Open Access An Exponential Reaching Law Sliding Mode Observer for PMSM in Rotating Frame

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1 Sen Orers for Reprints to The Open Automation an Control Systems Journal, 25, 7, Open Access An Exponential Reaching Law Sliing Moe Observer for PMSM in Rotating Frame Xiao Genfu * an Zhou Yanhui School of Mechanical an Electrical Engineering, Jinggangshan University, Ji an, Jiangxi, 3439, P.R. China Abstract: In orer to solve chattering problem of conventional sliing moe observer SMO, this paper proposes a exponential reaching law sliing moe observer ERL-SMO for a permanent magnet synchronous motor PMSM which substitutes exponential reaching law for constant reaching law. The ERL-SMO estimates the rotor position an the angular velocity from the back electromotive force EMF base on rotating frame instea of fixe frame. A phase locke loop PLL replaces low-pass filter which commonly leas to a phase elay. The stability of the propose exponential reaching law was verifie using the Lyapunov function. The simulation results show that ERL-SMO hasn t only achieve a precise ientification of PMSM spee an position, but also significantly weakene the signal chattering. Keywors: Exponential reaching law, PMSM, SMO, sensorless, vector control.. INTRODUCTION Recently, the usage of alternating-current ac motors instea of irect-current c motors has been increasing rapily in a wie range inustrial fiels. There are two types of ac motors: the inuction motor IM an the permanent magnet synchronous motor PMSM. The PMSM is very popular in ac motor applications for various spee controls ue to its high power ensity, higher output torue an high efficiency []. The IM has a simple structure, an it is easy to buil. However, it is not as efficient as the PMSM is consiere in terms of ynamic performance an power ensity. Since there is no brush in an ac motor, the size of the ac motor can be smaller with the same power an the lifetime is much longer than that of a c motor. However, an ac motor has a more complex control system than that of a c motor [2]. Precise position ata are necessary for an efficient vector control, since a PMSM receives a sinusoial magnetic flux from the PM of the rotor. Generally, the rotor position can be etecte by an absolute encoer. However, these sensors are expensive an very sensitive to environmental constraints such as vibration an temperature [3]. To overcome these problems, instea of using position sensors, a sliing moe observer SMO has been evelope for the estimate values of the position an velocity of the rotor [4-]. The SMO maintains goo robustness of variable structure control [2]. The control loop in an orinary observer is replace by a sliing moe variable structure [3], an when the system error reaches the sliing moe, the system s ynamic performance entirely epens on the sliing surface, which ensures goo robustness of the entire system to parameter variations [4]. Due to the iscrete switch control in the SMO, chattering becomes the inherent characteristic of *Aress corresponence to this author at the School of Mechanical an Electrical Engineering, Jinggangshan University, Jiangxi, 3439, P.R. China; Tel: ; xiaogenfu@63.com the sliing moe variable structure system. As chattering cannot be completely eliminate but only reuce, in the esign of the system, there shoul be a traeoff between chattering reuction an system robustness. For the traitional SMO, the switch function is use as the control function. Due to switch time an space lag, the SMO presents serious chattering [5]. The estimate signal of the SMO contains high-freuency oscillation components, so one or several filters are usually use to extract the reuire back electromotive force EMF signal. In a conventional SMO, a low-pass filter an an aitional position compensation of the rotor are use to reuce the chattering problem commonly foun in SMO using a signum function. However, the introuction of low pass filter usually causes phase elay, which shoul be compensate for accoring to the actual angular freuency. Thus, this cannot meet the control reuirements of high performance applications. An aaptive filter is propose, which still fails to completely compensate for phase elay [6]. This paper proposes a new SMO algorithm which uses an exponential reaching law base on rotating frame instea of fixe frame. At the same time, the phase locke loop PLL is use to estimate rotor position an spee, eliminating the nee for filters an phase compensator. The superiority of the propose algorithm has been prove by comparison with the conventional SMO through experiments. This paper consists of five sections, incluing the introuction. Section II introuces the conventional SMO, an section III proposes the new SMO. Section IV illustrates the experimental results, an section V raws the conclusions relating to the contributions of this paper. 2. CONVENTIONAL SMO 2.. PMSM Moel in Fixe Frame The PMSM consists of a rotor with a PM an a stator with a three-phase wining. It can be simplifie to a twophase moel though coorinate transformation /5 25 Bentham Open

