DYNAMIC FEEDBACK STABILIZATION OF NONLINEAR RC LADDER NETWORK

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1 ELEKRYKA Zeszyt (3 Ro LVI Paweł SKRUH, Jerzy BARANOWSKI, Wojcech MIKOWSKI Departmet of Automatcs, AGH Uversty of Scece ad echology DYNAMI FEEDBAK SABILIZAION OF NONLINEAR R LADDER NEWORK Summary. he goal of ths paper s to study stablzato techques for a class of R ladder etwors. he system s a electrcal crcut that cossts of olear resstors ad capactors. he crcut's dyamc behavor ca be modeled by olear dfferetal equatos. he problem s to determe the dyamc feedbac cotrol law that asymptotcally stablzes the system. It s show that wth a lear ad olear dyamc feedbac cotrol, the eergy of the closed-loop system asymptotcally decays to zero. he asymptotc stablty of the closed-loop system s proved by LaSalle's varace prcple usg approprate Lyapuov fucto. he results of umercal computatos are cluded to verfy theoretcal aalyss ad mathematcal formulato. Keywords: olear R ladder etwor, stablzato, dyamc feedbac, Lyapuov fucto SABILIZAJA NIELINIOWYH OBWODÓW DRABINKOWYH R ZA POMOĄ DYNAMIZNEGO SPRZĘśENIA ZWRONEGO Streszczee. W pracy rozwaŝoo zagadee stablzacj dla wybraej lasy uładów drabowych typu R. Wybraa lasa uładów obejmuje obwody eletrycze, sładające sę z rezystorów odesatorów o elowych charaterystyach. Dyama uładu jest opsywaa za pomocą elowych rówań róŝczowych. W pracy poazao, Ŝe zastosowae dyamczego sprzęŝea zwrotego asymptotycze stablzuje system do zera. RozwaŜoo zarówo lowe, ja elowe sprzęŝee zwrote. Własość asymptotyczej stablośc uładu zamętego została poazaa z wyorzystaem odpowedch fucjoałów Lapuowa oraz twerdzea LaSalle a. Wy teoretycze zostały zweryfowae przez oblczea umerycze symulacje omputerowe. Słowa luczowe: elowe obwody drabowe R, stablzacja, dyamcze sprzęŝee zwrote, fucjoały Lapuowa. INRODUION Almost all real systems are olear ad t s well ow that olearty requres advaced aalyss [,, 5, 4, 3]. he dyamcs of olear system s dffcult to aalyze

2 P. Sruch, J. Baraows, W. Mtows ad troduces to terestg pheomea such as bfurcatos, lmt cycles ad chaos. O the other had, olear electroc elemets have a wde rage of use may areas of electrcal egeerg. hey are corporated to crcuts order to desg electroc devces wth specfc features le parametrc amplfers, up-coverters, mxers, low-power mcrowave oscllators, electroc tug devces, etc. ypcal olear elemets clude olear capactors (varactor dode, jucto dode, olear ductors (saturable core ductor, Josephso juctos, ferroresoat power systems ad olear resstors (tuel dode, thyrstor, dead-zoe coductor, serally coected Zeer dodes, eo bulb, etc.. he propertes of electrcal ladder etwors have bee a subject of earler research. otrol problems for lear RL, R, L ad RL electrcal crcuts are wdely dscussed [7-9]. he dyamcs ad detaled characterstcs of olear electrcal crcuts are cosdered [6]. he papers [, ] cope wth lear ad olear stablzato techques for a olear RL crcut. o stablze the system, Authors have costructed varous forms of the feedbac. he asymptotc stablty of the closed-loop system has bee proved by LaSalle's varace prcple [4] usg specal Lyapuov fuctos. I [] stablzato problem of olear RL ladder etwors wth lear dyamc feedbac cotrol was cosdered. Research cluded ths paper was maly motvated ad spred by results obtaed [3, -3, 5-9]. hey have played a crucal role ad cleared the way to the ma results. he paper s orgazed as follows. I the ext two sectos, we shortly descrbe olear resstors ad olear capactors. I Secto 4, a olear R ladder etwor s troduced ad ts dyamcs s descrbed. I Secto 5, lear dyamc feedbac cotrol law s proposed ad t s proved that ths feedbac asymptotcally stablzes the system. Secto 6 s devoted to the system stablzato usg olear dyamc feedbac cotrol. I Secto 7, a smulato example s preseted. ocludg remars are gve Secto 8.. NONLINEAR RESISORS A resstor s a electrc devce that defes statc relato betwee voltage ad curret. hs statc relato s represeted by the equato r ( v, = voltage ad the curret ad r :R R R s a fucto., where v ad deote the A resstor s lear f the fucto r s lear. I that case, r ( v, ca tae several forms from whch the most famous s mpedece form: measured Ohms [ Ω ]. hs formula s also ow as Ohms' law. v = R, where R s the resstace A resstor s olear, f r s olear the relato r ( v, that ls the voltage v to the curret. We call a resstor dsspatve f for all real umbers v ad we have that ( v, = v r. (

