Speed Sensorless Variable Structure Torque Control of Induction Motor

Transcription

1 ISSN ATKAFF 51(1), 33 40(2010) Kael Jezenik, Asif Šabanović, Mian Roič Spee Sensoless Vaiable Stuctue Toque Contol of Inuction Moto UDK IFAC ; Oiginal scientific pape Inuction moto spee sensoless toque contol, which allows opeation at low an zeo spee, optimizing both toque esponse an efficiency, is popose. The contol is quite iffeent than the conventional fiel-oiente o iect toque contols. A new iscontinuous stato cuent FPGA base contolle an oto flux obseve base on sliing moe an Lyapunov theoy ae evelope, analyze an expeimentally veifie. A smooth tansition into the fiel weakening egion an the full utilization of the invete cuent an voltage capability ae possible. The efeence tacking pefomance of spee an oto flux is emonstate in tems of tansient chaacteistics by expeimental esults. Key wos: Discete-event iven system, FPGA, Inuction motos, Sliing moe contol, Toque contol Stuktuno pomjenjivo upavljanje momentom asinkonog motoa bez mjenog člana bzine vtnje. Peloženo je upavljanje momentom asinkonog motoa bez mjenog člana bzine vtnje, koje omogućuje a na malim bzinama i u stajanju te pitom optimia i oziv momenta i učinkovitost. Peloženo je upavljanje pilično ugačije o konvencionalnog upavljanja poljem i neposenog upavljanja momentom. Razvijeni su, analiziani i ekspeimentalno potvženi novi iskontinuiani egulato stuje statoa implementian u FPGA i estimato otoskog magnetskog toka zasnovan na kliznim ežimima i Ljapunovljevoj teoiji. Omogućeni su glatki pijelaz u poučje slabljenja polja i puna iskoistivost stujno-naponskih mogućnosti napojnog petvaača. Pefomanse sliježenja efeentne veličine bzine vtnje i otoskog magnetskog toka pokazane su ekspeimentalno obivenim pijelaznim pojavama. Ključne iječi: sustav vožen isketnim ogažajima, FPGA, asinkoni motoi, upavljanje zasnovano na kliznom ežimu, upavljanje momentom 1 INTRODUCTION Recent theoetical avances in the fiel of hybi an iscete event-systems, an significant incease of the computational powe available fo the contol of the powe electonic systems, invite both contol an powe electonics communities to aapt taitional contol schemes associate with powe electonics applications. In oe to aise the pefomance an efficiency of the ive applications, faste an moe sophisticate cuent contol schemes ae equie. The conventional cuent contol scheme consists of iscete-time PI cuent contolle an pulse-with moulato. It is eplace with the new sequential switching cuent contol stategy [1]. Hysteesis contolles can be a goo altenative fo such applications. They ae obust to system paamete vaiations, exhibit vey goo ynamics, equie simple implementation an enable iect contol of the bige tansistos without special moulatos. Thei main awback is a limite contol of tansisto switching fequency [2, 3]. In this pape, a novel switching sliing moe contolle fo a voltage-souce invete is pesente, which is suitable fo vey simple an inexpensive FPGA (Fiel- Pogammable Gate Aay) hawae implementation. Its main popeties ae obustness an fast ynamic esponse. When joine they euce invete switching fequency equie to keep the cuent eo within a toleance egion of given length. Moeove, a vey simple an low cost implementation is possible owing to the use of FPGA finite state automation implementation. In this pape, by combining the vaiable stuctue system an Lyapunov esign [4], a novel sliing moe algoithm of contolle/obseve fo inuction moto is evelope. This contol metho is base on estimation of the oto flux an spee of inuction moto (IM). Due to use of sliing moe pinciple it is obust against vaiation of loa toque, machine paametes an extenal istubances. The sliing moe pinciples, base on state vaiable eos, ae use as feeback contol to guaantee stability of contol AUTOMATIKA 51(2010) 1,