2 6 The Open Automation an Control Systems Journal, 25, Volume 7 Genfu an Yanhui The state euations, where the stator current is a state variable of the fixe frame voltage euation, can be represente as: i! t = R s L s i! L s e! + L s v! i t = R s L s i L s e + L s v Where i!, i!, e!, e!, an v!, v! represent the current, electromotive force, an voltage for each phase, respectively, in the fixe frame. R s an L s represent the stator resistance an inuctance, respectively. The electromotive force for each phase can be represente in the fixe frame as: * e! = f f sin e = f f cos 2 Where! f,! f an! represent the magnetic flux of the permanent magnet, the electric angular velocity, an the rotor angle, respectively. The rotor angle an spee can be represente in the fixe frame as:,! = arctan e.. * e =! / t 2.2. The Principle of Sliing Moe Control System If the system state variable is efine as:!xt = f x,u,t,x!r n,u!r m,t!r 4 The switch function is represente as: sx,s!r m 5 An the control variable is efine as: u + x sx > ux = u! x sx < Where, u + x! u x. If a system complies with the previous formula an meets the sliing moe existence, accessibility, stability, an goo ynamic uality, the system is calle the sliing moe control system Conventional SMO For the implementation of the sliing moe observer, the sliing surfaces below are selecte as: sx = î s!i s 7 Where î s is estimate current, i s is actual current. Accoring to the PMSM euation in the! coorinate system fixe frame, the conventional SMO is constructe as follows: î! t = R L î! + u! L L signî! i! î t = R L î + u L L signî i Where signs is sign function,! is switching gain. 8 Through a low pass filter, the high freuency components in signî! i are filtere an back EMF estimate value ê s is obtaine as follow: * ê! = ê = c s + c signî! i! c s + c signî i Where! c is cut-off freuency of the low pass filter. The angle of the rotor ˆ! can be estimate by back EMF, an the rotation spee ˆ! can be obtaine by ifferentiating the angle. ˆ! = arctanê ê ˆ! = ˆ t 9 The estimate angular position nees be compensate for the phase elay cause by a low pass filter. The final estimate value of the rotor position ˆ! z is the sum of ˆ! an compensation value with relate motor velocity.! = tan / c 2 ˆ! z = ˆ! +! 3 Fig. shows the conventional SMO where the sign function is use as the switching function, an the low pass filter LPF is use to eliminate the chattering effects from the switching, where! value is etermine by the Lyapunov function.

3 An Exponential Reaching Law Sliing Moe Observer The Open Automation an Control Systems Journal, 25, Volume 7 6 i iˆ R ˆ = i + u e t L L î λ sign iˆ i ê Fig.. The construct of conventional SMO. 3. EXPONENTIAL REACHING LAW SMO 3.. PMSM Moel in Rotating Frame The PMSM mathematical moel in the! coorinate systemrotating frame can be expresse as: i t =! R i L + u L + L e i L! E L i t =! R i L + u L + L e i L! E L J r t = T e! B r! T L T e = 3 2 p f i + L! L i i 4 Where, represent, -axis component respectively, an! e represents rotor electrical angular velocity, an E, E represent EMF in the - coorinate system, p is the number of pole pairs of the motor,! f is flux of permanent magnet, T e is electromagnetic torue, J is Moment of inertia, an! r represents mechanical angular velocity of rotor, an B is rotor friction coefficient Exponential Reaching Law Exponential reaching law is the sum movement of exponential e ηt an constant as follows:!s =! sgns! ks, >,k > 5 Where! an k are constants. With respect to the constant reaching law, exponential reaching law increases the initial spee of the system exponentially, an its parameters k an δ can be ajuste. Exponential reaching law not only can improve the uality of the sliing moe ynamic arrival process, but can also weaken the control signal chattering. Reaching conition of sliing moe control system can be expresse as: s!s < 6 Accoring to formula 5 s!s =!s[ks + signs] =!ks 2! signss =!ks 2! s 7 Due to k>,! >, we can know s!s <. We select formula 8 as Lyapunov function. V x = 2 s 2 + s The erivation of formula 8 is:! V x = s!s + s 2!s 2 9 Base on the above analysis, we know s!s < an! V x <, which can prove that exponential reaching law has asymptotic stability in view of Lyapunov Exponential Reaching Law SMO In orer to reuce the chattering of conventional SMO, switching function is replace by exponential reaching law. Exponential reaching law SMOERL-SMO can be expresse as follows: i t =! R L i + L u + L e L i! L sgns! s i t =! R L i + L u + L e L i! L sgns! s s = î! i s = î! i The back EMF of motor can be expresse as: e * =! sgns! s e * =! sgns! s The construct of ERL-SMO is shown in Fig Estimation of Rotor Position an Velocity 2 2 The low pass filter in conventional SMO will lea to a certain position estimation phase ifference. Phase locke

