2.3. SLIDING MODE BASED OUTER CONTROL LOOP FOR INDUCTION MOTOR DRIVES WITH FORCED DYNAMICS

Transcription

1 2.3. SLIDING MODE BASED OUTER CONTROL LOOP FOR INDUCTION MOTOR DRIVES WITH FORCED DYNAMICS Abstact: Though the loa toque estimation, the basic FDC base IM ive contol system pesente in the last two sections offes highe obustness with espect to paamete uncetainties an extenal loa toques than conventional shaft sensoless spee contol methos, ue to the use of the loa toque estimate in the contol law, as explaine in pevious sections. Thee is still, howeve, oom fo impovement when compae with the pefomance of some ives employing shaft sensos. To achieve a close appoach to this pefomance, a scheme is pesente using the sliing moe contol pinciple. It will be ecalle that the basic FDC system has a cascae contol stuctue consisting of an inne loop, which is a stato cuent contol loop, an a shaft sensoless FDC base spee contol loop, enabling a use specifie ynamic pefomance to be obtaine accoing to ynamic moes such as pesente an escibe in Chapte. This spee contol loop will now be efee to as the mile loop, since the obustness enhancement will be pouce by the aition of an oute sliing moe contol loop. Simulation an expeimental esults show goo coesponence with the theoetical peictions an emonstate the intene obustness impovements Intouction The new FDC base IM ive contol system aleay pesente base on feeback lineaisation [], pinciples of vecto contol [2] an sliing moe contol (SMC) [3] will be opeate in the fist oe linea ynamic moe fo this obustness enhancement investigation. As state peviously, the system opeates without shaft sensos, only the stato cuents being measue an the applie stato voltages being etemine by the compute switching algoithm of the invete with a knowlege of the instantaneous DC link voltage. The oto magnetic flux nom ynamics is also chosen as linea an of fist oe, as escibe in section 2..2.

2 Foce ynamic contol is applicable in many non-linea multivaiable automatic contol applications an was oiginally evelope fo electical ives employing IM [4] an expeimentally veifie in [5]. Vaious foce ynamics fo IM ives, incluing constant toque eman, secon oe spee ynamics, an the possibility of iectly contolling the ive acceleation, wee intouce in [6] an veifie by simulations. The expeimental esults fo all these thee pescibe ynamics wee pesente in [7]. Such a contol system is suitable fo sensoless contol of electic ives employing IM with moeate accuacy ( 5% ). The oiginal basic sensoless IM ive contol system with foce ynamics is shown in Fig It contains a set of thee obseves fo estimation of the oto magnetic flux, the oto angula spee an the loa toque. emane oto spee ^ Maste contol law α Γ^ L Filteing obseve emane α_β cuents I α I β β α Roto flux estimato β 2/3 tansfom U α U β Slave contol law Signum slave CL 3/2 tansfom U 3 U 2 I α U c emane theephase voltages U I β Sliing-moe obseve Powe electonic ive cicuit α β tansfom v α eq V β eq I a Roto spee I b Measue cuents Angula velocity extacto Extenal loa toque Γ L Inuction moto Fig Oiginal contol system of shaft sensoless spee contolle electic ive with IM an foce ynamic Since the contol law itself, the magnetic flux obseve, the oto angula spee obseve an loa toque obseve ae all moel-base, i.e., epenent on estimates of the moto paametes, then some sensitivity to the eos in these estimates is expecte. It means that the close-loop pefomance of the whole contol system is

3 a affecte by eos in the estimates. To euce this sensitivity the intention hee is to close an oute contol loop aoun the oiginal close-loop system an to impove the obustness of the whole contol system. Moel efeence aaptive contol (MRAC) an SMC base oute contol loops wee theoetically suggeste in [8]. The fist expeimental esults fo the MRAC oute loop wee epote in [9]. The pupose of this chapte is the expeimental veification of the SMC base oute contol loop. A schematic block iagam of the oveall contol system, incluing the SMC oute contol loop, is shown in Fig L L 2 L 3 POWER ELECTRONIC IM ˆ& ˆ& S a S b S c U c OUTER LOOP CONTROLLER ' MIDDLE LOOP CONTROLLER INNER LOOP CONTROLLER i a U c i b Fig Moifie foce ynamic ive contol system with the SMC oute loop Thus, the oiginal FDC base spee contolle becomes the mile loop contolle of the new scheme Contol System 2a) Moel of Moto an Loa The α, β fame base moel of IM in matix notation is as follows: & L 5 L (2.3.) J J & P( ) + c I (2.3.2) T T ( Γ Γ ) ( c T I Γ ) 4 [ c P( ) I U] I & c 2 a +, (2.3.3)

