Minimum Energy Forced Dynamic Position Control of PMSM Drives

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Tobias Bridges
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1 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 Minimum Enegy Foe Dynami Poition Contol of PMSM Dive STEPHEN J. DODDS, GUNARATNAM SOORIYAKUMAR, ROY PERRYMAN Shool of Computing an Tehnology, Univeity of Eat Lonon, Univeity Way, Lonon E6 RD Unite Kingom Abtat: A new loe loop ontol ytem fo pemanent magnet ynhonou moto (PMSM) ive i peente that minimie enegy lo fo hange of poition of a balane mehanial loa in a peifie manoeuve time with zeo initial an final veloitie. An inne foe ynami ontol (FDC) loop epon to an angula aeleation eman, eating a ouble integato plant fo an oute poition ontol loop implemente by a peial liing moe ontol law with a pieewie linea-paaboli withing bounay having a bounay laye to pevent ontol hatte. Veto ontol i ahieve by an FDC iet axi uent ontol loop with zeo efeene input, auming a non-alient PMSM. The ontol pofile onit of ontant aeleation an eeleation epaate by a peio of ontant veloity. It paamete ae et to minimie the enegy expenitue in the mehanial loa. The effetivene of the ontol heme i emontate by imulation. Key-Wo: Foe ynami ontol, liing moe ontol, eleti ive, minimum enegy manoeuve. Intoution A a ontibution towa impoving the envionment, a new loe loop ontol ytem i peente aime at minimiing the enegy lo in poition ontol appliation with zeo initial an final veloitie an a peifie manoeuve time. Maximiing the manoeuve time in geneal minimie the enegy loe beaue the powe lo in iven mehanim ineae with the uae of the veloity. An example of an appliation i a et of poitioning mehanim on a poution line in the manufatuing inuty that often nee ooination. If a given mehanim eahe the eie poition befoe it i neee, then it will onume moe enegy than if it i ontolle to eah the eie poition jut in time. The ontant aeleation-veloity-eeleation pofile ha been hoen beaue a) it yiel a ubtantially lowe enegy expenitue fo a peifie manoeuve time than poible with onventional linea ontol tehniue an b) it ha a peiely efine an tuly finite ettling time that an be eaily ajute fo ooination with othe ontolle egee of feeom in a motion ontol ytem.. Moel of PMSM an Loa The tate iffeential euation i t = Ai + Bω i + Fu () i t = Ci + E ω Di + Gu () ω t = ( H+ Ki) i MΓL F v ω (3) θ t =ω (4) moel the PMSM in the ynhonouly otating - o-oinate ytem, togethe with it mehanial loa, whee i, i an u, u ae the tato uent an voltage omponent, ω an θ ae the oto angula veloity an angle, an i the loa toue whih Γ L epeent the mehanial loa a well a any extenal loa toue []. The ontant oeffiient ae: A = R L ; B= pl L ; C= pl L ; D= R L ; E = pψ L ; F= L ; G = L ; PM H = 3pΨ o PM ( J ) ; K = 3p o ( L L ) ( J ) H = 3pΨ PM ( J ); K = 3p( L L ) ( J) whee Ψ i the pemanent magnet flux, PM R i the tato eitane, L an L ae the iet an uaatue axi inutane, p i the numbe of pole pai, J i the oto moment of inetia an J = J + J L, whee J L i the loa moment of inetia efee to the oto. It i evient fom (3) that the ontol toue elivee by the moto i: Γ = J( H+ Ki) i (5) 3 Inne Foe Dynami Contol Loop 3. Oveview ISSN: Page 73 ISBN:

2 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 The geneal theoy of foe ynami ontol (FDC) Subtituting fo i t an i t in () uing, i given in [] an the ontol ytem to be fomulate employ FDC fo ontol of the iet epetively, () an () then yiel: axi uent, i, an the angula aeleation α t = ( H + Ki )( Cωi Di Eω + Gu) () α= ( H+ Ki) i MΓL (6) + Ki( Ai + Bω i + Fu) M ΓL t with u an u a the ontol vaiable. To keep the Sine both u an u now appea on the RHS of uent an magneti flux veto mutually pepeniula, fo maximum toue effiieny (i.e., the veto ontol onition) the iet axi tato uent omponent i ontolle with the emane value et to i =. The FDC angula _em aeleation ontol loop i teate a an inne loop fo the oto angle ontol fo whih a minimum enegy ontol law i fome with the emane angula aeleation, a em, a the ontol vaiable, a peente in ubetion 3.6. The enegy expenitue i minimie by etting the longet aeptable manoeuve time an then maximiing the magnitue of the ontant aeleation an ontant eeleation. Thi automatially minimie the peak angula veloity magnitue uing the manoeuve. (), the ank of the plant with epet to ω i the oe of the eivative on the LHS of (), whih i. The eie loe loop iffeential euation i theefoe of fit oe an if thi i again hoen to be linea with a ettling time of T a (5% iteion), then α t = ( 3 T a )( α _em α ). () The ontol law i omplete by euating the RHS of (), () an olving the eulting euation fo u, noting that u i now known fom (8): ( H+ Ki)( Cω i Di Eω + Gu ) ΓL 3 ( ) ( em ) + Ki Ai + Bω i + Fu M = α α t T a ( 3T )( α α) α em 3. Diet axi tato uent ontol + Ki( Ai Bω i Fu ) + M ΓL t By inpetion of (), the ontol vaiable, u, u = appea on the RHS an theefoe the ank of the G H+ Ki plant with epet to i + ( Cω i + Di + Eω i jut an o the eie ) loe loop iffeential euation i of fit oe. (3) Chooing thi a linea with a ettling time of Ti to It houl be note that α may be alulate uing (6) 95% of the teay tate tep epone (i.e., the with the known moto paamete an uent laial 5% iteion) yiel: meauement, an etimate of the loa toue, Γ L i t = ( 3 Ti)( i_em i ), (7) being obtaine fom the obeve peente in etion 3.4. i i then foe to have the ynami of (7) by euating the RHS of () to the RHS of (7) an then olving the eulting euation fo u. Thu the foe ynami iet axi uent ontol law i: u = ( 3/Ti )( i_em i ) + Ai Bωi F (8) The veto ontol onition i epete euiing i = (9) _em 3.3 Roto angula aeleation ontol The ank of the plant with epet to α of (6) may be etemine a the lowet oe time eivative of α with iet algebai epenene on the ontol vaiable, u o u. It i lea fom (6) that thee i no iet algebai epenene of α on u o u. Hene (6) i iffeentiate: α t = H + Ki i t + Ki i t M Γ t () L 3.4 Obeve The main pupoe of the obeve i to etimate the ΓL an Γ L t but it alo poue etimate of θ an θ t. Γ L an ΓL t ae teate a tate vaiable in the eal time plant moel of the obeve an in the abene of knowlege of the time vaiation of Γ L, it i aume that Γ L t =. Thi oe not impoe a eiou limitation povie the obeve eigenvalue ae uffiiently lage. The eal time moel onit of (3) an (4) togethe with the two loa toue tate euation, L t = L an L t =, whee L = Γ L an L = Γ L t. The obeve blok iagam i hown in Fig.. ISSN: Page 74 ISBN:

3 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 K 4 e o K 3 K K M Γ ˆL L = t L ˆ ˆ =Γ L ( + ) ˆω Ho Koi i Γ L Fig. : Obeve fo etimation of an ΓL t. The obeve an be eigne by eigenvalue plaement. The haateiti euation i K + K K3M K4M = (5) Applying the Do 5% ettling time fomula [3] fo the obeve ettling time, i.e., To =.5( + n) To, whee n i the ytem oe, fo all the oetion loop eigenvalue plae at = T o, yiel the eie haateiti euation: n ( ) n T = + 5 T = n= = (6) whee = 5 ( T o ). Compaing the LHS of (4) with that of (6) then yiel the euie obeve oetion loop gain: K = 4, K = 6, K = 4 3 M, K = 4 M (7) { 3 4 } The malle T i mae, then the moe auate Γ ˆ L ( t) an ( Γ L () t t) will be. The mehanial loa i exlue fom the eal time moel of the obeve but it peene i eflete a a time vaying loa toue omponent whih i ompenate by the FDC algoithm though the tem ontaining Γ ˆ L ( t) (ef., (6)) an Γ ˆ t (ef., (3). Thi yiel obutne [6, 7]. L () 4 Minimum Enegy Poition Contol 4. Theoy Fig. how the angula veloity-time pofile fo the poition ontol. ω () t aeleation, α eeleation, α b ω p θ a t T α T T T m α Fig. : Veloity-time pofile fo poition ontol. ˆθ + The toue euie to oveome viou fition i Γ t = Bω t, whee B i the viou fition f oeffiient. Then, efeing to Fig., the enegy lo uing a manoeuve of uation,, i: T T L = ω + ωp α () T α W B t t B t T { p } = B α t t+ ω t 3 Tα T = B α t +ω p [] t 3 W L = Bω p{ ( 3) Tα + T } (8) ine the peak veloity i given by ωp =αt α (9) It i evient fom Fig. that the oto angle at the en of the manoeuve i T θ = m ω () t t =ω p( T + ) () α T whee Tm = Tα + T () i the manoeuve time. Fom () an (): ω p = θ ( Tα + T) () Tα = ( T m T ) (3) Then ubtituting fo ω p in (8) uing () yiel W L = Bθ 3 Tα + T Tα + T (4) Subtituting fo T α an (4) uing (3) yiel W L = 4 3 Bθ Tm + T Tm + T (5) Sine W 3 L T = 8 3 Bθ T Tm + T, it i evient that WL ha a maximum at T = an i minimie by eleting the veloity-time pofile with the maximum value of T, whih will be enote by Tmax. Subtituting fo ωp in () uing (9) yiel α=θ Tα( Tα + T) (6) an ubtituting fo T α in (6) uing (3) yiel α= 4θ ( Tm T ) α max = 4θ ( Tm Tmax) (7) i.e., uing the maximum poible angula aeleation an eeleation magnitue, α max, yiel T max an theefoe minimie the enegy lo. It emain to eive an euation fo ω p in tem of the emane oto angle, θ em, an the euie manoeuve time, T m, ine thi i euie fo the loe loop poition ontol law. Fo thi pupoe, θ in () an (7) i eplae by θ em. Fom (7): max = m 4θem αmax T m T T (8) in () uing (3) yiel ω = θ T + T. (9) Subtituting fo T α p em m ISSN: Page 75 ISBN:

4 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 Then ubtituting T T in (9) uing (8) yiel = max ω p = θ em Tm + Tm 4θ em α max 3.3 effetively eate a ouble integato plant { θ = ω, ω = u whee u =α em i the ontol vaiable. (3) 4. Oute loop poition ontol law The inne loop foe ynami ontol law of etion (3) 4.. Bai bang-liing moe ontol law A loe loop ealiation of the ontol pofile of the peviou etion an be ahieve by mean of a peial liing moe ontol law having a withing bounay in the phae plain ompiing two paaboli egment meeting at the oigin of the phae plain, imila to the time optimal ontolle [], an omplete with two hoizontal taight line egment a hown in Fig. 3, with ω p alulate aoing to (3). Below the withing bounay, u =+αmax, an above, u = αmax, a hown, yieling the loe loop phae potait ompiing two familie of paabola with oppoitely igne aeleation paamete, a hown in Fig. 3. ω u = α max b +ω p +θ a θ θ θ em =θe ω p u =+α max Fig. 3: Swithing bounay fo minimum enegy lo ontol law an the loe loop phae potait. The tajetoie ae iete towa the hoizontal taight line egment of the bounay fom both ie, whih i Utkin onition fo liing motion [4]. The liing motion maintain ω = ω p along egment b- of the loe-loop tate tajetoy, a euie. The ontol atuate at u =+αmax along egment a-b of thi tajetoy, an at u = αmax along egment -. If the emane oto angle, θ em, i ontant, then θ e =θ θ em =θ an theefoe (3) an be ewitten a: { θ e = ω, ω = u (3) The loe loop ontol law yieling the behaviou efine in Fig. i bae on the olution to the tate tajetoy iffeential euation fome by iviing the fit of euation (3) by the eon: θ ω =ω u (33) The geneal tate tajetoy euation fo α em ontant i obtaine by the metho of epaation of vaiable: uθ e = ωω θ e =θ e ( ) +ω ( u ) + A(34) whee A i the abitay ontant of integation that epen upon the initial tate. The euation of the two paaboli egment of the withing bounay meeting at the oigin of the phae plain in Fig. ompie the euation of the two tate tajetoie that teminate at the oigin, a follow: ω αmax fo ω > θ e =,fo ω = +ω αmax fo ω < whih an be expee a a ingle euation: θ e = ω ( αmax ) gn( ω ) (35) whee gn ( ω ) { + fo >, fo ω <, fo ω = } (36) i the ignum funtion. Thi i the time optimal withing bounay [5] fo plant (3), but i only ue fo θ e <θ. To ealie the minimum enegy withing bounay, fo θ e >θ, anothe bounay inluing the two hoizontal taight line egment i ue, whih i expee by: ω = ωpgn ( θe) (37) The euie withing bounay i obtaine by ombining (33) an (34) but in oe to ahieve thi, the altenative veion of (33) i neee in whih ω i the ubjet, a in (34): ω = αmax θe gn( θ e ) (38) The ombination of (36) an (37) will be one at the tage of foming the withing funtion. Let the euie withing bounay be: S ( θe, ω ) = (39) Then the withing funtion i S ( θ e, ω ) an i aange uh that if ω i ineae beyon the value on bounay (39) without hanging θ e, then S ( θe, ω ) >, an vie vea o that the euie ontol law beome: u = αmax gn S ( θe, ω ) (4) The withing funtion oeponing to (36) i: S ( θe, ω ) =ω +ωpgn( θ e) (4) an that oeponing to bounay (37) i: S ( θe, ω ) =ω + αmax θe gn ( θ e) (4) then the euie withing funtion i obtaine by ombining (4) an (4) a follow: S= { S fo θ e >θ, S fo θe θ } (43) whee θ i the thehol efine in Fig. 3. Thi an be witten a one euation: ISSN: Page 76 ISBN:

5 { } e e S= + ig θ θ S + ig θ θ S (44) whee ig( ω ) { + fo >, fo ω }. (45) Thi ignum funtion i ue intea of (36) in view of (43) to avoi u = in the ae event of θ e =θ, but whih annot be ignoe with the finite wo-length of the igital poeo ue fo the ontol law implementation. So the bai oute loop ontol law i given by (4), (4), (4) an (44). 4.. Intoution of a bounay laye A it tan, the oute loop ontol law eive in etion 4.. woul uffe fom ontol hatte []. Thi may be avoie by eplaement of the withing bounay with the well-known bounay laye [,4] y eplaing the ignum funtion of (4) by the atuation funtion: at ( S,K ) { KS fo S < K, gn ( S) fo S K} (46) In aition, it i neeay to intoue amping at the oigin of the phae plane by mean of the tem involving the amping time ontant, T, in the final veion of the poition law: S ( θe, ω ) =ω +ωpgn( θe) S =ω + α maxt + αmax θe αmaxt gn( θe ) θ = Tω p +ωp ( αmax) S= { + ig( θe θ ) S + ig( θe θ ) S} u = αmax at S ( θe, ω ),K (57) In fat, the ytem appoahe fit oe linea ynami with time ontant, T, a the emane tate i appoahe. Fig. 4 how a keth of the oiginal an moifie withing bounaie togethe with the bounay laye an a typial tate tajetoy oeponing to the veloity-time pofile of Fig.. K 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 u =+α max Bounay laye State tajetoy ω θ ω p u = α Slope = /T max +ω p Moifie withing bounay +θ θ θ em =θe Bai withing bounay Fig. 4: Bounay laye an typial tate tajetoy. 5 Simulation The paamete of the non-alient PMSM ae a follow: Rate voltage: V = 43V ; Powe Rating: P = W ; Bae pee: ω b = 5a / ; Max. toue: Γ max = 4Nm ; Ψ PM =.38 Wb ; L = L = 5.4e 3H ; R =.Ω ; p= 5; J = 3e 4 Kg m J = 3e-4 kg m. The paamete of the mehanial loa efee to the PMSM oto ae: JL = 4J ; oeffiient of viou fition euivalent to 8% maximum loa powe: Fe =.8P ω b. The FDC inne loop paamete ae a follow: Ti = 5e 3; T α = e 3; To = e 4. The oute poition ontol loop paamete ae a follow: maximum aeleation magnitue: α max = HP V ; bounay laye gain: K = ; liing moe time ontant: T =.5. Sine fo mot appliation the enegy tanmitte to the mehanial loa will fa exee the loe in the PMSM an it ive, fo thi initial invetigation the powe eletoni i aume ieal, the pae veto moulation not being imulate, an the ion loe in the PMSM ae ignoe. Alo the meauement intumentation i aume ieal. The total enegy onume i alulate a W=.5ui ( + ui ) (58) an theefoe thi inlue the PMSM oppe loe. Fig. 5 to 9 how the eult fo a et-to-et manoeuve with a fixe emane angle of θ = 6a. Thi euie nealy oto evolution, whih i ealiti fo motion ontol appliation. The manoeuve time i Tm = Tmmin +ΔT m (59) whee T mmin i the minimum manoeuve time fo the ouble integato plant of the oute loop. Fom (7) with θ =θ : em m = θ em α max +Tmax T 4. (6) Hene fo the minimum manoeuve time, Tmax = : Tmmin = θem αmax (6) The enegy expenitue penalty fo euing the ettling time i evient in Fig. 6. The phae plane tajetoie of Fig. 7 ae a peite in etion 4.. Fo ompaion, Fig. 8 how the enegy expenitue gaph fo a onventional tate feebak ontol law u = g θ θ ˆ + g ω ˆ (6) em with the gain etemine by oinient pole plaement to yiel the ame manoeuve time fo itial amping uing the Do % ettling time fomula [3]: n + g+ g = n 5Tm n= g = 784 5T ; g = 56 5Tm (63) m ISSN: Page 77 ISBN:

6 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 Roto Angle [a] Δtm [] time with onventional linea tate feebak. Compaion with Fig. 6 iniate ignifiantly highe enegy expenitue, the peentage ineae woening with ineae of the manoeuve time, being about 4% fo the longet manoeuve time. The eaon fo thi i that the itially ampe linea eon oe epone have ignifiantly lage initial angula veloity magnitue than thoe of the new ontol law. The ovehoot in Fig. 6 an 8 ae ue to ome of the kineti enegy toe in the oto/iven mehanim being etune to the powe upply uing eeleation. Enegy Expenitue [J] Roto Angula Veloity [a/] t [] Fig. 5: Poition epone fo iffeent manoeuve time Δtm [] t [] Fig. 6: Enegy fo iffeent manoeuve time Δtm [] Roto Angle Eo [a] Fig. 7: State tajetoie fo iffeent manoeuve time. Enegy Expenitue [J] Δtm [] t [] Fig. 8: Enegy expenitue fo iffeent manoeuve 6 Conluion an Reommenation A new loe loop poition ontol tehniue ha been peite to yiel lowe enegy expenitue fo a given emane poition hange an manoeuve time than poible with onventional linea ontol tehniue. The ytem i potentially vey patiable in view of it ability to peiely ealie a emane manoeuve time fo a given poition eman without the nee to moel the mehanial loa. Ue on a lage ale thoughout inuty, it woul bing about ignifiant enegy aving. Expeimental tial ae eommene a a next tep. Alo, the tue optimal ontol that abolutely minimie the enegy expenitue houl be etemine to povie a tana of ompaion fo the popoe ontol ytem. Refeene: [] Vittek, J. an Do, S. J.: Foe Dynami Contol of Eleti Dive, EDIS, Žilina Univeity Publihing Cente, 3, ISBN [] Do, S. J., Foe Dynami Contol: A Moel Bae Contol Tehniue Illutate by a Roa Vehile Contol Appliation, Poeeing of AC&T 6, SCOT, Univeity of Eat Lonon, UK, ISBN [3] Do, S. J., Settling Time Fomulae fo the Deign of Contol Sytem with Linea Cloe Loop Dynami, to be publihe in the Poeeing of AC&T 8, SCOT, Univeity of Eat Lonon, UK. [4] Utkin, V. I., (99). Sliing Moe in Contol Optimization, Spinge-Velag, 99, ISBN [5] Ryan, E.P., Optimal Relay an Satuating Contol Sytem Synthei, Pete Peeginu, ISBN: & [6] Do, S. J., Obeve Bae Robut Contol, Poeeing of AC&T 7, SCOT, Univeity of Eat Lonon, UK, ISBN [7] Stale, P. A., Do, S. J. an Wil, H., Obeve Bae Robut Contol of a Linea Moto Atuate Vauum ai Beaing, Poeeing of ISSN: Page 78 ISBN:

7 3 IASME/WSEAS Int. Conf. on Enegy & Envionment, Univeity of Cambige, UK, Febuay 3-5, 8 AC&T 7, SCOT, Univeity of Eat Lonon, UK, ISBN ISSN: Page 79 ISBN:

A Novel Discrete-Time Predictive Current Control for PMSM

A Novel Discrete-Time Predictive Current Control for PMSM A Novel Dicete-Time Peictive Cuent Contol fo SM Jung-Won Sun, Jin-Woo Lee, Jin-Ho Suh, Young-Jin Lee, an Kwon-Soon Lee Depatment of Electical Engineeing, Dong-A Univeity 840, Haan-ong, Saha-gu, Buan, 604-714,