 0 = (15) For t=, equatons (14) and (15) can be rewrtten as k k 3 () () () = (16a) f f Tus, from (17), usng (16a), we can solve for (t) recursvely usng frst Taylor appromaton. Smlarly, to obtan a more accurate soluton, we use second Taylor appromaton as n (6) to get () = (0) + (0) + (0) {( + 1)} = () + ()() +! (18) Tus, from (18), usng bot (16a) and (16b), we can solve for (t) recursvely usng second appromaton. Also, equatons (17) and (18) may be solved wt a fed value of te step or a dynamc value. Tat s, can be canged durng recurson. B. Eample [5] Consder nonlnear tme nvarant system = + From above equaton we fnd, = 0. were, (0) = 0 (19) (0) Ten usng (19) and (0), we fnd soluton at any pont of tme n any nterval by usng te followng recurson {( + 1)} = () + ()() +! = () + () + () () ()!
Gos, Deb and Sarkar 6 Te eact soluton s obtaned usng Volterra ntegral equaton as { } -1 (t) = 5.1095ep(1.3416t) 0.7453 + 0.1708 soluton usng frst as well as second Taylor appromaton, and te eact soluton, are sown togeter for better clarty. For m=0, T=4, te dscrete soluton ponts are presented n Table 3. Soluton by 1st Taylor seres s represented by Taylor1 and nd Taylor seres by Taylor n grap. Fgure 5 sows te recursve solutons obtaned va frst Taylor appromaton wle fgure 6 sows te solutons obtaned usng second appromaton, wt m=0 n eac case. Table 3: Recursve soluton of te state obtaned va frst and second Taylor appromaton compared wt te eact soluton (for T=4 s, m=0 and =0. s). Tme (Sec.) Eact soluton (t) Pontwse recursve soluton of (t) wt 1st appromaton Pontwse recursve soluton of (t) wt nd appromaton Fg. 5. Frst Taylor appromaton for m=0. 0 0 0 0 0.000 0.036 0.0400 0.0360 0.4000 0.0654 0.0717 0.0651 0.6000 0.0887 0.0963 0.088 0.8000 0.1071 0.115 0.1066 1.0000 0.115 0.195 0.110 1.000 0.138 0.140 0.133 1.4000 0.1416 0.1483 0.1411 1.6000 0.1483 0.154 0.1479 1.8000 0.1536 0.1586 0.1531.0000 0.1576 0.1619 0.157.000 0.1607 0.164 0.1603.4000 0.1630 0.1660 0.168.6000 0.1649 0.1673 0.1646.8000 0.1663 0.168 0.1661 3.0000 0.1673 0.1689 0.167 3.000 0.168 0.1694 0.1680 3.4000 0.1688 0.1698 0.1687 3.6000 0.1693 0.1701 0.169 Fg. 6. Second Taylor appromaton for m=0. 3.8000 0.1696 0.1703 0.1695 To compare te valdty of tese results, bot te fgures sow te eact soluton. As epected, te second appromaton s muc better tan te frst appromaton. In fgure 7, all te results,. e., recursve Fg. 7. Comparson between frst appromaton and second appromaton for m = 0.
Gos, Deb and Sarkar 7 And also, te recursve metod requres muc less memory. Te smplcty of soluton usng Taylor seres va recurson makes t powerful tool over oter metods. Also, solvng nonlnear system s a more dffcult task compared to solvng lnear systems. And t always calls for specal tecnques nvolvng muc more matematcal complety. In ts paper we ave used te same tool to solve lnear as well as nonlnear systems. We tnk ts s really a specalty of ts work. Fg. 8. Frst Taylor appromaton for m = 110. In Fgure 3 and 6, 1st soluton and nd soluton are solved wt te same sub- nterval, but for te sake of clarty, te grap for Taylor soluton s plotted wt an ntal samplng perod of /, and after tat te plot s contnued wt te same samplng perod, so tat te sample ponts do not mngle. In Fgure 8, no of ntervals taken s 110. But for te sake of clarty, all of 110 ponts are not plotted. Instead, only a few ponts are ndcated n te grap. IV. CONCLUSION We ave presented recursve metod for solvng lnear as well as nonlnear state equatons of a control system based upon Taylor seres epanson. We ave used bot frst and second epanson usng dfferent step szes. It s observed tat te results obtaned n ts manner are reasonably relable. For frst appromaton, usng 0 steps (m=0), te devaton of te soluton from te ea ct results are qute notceable. But for second Taylor appromaton te presented recursve metod offers gly dependable solutons and almost overlaps wt te eact soluton. Tree tables (tables 1, and 3) and egt fgures (fg. 1 to fg. 8) are presented to compare te results n bot quanttatve and qualtatve manner. It s clear from te plots of dfferent fgures, ow close te Taylor seres based solutons are. In fact, solutons obtaned va Taylor and also, wt Taylor 1 (wt smaller subnterval of course), are very close to te eact solutons. Tree tables (Tables 1, and 3) also reflect ts fact. Soluton of dfferental equaton usng Taylor seres s not new. But te nnovatve dea n ts work s to analyze a dynamc system by solvng state equatons usng Taylor seres, especally n a recursve manner. A recursve metod s consdered to be faster tan metods nvolvng Kronecker products and nverson of large matrces. REFERENCES [1] Perko Lawrence, Dfferental Equatons and Dynamcal Systems, 3rd ed., Sprnger, Berln, 006 [] Krasnov, M., Kselev, A., Makarenko, G. and Skn, E., Matematcal analyss for engneers (vol -), Mr Publsers, Moscow, 1990. [3] Eslac, M. R. and Degan Med, Applcaton of Taylor seres n obtanng te ortogonal operatonal matr, Computers & Matematcs wt Applcatons, Volume 61 Issue 9, pp. 596-604, 011. [4] Yang, Cng-yu and Cen, Ca'o-kuang, Analyss and parameter dentfcaton of tme-delay systems va Taylor seres, Int. J. Sys. Sc., Vol. 18, No, 7, pp. 1347-1353, 1987. [5] Rao, G. P., Palansamy, K. R., and Srnvasan. K., Etenson of computaton beyond te lmt of ntal normal nterval n Wals seres analyss of dynamcal systems, IEEE Trans. Autom. Control, vol. AC-5, pp. 317-319, 1980. [6] Beaucamp, K. G., Applcaton of Wals and Related Functons, Academc Press, London, 1984. [7] Beaucamp, K. G., Wals functons and ter apllcatons, Academc Press, London, 1975. [8] Jang, J. H. and Scaufelberger, W., Block pulse functons and ter applcaton n control system, LNCIS 179, Sprnger Verlag, Berln, 199. [9] Deb, Ans; Sarkar, Gautam; and Sengupta Anndta; Trangular ortogonal functons for te analyss of contnuous tme systems, Antem Press, London, 011. [10]. Deb Ans, Sarkar Gautam, Ganguly Anndta and Bswas Amtava,Numercal Algortm for te Soluton of Trd Order Dfferental Equatons n Ortogonal Hybrd Functon (HF) Doman Publs ed n IEEE Eplore, INDICON 011, 16-18t December, Hyderabad, 011. [11]. Ganguly Anndta, Deb Ans and Sarkar Gautam, Numercal Soluton of Second Order Lnear Dfferental Equatons usng One- Sot Operatonal Matrces n Ortogonal Hybrd Functon (HF) Doman Proceedngs of te conference ACODS 01, 16-18t February, Bangalore.