Åström, Karl Johan

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LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Recursive Formulas for the Evaluation of Certain Complex IntegralsÅström, Karl Johan1968Document Version:Publisher's PDF, also known as Version of recordLink to publicationCitation for published version (APA):Åström, K. J. (1968). Recursive Formulas for the Evaluation of Certain Complex Integrals. (Technical ReportsTFRT-7001). Department of Automatic Control, Lund Institute of Technology (LTH).Total number of authors:1General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.'{'JootRECURSIVE FORMULAS FOR THE EVALUATIONOF CERTAIN COMPLEX INTEGRALSo ..K-J ASTROMREPORT 6804 SEPTEMBER 1968; TUND INSTITUTE OF TECFINOTOGYDIVISION OF AUTOMATIC CONTROL1RECURSIVE FORMULAS FOR THE EVALUATTON OF CERTAIN COMPLEX INTEGRALSby K-J Astr.örnAbstractThis paper pnesents recunsive formulas which admits simul-taneoustest of stability and evaluation of quadnatic lossfunctions forlinear discrete time dynamical systems. The method admits a sig-nificant neduction of the number of computations in compa::isonwith previousÌy known methods.I. Int::oductionI¡,le wil-l consider the evaluation of integnals of the type-lí1r B(z) B(z ð.2_1A(z) A(z -) zB are polynomials with real coefficients) (1)2¡riwhei:eA(z) =B(z) =A andna +,n+anuL"bL'n-ln-I(2)(3)++++oob bnand f denotes the integral along the unit cincle in the positivedirection.Integnals such as (1) occun in many controf and communicationpr.oblems. The sum of squaues of the values of the impulse nes-ponse of a dynamical system wittr the pulse tnansfer functionB(ù/A(z) is ê.8. given by (f ). Evaluation of quadratic foss-functions, genenation of quadnatic Lyapunov functions for' linearsystems and investigation of the accuracy of parameter estimationin linear systems also lead to integnals of type (1)e see e.g.tr I and tgl .Closed fonm solutions of (f) for polynomials of low oirder aneavailable in litenatune. See e.g. Juny Is, p. 2gT-2gg ] .For la::ge n, say.n >, 4) the cl-osed form solutions are, however,very cumbersome to use. It is also wellknown that "oI, where Iis tlne integr.al defined by (1) can be obtained as the first com-ponent of the vector. x which satisfies the following lineanequation.2atr- uo*u2u2See e,g fs,-r ^2-300p. 16B- l]zfban= n-lnTi=0n- l-Li=0b(3)*2xn+1l_t_on,'l 2ibbu2*u4tr*.3 2 i+1-3 -o*-4 -1*-s 2 n-¿ x1J 2t"b í+2a-a0n-l n naao2 L b.bI a i+n-1ilan 02bbIn this papen \^7e pnesent necunsive equations fon the integnalwhich leads to considenably fewen computations than a dinectsolution of (3). The necu::sive equations are denived using ele-mentary results fnom the theony of analytic functions. The ne-cunsive equations can be conveniently used both fon hand- andmachine computation. Analogous nesults fon continuous time sys-tems have been obtained by Nekolnf and Beneã [+].2, P::etiminanies and notations\,rle finst obsenve that the integnal (1) will always exist if thepolynomial A(z) has al-l- its zet'os inside the unit cincle. Insuch a case r¡re can always find a stable dynamical system withthe pulse tnansfe:: function B(z)/A(z) and the integral (1) isthen simply the sum of squanes of the o:rdinates of the impulsenesponse of the system.If A(z) has zeros on the unit cir-cle the integnal divenges.If A(z) has ze?os both inside and outside the unit cincle butnot on the'unit cincl-e: the integr-al (1) sti1l exists. In sucha case h7e can always find a polynomial A'(z) with al-l- its zerosinside the unit circl-e such thatA(z) e(z-l) = A'(z) A'(z-r)and the integnal then repnesents the sum of squares of the im-pulse r?esponse of a stable dynamical system whose pulse tnans-fen function is B(z) /A'(z) .In many pnactical cases , howeve::, \^re obtain the integnal as aresult of an analysis of a dynamical system whose pulse tnans-fen function is B(z) / A(z) , In such a case it is natunally ofgreat importance to test that A(z) has all its zeros inside theunit ci::cle because when this is not the case the dynamical sys-tern witl be unstable although the integr-al (1) exists.In orden to pnesent the nesultintnoduce some ¡:otations. Let Abya simple form we will finstdenote the polynomial definedn (4)an(s)l_nA" (z) =FuntherAO(z) =B¡.