3 Dyamc feedbac stablzato he product v represets the power suppled to the compoet; therefore a dsspatve resstor s characterzed by the property that o voltage-curret par ca produce egatve power. Most resstors electrocs are dsspatve compoets but o-dsspatve resstors certaly exst ad are owadays assembled usg semcoductors. I ths paper, t s assumed that the voltage drops across resstors preseted Fg. ca be modeled as where deote the currets the crcut, resstaces, =,, K,. v R dp = ( R( = R( p&, p represet the electrc charges, R stad for the 3. NONLINEAR APAIORS A capactor s defed as a electrc compoet whose charge s a fucto of voltage. Its capactace s defed as the dervatve of the charge q wth respect to the voltage v ( q ( v dq =. (3 dv he curret flowg through a capactor s smply the tme dervatve of the charge ( v dq =. (4 If a capactor s lear ts charge s v q =, (5 ad so the curret through the capactor s ( v d dv = =. (6 Modelg a olear capactor by replacg wth ( q (6 s geeral a ot good approach, because ( q vares wth tme. However, f s ot a strog fucto of q ad q does ot vary sgfcatly wth tme, we ca use t as approxmato our aalyss. I ths paper, we assume that the voltage drops across capactors preseted Fg ca be wrtte as v = j t = d q, (7

4 P. Sruch, J. Baraows, W. Mtows where j deote the currets the crcut, q represet the electrc charges, capactaces, =,, K,. stad for the 4. SYSEM DESRIPION Let us cosder a olear aalog crcut show Fg.. he crcut cossts of a set of resstors ad capactors that are coected together to form a etwor. he resstors ad capactors have geeral olear characterstcs. Fg.. R ladder etwor Rys.. Obwód drabowy typu R Accordg to Krchhoff's voltage law, the currets the crcut are modeled by the equatos: R R ( + j = d ( t u + j j =, (8, (9 ad so o R ( + j j =. ( Applyg Krchhoff's curret law to the crcut yelds j + =, ( j =, ( + 3 ad cosequetly j = +, (3 j =. (4

5 Dyamc feedbac stablzato 3 Itroducg the otato = [ K ( ], (5 t = [ j j K j ( ] j, (6 t = [ p p K p ( ] p, (7 t = [ q q K q ( ] q, (8 t dp dq =, j =, (9 ad substtutg the varables j (8-( by the expressos (-(4 we get R R ( p p + dp p R ( p& = u, ( p + 3 dp p p p & =, ( ( p p + dp p p & =. ( Wthout loss of geeralty t ca be assumed that ( p& R ( p& R =, (3 for =,, K, ad ( p, p ( p = + =, (4 for =,, K,, ad ( p ( p = =. (5 he, the crcut s dyamc behavor ca be modeled by the followg equato: ( p p& + ( p p Bu R & =, (6 where R ( p& dag( ( p& R ( p& K R ( p& =, (7 R

6 4 P. Sruch, J. Baraows, W. Mtows d e K f d e3 K f3 d3 K ( p =, (8 M M M O M M K d e K f d where d =, (9 ( p d = + ( p ( p, (3 for =, 3, K,, e = f =, (3 ( p [ K ] B =. (3 he tal codto ( p = p s gve as well. he objectve of the paper s to study the R ladder etwor (6 uder the followg codtos: (A he fuctos R (, ( the set Ω, where (A R ( ξ >, ( ξ > Lemma, =,, K, are cotuous wth cotuous dervatves Ω R s a eghborhood of zero; for ξ Ω ad ξ =,, K,., ( ξ ξ = ξ ( ξ ξ > for Ω\ { } Proof. he lemma s equvalet to the statemet that for every ξ. (33 ξ Ω the matrx ( ξ s postve defte. It ca be observed, that the matrx ( ξ s a trdagoal matrx ad ts determat ca be computed by the recursve formula [4, p. 54]. Usg ths formula, the leadg prcpal mors M ca be expressed as follows: M = K, (34 ( ξ ( ξ ( ξ for =, 3, K,. he leadg prcpal mors =,, K,. hs meas that the matrx ( ξ s postve defte. ξ for M are postve, because ( >

