Control of Electromechanical Systems using Sliding Mode Techniques

Transcription

1 of Eectroechanica ing Siding Mode Techniqe Heide Brandttädter and Martin B Abtract Thi artice propoe a iding ode contro for eectroechanica yte, for intance a DC otor with an inverted pend a oad i conidered. In contrat to conventiona cacade contro trctre not ony the variabe of the echanica yte bt ao the eectrica variabe are part of the contro aw and votage i ed a dicontino contro inpt. The new contro approach offer better perforance, iniia ipeentation copexity, provide robtne, and a decreae of power conption. The perforance of the preented approach i deontrated via nerica iation and a rea experient. I. INTRODUCTION It i coon to deign contro for echanica yte with torqe or force a the contro action. It i aed that there exit a fat inner contro oop providing a deired crrent i. Therefore, for the peed controer in the oter oop the crrent contro oop i treated a an idea crrent orce, which ean a given reference crrent i wi be rag repaceent tracked ideay. An exape for ch a contro chee i hown in fig. for a 2nd order echanica yte driven by a DC otor. There i a ow-eve feedback oop ing a pe width odated (PWM) igna or inear apifier providing the deired crrent or torqe. The deired crrent i given by the contro aw for the echanica yte. i PWM crrent feedback oop i Mech., Fig.. Conventiona cacade contro trctre for a 2nd order echanica yte driven by a DC otor For high perforance yte there are iit of thi contro chee. The bandwidth of the inner contro oop i iited, becae the deired crrent i i an integrated PWM igna. Hence, the perforance of the oter contro oop, the perforance of the echanica yte, i iited by thi bandwidth of the crrent contro oop. Particary de to a very owy oving echanica yte, poition enor data are iited and therefore a fficient etiation of the peed igna i not poibe and the accrate contro of the echanica yte i retricted. An additiona enor for the eectrica variabe of the yte cod offer better Both athor affiiation: Intitte of Atoatic Engineering, Techniche Univerität München, D-829 München, Gerany http : // heide.brandttaedter@t.de b@t.de etiation of the not eared peed variabe of the echanica yte. If a PWM igna i ed, crrent contro i baed on an exacty defined witching freqency. In coneqence, independent of the contro objective thi extreey high witching freqence ing axia power i away appied. If the echanica variabe and are ao taken into accont for deign of contro aw, redction of energy can be achieved. To overcoe the above decribed probe, thi artice invetigate a contro chee for eectroechanica yte ing votage a the dicontino contro inpt. Additiona to the echanica variabe, the variabe of the eectrica yte, the crrent, infence the contro aw. Siding ode contro i choen becae it offer robtne a we a fat dynaic. Frtherore ipeentation of iding ode contro a fat witching contro i ao practicabe. Severa appication of iding ode contro of DC otor, indction otor and ynchrono otor have been propoed, e.g. in [].In the oter contro oop of the contro chee hown in fig. PD or PID contro i ay ipeented. Moreover, in order to iprove robtne, tracking probe for poition and angar peed of a DC otor ao have been oved baed on a iding ode contro approach [2]. Iproveent of robtne by adding iding ode contro of echanica yte for an indction otor drive with forced dynaic ha been hown [3]. Neverthee, thi approach ti i baed on a cacade contro trctre aing fat idea ow-eve feedback oop. The contro chee for an eectroechanica yte which i propoed in thi artice e advantage of iding ode contro and at the ae tie chattering effect are decreaed becae the variabe of the eectric drive are part of the contro aw. The eectric otor act a fiter for high-freqency igna. Copared to the hown cacade contro ing iding ode contro trctre yte dynaic becoe fater. The yte perforance i characterized by inenitivity to paraeter variation and rejection of ditrbance de to iding ode contro characteritic a we a iniia ipeentation copexity and redction of energy. Thi artice anayze iding ode contro of eectroechanica yte. In ection II the propoed contro chee expained for arbitrary eectroechanica yte receive priority conideration. Afterward it i deveoped for a ape yte, an inverted pend driven by a DC otor. Section III dice the ret of nerica iation and the experienta ret. Perforance of the propoed iding ode contro i copared to that of a conventiona contro ethod.