(z) =ñ -lz"A(z -) =introducekk kuo, * uI,+ ^I" + ...polynomials+ ... + rIKatJthek-1L+-kD.K_k k .k k-lhz+D_z-ôl + +which are defined recunsively by(6){Ao(z)4Af\(z) jokkk(z) jIAO_r(z) (7)(B)(e)(10)( 11)(l.2)*1B (z) {Bo(z)k-1wheneßandz ß kva-J(hJ!"kokkw/d. o1-/aoa'ìAn(z)(z)= A(z)= B(z)BnIf these equations should have a meaning \nre must natunally ne-vqui::e that all ai ane diffenent fnom zene. To see the irnplica-tions of this we will make use of the following theor.em.Theonem IThe polynomial A(z) has all- its zeros inside the unit circle ifand only if -.k > o fon al-l k.oThis theonem is essentially the Schun-Cohn stability cnitenionfon l-inean discrete time dynamical systems. See ê.g. IZ] )[g, p 126], [ol . lfe also have the following nesult which willbe used in the pnoof of our main result.Theorem 2Let the polynomial AO(z) have all- its zeros inside the unit cinc-Ie, then AO_r(z) al-so has all its zeros inside the unit cincle.This theonem is also givqn by Schun [6]. A simple pr.oof is givenny nüãiðra [s].i¡/e thus find that the polynomials AO(z) can always be find if theoniginal polynornial- has all- its zeros inside the unit cincl-e. TfS no \^7 l- a ways ge a s at same step j-nothe neduction. The equations (7) and (B) can thus be pnofitableexploited as a stability critenion.3. The main nesultI,rle will- now show that the integnal (f) can be computed necu::sively.Fon this purapose \^re intnoduce the integnal* Ik defined bytr)-11 f BO(z) BO(z dzï k (13)12ni A. (z) A. (z-KK )zIt follows from (1) that f = T . I,r/e now havenTheonem 3Let the polynomial A(z) have all its zeros inside the unit cincle.The integnal" Ik defined by (13) then satisfies the following ne-cunsive equations2kßïL2-k2ß'o{rI(14)(15)PnoofAs A(z) has al-l its zeros inside the unit cincle it follows fnomTheonem 1 that al-l "I are diffenent fnom zero. ït thus followsUfnom (9) and (10) that al-l- polynomials At and Bt can be defined.Funthenmone it follows fnom Theonem 2 that all polynomials Athave all- zeros inside the unit cincle. All integrals In thusexist.To pnove the theonem we will make use of the theony of analyticfunctions. The integral (13) equals the sum of nesidues at the_1 _'tpoles of the function Bn(z) BO(z -)/{z AO(z) AO(z -)} inside theunit cincle. As the integral is invaniant unden the change ofva::iables z + L/2, \^re also find that the integnal equals the sumof nesidues of the poles outside the unit ci::cl-e.Now considen-1] Tr-r1 fBO_r(z) Bo_r(zð,2)-1Ik-1 ZlIL ¡\i_t(z/ f{È_l(z ) zbThe poles of the integnand inside the unit cincle are z = 0 andthe zeros z. of the pol-ynomial AO_r(z). ft follows from (7) and(4 ) thatA, (2.) = o.A.'(2.) =K I KK ]-_'lA. - (2. -) = z. {A" (2.K--LaaKlkd. z.Kl-IA. (2.-KI )-t- rr _ I- u. A. (2. -)ÌKK ]Hence-k- a. z.--4. (2.) jK]-KA- .,iA. (2. -)K'l-l _1AL_t(ri -) = ,i{AO(2, -)(1 - 0 )zFu::the:: it fol-lows f::om (4) and Q) thatAO_r(z) = AO(z) - oOAO(z)2k)aHenceA,- . (0) = 4,,(0)J(- -L J(The functions_'lBO_r(z) Bo_a(z -)Ao_r(z) Ar_r(andBO_r(z) BO_r(zor.[ = .f cr - "fl1ko.A. (0)KJCBal-!zo(z) B (z)k-1 1- 'lñ-r !z) Ao_r(z) AO_a(z)-lIZBO_r(z) Bo_r(z)2!1A,- . (z) {z(L-aJ(_ I ) A" (z-K )Ì Ai-l\ (z){(l-cv, )Ao(z) Ì2khave the same poles inside the unit cincle and the sameresidues at these poles. Hence-1 kdz-1BO_r(z) BO_a(z dzI-I- cxr-K_'l )A, - (z) A. (z ') z'K-I. KIIffrL-t2níBo-t(z) Bt-t _l(z -)t_--7r- ok(16)-12rí A. (z) A. -(zK K-I )where the second equality is obtained by making the variablesubstitution , * ,-I. The integnand has poles at the zeros of7AO(z).Ao_r(z- 1) = z{Ao- -Í A_ or^k_t(z ') - okAkfnom (7) that_'t(z ') Ìït foll-ows , howeven,such that_v"AO(z) Ìa-zKAO (z )Hence fon z.a_'tAr_r(ri -) =0 we getl--1- L (-L. L\a \ L.l_K r- )The functions-fÞ"k-l (z) Bk-1 (z )-1Ao(z) Ao_a(zand)-1B. .