7 Dyamc feedbac stablzato 5 5. LINEAR DYNAMI FEEDBAK ONROL LAW Let us cosder the system (6 wth lear dyamc feedbac gve the followg form: where R w, a >, b >, >. he resultg closed-loop system becomes u + aw = bu, w ( w w & =, (35 ( ( w + p = w + B p =, (36 ( p p& ( p ( + BB p + Bw( = R & + t, (37 + ( a + b w + bb p = w&. (38 heorem. Suppose the assumptos (A-(A hold. he the closed-loop system (37, (38 s locally asymptotcally stable. Proof. he proof reles o mposg a sutable Lyapuov fucto for the closed-loop system (37, (38. Followg caddate s cosdered: p a V p,w = w t + b ( ( ( ( ξ ξ d ξ + + ( w B p. (39 he tegral the formula (39 deotes a le tegral alog the straght le the space from the begg pot to the edg pot p. It ca be prove by usg Lemma that ths tegral s postve for ξ ad equals zero for ξ =. he dervatve of the fuctoal (39 becomes V& a b a = w t b ( p,w = w w& + ( ( p p p& + w + B p ( ( w& + B p& (. ( w& + p ( p p& u w& + B p& R (4 Evaluatg the tme dervatve of V alog the soluto of the system (37, (38 gves V& a b ( p,w = w ( bu aw + p ( p R( p& ( Bu ( p p u ( bu aw u B R( p& Bu ( p p (. (4 After some elemetary calculatos V& b ( p,w = ( bu aw ( Bu ( p p R( p& Bu ( p p (, (4

8 6 P. Sruch, J. Baraows, W. Mtows t ca be see that V& b ( p,w = w& p& R( p& p&. It should be oted that V ( p,w > for col ( p w Ω \ { }, V (, = ad V ( p,w ( p, w Ωc, c col, where Ω c s a compact set defed as follows: Ω { z = ( p,w : p Ω, w R,V ( z c} c = < c s a real postve umber, col(, w [ p w ] (43 & for col, (44 p =. he ext part of the proof s based o the LaSalle s varace prcple [4]. Accordg to ths prcple, the trajectores of the closed-loop system (37, (38 startg largest varat set S, where { z Ω V& ( =} Ω c eter the S = : z. (45 From V & ( z = t follows that w& = ad p& =. hs mples that w ( t p p =. Usg ths result (6 ad (35 yelds c a ( p p B w = b w = ad. (46 It s easy to show that the equato (46 has soluto oly for w = because a >, b > ad Lemma s vald. hus w = ad = p for all > t. hs meas that S = { } cotas oly the zero soluto, ad by LaSalle s prcple, the org asymptotcally stable ( the Lyapuov sese. + R s 6. NONLINEAR DYNAMI FEEDBAK ONROL LAW Let us cosder the system (6 wth olear dyamc feedbac gve the followg form: u = K + aw = bu, w ( w w & =, (47 ( w + p ( w + B p =, (48 K where K ( K + w + B p( = γ, (49 t ad K >, γ >.

9 Dyamc feedbac stablzato 7 he closed-loop system s gve by the followg equatos: R( p& p& + ( p + BB p + Bw =, (5 K K b b w& + a + w + B p = K K. (5 heorem. Suppose the assumptos (A-(A hold. he the closed-loop system (5, (5 s locally asymptotcally stable. Proof. he proof ca be dvded to two parts. Frst, t ca be proved that V p a K t = d l b ξ + λ K ( p,w w + ( ( ξ ξ ( (5 s the Lyapuov fucto for the system (5, (5. he, followg the method as the prevous secto, t ca be cocluded by LaSalle s theorem [4] that the trajectores ted to the org { } as t goes to fty. 7. SIMULAION EXAMPLE R crcuts are useful, smple ad robust passve electrc crcuts. hey play tegral roles everyday electroc equpmet such as traffc lghts, pacemaers, audo ad rado equpmet. Whle ther applcatos are umerous ad vared, they are mostly used for ther sgal flterg capabltes ad precse tmg abltes (for example: We-brdge oscllator, phase-shft oscllator, hgh pass ad low pass flters, etc.. I ths secto, a smple R crcut wth olear elemets s aalyzed to verfy mathematcal formulato from the prevous sectos. Let us cosder the crcut preseted Fg.. Fg.. R ladder etwor wth = Rys.. Obwód drabowy typu R dla =