2 system. The popose metho is investigate an veifie expeimentally using hawae in the loop simulation. The pape is oganize as follows. In Sect. 2 ynamic moel of IM, togethe with contol system esign is pesente. Sect. 3 pesents popose vaiable stuctue contol (VSC) cuent an flux contol schemes with evelope new ecision logic-base event-iven contolle. Sect. 4 escibes the new oto flux obseve base on continuous sliing moe contol. Acquie moels ae expeimentally confime using in house evelope DSP/FPGA boa [5]. Mapping of the obtaine moels into FPGA executable coe is pesente in Sect. 5. Expeimental esults ae pesente in Section 6. The finings an the comments of pesente appoach ae iscusse in the conclusion. 2 CONTROL SYSTEM DESIGN 2.1 Machine Dynamics The ynamics of IM consist of mechanical motion, ynamics of stato electomagnetic system an the ynamics of the oto electomagnetic system with electomagnetic toque evelope by machine: ω t = 1 J (T e T L ), (1) i s ( s t = 1 / u s L s L 2 s R s i s s L m m L ( R L jpω Ψ s L t Ψ s = R L m i s s t L ( i s 1 s = / Ψ s L s L 2 s L m Ψ s m L L ), (2) ) Ψ s, (3) ), (4) T e = 3 2 pl m L (Ψ s i s s), (5) whee ω is mechanical oto angula spee, the two imensional complex space vectos Ψ s s = [Ψ s sa Ψ s sb ], Ψ s = [Ψ s a Ψ s b ], u s s = [u s sa u s sb ], i s s = [i s sa i s sb ] ae stato an oto flux, stato voltage an cuent, espectively, T e is electomagnetic moto toque, T L is loa toque, J is inetia of the oto an p is the numbe of pole pais of the machine. R stans fo esistance an L fo inuctance. The meaning of the subscipt an supescipt ae as follows: s is stato, is oto, an m is mutual (L m is mutual inuctance). The subscipt enotes the location of vaiable an supescipt enotes the fame of efeences. One of the most impotant issue in implementing contol of inuction machine, eithe iect toque contol (DTC) o fiel oiente contol (FOC) stategies is to obtain eal-time instantaneous flux magnitue an its position with sufficient accuacy fo the entie spee ange [2, 6, 7]. The ifficulty of flux estimation lies in the nonlinea inuction machine ynamics, which is chaacteize by spee epenent an time vaying paametes [8]. 2.2 Sliing Moe Contol System Design The esign of sliing moe system can geneally be pesente as two-step poceue: esign of the switching manifol an esign of the contol that enfoces sliing moe. The switching manifol must be selecte in such a manne to guaanty esie system. The poceue may be pesente as follows [9]: 1. Fom the system specification select the vecto function σ = σ ( x,x ) whee x epesents esie values of the contolle vecto vaiable x, so that esign specifications ae met if the sliing moe is enfoce in manifol: S = { x : σ = σ ( x,x ) =0 }. (6) 2. Select the contol input so that the attactiveness an the stability of the solution σ = σ ( x,x ) = 0 is guaantee. Since fulfillment of the esign equiements is euce to the stability of (6) natual selection of the Lyapunov function caniate is pesente in the following fom: V = σ T σ/2 an V = σt σ. (7) This selection guaanties that V σ 0 > 0 an V σ=0 =0. The Lyapunov stability equiement is fulfille, if such a contol can be selecte that V < 0 σ 0 an V =0. This objective can be achieve, if the σ=0 following is satisfie: a) The eivative V of the Lyapunov function is function of contol (eivative of function σ = σ ( x,x ) =0 must be function of contol); b) Thee is such a function Φ(σ) so that σ T Φ(σ) σ 0 < 0; an c) Thee is a unique solution fo contol fo the following equality: V = σ T σ (u) =σ T Φ(σ) < 0. (8) Thee ae many iffeent ways fo selecting Φ(σ). Fo the application in electical ive contol, two paticula foms as given in (8) may be of inteest: V = σ T Φ(σ) < 0 { Φ(σ) =Γsign (σ), Φ(σ) =Dσ, D 0, whee Γ is positive efinite iagonal o matix with peominant iagonal, sign (σ) = (9) 34 AUTOMATIKA 51(2010) 1, 33 40