2 II. DESCRIPTION OF THE CONTROL ALGORITHM t be ffied.[4] frag repaceent In thi artice noninear n-dieniona contro affine yte A. Matheatica Mode of the conidered Eectroechanica PSfrag repaceent i ẋ = f(x) + g(x) x R n, R () The dynaic of the eectrica yte are given by PWM are conidered. For eectroechanica yte the yte i DC tate Motorx contit of the tate variabe of the eectrica bytei x eectr and the variabe of the echanica byte PWM L i = a R a i K n (4) x ech. For the yte () the idea of the propoed iding = K i T (5) ode contro chee i hown in fig. 2. In contrat to fig. for, contro where i i theiaratre crrent, a the ppied votage, R a deign of the echanica yte additionay to the aratre Mech. reitance and L the aratre indctance; K crrent feedback theoop echanica dynaic, the dynaic of the DC otor are repreent the torqe contant and K n the indction contant taken into accont. of the DC, otor; i the angar peed, i the oent crrent offeedback inertia of oop the yte and T i the oad. SMC Eectric Mech. SMC Actator Eectric x eectr Actator x ech x ech = x eectr Fig. 2. Scheatic diagra of the propoed contro chee for eectroechanica yte. Foowing the idea of iding ode contro, the contro inpt i eected a a dicontino fnction of the yte tate (x) = { + (x) if (x) > (x) if (x) < (x) R. (2) in order to enforce deired dynaic of the echanica yte which are given by (x) = x (t) x(t) = with x (t) a reference inpt. The contro chee wi be derived for a ape yte which conit of an inverted pend driven by a DC otor (fig. 3). The contro inpt of the eectroechanica yte of third order i a caar variabe, the appied votage to the DC otor a. The iding anifod given by (x) = divide the tate pace into two bpace. The contro inpt (x) = a, which i defined everywhere in the tate pace except for the iding anifod, i deigned in ch a way that the tangent vector of the tate trajectory point toward the iding anifod. Hence, there exit a neighborhood of the iding anifod, ch that once the trajectory enter the neighborhood, it tay within thi neighborhood for a beqent tie. More preciey, finite aping tie i aed within thi neighborhood, the yte tate ove fro one ide of the iding anifod to the other. Finay, witching at high freqencie, theoreticay infinitey fat witching ead to a iding otion of the yte aong the iding anifod. The yte foow the dynaic given by (x) =. To ake re that the yte reache the iding anifod independent of the initia condition, the iding ode condition i ṡ(x) < and i ṡ(x) > (3) (x) + (x) otor Fig. 3. Scheatic diagra of the conidered eectroechanica ape yte, an inverted pend actated by a DC otor. The echanica yte hown in fig. 3 define the oad on otor ide T = c g in( ) (6) where c i a frictiona contant ( N rad ), repreent the gear redction of the otor, i the pend a, g the gravitationa contant, the pend ength and the ange of the pend with the pright poition being zero. Ange and angar peed are conidered to be on the otor ide before the gear. With [x, x 2, x 3 ] = [,, i] and the anipated variabe = a, the votage ppy, fro (4), (5) and (6) the tate decription of the yte i given by ẋ = x 2 (7) ẋ 2 = (K x 3 c x 2 + g in(x )) (8) ẋ 3 = L ( R ax 3 K n x 2 ) (9) For dynaic anayi of the dynaic of the yte a inear tate decription of the yte i deired. With a = g the foowing cacation can be done:

3 PSfrag repaceent a in( x ) = a with ξ = in( x ) x x = a ξx in( x ) x = i( x ) () The fnction ξ(x) = i(x) i bonded. x R : ξ(x).272;. That ean the noninear part of the differentia i eqation (8) can be approxiated by a inear ter with PWM a bonded coefficient: a in( x ) = (a + )x with.273 a Mech. () ;, crrent feedbackthe oopcontroer for the yte i eected a SMC Eectric = ign((x)) (2) Actator with x (x) = k x + k x 2 + k 2 x 3, k, k, k 2 R. In order ech to x e thi controer, it t be garanteed that the iding eectr ode condition (3) i ffied. B. Oberver Deign Since the cacation of oad torqe i not accrate enogh, de = to nknown friction torqe of the otor and gear, a non negigibe error occr in the angar peed. In order to rectifythi error a we a to redce the nber of enor a iding ode oberver i ed to etiate the angar peed otor (fig. 4). a Fig. 4. DC otor ode of the DC otor î i ign() Scheatic diagra of the oberver Aing that the crrent i i earabe, baed on one of the eqation of the DC otor (4), the oberver i deigned a foow: L dî dt = a R a î (3) The ter, which repreent the nknown ter K n, i defined a a witching