10 8 P. Sruch, J. Baraows, W. Mtows he resstaces R ad R are lear [ Ω], R = [ Ω] R =. 3. (53. he characterstcs of the capactaces ad are olear ad gve the aalytc form: ( ( = exp(. p ( = 5exp. ( p p p, (54 3 p. (55 5 he dyamcs of electrc charge flow the crcut ca be descrbed by the equato ( p p& + ( p p Bu R & =, (56 where R R ( p& =, (57 R ( p = ( p ( ( ( ( p p p + p [ ], (58 B =, (59 ( = [ p p ] p. (6 t Here p represets the vector of electrc charges, u s the cotrol voltage, R u R, t > p,. he voltage of the power source s measured volts [V], the resstace of the resstors s measured ohms [Ω], the capactace of the capactors s measured farads [F] ad the charge across the capactors s measured coulombs []. he followg tal codtos are used for the dfferetal equato (56: ( 3., p ( p = =. (6. Let us troduce oe-dmesoal parallel compesator +. w = 5. u, w( = w&, (6 ad desg the cotroller. 5( w + p u =. (63 It s easy to chec that the assumptos (A-(A hold as well as Lemma s vald. Accordg to heorem the closed-loop system (56, (6, (63 s asymptotcally stable. he trajectores of the ope-loop system (dot le ad closed-loop system (sold le are show Fgs. 3-5.

11 Dyamc feedbac stablzato 9 Fg. 3. he electrc charge p the ope-loop crcut (dot le ad closed-loop crcut (sold le. he same smulato results are show dfferet tme wdows p t w uładze otwartym (la przerywaa oraz w uładze zamętym (la cągła. e sam przebeg jest poazay w róŝych oach czasowych Rys. 3. Dyama zma ładuu eletryczego ( Fg. 4. he electrc charge p the ope-loop crcut (dot le ad closed-loop crcut (sold le Rys. 4. Dyama zma ładuu eletryczego p w uładze otwartym (la przerywaa oraz w uładze zamętym (la cągła

12 3 P. Sruch, J. Baraows, W. Mtows Fg. 5. he state varable w of the compesator. he same smulato results are show dfferet tme wdows w t ompesatora dyamczego. e sam przebeg jest poazay w róŝych oach czasowych Rys. 5. rajetora ( 8. ONLUSIONS I the paper, lear ad olear techques for stablzato of a class of olear R ladder etwors have bee vestgated. It has bee show that the system s asymptotcally stable whe lear dyamc feedbac s appled. he asymptotc stablty of the closed-loop system has bee proved by LaSalle's varace prcple usg approprate Lyapuov fucto. he smlar results have bee obtaed wth olear dyamc feedbac. Numercal calculatos ad computer smulatos have bee performed the MathWors MALAB /Smul evromet to show the effectveess of the proposed methods. AKNOWLEDGEMEN hs wor was supported by Mstry of Scece ad Hgher Educato Polad the years 8- as a research project No N N

13 Dyamc feedbac stablzato 3 BIBLIOGRAPHY. Gucehemer J.M., Holms P.: Nolear Oscllatos, Dyamcal Systems ad Bfurcatos of Vector Felds. Sprger, Berl Hayash.: Nolear Oscllatos Physcal Systems. McGraw-Hll, New Yor Kobayash.: Low ga adaptve stablzato of udamped secod order systems. Archves of otrol Sceces, Vol., No. -, p LaSalle J., Lefschetz S.: Stablty by Lapuov's Drect Method wth Applcatos. Academc Press, New Yor, Lodo Morsy N.: he heory of Nolear otrol Systems. PWN, Warszawa Mtows S.: Nolear Electrc rcuts. Wydawctwa AGH, Kraów Mtows W.: Stablzato of Dyamc Systems. WN, Warszawa Mtows W.: Dyamc feedbac L ladder etwor. Bullet of the Polsh Academy of Sceces: echcal Sceces 3, Vol. 5, No., p Mtows W.: Stablzato of L ladder etwor. Bullet of the Polsh Academy of Sceces: echcal Sceces 4, Vol. 5, No., p Mtows W.: Aalyss of udamped secod order systems wth dyamc feedbac. otrol ad yberetcs 4, Vol. 33. No. 4, p Mtows W., Sruch P.: Stablzato of secod-order systems by lear posto feedbac. Proc. of the th IEEE Iteratoal oferece o Methods ad Models Automato ad Robotcs, Mędzyzdroje, Polad, 9 August September 4, p Mtows W., Sruch P.: Stablzato methods of a o-lear oscllator. Proc. of the th IEEE Iteratoal oferece o Methods ad Models Automato ad Robotcs, Mędzyzdroje, Polad, 3 August September 5, p Mtows W., Sruch P.: Stablzato results of secod-order systems wth delayed postve feedbac. I: Modellg Dyamcs Processes ad Systems, Edted by W. Mtows ad J. Kacprzy. Seres Studes omputatoal Itellgece 9, Vol. 8, p. 99-8, Sprger, Berl, Hedelberg Moo F..: haotc Vbratos: A Itroducto for Appled Scetsts ad Egeers. Joh Wlley & Sos, New Yor Sruch P.: Stablzato of secod-order systems by o-lear feedbac. Iteratoal Joural of Appled Mathematcs ad omputer Scece 4, Vol. 4, No. 4, p Sruch P.: Stablzato of lear fte dmesoal oscllatory systems. PhD dssertato, Aadema Górczo-Hutcza, Departmet of Automatcs, Kraów Sruch P.: Stablzato methods for olear secod-order systems. Archves of otrol Sceces 9, Vol. 9, No., p. 5-6.