3 Fig. 1. Block iagam of event-iven sliing moe contol of IM [sign (σ 1 ),...,sign (σ m )] an D is positive efinite matix. It shoul be state hee that the secon choice in (9) actually guaanties asymptotical stability an thus oes guaantee sliing moe motion. Howeve fo time t > t 0 the motion stays in the ɛ vicinity of the sliing moe manifol. Fo vecto function Φ(σ) = Γsign (σ) esulting contol is iscontinuous an manifol (6) is eache in finite time. Many systems, fo example estimatos an obseves, o not suppot switching opeation well. Fo this application the choice fo Φ(σ) is selecte as Φ(σ) = Dσ, D 0 which esults in the continuous contol with asymptotic stability of the solution σ = σ ( x,x ) =0. The continuous contol function is then: u(t) =u equ +(GB) 1 σ (10) whee u equ is so-calle equivalent contol an is continuous function [9]. Since the contol pesente in this pape is switching by its natue, Φ(σ) =Γsign (σ) was use. 3 PROPOSED VSC CURRENT AND FLUX CON- TROL SCHEME The aim of this pape is to pesent an application of switching contol in switching powe convetes, which use the finite state machine pinciple embee in FPGA envionment. With the use of logic event iven appoach we inten to pesent the opeation of switching convetes, which ae epesente as the set of enegy stoage elements with thei inteconnections ynamically change by the opeation of the switching matix [10]. The switching matix explains the ole of switching elements that etemine the powe exchange between enegy stoage elements. By the intouction of switching into the stuctue of the system the esign in the famewok of switching contol is mae a natual choice. Consieing the opeation of an IM togethe with the invete, a iscete event-iven appoach can be intouce [3, 10] (Fig. 1). The invete is connecte to the DC-link voltage an its opeation is base on the tansisto switching patten. Instantaneous ynamics of the contol ae etemine. Diffeent tansisto switch patten esults in iffeent plant ynamics. Applying pope switching among these plant ynamics, the system can tack the esie efeence tajectoy. Switching patten of each tansisto is consiee as a iscete state of the system an the changing of the tansisto switching patten is consiee as a tansition of the system among iscete states. The tansition of the iscete system between iscete states can be consiee as occuence of an event. To contol tansitions of the system among the iscete states, some aitional conitions ae intouce. Consieing the vecto cuent eos as conitions fo the tansitions, a iscontinuous cuent contol Φ(σ) = Γsign (σ) (9) is achieve, which is fast, obust an simple fo implementation. To achieve safe an manageable ive opeation, monitoing an potection functions shoul be inclue. They ae event-iven inheently, as they eact on the change of logic conitions. If they ae combine with the popose iscontinuous cuent contol an oto flux obseve, ove- AUTOMATIKA 51(2010) 1,

4 all functionality of the ive can be systematically escibe by Fig. 1. To contol the stato cuent, the secto of momentay invete voltage u s is ecognize fist. The new output voltage vecto (the tansisto switching patten) fo the stato cuent contol is selecte especting the cuent contol eo an Lyapunov stability conition. Consieing space vecto epesentation of the invete stato voltageu s, it is epesente as vecto otating aoun the oigin (Fig. 2). Six active switching vectos of the thee phase tansisto bige esult in six output voltage vectos enote as V 1,...,V 6. Accoing to the signs of the IM teminal phase voltages u s1, u s2 an u s3, the phase plane is ivie into six sectos enote with Su1,...,Su6. Fig. 2. Stato voltage u s secto allocation spaces The situation pesente