4 62 The Open Automation an Control Systems Journal, 25, Volume 7 Genfu an Yanhui i î δ sign iˆ i iˆ R Lωe = iˆ + u + iˆ e t L L L L iˆ R L e = iˆ ω + u + iˆ e t L L L L ê Fig. 2. The construct of ERL-SMO. e a e b e c E s E s θˆ ωˆ ˆ θ = ˆ θ ± π / 2 z Fig. 3. The construct of PLL. loop PLL in ERL-SMO will replace low pass filter to gain estimate value of rotor position an velocity. EMF of three phase stator winings is expresse as follows: e a = E cos!t e b = E cos!t 2! e c = E cos!t +2! 22 Where! = t, an! = 2 f = pn / 3, p is the number of pole pairs of the motor, an n is the velocity of the motor. Accoring to the synchronous rotating coorinate transformation theory, the transformation matrix of - coorinate system is expresse as follows: cos ˆ!t cos ˆ!t 2 o cos ˆ!t +2 o T = 2 sin ˆ!t sin ˆ!t 2 o sin ˆ!t +2 o 23 3 / 2 / 2 / 2 Where ˆ! = ˆt is the phase angle calculate by PLL, an! = ˆ is the error of phase lock. The target of PLL is! =. The rotating coorinate transformation of three-phase inuce electromotive force can be expresse as follows:! E E! sin ˆ! = 3 M cos ˆ sin sin ˆ = + 3 M cos sin ˆ + cos ˆ + cos ˆ + 24 The element of matrix M2*4 is: M =, M 2 =3E, M 3 =, M 4 =, M 2 =-M 2, M 22 =M, M 23 =-M 4, M 24 =M 3. When phase is locke by PLL, we know ˆ! =!, an above formula can be simplifie to:!! E E =! M M 2 3 M 2 M! = E s E s! E s E s sin cos 25 Accoring to the efinition of synchronous rotating - coorinate system an formula 25, we may know E s = M cos! =. The PLL can be constructe to gain PMSM spee an rotor position through former formula as shown in Fig. 3.

5 An Exponential Reaching Law Sliing Moe Observer The Open Automation an Control Systems Journal, 25, Volume 7 63 U c ω ψ + L i e f ω L i e Ê i i u u U α U β θ i, i, i u a, ub, u a b c c Ê ω e Fig. 4. Block iagram of sensorless control system. Table. PMSM parameters. Parameters Unit Value Stator phase resistance Ohm.92 InuctancesL H.5 Flux linkage V.s.228 Inertia Jkg.m^2 6 Friction factor N.m.s 6 Pole pairs P 3 2 Speem/s act value est value 2 Fig. 5. Spee estimation of conventional SMO Times

6 64 The Open Automation an Control Systems Journal, 25, Volume 7 Genfu an Yanhui.5 errorm/s Fig. 6. Error of conventional SMO spee Times 2 Speem/s act value est value 2 Fig. 7. Spee estimation of ERL-SMO Times errorm/s Fig. 8. Error of ERL-SMO spee estimation Times

7 An Exponential Reaching Law Sliing Moe Observer The Open Automation an Control Systems Journal, 25, Volume act value est value 5 Thetara Times Fig. 9. Position estimation of ERL-SMO Errorra Times Fig.. Error of position estimation. 4. SIMULATIONS AND ANALYSIS OF RESULTS In orer to verify the effectiveness, the ERL-SMO is simulate in Matlab/Simulink. The PMSM parameters are shown in Table. The vector control metho is aopte, using a step function as spee input. Because of the mutual coupling of, - axis variable, fee-forwar ecoupling control strategy is applie. Sensorless control system block iagram is shown in Fig. 4. Figs. 5, 6 are the PMSM spee estimate value an the spee error of the conventional SMO. As can be seen, because of the switch function, there is a big chattering. Figs. 7, 8 are the PMSM spee estimate value an the spee error of the ERL-SMO. As can be seen, in the ynamic process, ERL-SMO can track actual spee value uickly. After entering the steay state, the estimate error is close to zero. Figs. 9, are the PMSM rotor position estimate value through the ERL-SMO an the estimation error of the rotor position. As can be seen, error of position estimation is small an the estimation of the rotor position is accurate. CONCLUSION This paper has propose an ERL-SMO for a sensorless system for a PMSM. The stability of the new SMO has been prove with the use of a Lyapunov stability analysis. The chattering problem in the conventional SMO was resolve by using the exponential reaching law instea of the conventional sign function. A PLL was employe to reuce the estimate error associate with low pass filter. The superiority of ERL-SMO has been confirme through simulation. The ERL-SMO improves the reliability an accuracy of preiction of the rotor position, having significance in terms of the practical application an theory research.