14 3 P. Sruch, J. Baraows, W. Mtows 8. Sruch P.: Feedbac stablzato of a class of olear secod-order systems. Nolear Dyamcs, Vol. 59, No. 4, p Sruch P.: Feedbac stablzato of dstrbuted parameter gyroscopc systems. I: Modellg Dyamcs Processes ad Systems, Edted by W. Mtows ad J. Kacprzy, Seres Studes omputatoal Itellgece, Vol. 8, p , Sprger, Berl, Hedelberg 9.. Sruch P.: Stablzato of olear RL ladder etwor. Proc. of the 7th oferece o omputer Methods ad Systems, 6-7 November 9, Kraów, p Sruch P., Baraows J.: Lear feedbac cotrol of a olear RL crcut. Proc. of the 3th Iteratoal oferece o Fudametals of Electrotechcs ad rcut heory I-SPEO 9, Glwce Ustroń, Polad, -3 May 9, p Sruch P., Baraows J.: Nolear feedbac cotrol of a olear RL crcut. Proc. of the 3th Iteratoal oferece o Fudametals of Electrotechcs ad rcut heory I-SPEO 9, Glwce Ustroń, Polad, -3 May 9, p Nayfeh A.H., Moo D..: Nolear Oscllatos. Joh Wley & Sos, New Yor urowcz A: heory of Matrces, 6 th Edto. Wydawctwa AGH, Kraów 5. Wpłyęło do Redacj da weta r. Recezet: Dr hab. Ŝ. Zbgew Goryca, prof. Pol. Radomsej Omówee W pracy rozwaŝoo zagadee stablzacj dla wybraej lasy uładów drabowych typu R. Wybraa lasa uładów obejmuje obwody eletrycze, sładające sę z rezystorów odesatorów o elowych charaterystyach. Wartość rezystacj dla rezystorów elowych jest fucją prądu. Nelowy odesator charateryzuje sę tym, Ŝe jego pojemość zaleŝy od apęca występującego a oładzach odesatora lub od ładuu zgromadzoego a tych oładzach. W rzeczywstośc wszyste obwody eletrycze są elowe, gdyŝ wszyste elemety rzeczywste wyazują cechy elowośc. Dyama uładu drabowego R moŝe być modelowaa matematycze za pomocą elowych rówań róŝczowych zwyczajych. Na podstawe tego modelu sostruowao dyamcze sprzęŝea zwrote, tóre asymptotycze stablzują system. Zapropoowao zarówo lowe, ja elowe regulatory. Własość asymptotyczej stablośc uładu zamętego została poazaa z wyorzystaem odpowedch fucjoałów Lapuowa oraz twerdzea LaSalle a. Wy teoretycze zostały zweryfowae przez oblczea umerycze symulacje omputerowe.

15 Dyamc feedbac stablzato 33 Obwody elowe są powszeche stosowae w urządzeach eletryczych eletroczych. Dzę elemetom elowym jest moŝlwe realzowae tach czyośc, ja prostowae, stablzacja apęca prądu, modulacja detecja sygałów, wytwarzae sygałów o róŝych ształtach tp. Aalza uładów elowych jest truda bardzo często przyblŝoa. Metoda badaa tach uładów polega zazwyczaj a learyzacj poszczególych elemetów lub aalze umeryczej. Dlatego teŝ a szczególą uwagę zasługują rozwązaa aaltycze, tóre uwzględają elowośc występujące w uładze.