in Fig. 2, whee the stato voltage space vecto u s is in secto 1, is the following. In this secto voltage vectos V 0, V 7, V 1, V 2 an V 6 ae selecte fo the IM cuent contol. V 0, V 7 ae two zeo vectos, while V 1, V 2, V 6 ae neaest ajacent live output voltage vectos to this secto. With the use of the iscete event system theoy, five output voltage vectos V 0, V 7, V 1, V 2 an V 6 ae ecognize as possible iscete states of the system. The esulting events epesent allowe tansitions among the iscete states i.e. allowe switching pattens. The stuctue of the popose stategy is epesente by Peti Net gaph (Fig. 3) [11]. Switching among the available output voltage vectos in each secto is etemine by conitions that oiginate fom the eivative of the Lyapunov function. Fo the Lyapunov function caniate V = 1 2 σt σ = 1 ( ) i T ( ) 2 s i s i s i s, (11) the stability equiement is fulfille if such a contol can be selecte, that the eivative of the Lyapunov function caniate is negative V = σ T σ 0 Deivatives of cuent contol eo may be expesse with the voltage equation: ( ) i 1 t s i s = t i s / (u L s L 2 s R s i s e ), m L (12) V 0 V 6 T 7 T 2 T 8 T 1 T 10 T 9 V 7 Fig. 3. PN-gaph of the switching sequence in Secto 1 whee i s, i s ae esie an actual stato cuent of moto, u s is voltage contol input, R s i s is esistive voltage op an e is electomotive foce (EMF) of the moto. The conitions fo the sequential switching of the powe invete ae selecte as: S R = 1 2 (1 sign(a)), S S = 1 2 (1 sign(b)), (13) S T = 1 2 (1 sign(c)), whee A = ( ) i sa i sa, B = ( ) ( ) i sa i sa /2 3 i sb i sb /2, C = ( ) ( ) (14) i sa i sa /2+ 3 i sb i sb /2, which is evolve fom the Lyapunov function eivative. When U DC has enough magnitue that V 0, than V 0 an i s i s. Notice that if S R, S S, S T equal to zeo simultaneously, no cuent is elivee to the moto. The main isavantage of the sliing moe contol is in the significant tansistos switching fequency vaiations ue to the changing opeating point o loa vaiations, as well as in the ifficulties to fol back o even limit the shot cicuit output cuent with sliing moe output voltage contol [3]. The popose logic event-iven stato cuent contol is simila to DTC an can be ealize in the fom escibe in Table 1, whee states of stato cuent contol eo ae pesente by sign(d i ) (D i = S R,S S,S T ) an cuently active voltage secto is pesente by signu (U = u S1,u S2,u S3 ). Fomulas fo S R, S S an S T ae pesente in (13), wheeas allocation of voltage secto is pesente in Fig. 2. To futhe impove the pesentation, active voltage vectos ae make in Table 1 by gay backgoun. Because the tansition between invete switch states is pefome by switching only one invete leg simultaneously, the chatteing in stato cuent cause by invete switching (an consequently toque chatteing) is euce. T 6 V 1 T 3 T 5 T 4 V 2 36 AUTOMATIKA 51(2010) 1, 33 40

5 Table 1. Look-up Table signu Su1 Su2 Su3 Su4 Su5 Su6 signd i Si0 000 V7 V0 V7 V0 V7 V0 Si1 100 V1 V1 V7 V0 V7 V1 Si2 110 V2 V2 V2 V0 V7 V0 Si3 010 V7 V3 V3 V3 V7 V0 Si4 011 V7 V0 V4 V4 V4 V0 Si5 001 V7 V0 V7 V5 V5 V5 Si6 101 V6 V0 V7 V0 V6 V6 Si7 111 V7 V0 V7 V0 V7 V0 The efeence cuent can now be calculate simply as: i s = + j 2 L T e L m 3p L m. (15) The oientation value equie fo the q-ab tansfomation use in conventional cuent contol is pefome applying the components of the oto flux, which ae iectly employe: ˆΨ b e jθ = cos Θ+j sin Θ = ˆΨ a +j = 1 ˆΨ. (16) The avantage of popose tansfomation is that the sin an cos function ae eplace with multiplication in the applie FPGA algoithm. 