4 whee Τ [ α β ], Ι Τ [ι α ι β ], U T [u α u β ], an c L /(L s L -L 2 m), c 2 L m /L, c 3 R /L /T, c 4 L m /T, c 5,5.p.L m /L an a R s +(L 2 m/l 2 )R. Also: P c p p c ( ) 3 3 an T. (2.3.4) 2b) The Mile Loop Contolle 2b) The foce ynamic contol law On the basis that the inne loop contolle maintains zeo stato cuent eos, equation (2.3.3) is eliminate fom the contol poblem. The feeback lineaisation pinciples [] ae use fo the foce ynamic contol law evelopment. The lineaising functions, which foce the system vaiables to obey specifie linea close-loop iffeential equations, ae fomulate fo the oto spee an the magnetic flux nom. In this application, they ae fist oe with time constants T fo the oto spee an T fo the emane oto flux nom. These two vaiables theefoe satisfy equations: & ( ) (2.3.5a) T ( ) &, (2.3.5b) T 2 2 α + β whee. The lineaising function fo & is then obtaine by equating the ight han sies of equations (2.3.) an (2.3.5a): T T J T I ( ) + ΓL. (2.3.6) c5 T The eivative of the oto magnetic flux nom can be poven to be given by (2.3.7a) an equating the ight han sies of equations (2.3.5b) an (2.3.7a) yiels (2.3.7b): T ( I) & 2 c 3 c 4 (2.3.7a)

5 ( ) T c3 I +. (2.3.7b) c 2c T 4 4 The equie contol law is then obtaine by combining equations (2.3.6) an (2.3.7b), but fist, the state vaiables (x) ae eplace by thei estimates ( ˆx ) fom the obseves. Also, the constant moto paametes (p) ae eplace by thei estimates ( ~ p). Futhemoe, the fictitious contol vecto, I, is eplace by the emane cuent vecto, I, which is the inne contol loop efeence input. Thus, the foce ynamic contol law is: I α β α β ~ c ~ c 3 4 ~ c 5 ~ J T + 2 ~ c T ( ˆ ) 4 + Γˆ L. (2.3.8) It shoul be note that the constant extenal istubance toque, Γ L, is teate as a state vaiable an estimate in the obseve togethe with the othe state vaiables. The nominal close-loop system is then govene by the tansfe function: () s () s + st. (2.3.9) This featue a) automatically yiels soft stating an b) enes the system ieally suite fo inclusion as an ajustable linea element in an oveall contol system such as in motion contol, whee an oute linea position contol loop woul be ae. 2c) State Estimation Detaile esciption of this pat was aleay given as section 2..3 an in abbeviate fom as section an eae is efee fo stuy of state estimation an obseves esign to the coesponing sections fo this.

