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1 Univeity of Eat London Intitutional Repoitoy: hi pape i made available online in accodance with publihe policie. Pleae coll down to view the document itelf. Pleae efe to the epoitoy ecod fo thi item and ou policy infomation available fom the epoitoy home page fo futhe infomation. utho(): Pedeen, Jacob L.; Dodd, Stephen J. itle: oced dynamic contol of non-minimum-phae plant via tudy of the claical inveted pendulum Yea of publication: 00 Citation: Pedeen, J.L. and Dodd, S.J. (00) oced dynamic contol of nonminimum-phae plant via tudy of the claical inveted pendulum Poceeding of, (C&) he School of Computing and echnology 5th nnual Confeence, Univeity of Eat London, pp.5-5.

2 5 ORCED DYNMIC CONROL O NON-MINIMUM-PHSE PLNS VI SUDY O HE CLSSICL INVERED PENDULUM Jacob L. Pedeen, Stephen J. Dodd Delphi Dieel Sytem Ltd, Pak Royal, London, United Kingdom CIE, Univeity of Eat London, United Kingdom Jacob.pedeen@delphi.com, tephen.dodd@pacecon.co.uk btact: he geneal poblem of contolling a non-minimum-phae plant i tackled via tudy of the claical inveted pendulum (IP). full nonlinea model of the IP i ued fo imulation and a lineaied veion i ued fo the contolle deign. he tolley poition i contolled while keeping the pendulum inveted by ue of an input/output feedback lineaiation method called oced Dynamic Contol (DC). hi i geneally moe taightfowad to apply than conventional technique uch a linea tate feedback with pole aignment but in it baic fom yield ight half plane zeo cancellation which ceate an untable cloed loop mode. hi i cicumvented in thi pape by ceating an atificial contolled output that i a weighted um of the tate vaiable uch that the ight half plane zeo do not eit in the tanfe function. uthemoe a non-ocillatoy epone with a pecified ettling time i achieved with the aid of the Dodd ettling time fomula (Dodd, 008). he computational delay intoduced to eliminate the algebaic loop in the nonlinea model i hown to have a negligible effect. Simulation ae peented that demontate the coect opeation of the contol ytem and detemine diffeence between the ideal and actual tep epone due to the nonlineaitie, paametic eo and etenal ditubance.. Intoduction It i well known that linea tate feedback contol (LSC) law fo linea non-minimum phae plant, can be deigned by pole aignment, an acceptable tanient epone being attainable by balancing the ight half plane zeo by mio image pole (anklin et al., 00). hi i applicable, fo eample, to the inveted pendulum when it motion i eticted to mall petubation about an opeating point. he motivation fo applying foced dynamic contol (DC), howeve, i that it i alo applicable to nonlinea plant and i quicke to apply than conventional LSC. Since DC i a time domain method, a diffeent appoach to deal with non-minimum phae plant i needed. It hould be noted hee that the tem non-minimum phae, tictly applie to linea plant and that the equivalent tem coveing nonlinea plant in addition i untable zeo dynamic (Stadle, 008). he diect application of DC to a plant of ank n whee n i the plant ode will leave the zeo dynamic of ode n uncontolled in the cloed loop ytem. he pupoe of thi pape i to invetigate the olution of thi poblem by appending the plant tate pace model with an atificial output equation uch that the new plant ceated i of full ank and the application of DC to thi achieve full tate contol and theefoe avoid the cloed loop intability. he pape addee the choice of the cloed loop dynamic uing the Dodd ettling time fomula (Dodd, 008). In thi pape, the DC deign i baed on a lineaied model of the IP but the