4 ROTOR FLUX OBSERVER The oto flux obseve, pesente in this wok, is base on the stato equation, whee the eivative of the estimate stato flux is calculate fom measue stato voltage an cuent [7, 12]. The obseve equation (17) epesents the fist oe vectoial iffeential equation. The stato voltage u s an cuent i s seve as contol input to the estimate stato flux ˆΨ s. In oe to avoi voltage eo influence ue to invete non-linea behavio, the measue value of the stato voltage is use instea of the commonly use efeence voltage: t ˆΨ s =û s ˆR s i s +û k, (17) Non-moele ynamic is set as a emaining signal û k, calculate fom the magnitue eo of the oto flux ΔΨ. The stato paametes of the IM appea in the oto flux obseve; i.e. stato esistance ˆR s an stato inuctanceσ ˆL s. The influence of the stato inuctance vaiation is small, but stato esistance changes significantly uing the opeation. The stato esistance paamete mismatch influences both, the amplitue an the oientation eo of the estimate oto flux. It has the highest influence in the lowspee egion, combine with the applie loa toque. This effect is use to obtain stato esistance eo fom the oto flux magnitue eo. It is iectly employe in the popose obseve stuctue, poviing low sensitivity of the obseve to the stato esistance vaiation. The influence of the stato esistance vaiation is compensate by intoucing a non-linea magnitue an oientation feeback compensato in the obseve. û k = u m + ju p. (18) The switching function of the VSC flux magnitue contolle C m is set to the eo between the efeence an estimate oto flux magnitue: σ m = D m σ m =0, σ m =Ψ t ˆΨ. (19) The iscete pat of the esulting unknown offset voltage u m is: C m : u m (k) = K m ((1 + T s D m ) σ m (k) σ m (k 1)). T s (20) The vaiation of the stato esistance Δ ˆR s impacts the toque vaiation of the IM, expesse by the vaiation of the oto flux eo an toque: ΔT e = σ m Te / Ψ, (21) σ p =ΔT e sign (ˆωT e ). (22) The switching function of the VSC oientation contolle C p takes into account the toque vaiation: σ p = D p σ p =0. (23) t When the toque is applie, the souce of the connection between the toque eo an the oto flux eo is the influence of the stato esistance eo. The iscete fom of the oientation coection signal becomes: C p : u p (k) =u p (k 1)+ + K p ((1 + T s D p ) σ p (k) σ p (k 1)) (24) T s Coection input signals influence the magnitue an oientation of the magnetic flux eo, an thus make the popose obseve obust to the paamete vaiations. The esulting block iagam, which is suitable fo poviing the AUTOMATIKA 51(2010) 1,

6 u m Cm Ψˆ Ψ u sa u sb Cp u p Rˆs Rˆs σlˆs σlˆs i sa i sb Ψˆ L Lm L Lm Fig. 4. Block iagam of the oto flux obseve estimate stato an oto fluxes of the IM, is shown in Fig. 4. The estimate synchonous spee is now expesse with the following coss pouct: ˆω s = 1 2 s ˆΨ s t Ψˆ a Ψˆ b ˆΨ s. (25) The estimate spee is compute entiely fom the estimate oto flux an its time eivative: ˆω = 1 ˆΨ ˆΨ t p 2. (26) The esulting estimate spee is subjecte to high noise levels ue to the eivative tem that can be euce by employing low pass filtes. The stability is compute only fo magnitue oto flux eo, which is a common input fo both, the magnitue an oientation eos of oto flux obseve. A Lyapunov stability citeia is use to veify the stability of the obseve. The squae magnitue of the oto flux eo is set as a Lyapunov caniate function. In the selecte case, a Lyapunov caniate function is, as equie, always positive. The notation is aopte fo complex numbes. {( ) } V = Re ˆΨ Ψ )(ˆΨ Ψ /2. (27) The eivative of the Lyapunov function caniate must be negative: {( ) ( ) } V = Re ˆΨ Ψ ˆΨ Ψ /t < 0. (28) The eivative of oto flux eo is obtaine using (2) an (3) ( ) ˆΨ Ψ i s =Δu s ΔL s t t ΔR si s + ˆΨ u m. (29) By using tigonometic functions fo oto flux esciption, when the amplitue of the oto flux is known, the conition becomes: ( )( ( )) ˆΨ Ψ ˆΨ Ψ cos ˆΘ Θ < 0. (30) If oto flux angle eo is sufficiently small than the conition is negative; theefoe the obseve is stable within given estictions. 