6 2.3.3 Sliing Moe Oute Loop Contolle Sliing moe base contol law. Sliing moe contol [3] is a fom of bang-bang contol in which the plant state is foce towas an maintaine within a close vicinity of a bounay etemine by the contol system esigne. If, fo a single input, single output plant, the state vaiables ae chosen as the contolle output an its eivatives up to an oe equal to -, whee is the plant ank, then if the state is maintaine pecisely on the bounay, the close loop ynamics is etemine by the bounay alone an is inepenent of the plant paametes an any extenal istubances. The state point then appeas to slie in the bounay. Hence the tem sliing motion is use. Since the bounay is an - imensional hype-suface in an imensional output eivative space, the oe of the close-loop system in the sliing moe is always -. This fact is mae use of in the oute loop contolle. Consie now the close-loop system ceate by the inne an mile contol loops. This will have the nominal tansfe function: ˆ () s () s + st. (2.3.) Since the IM is inheently nonlinea, eos intouce by uncetainties of its paamete values will cause the linea ynamics of (2.3.) to become nonlinea. It may be shown that the sliing moe oute loop can compensate fo this, but to illustate its action moe simply, these paametic uncetainties, extenal loa toque an impefect opeation of the mile contol loop ue to the non-zeo iteation inteval will (oughly) be epesente by a change of time constant an DC gain. Then the combine inne an mile loop ynamics esulting fom the afoementione eos an istubances may be epesente by the tansfe function: ˆ () s K () s ' + st whee > & an > & T. K, (2.3.) T ' In oe to ceate a sliing moe contol loop that oes not euce the system

7 oe (equal to the ank without tansfe function zeos), which is equie to yiel the close-loop system ynamics of (2.3.), a pue integato is intouce at the spee efeence input of the mile contol loop to incease its oe by one befoe fomation of the sliing moe oute loop. Fig shows this, teate as a new plant with contol vaiable, u, an with the oute loop contolle. S +u m u u S m u s Inne & mile loop K + st ˆ Slope, K T ˆ& s Fig Sliing moe base oute contol loop The output eivative, ˆ &, may be obtaine without iffeentiation by using the secon of the filteing obseve equations. If the gain, K, is infinite, then the tansfe chaacteistic between S an u yiels the bang-bang contol law use in classical sliing moe contol: whee: ( S) ' u u sgn, (2.3.2) m S ˆ T ˆ &. (2.3.3) The opeation of such a contol loop may be examine in the phase-plane, i.e, the gaph of ˆ & against ˆ. The contol, u, switches between + u m an u m, when S. Setting this conition in (2.3.3) then yiels the switching bounay. This is shown in Fig togethe with a family of state tajectoies commencing fom iffeent stating points (efee to as a phase potait).

8 ˆ & Switching bounay equation - ˆ & T ˆ ( ˆ ) Fig Close-loop phase potait an switching bounay fo sliing moe base oute contol loop It is evient that ove most of the bounay shown, the tajectoies of the phase potait ae iecte towas the bounay fom both sies, meaning that once the bounay is eache, the tajectoy is maintaine close to it while the contol, u, apily switches (contol chatte). This is the conition fo sliing motion an the contolle escibe is the classical sliing moe contolle. Une these cicumstances, the close-loop system obeys the switching bounay equation shown in Fig an this coespons to tansfe function (2.3.) with K an T T, e. i. ˆ () s () s + st. (2.3.4) This means that without paamete uncetainties o extenal loa toque, the oute loop contolle makes no iffeence to the close-loop ynamics. It meely compensates fo the effects of the paamete mismatches an loa toque, theeby yieling the equie obustness. It is impotant to note, howeve, that with this shaft sensoless system, the contol can only be as accuate as the spee estimate, ˆ. In the final contol system esign, the contol chatte, which woul inteact in an unesiable way with the switching of the powe electonics, is eliminate by eucing the slope, K, in Fig to a finite, but elatively lage value. Then it may be shown that as K, S, an theefoe the esulting pefomance is simila to the classical sliing moe contolle, yieling simila obustness povie u <u m.

9 Since an angula acceleation estimate fom filteing obseve woul be likely to contain substantial noise contamination, an oppotunity to use an equivalent oute loop contol algoithm avoiing this was taken (although this is available fom the filteing obseve). With efeence to Figue 2.3.3, the integato effectively cancels the iffeentiato in the inne feeback loop. The esulting block iagam is shown in Fig an this yiels the SMC base contol algoithm of equation (2.3.5). SM [ ( ˆ ) t T ˆ ] K. (2.3.5) - s T - K sm Implemente Inne/Mile Contol Loop with nominal tansfe function + st $ $ Fig SMC base oute contol loop implemente in expeiments Simulation an Expeimental Results The following set of simulations an expeiments emonstate the pefomance of the contol system by simultaneously applying i) a step angula velocity eman of a/s, with time constant T,2 s, ii) a step oto magnetic flux nom eman of 5e-3 (Vs) 2, with pescibe time constant T 2,5 ms an with zeo initial state vaiables thoughout the system. The simulations wee caie out in the Matlab envionment, the iffeential equations of the moto an obseves being integate using the explicit Eule metho. The simulations pesente inclue the achieve sampling inteval an the finite wo length use in the expeiments. The paametes of the inuction moto elevant to both the simulations an the expeiments ae liste in the Appenix. Fo the expeiments, the contol algoithms wee implemente on a PC equippe with a PC Lab ca. The stato cuents wee measue though LEM