3 6 pefomance of thi deign i aeed fo the tue nonlinea IP a well a the lineaied model by imulation. It i intended that the wok peented hee hould pave the way to geneally applicable DC of plant with untable zeo dynamic.. Inveted pendulum modelling. Nonlinea model he input foce,, i ued fo contolling the movement of the IP, a hown in igue. Y P y P l m, J y olley ma = M olley 0 X P Pivot point igue Inveted pendulum ytem he coodinate, θ (angle of the pendulum) and (poition of the tolley), ae output fom the ingle input, multiple output (SIMO) plant. Lagange method i ued to deive the following equation of motion: b ml co ml M m mgl in ml co m l J in () () b m l M m M m M m mgl ml J l m J l m () () he tate of the IP ae choen a,, and. Equation () and () may then be eplaced by the following tate diffeential equation: B u a 0 0 a b a 0 0 a b he meauement equation i whee y C (5) (6) a M m m g l q 0, a m l b q, a m g l q, a J l m b q, b m l q, b J l m q and 0 q J M m l M m, q J M m l m M he coeponding tate vaiable block diagam ued fo the linea Matlab-Simulink imulation i hown in igue.. Linea IP model Lineaiing () and () about the opeating point, 0 yield:

4 7.... X 5. X. X igue : Linea IP Simulink model X theta. oced Dynamic Contol (DC) he geneal DC method i fully decibed by (Vittek and Dodd, 00). Hee, the goal i to contol the tolley poition while keeping deviation of the pendulum poition about the vetical within acceptable bound. It i evident by inpection of () and () that thi plant i of ode,. he ank w..t. the contolled output,, i due to the diect dependence of on the contol foce,. Diect application of DC would theefoe yield a cloed loop ytem of ode and theefoe only two of the plant tate vaiable, i.e., and, would be contolled. he othe two tate vaiable, and, ae aociated with the zeo dynamic ubytem (). he input of thi ubytem i but thi i not ued to contol the pendulum angle,, which will vay a a ide effect. In the lineaied model, the zeo dynamic i decibed by () and the natual motion of the unfoced ytem, obtained by etting 0, i untable ince the oot of the chaacteitic equation, mgl J l m 0 ae, mgl J l m. o cicumvent thi poblem, the appoach taken i to augment the plant model by ceating an atificial contolled output z C C C C (7) whee the contant coefficient ae choen uch that a) the plant i of full ank, i.e., the ank with epect to z i, and b) contolling z to each a contant demanded value eult in eaching the ame value. igue how a block diagam of thi augmented plant. o conciene, the coefficient ae defined a M m, b, ml, ml J l m and 5 mgl J l m. he tandad DC method will now be applied to the augmented plant of igue. Since the plant ha to be of full ank, C, C, C and C ae choen o that z, z, z and z ae tate vaiable, which i achieved by.... X 5.. X igue : Linea IP with atificial output z. enuing that none of thee vaiable ha diect algebaic dependence on the contol vaiablale,. Diffeentiating (7) yield z C C C C (8) X C C C C Subtituting fo the deivative of the tate vaiable appeaing on the ight hand ide of (8) uing (5) and (6) yield: z C C C C (9) om igue, 5 5 (0) 5 Z

5 8 Subtituting fo z C C C C C in (9) uing (0) yield C 5. () he tem in () mut vanih in ode fo z to be a tate vaiable. Hence C C and C 0 C 0 which yield z C C z C C. () Repeated ubtitution fo and uing the equation implied by igue yield z C C 5 z C C C 5 z C 5 C C 5 () o to vanih, C C, yielding z C 5 and diffeentiating again yield z C 5. () C ha be choen uch that z in the teady tate auming cloed loop tability. hen, cont, 0, 0 and 0 and theefoe the teady tate output atify z C C C. Summaiing, the equied contant ae C, C and C 0. hen (7) yield C 0, z C C (5) z (6) z 5 (7) z 5 (8) he output deivative equation fo DC i then obtained by a futhe diffeentiation: z C 5 Subtituting fo z z C 5 uing (0) yield Solving fo then yield the geneal DC law in which z ha to be choen to achieve the equied cloed loop dynamic: z 5 5 (9) he Dodd 5% ettling time fomula,.5 n (Dodd, 008), will now be c ued to obtain the deied non-ovehooting cloed loop tep epone, zt. hu.5 n z C S z.5 n C n S n (0) Whee z i the efeence input and c i cloed-loop time contant fo the n-ode ytem. o n =, (8) become z a 5 whee a z a fom which S a 6a a a z a z o in the time domain: z a z 6a z a z a a z z a z z a z 6a z a z Uing () to (6) and etting z give