5 IMPLEMENTATION The popose appoach is base on fast paallel pocessing an suitable fo a Fiel Pogammable Gate Aay (FPGA) implementation. In such implementation it is possible to epouce an ieal sliing moe pocess. Howeve, with FPGA implementation, esigne has the ifficult task to chaacteize an escibe the hawae achitectue that coespons to the chosen contol algoithm. FPGA esignes must follow an efficient esign methoology in oe to benefit fom the avantages of the FPGAs an thei poweful CAD (Compute Aie Design) tools. Fom a softwae point of view, Hawae esciption language (HDL) moeling system is base on using the vaiables that equest logic values as well. FPGA implementation of Table 1 is pesente in Fig. 5, whee signdi1, signdi2 an signdi3 pesent S R, S S an S T, espectively. Voltage secto states ae pesente by signu1, signu2, signu3. Invete leg switching outputs ae enote as follows: TOP1 an BOT1 pesent states of top an bottom tansisto in invete leg R, TOP2 an BOT2 ae pesenting states of tansistos in invete leg S, wheeas TOP3 an BOT3 pesent states of tansistos in invete leg T. 6 RESULTS The sliing moe toque an flux algoithm is implemente onto the in-house evelope DSP/FPGA boa. The DSP/FPGA boa contains Texas Instuments TMS 320C32 igital signal pocesso an Xilinx Spatan family fiel pogammable gate aay. DSP seves fo A/D convesion an geneating of the efeence cuent. Replacing usual sequential calculation of algoithms on the DSP by paallel executable FPGA hawae inceases the calculation spee. A/D convesion, which takes 5 μs is the most citical opeation egaing time. Accoing to the fact, that A/D convesion takes most of the calculation time, switching fequencies up to 200 khz ae theoetically possible. Fig. 6 an Fig. 7 show tansient esponse of estimate moto spee ω an toque T e. Fig. 8 shows tansient of moto phase cuents i s1, i s2 an i s3. Desie spee 38 AUTOMATIKA 51(2010) 1, 33 40

7 Fig. 5. FPGA schematic cicuits [ ] i1, i2, i3 A Fig. 6. Spee sensoless efeence tacking Fig. 8. IM phase cuents [ Nm ] T e Fig. 7. Estimate toque tansient in expeiment vaies following the amp signal an aitional loa is intouce at the time of 0.3 s. Satisfactoy tacking of spee, even in the case of slow tansient though zeo, is shown in Fig. 6. Estimate toque pouce by the moto is shown in Fig. 7, pesenting the impact of toque, equie fo acceleation, eceleation an loa toque. Phase cuent, pesente in Fig. 8 shows sinu- soial signals, contaminate with satisfactoy cuent ipple cost by switching. Likewise, toque ipple is within esie limits. 7 CONCLUSION Oveall functionality of the invete an the pefomance of the popose event-iven cuent contol was checke by simulations fist [10], an expeimentally confime in this pape. Special attention is pai to the mapping of the popose esign appoach into the schematic fom fo the FPGA implementation. Sliing moe oto flux obseve base on IM moel is use to obtain oto position an spee. This obseve is sensitive to stato paamete changes an is impove with stato esistance aaptation. The popose metho shows goo convegence an close loop stability of the obseve ove a wie spee ange, une moel uncetainties, in paticula those cause by stato esistance eo. Solutions base on specific hawae, that allow high concuency, ae suitable to be use in powe electonics an motion contol applications. AUTOMATIKA 51(2010) 1,