fnction of the tracking error of the oberver = o ign (î i) (4) where o i a contant factor. The contant o i choen in ch a way that iding ode i enforced in the anifod i = î i =. Once iding ode i enforced, the differentia eqation (4) and (3) have the ae otion and the ter R a i and R a î are eqivaent. Therefore the ter and K n have to be eqa. The nknown angar peed can be etiated. Siding ode i then achieved in a finite tie if the iding ode condition (3) i atified, which ean i and ṡ i have to have oppoite agebraic ign and i for i =. Therefore for a tabiity anayi the differentia eqation Lṡ = K n ign( i ) R i (5) ha to be anayzed. The condition ṡ i for i = i atified. For i the wort cae wi be conidered, that i, R i o a that it can be negected. The foowing condition are then obtained for o : o > K n if i > o > K n if i < To arize, o > K n enre ayptotic behavior of the oberver. To deterine the vae of o wort cae i conidered again and the bigget poibe vae i taken for. Dring appication a ow pa fiter ha to be ed to gain the average vae = K n of the dicontino tie fnction = o ign (î i). So far the ter K n i oberved, to gain the haft peed the ret of fitering ha to be divided by the factor K n. C. Paraeter Deign In genera the deign of a iding ode contro can be divided into two probe. At firt the witching anifod with iding ode in order to deign the deired dynaic of the otion eqation i eected. Second objective i to find a dicontino contro fnction ch that the tate reache the anifod and iding ode exit in that anifod. In thi ection an idea of contro paraeter deign for the eected iding ode contro (2) i preented. Once a iding ode anifod i defined and the yte behavior on it i anayzed, finding optia contro paraeter ha be aied. Afterward the exitence of iding ode i proved. a) Definition of the Siding Manifod: Dynaic of the yte can be defined ing a iding rface = din the foowing way: = k k 2 x + k k 2 x 2 + x 3 = (6) with k, k, k 2 >. Then, withot o of generaity et k 2 =. In the cae of x 3 =, for a x = an angar peed x 2 = with oppoite ign i aigned. In coneqence the yte ove toward the ntabe eqiibri poition [x, x 2 ] = [, ] =. b) Behavior on the Siding Manifod: Once iding ode i enforced, the yte oe it own dynaic and the new dynaic are ony defined by the definition of the iding anifod. In or cae thi i the poition of the iding pane in three dieniona tate pace. Characteritic of ao are not reevant to dynaic of the controed yte on the iding rface.

4 Oberving at the dynaic on the iding anifod can hep to find the optia iding ode contro paraeter. Soving = ing (9) the crrent providing the deired dynaic for the echanica yte i given by x 3 = k x k x 2 (7) Repacing x 3 in (8) and () ead to a tate decription of a 2nd order yte x = x 2 x 2 = ( a ξ K k )x ( c + k (8) )x 2. The noninear fnction a ξ can be negected in copariion to Kk for k >.3 ince a Kk ξ <<. The characteritic poynoia of the yte i then given by 2 + k ( K + c + k ) K (9) Beide, the tandard for of a characteritic poynoia of a 2nd order yte i given by 2 + 2Dω + ω 2 where D i the daping ratio and ω i the characteritic freqency. It ean if iding ode i enforced according to (9) dynaic of the yte can be deigned a dynaic of a 2nd order yte. By tning the paraeter D and ω deired dynaic can be choen by k = K ω 2 k = 2Dω c K (2) paraeter k i proportiona to the econd power of the freqency ω of the yte. Once k ha been choen, k baed on the deired daping of the yte can be aigned. c) Exitence of Siding Mode: Uing the anipated variabe = ū ign() the yte tate reache the iding anifod (6) tarting fro every initita condition in finite tie, becae ṡ = (k ẋ + k ẋ 2 + ẋ 3 ) = ( k a ξ x + (k k c K n L )x ( k K R a L )x 3) ū ign() < L (2) With known initia vae x, x 2 and x 3 and defined paraeter k and k the firt three ter of (2) are bonded. Since ign() = >, there away exit a ū > ffiing ineqation (2). The ineqaity (2) eventay i garanteed if the of the firt three ter i aer than L. Large contro inpt acceerate yte dynaic. Therefore arge apitde of the anipated variabe infence the perforance of the yte before reaching iding ode. The experienta etp aow apitde vae of ū = ±24. D. Defining a Benchark In thi artice the propoed iding ode contro i copared to a contro chee decribed in fig.. In the oter contro oop for the echanica yte ony the tate variabe repreenting the ange and angar peed are part of the contro aw. The third variabe, the