6 9 z a a () 5 5 6a a Subtituting fo z in (7) uing (9) yield the equied DC law: 5 a a 5 5 6a a 5 () o ummaie, an oveall block diagam of the contol ytem i hown in igue. Refeence input: oced dynamic contol (DC) State: Y,,, y P l m, J y olley ma = M IP plant / plant model olley 0 P X igue : DC and IP plant Regading futue pactical implementation, a apid pototyping ytem uch a dspce could be ued. ll the plant tate ae equied but in the eal wold only the tolley poition and the pendulum angle would be meaued. he othe tate, and could be calculated uing oftwae diffeentiation. fomulated above, the contol vaiable i the foce [Nm] but in the eal ytem thi would be implemented a a contol ignal equivalent to the foce fom the poceo. o eample, if the actuato i a DC moto, the toque would be m Km ia R. whee i a i the amatue cuent, K m i the moto toque contant, R i the geabo atio and P Pivot point i the tuck wheel adiu fom which i R. K. hi would then be the a m cuent demand of a elatively high bandwidth cuent contol loop a hown in igue 5. oce equivalent cuent (fom the DC) Cloed loop cuent contol i a DC moto m Geabo & wheel igue 5: Contol foce implementation.. Simulation Paamete: he paamete ued ae a follow unle othewie pecified: Gavitational acceleation: g 9.8 m ec ; olley ma: M 0. kg ; Pendulum ma: m 0. length: l 0.5 the pendulum pivot point: kg ; Pendulum m ; Moment of inetia about J ml kg m. Simulation with linea plant model: In all the imulation peented below, all the tate vaiable tat at zeo, the 5% ettling time i et to, a poitive tep tuck poition efeence, t i applied at t and an equal and oppoite tep efeence input i applied at t 7. Inpection of igue eveal an algebaic loop,. It i well known that Matlab/Simulink doe not accept thi. imple olution of thi poblem i to inet a time delay of one numeical integation tep, h, in the loop to ende the imulation a caual ytem. But thi doe intoduce a modelling inaccuacy. he autho theefoe conideed it neceay to have a linea IP model without uch a delay to validate the DC pefomance and accuacy. hi i deived in the appendi (0). igue 6 how

7 50 the epone to two conecutive oppoitely igned tep of m magnitude of the DC applied to the afoementioned model upeimpoed on the tep epone of the nominal cloed loop ytem with tanfe function (8) fo n, which i ued a a benchmak. hee epone ae not ditinguihable fom one anothe and pa though the 0.95 line at t, confiming thei coectne. igue 8 how the coeponding tolley poition, indicating the undehoot following the tep change in that ae typical with non-minimum-phae plant. igue 6: Validation of DC igue 8 olley poition t le than 0%, of the tep efeence input magnitude, thi undehoot i conideed to be acceptable. he coeponding pendulum angle plotted in igue 9 i kept within deg. of the vetical poition, confiming the effective contol of all the plant tate. no eo between the two epone i viible in igue 6, igue 7 how thi eo on a viible cale, poving that it i negligible. he peudo andom behaviou i due to the numeical integation opeating at a finite wod-length. igue 7: Eo between z of imulated DC without the plant delay and nominal z. igue 9: Pendulum poition (Vetical = 0) igue 0 how the eo coeponding to igue 9 when the plant model without the algebaic loop (igue in the ppendi to thi pape) i eplaced by the baic one of igue but with a time delay of h m intoduced between and the gain,. lthough thi ha inceaed the eo magnitude by a facto of