10 cuent tansfomes an evaluate using a PC Lab Ca PL82. A six-tansisto Semikon IGBT moule was use as the thee-phase invete. All the expeiments pesente wee caie out with a DC supply voltage of U DC 6 V an a step oto spee eman of em a/s an a time constant of T,2 s. Data logging of the expeimental vaiables was caie out fo a,82 s time inteval. The ey cuent bake seve as a loa uing the expeiments. In all the gaphs pesente, the stato cuent an oto magnetic flux components as functions of time uing the stating inteval t (-,) s ae shown in subplots (a). The estimates of the oto spee fom the pseuo-sliing moe obseve ae shown in subplots (b) as functions of time fo the whole ata logging inteval. Plots (c) show the loa toque estimate, ˆΓ L, fom the filteing obseve togethe with the oto magnetic flux nom estimate. Finally, subplots () show the ieal oto spee, i, togethe with eal oto spee,, an its estimate, ˆ, fom filteing obseve fo the whole ata logging inteval. The simulation an expeimental esults ae plotte, espectively, in the left an ight columns. The simulation an expeimental esults obtaine with the mile an inne contol loops, excluing the SMC oute loop, ae shown in Fig The eos between the ieal oto spee an the eal an estimate oto spees can be clealy seen in subplots (). These eos, which ae attibute to impefections in the constant paamete estimates, an the finite gains of the obseves impose by the non-zeo sampling inteval, pesist even in the steay-state. The simulation an expeimental esults coesponing to those of Fig , but incluing the SMC oute loop, ae shown in Fig The significant euction of the afoementione eos bought about by the SMC oute contol loop, both uing the acceleation of the ive an in the steay state, is evient by compaison of these two figues. The expeimental esults confim that the aition of a SMC base oute contol loop to the foce ynamic shaft sensoless spee contol of electic ives with inuction motos consieably impoves thei pefomance. The ive follows the ieal spee esponse much moe closely uing spee-up an also in the steay-state. Even a substantial incease of the loa toque was foun not to eteioate the ive pefomance significantly afte the aition of the SMC oute loop an this impovement is even enhance at highe oto spees.

11 .5.5 [A], [Vs] i α i β α β [A], [Vs] i α i β ψ α ψ β a) Stato cuent an oto magnetic flux components - simulation [a/s] i b) Estimate, unfiltee oto spee fom pseuo-sliing moe obseve [Nm], [(Vs) 2 ] ψ Γ L c) Estimate oto magn. flux nom an loa toque fom filteing obseve 8 i 6 [a/s] obs a2) Stato cuent an oto magnetic flux components - expeiment [a/s] i b2) Estimate, unfiltee oto spee fom pseuo-sliing moe obseve [(Vs) 2 ], [Nm] ψ Γ L c2) Estimate oto magn. flux nom an loa toque fom filteing obseve [a/s] i ) Ieal spee esponse, eal an estimate oto spee obs ) Ieal spee esponse, eal an estimate oto spee Fig Spee esponse an coesponing state vaiables with inne an mile contol loops alone