8 REFERENCES [1] J. Holtz, Sensoless contol of inuction machines with o without signal injection?, IEEE Tansactions on Inustial Electonics, vol. 53, no. 1, pp. 7 30, [2] J. Holtz, Estimation of the funamental cuent in low switching fequency high-ynamic meium voltage ives, in Poceeings of the 42n IAS Annual Meeting, pp , Septembe [3] A. Tilli an A. Tonielli, Sequential esign of hysteesis cuent contolle fo thee-phase invete, IEEE Tansactions on Inustial Electonics, vol. 45, no. 5, pp , [4] V. I. Utkin, Sliing Moes in Contol an Optimization. Belin: Spinge Velag, [5] D. Hecog, B. Gegič, S. Uan, an K. Jezenik, A spbase emote contol laboatoy, IEEE Tansactions on Inustial Electonics, vol. 54, no. 6, pp , [6] G. Buja an M. Kazmiekowski, Diect toque contol of pwm invete-fe ac motos a suvey, IEEE Tansactions on Inustial Electonics, vol. 51, no. 4, pp , [7] C. Lascu, I. Bolea, an F. Blaabjeg, Diect toque contol of sensoless inuction moto ives: A sliing-moe appoach, IEEE Tansactions on Inusty Applications, vol. 40, no. 2, pp , [8] E. Mitonikas an A. Safacas, An impove sensoless vecto-contol metho fo an inuction moto ive, IEEE Tansactions on Inustial Electonics, vol. 52, no. 6, pp , [9] A. Šabanović, K. Jezenik, an N. Šabanović, Vaiable Stuctue Systems: Towas the 21st Centuy, vol. 274/2002 of Lectue notes in contol an infomation sciences, ch. Sliing moes applications in powe electonics an electical ives, pp Spinge Belin / Heielbeg, [10] A. Polič an K. Jezenik, Matix base event-iven appoach fo cuent contol esign of vsi, in Poceeings of the 2005 IEEE Intenational Symposium on Inustial Electonics, (Dubovnik, Coatia), pp , June [11] A. Polič an K. Jezenik, Close-loop matix base moel of iscete event systems fo machine logic contol esign, IEEE Tansactions on Inustial Infomatics, vol. 1, no. 1, pp , [12] C. Lascu, I. Bolea, an F. Blaabjeg, Compaative stuy of aaptive an inheently sensoless obseves fo vaiablespee inuction-moto ives, IEEE Tansactions on Inustial Electonics, vol. 53, no. 1, pp , Kael Jezenik eceive B.Sc. (1968), M.Sc. (1974) an D. Eng. (1976) egees in electical engineeing fom the Univesity of Ljubljana. In 1976 he joine the Univesity of Maibo an in 1985 he became a Full Pofesso an Hea of the Institute of Robotics. His eseach an teaching inteests inclue automatic contol, obotics, powe electonics an electical ives. Cuent pojects in these aeas ae high pecision tacking contol in machine tools, DD obots an obust toque contol in EV s. He consults on inustial sevo contol systems an othe contol applications. Asif Šabanović eceive B. Sc. (1970), M. Sc. (1975), an D. Sc. (1979) egees all in electical engineeing fom Univesity of Saajevo Fom 1970 to 1991, he was with Enegoinvest Institute fo Contol an Compute Sciences, Saajevo. An then he was with the Depatment of Electical Engineeing, Univesity of Saajevo. Since 2000, he has been with the Faculty of Engineeing an Natual Sciences, Sabanci Univesity. He has been Visiting Reseache with the Institute of Contol Science, Moscow, Russia ( ) an Visiting Pofesso at the Califonia Institute of Technology, Pasaena ( ), Keio Univesity, Yokohama, Japan ( ), Yamaguchi Univesity, Ube ( ). He was hea of CAD/CAM an the Robotics Depatment at TUBITAK-MRC, Istanbul, Tukey ( ) an Hea of the Engineeing Depatment of B. H. Engineeing an Consulting ( ). His fiels of inteest inclue contol systems, motion contol systems, obotics, mechatonics, an powe electonics. Mian Roič eceive the B.Sc., M.Sc., an Ph.D. egees fom the Univesity of Maibo, Maibo, Slovenia, in 1994, 1996, an 2000, espectively. He is cuently a Teaching an Reseach Assistant in the Institute of Robotics, Faculty of Electical Engineeing an Compute Sciences, Univesity of Maibo. His pime fiel of inteest is the contol of sevoives, especially contol of ac ives. AUTHORS ADDRESSES Pof. Kael Jezenik, Ph.D. Asst. Pof. Mian Roič, Ph.D. Faculty of Electical Engineeing an Compute Science, Univesity of Maibo, Smetanova 17, SI-2000 Maibo, Slovenia s: kael.jezenik@uni-mb.si, mian.oic@uni-mb.si Pof. Asif Šabanović, Ph.D. Faculty of Engineeing an Natual Sciences, Sabanci Univesity, Ohanli-Tuzla, Istanbul, Tukey, asif@sabanciuniv.eu Receive: Accepte: AUTOMATIKA 51(2010) 1, 33 40