8 5 appoimately 0, the peak eo i le than 0 m in magnitude, which can be conideed acceptable when compaed with the poition change of m. of the m tep efeence magnitude, which i conideed acceptable. Vaiation of the tep epone with inceaing efeence input magnitude i hown in igue. Depite the plant nonlineaity, the tep epone hape doe not vay viibly with the efeence input magnitude. igue 0 Eo between z of imulated DC with the plant delay and nominal z. Simulation with nonlinea plant model: igue how a tep epone eo coeponding to igue 0 but with the nonlinea plant model uing the ame DC contol law a applied above to the linea plant model. igue : Step epone with nonlinea plant model and inceaing efeence input magnitude. Net the obutne of the ytem to plant paamete vaiation fom the nominal value ha been invetigated with +/- 0 % mimatche of the pendulum length, pendulum ma, tolley ma and the wheel beaing fiction, in all combination. igue how the nominal eo with no plant paamete mimatche, identical to that of igue, togethe with the eo with the maimum poitive and maimum negative peak taken fom the complete et of imulation to how the wot cae. igue : Eo between nominal z and z of DC with nonlinea plant model and delay. can be een, the eo inceae futhe in magnitude by a facto of appoimately 00 but it till only peak at appoimately 0.8%

9 5 igue : Nominal eo and wot cae eo envelope fo +/- 0% plant paamete vaiation. he peak wot cae eo of m i acceptable. igue how the coeponding wot cae pendulum angle, t, taken fom the complete et of imulation upeimpoed on the pendulum angle with no paamete mimatching. It i evident that the paamete mimatching ha a negligible influence on. output with full ank i applied. hi technique may pove effective fo many othe nonminimum-phae plant and theefoe futhe invetigation in thi diection ae ecommended. Regading the IP, application of DC uing the full nonlinea plant model i ecommended a thi would be a ueful peliminay tudy fo contol of othe nonlinea non-minimum-phae plant fo which the contol ytem pefomance may moe citically depend on ue of the nonlinea model. he main poblem to be olved fo the IP i the fomulation of the full ank fictitiou output equation fo the nonlinea cae. lo, a peliminay tudy of DC of the IP with the tolley on an inclined lope ha been caied out by the autho and thi indicate a teady tate tolley poition eo. modification of the DC method to eliminate thi eo would be a wothwhile futhe invetigation a the geneal method could be applied to othe plant with ignificant contant o lowly vaying etenal ditubance. 6. Refeence Dimaogona. D., Haddad S., Vibation fo Enginee, Pentice Hall, Englewood Cliff, NJ, 99. Dodd S. J., "Settling ime omulae fo the Deign of Contol Sytem with Linea Cloed Loop Dynamic, Poceeding of C&, Univeity of Eat London, 008. igue : Wot cae pendulum angle ecuion fo +/- 0% plant paamete vaiation. 5. Concluion and Recommendation he invetigation how that DC i vey effective in contolling the non-minimum phae plant coniting of a tolley uppoting an IP when the technique of ceating a fictitiou plant anklin G.., Powell D. P. Emami-Naeini., eedback Contol of Dynamic Sytem (outh Edition), Pentice Hall, Uppe Saddle Rive, NJ, 00. Stadle P.., Dodd S. J., "Modelling and Contol of a Vacuum i Beaing Linea Dive in the Nanomete Range", Poceeding of C&, Univeity of Eat London, 008.

10 5 Vittek J., Dodd, S. J., oced Dynamic Contol of Electic Dive, Zilina Univeity Pe, Slovakia, 00. Wood R. L., Lawence, K. L., Modeling and Simulation of Dynamic Sytem, Pentice Hall, Englewood Cliff, NJ, ppendi Plant model without algebaic loop Uing Maon ule on igue : () 5 () 5 5 () b 5 () a a a () whee b0, 0 0 a, a 5 and a0 5. o the output : a a a 0 b a a a 0 () whee: b. igue how the coeponding Simulink block diagam which i the tate vaiable block diagam in the contol canonical fom ealiing tanfe function () and (). a a a0 X b0 5 b b X_dot X theta_dot theta igue : Linea IP model without algebaic loop