12 2.5.5 [A], [Vs] i α α β iα [A], [Vs] ψα ψβ iβ i β a) Stato cuent an oto magnetic flux components - simulation 2 5 [a/s] a2) Stato cuent an oto magnetic flux components - expeiment [a/s] i 5 5 i b) Estimate, unfiltee oto spee fom pseuo-sliing moe obseve [(Vs) 2 ], [Nm] Γ L b2) Estimate, unfiltee oto spee fom pseuo-sliing moe obseve [(Vs) 2 ], [Nm].. Γ L c) Estimate oto magn. flux nom an loa toque fom filteing obseve [a/s] i obs c2) Estimate oto magn. flux nom an loa toque fom filteing obseve 2 [a/s] 8 6 i ) Ieal spee esponse, eal an estimate oto spee -2 obs ) Ieal spee esponse, eal an estimate oto spee Fig Spee esponse an coesponing state vaiables with sliing moe base oute contol loop

13 2.3.5 Conclusions an Recommenations The peliminay expeimental esults confim that the aition of a SMC base oute contol loop to the foce ynamic shaft sensoless spee IM ive contol system consieably impoves its pefomance. Suggestions fo futue eseach wok ae: a etaile investigation of obustness with espect to moto an loa paamete uncetainties an compaison of the effectiveness of both the MRAC an SMC oute contol loops, implementation of sliing moe base oute contol loop fo foce ynamic contol of synchonous moto electic ives, a futhe set of expeimental tials fo IM ives with highe powe atings togethe with an investigation of the vaiation of the filteing obseve pefomance with the pole locations Refeences [] ISIDORI, A.: Nonlinea Contol Systems. 2 n eition. Spinge-Velag Belin, 99. [2] BOLDEA, I., NASSAR, A. S.: Vecto Contol of AC Dives. CRC Pess Lonon, 99. [3] UTKIN, V. I.: Sliing Moes in Contol an Optimisation. Spinge- Velag Belin, 992. [4] DODDS, S. J., VITTEK, J., UTKIN V. A.: Sensoless Inuction Moto Dive with Inepenent Spee an Roto Magnetic Flux Contol - Pat I- Theoetical Backgoun an Pat II-Simulation an Real Time Implementation. Jounal of Electical Engineeing, Batislava, vol. 49, y. 998, No. 7-8 an No. 9-. pp an pp [5] DODDS, S. J., VITTEK, J., MIENKINA, M.: Implementation of a Sensoless Inuction Moto Dive Contol System with Pescibe Close- Loop Roto Magnetic Flux an Spee Dynamics. Poceeings of EPE 97 Confeence, Tonheim, Noway, Sept. 997, pp

14 [6] DODDS, S. J., VITTEK, J., PERRYMAN, R.: Foce Dynamic Contol of Shaft Sensoless Inuction Moto Dive. Poceeings of SPEEDAM 98 Confeence, Soento, Italy, June 998, pp. A-9 - A-4. [7] VITTEK, J., ALTUS, J., DODDS, S. J., PERRYMAN, R.: Inuction Motos Electic Dives with Foce Dynamics. Poceeings of IASTED 99 Contol an Applications Confeence, Banff, Canaa, July 999, pp [8] Vittek, J., Dos S. J.: Robust Cascae Foce Dynamic Contol of Shaft Sensoless Inuction Moto Dives. Poceeings of EDPE 99 Confeence, Staa Lesna, Slovakia, Oct. 999, pp [9] VITTEK, J., ALTUS, J., BUDAY, J., MIKLO, J.: MRAC Impoves Pefomances of Inuction Moto Dive with Foce Dynamics. Poceeings of IASTED CA 99 Intelligent Contol an Systems Confeence, Oct. 999, Santa Babaa, CA, USA, pp Appenix Thee-phase inuction moto AM 3~ 4AP8-2 MEZ SIEMENS paametes ae as follows: Inuction moto paametes Equivalent cicuit paametes Rate powe P n. kw Mutual inuctance L m,474 H Rate spee n a/s Stato inuctance L s,482 H Teminal voltage Y/Δ 4/23 V Roto inuctance L,482 H Rate cuent Y/Δ 2.4/4.2 A Stato esistance R s 7,5 Ω Moment of inetia J.35 kgm 2 Roto esistance R 6,5 Ω Acknowlegements The authos wish to thank the Euopean Commission, Bussels, fo funing the INCO Copenicus pogamme No.9669 UCODRIVE an the Slovak Ministy of Eucation fo aitional financial suppot of this pogam by gant No. PL9669.