crrent, i conidered a anipated variabe. In the inner eectrica contro oop the crrent i controed with PWM. aw for the echanica yte i ipeented a a feedback inearization contro. The redced yte i then given by ẋ = x 2 ẋ 2 = a ξx c x 2 + K i y = x and the feedback inearization by (22) y = x ẏ = x 2 (23) ÿ = ẋ 2 The behavior of the yte wi be conideraby ipified throgh the feedback inearization. If the Inpt-Otpt behavior i anayzed, a ipe integrator chain i een, then it i poibe to e a conventiona feedback oop contro. In thi cae the ITAE (Integra Tie Mtipied Abote Error) criterion wa epoyed to deign the feedback gain, the poe of the yte. Poe are deigned to be p /2 = ω (.77 ± j.77), where ω i a caing factor. It hod be taken into accont that thi factor ha an ipact on the tranfer fnction of the yte. Therefore feedback ha to be tipied by a factor K, which are teady tate accracy. The redced yte i a 2nd order yte whoe tranfer fnction i repreented by Hence, K = ω 2 i obtained. K F () = ω + ω 2 (24) III. EALUATION OF THE PROPOSED CONTROL SCHEME The perforance of the derived iding ode contro i vaidated in nerica iation and appied to an experienta etp. A. Nerica Siation Nerica iation are done with Matab/ Siink ing variabe tepize, inia tepize i T a = 8. In order to provide coparabiity of iation ret of the iding ode contro yte to thoe of the benchark contro yte, the ae dynaic for both contro ethod were pecified. That ean for both contro trategie the dynaic of the coed oop were characterized by ω = 5 and D = 2. Uing (9) a paraeter of the iding ode contro k = 76 and k =.4 were obtained. In the

5 ech benchark yte the inner contro oop for the crrent i iated a PWM contro baed on 2 khz aping rate. For both yte the contro objective i = and the initia ange i =.3 rad. A ditrbance of 2 N i added at the Sfrag repaceent tie t D =.2 for a tie period of.4. A zero order hod, aping tie T S =. wa added in order to iate x ech i x Fig. 5 how the trajectorie of ange and angar peed for PWM eectr iation of the iding ode contro and the benchark contro. Detaied trajectorie of ange and angar peed can be een in fig. 6. Fig. 7 deontrate infence of the contro paraeter i k and k on the tranient repone of the yte Mech. ing = iding ode contro., iding ode contro crrent feedback oop benchark contro.5 otor SMC Eectric a DC Actator otor xign() ech ode x eectr of the DC otor î 5 in rad in = otor Fig. 5. a Siation ret: Siding ode contro of the eectroechanica DC yte otor and benchark contro with ditrbance. ign() ode of the DC otor î in rad in..5 iding ode contro benchark contro Fig. 6. Siation ret: Siding ode contro of the eectroechanica yte and benchark contro with ditrbance- Coe p. x eectr = otor a dicrete contro. The oberver i not ed. DC otor ign() i DC otor: pend: ode of PWM R a =.36 Ω in = kg L =.8 H ax =.3 kg the DC otor K =.32 N g = 9.8 î A K 2 n = 6/37 in = i =.34e 5 kg 2 5 ax =.5 iding ode contro Mech. A = 24 c =.3 N (etiated) benchark contro rad, = 9 crrent feedback Tabe oop. Paraeter of the experienta etp ed for contro deign and frag repaceent insmc nerica iation. By changing a and ength of the pend Eectric different oad can be reaized for the experienta etp. In nerica -5 Actator iation = ax and = ax are ed in rad in.5 ω = ω = 5 ω = 9 Fig. 7. Siation Ret: Siding ode contro of the eectroechanica yte with ditrbance. Tranient repone for different deign paraeter. Increaing paraeter k and k go together with increaing ω and fater tranient repone. B. Experienta Setp The propoed contro agorith wa appied to the ape yte (fig. 3). The actator i a 5 W Maxon DC otor with gear. The a and ength of the pend can be odified and o different oad can be reaized. An H- Bridge provide the reqired dicontino contro inpt for the iding ode contro. In the benchark contro yte the DC otor i powered by a Copey PWM-apifier. A fraework for the contro yte Matab/ Siink Rea Tie Workhop i ed. The controer rn nder RT-Linx with a aping tie of.. Detaied paraetric vae of the echanica and eectrica byte can be een in tabe. Fig. 8 and 9 preent the trajectorie of ange a we a eared crrent if the propoed iding ode contro i appied to the eectroechanica yte. Meareent were repeated for different oad. C. Ret The efficiency of the controer wa proved by ean of nerica iation a we a experient. The derived iding ode contro of the eectroechanica yte offer robtne, fat dynaic and copared to a benchark contro it e e power. d) Fat Perforance: A hown in fig. 6 baed on iation the iding ode contro i fater than the benchark contro yte. Uing iding ode contro the contro objective = i achieved withot overhooting after 8 wherea the benchark contro need (ee fig. 6) Thi i a 25% fater tranient repone. Deayed reaction of the conventiona controed yte i caed by