8 66 The Open Automation an Control Systems Journal, 25, Volume 7 Genfu an Yanhui CONFLICT OF INTEREST The authors confirm that this article content has no conflict of interest. ACKNOWLEDGEMENTS This work was supporte by Science an Technology Support Project of Jiangxi Province of China No.24-2BBE557. REFERENCES [] A. Wang, an X. Jia, A new exponential reaching law of sliing moe control to improve performance of permanent magnet synchronous motor, IEEE Trans. Magn., vol. 49, pp , 23. [2] P. Pillay, an R. Krishnan, Application characteristics of permanent magnet synchronous an brushless c motor for servo rive, IEEE Trans. In. Appl., vol. 27, pp , 99. [3] F. Parasiliti, R. Petrella, an M. Tursini, Sensorless spee control of a PM synchronous motor by sliing moe observer, Proc. IEEE ISIE, vol. 3, pp. 6-, 997. [4] P. Vaclavek, an P. Blaha, Lyapunov function-base flux an spee observer for ac inuction motor sensorless control an parameters estimation, IEEE Trans. In. Electron., vol. 53, pp , 26. [5] S. Ichikawa, M. Tomita, S. Doki, an S. Okuma, Sensorless control of permanent magnet synchronous motors using online parameter ientification base on system ientification theory, IEEE Trans. In. Electron., vol. 53, pp , 26. [6] S. Chi, Z. Zhang, an L. Xu, Sliing moe sensorless control of irect rive PM synchronous motors for washing machine applications, IEEE Trans. In. Appl., vol. 45, pp , 29. [7] H. Kim, an J. Son, A high-spee sliing-moe observer for the sensorless spee control of a PMSM, IEEE Trans. In. Electron., vol. 58, pp , 2. [8] Z. Qiao, an T. Shi, New sliing-moe observer for position sensorless control of permanent-magnet synchronous motor, IEEE Trans. In. Electron., vol. 6, pp. 7-79, 23. [9] H. Lee, an J. Lee, Design of iterative sliing moe observer for sensorless PMSM control, IEEE Trans. Control Syst. Technol., vol. 2, 23. [] B. Thiago, an F. M. Vinicius, Discrete-time sliing moe observer for sensorless vector control of permanent magnet synchronous machine, IEEE Trans. In. Electron., vol. 6, pp , 24. [] Y. Feng, X. Yu, an F. Han, High-orer terminal sliing-moe observer for parameter estimation of a permanent magnet synchronous motor, IEEE Trans. In. Electron., vol. 6, no., pp , 23. [2] S. Wang, an X. Li, Computer-controlle variable structure systems: The state of the art. IEEE Trans. In. Informat., vol. 8, pp , 22. [3] G. Foo, an M. F. Rahman, Rotor position an spee estimation of avariable structure irect torue controlle IPM synchronous motor rive at very low spees incluing stanstill, IEEE Trans. In. Electron., vol. 57, pp , 2. [4] M. L. Corraini, G. Ippoliti, S. Longhi, an G. Orlano, A uasi sliing moe approach for robust control an spee estimation of PM synchronous motors, IEEE Trans. In. Electron., vol. 59, pp. 96-4, 22. [5] G. Tarchala, Influence of the sign function approximation form on performance of the sliing-moe spee observer for inuction motor rive, In: Proceeing of IEEE International Symponysim. In. Electron., 2, pp [6] G. L. Cascella, N. Salvatore, an L. Salvatore, Aaptive sliingmoe observer for fiel oriente sensorless control of SPMSM, Proc. IEEE In. Appl. Conf., vol. 2, pp , 23. Receive: September 6, 24 Revise: December 23, 24 Accepte: December 3, 24 Genfu an Yanhui; Licensee Bentham Open. This is an open access article license uner the terms of the Creative Commons Attribution Non-Commercial License which permits unrestricte, non-commercial use, istribution an reprouction in any meium, provie the work is properly cite.