(z) B. -(zK-t Ì(-l Bo_r(z) BO_a(z))_'lAo(z) Ao(z -) AO(z) AO(z)IZthus have the same poles inside the unit cincle and the samenesidues at these poles. The integnal of these functions anoundthe unit circle ane thus the same. The equation (16) now givesBo-r(z) Br-r Q-I) dzI1- "t_'tAo(z) Ao(z -)I fïr- lltLNow introduce (B) and we findI z-1 ,r -lßOAO(z -)) ð,22 f{Bo(z) ßrAL (z) j {BO ( z(1-cr )rk k-1I2ní -1A. (z) A, (zKK )-1 -1BO(z) Bo(z dz BO(z) AO(z) ß )I f -1 k f dzIô_.¿'fi L AO(z) Ao(z 2rí k (z) A. (z-KK)z )ir -lAO(z) Bo(z -)k f2rí A. (z) A. (z-Kj<?*32ní-tß dz Ao(z) Ao(z dz)f (r7 )1 -1)z AO(z) Ao(z )zThe first integnal equals Ik. The second integnal can be neducedas followsÕ-1ß BO(z) Ao(z dz f Bo(z) Ao(z)) ß kk2ljkff,g=-1¡1,K22ni Ao(z) AO(z )z2rí Ao(z)dz(z) zkß2ni fB. (z)KOZ BL(o)aå--"k euro>t.kK-rao-.T-A. (z)Kß ß kkI,r/here the finst equality follows fnom ( 4 ) , the thi:rd fnom theresidues theonem and the fifth fnom (10 ) . Similanly we find thatthe thind integnal of the right numben also equals ß1.using (4) the fou::th tenm of the night memben of (17) can beneduced as followst 2AO(z) A (z dz kAo(z) At(z-L) z 2triß f2niSummarizing we find (14).2kþVlhen k = Q we get fnom (13)zI fIo ITL ( boo(Jao )2 z 2dz þwhich completes the pnoof os the theonem.94. The alsorithmHaving obtained the necunsive fonmula (f4) we will now developa computer algor.ithm fon the evaluation of (f ). Ide first obsenvethat if we carry out the reduction of the polynomials At and Btgiven by the equations (7) and (B), Ide obtain oO and ßk in des-cending onder wheneas a direct evaluation of (14) nequires thecoefficients in ascending onder. A dinect application of (7),(B) and (14) will thus nequire temporany sto::age of oO and ßk.To avoid this we obse:rve that (14) is a linean difference equa-tion. The solution can thus be wnitten askwherect-rDL-tandI otToD+î"k DtI n (18)(re)(20)(2L)(22)(23)I'"k--I-o¿-K2D ßk k2kIn0llFork= 0 we get+DoHence22Iaì =QI ß onß DT_T-I-I- n (\ot0The1.,integnal (1) can now be computed recursively as followsSetk=fl:cr,=landDn=oTest if A(z) is stable by checking ": > 0. Stop compu-tation if unstable othenwise pnoceed.Compute oO and ßk fnom (9) and (10) and afrom (7) and (B).Compute Ct_t and DO-, fnom (f9) and (20).Repeat steps 2; 3 and 4 until k = fCompute I = I' from (23)Jk-1 k-1iand bl-456The number of algebnaicthis procedure is showna considerable savingsol-ution of (3) for xoper.ations required to compute I byin table 1a.As seen by this we obtainin companison with a stnaight forwardusing gaussian neduction.IFor. compa:rison we show in tabler.equined for a stnaight forwandgaussian redu.iiottTable la Numberpute Ilb the number of oPerationssolution of (3) for xr usingof ar.ithmetic operations required to com-using the recunsive formulasOr-der of system Add/Subtract Multiply Divide1022424n+22E.t0nT346r4ln2 * 4r,* If14I131n2+ 3n+fTable 1b1INumber- of ar.ithmetic operations nequined to com-pute the integr-al fo::m (3) using gaussian neduc-tionOnden of system Add/Subtnact Multiply Divide2EJl_0nIB24763*1sr,2+2on+12(g)4nL2t3B14613*0r,2*l-6n+62n641666n2 *n*22IrlL25. Refenenceslz)Aström, K.J. rrOn the achievable accuracy in identificationproblemsr', Paper t.B in IFAC Congness on Identifica-tion in Automatic Contnol- Systems Pr:ag 1967.Cohn, A. ttüben die Anzahl der \,tlunzeln in einer Kreisett,Math. Zeitschrift 14 (r922) 110-148.Ig] Ju:ry, E.f . ltTheony and Application of the z-transformMethodrt , \,r/iley, New York 19 6 4 .Nekolnf , J. and BeneÉ, J. rrsimultaneous Control of Stabi-lity and Quality of Adjustment-Application of Statis-tical- Dynamicstt, in Coales et.al . trAutomatic and Re-mote Control" (Proceedings of the finst IFAC CongnessMoscow f 960 ) , Butter.woirths , London t9 61 ,, VoI. 2 ,p 734-744.[+]Is] ðka, J. 'tAlgebraická kritér.ia stability inpulsnichsoustavrr, (Algebraic c::iteria fon stability of samp-led data systems), Strojnicky ðasopis XIIf, ó 5.395-403.Kuz l-Io] Schun, ,J. ttUben Potenzneihen die im fnnennkneises beschränkt sindtt, Journal für'angewandte Mathematik, L47 (1917) pp.(1918) pp. L22-14s.des Einheits-die neine und205-232, 149

Recursive Formulas for the Evaluation of Certain Complex Integrals

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Recursive Formulas for the Evaluation of Certain Complex Integrals

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