6 ode of the DC otor = î iding ode contro benchark contro otor in a DC otor ω ign() = ode ω = 5 of the DC ω otor = 9 î iding ode contro benchark contro I in A in rad in in rad ω = ω = 5 ω = 9 ange reference oad.5 oad I in A ange reference oad oad Fig. 8. Experienta ret: Siding ode contro of the eectroechanica yte. Poition contro with different, not aignabe oad. in rad ange reference oad oad.5 Fig. 9. Experienta ret: Siding ode contro of the eectroechanica yte. Poition contro with different, not aignabe oad- Coe p. the inner contro oop providing the deired crrent given by the contro in the oter contro oop, which i not away of axia apitde. The tie contant of the DC otor ed in the experienta etp i T = L R a.25. Therefore in order to prove the advantage of copete tate feedback contro, aping rate of at eat T S =. are neecary to be abe to deontrate fater contro perforance of the iding ode contro. Experienta ret nderine the accrancy e) Robtne: The iding ode controed yte proved to be very robt againt paraeter variation. Changing the oad abot approxiatey 5% doe not effect perforance conideraby a deontrated in fig. 8 and 9. The contro objective i achieved withot teady tate error and the abote accracy of the eared ange i ±.5e 3 rad. In contrat, feedback inearization a we a poe paceent with the hep of the ITAE criterion reqire we known ode. Therefore whie ing iding ode contro, it i eaier to copenate a pertrbation of the torqe oad. The benchark contro can generate a teady tate error. f) Sipe Ipeentation: Ipeentation of the propoed iding ode contro i ipe and tabiity anayi can be ade withot probe ing the iding ode condition (3). Neverthee, fat hardware i reqired: A contro nit offering at eat khz aping rate i neceary to get acceptabe ret concerning crrent and ange rippe. g) Chattering Probe: Since witching i away reqired, the idea iding ode ( = ) i never hod. Regarding (6), k 2 i the feedback gain of the aratre crrent. If k 2, which eqa to k, k, the iding rface wi neary ay on the pane pread by x and x 2. Thi wi cae chattering. Hence, k 2 cannot be infiniteia a and therefore, aide fro phyica contraint, ing thi ethod the yte cannot be ade infinitey fat. De to a aping rate of ony. copared to the tie contant of.25 of the eectrica yte in fig. 8 the indctance of the otor cannot totay fiter ot chattering before it reache the echanica byte and a crrent rippe of ± A i oberved. h) Power Redction: baed on PWM igna i not a fexibe a iding ode contro becae perforance depend on the defined aping rate of the PWM nit. High witching freqencie cae high crrent rippe. Thi incde arge power oe. For the zenario preented in fig. 5 energy conption of the benchark contro in teady tate i 36 W wherea it i ony 4 W for the iding ode contro. I. CONCLUSION In thi artice a iding ode contro for eectroechanica yte wa deveoped and experientay teted. Ret how that the propoed contro chee iprove perforance of eectroechanica yte copared to conventiona contro chee. If the dynaic of the eectric drive are taken into accont, the phenoenon of chattering can be avoided and power conption can be decreaed. Becae the inpt votage of the eectroechanica yte i witched depending on the echanica and eectrica variabe the contro i very robt with regard to ditrbance and different initia tate. Firt experienta ret indicate, that the propoed approach iprove perforance of echanica yte driven by different eectric actator, ch a DC otor, ynchrono and indction achine with not excacty known paraeter and which are operating nder nknown condition. REFERENCES []. Utkin, Siding Mode Deign Principe and Appication to Eectric Drive, IEEE Tranaction on Indtria Eectronic, vo. 4, no., Febrary 993. [2] A. Cavao and F. aca, with Siding Mode Switching Modator, Internationa Conference on Indtria Eectronic, and Intrentation, vo. 3, pp , Septeber 994. [3]. ittek, S. Dodd,. At, and R. Perryan, Siding Mode Baed Oter Loop for Indction Motor Drive With Forced Dynaic, IASTED conference on and Appication, 2. [4]. Utkin,. Gdner, and. Shi, Siding Mode in Eectroechanica. London: Tayor & Franci, 999.