Monovariable H Robust Controller for a Permanent Magnet Synchronous Machine (PMSM)

Monovariable H Robust Controller for a Permanent Magnet Synchronous Machine (PMSM) J. Kheri # # Automatic Control Unit, National School of Engineers of Sfax, ENIS Roa of soukra, km 3.5 - BP.W, 3038, Sfax, unisia kherijamel@.yahoo.fr Abstract his paper eals with the moelling, analysis esign an experimental valiation of a robust control metho for a permanent magnet synchronous machine (PMSM) supplie with a PWM inverter base on a monovariable H infinity stanar controller. Uner the influence of uncertainties an external isturbances, by a variation of ±0% of all motor parameters from the nominal values, the robust performance control problem is formulate as a monovariable H infinity stanar scheme an solve by a suboptimal iterative H infinity strategy. his new esign metho is able to ensure the stabilization of the augmente system forme of the perturbe system an weighting filters with improve performance in face of parameter variation an external isturbances. A Simulation an experimental stuy was carrie out to illustrate the effectiveness of the propose metho. Keywor- Robust Performance, Monovariable Stanar H infinity controller, Uncertainties, PMSM I. INRODUCION In recent years, growth in magnetic power evice, an control theory have mae the PMSM play a crucial important role in high performance inustrial rives in the low-to-meium- power range. he esirable iiosyncrasy of the PMSM is its compact simple structure, high-energy efficiency, high air-gap flux ensity, reliable operation, high torque capability an high power ensity. Otherwise, compare with inuction motors, PMSM have the avantage of higher efficiency, ue to the absence of rotor losses an lower vacuum current below the rate spee an its ecoupling control performance is much less sensitive to the parameter variations of the motor [], [2]. Nevertheless the control performance of the PMSM moel is still influence by the uncertainties of the plant, which all along are compose of unpreictable plant parameter variations an external isturbances. hus, many approaches such as optimal control [3], aaptative control [4], [5], nonlinear control [6] an robust control [7] have been evelope for the PMSM to eal with uncertainties. During the past ecae the H control problem have been extensively celebrate for its robustness in counter-acting parameters variations an external isturbances. he main objective of the H control is to synthesize a controller making the close-loop system to satisfy a prescribe H norm constraint which representing esire stability or tracking requirements. However.to ensure goo performance robustness uner uncertainty perturbations, the H approach give usually a solution with high control gain. Motivate by this iea, our main contribution in this paper, is to evelop an efficient robust metho calle monovariable H control that is insensitive to these variations. he organization of the paper is as follows. First the problem notification is presente an the nominal an uncertain plant of the PMSM is swiftly escribe.secon a robust monovariable H controller is evelope. Besies filters weightings incorporate in the system to ensure H performance are mentione. Finally the control law esign is carrie out to the regulation of the spee an -axis current of the PMSM. he experimental set-up assemble in the LAII laboratory is mae of a Dspace 04 numerical prototype car with its interface, a three-phase IGB base inverter using a vector PWM technique an a permanent magnet synchronous servo-motor (PMSM) equippe with position sensors [8]. II. PROBLEM FORMULAION Before conceiving the robust monovariable H controller, let s note that the process has to be surrogate is uner the stanar configuration below where the state space approach is aopte for resolution [9], [0], [], [2], an [3]: ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 553

robust output z output y reference e Augmente Plant P H controller isturbance w input u Fig Stanar robust feeback control system Where w is the isturbance input, u is the regulate variable, z is the robust output that we want to minimize an y is the measure variable that we use to monitor the system. P is the augmente plant that inclues weighting functions which will be efine later. Let s note that the H controller is referre by K. hus we can write: z w = P y u = P P 2 w P2 P () 22 u Consequently, we can express the epenency of z on w as: z=f(p,k)w (2) As mentione above, the objective of H control is to fin a controller K stabilizing such that the H norm of F(P,K) is boune by a constant γ which represent the esire performance level of the close loop system: F(P,K) < γ (3) Let s give the H structure of the control plant as shown in Fig (2). his Fig shows the interconnection of the augmente plant which inclues the nominal moel of the system G nom, the controller K, the input reference r, the unmeasure isturbances p on the control u, the output reference e, an the noise measure b on the output y. inputs an outputs are weighte by weighting functions W, W 2 an W 3 W is known as the tracking error performance, W 2 is the robust performance an W 3 is chosen to set uncertainties norm that the close-loop system must allow. W 2 z 2 W z W 3 w 2 r e u + + y K Gnom b + + Fig 2 H stanar configuration ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 554

Due to the control low structure u=ky, one confuse the input b an r. hus we can aopt the new H structure: W z W 2 z 2 r e u + y - + p G nom w 2 W 3 P K Fig 3 H configuration of the control plant On referring to the Fig (3), the inequality (3) can be expresse in terms of weighting functions as follow: WS WGSW 3 < γ (4) WKS WW 2 2 3 Where the sensitivity function GK S= an it complementary = are the ratio of output to the +GK +GK isturbance an the ratio of output to input respectively State Moel of the Permanent Magnet Synchronous Machine (PMSM) ake into account the classical simplifying hypothesis, the nominal moel of the permanent magnet synchronous machine, in the Park lanmark [4], can be given by the following set of equations [5], [6], [7]: I R Lq = I + w r I q + v t L L L Iq R L Φ = Iq wri wr + vq t Lq Lq Lq Lq wr ( ce cr) fc = p wr (5) t J J c = p[( L L ) I + Φ] I e q q 2 r fc = [( L Lq) I + Φ] Iq wr cr w p p t J J J Where v -q : -q axis stator voltages I q : -q axis stator currents L -q : -q axis inuctances R : the stator resistance Φ : the flux linkage of the permanent magnets w r : the rotor spee f c : the friction coefficient ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 555

J : the moment of inertia c e-r : the electromagnetic an the loa torques p : the number of pole. o conceive such control law, we shoul linearise the nominal moel aroun an operating point, 0 0 s 0 q0 r0 0 q0 0 0 x s,u [8]: x = I I w,u = v v (6) herefore, the linearize nominal moel is as follow: δx = A δx +Bδu+D v s sl s sl sl y=c δx where sl s δx s = δi δiq δw r,δu= δv δv q (8) R Lq Lq - wr0 Iq0 0 L L L L L R Φ L A sl = - wr0 - - - I 0,B sl = 0, Lq Lq Lq Lq Lq 2 2 p p fc 0 0 (L - L q )I q0 [Φ+(L - L q )I 0 ] - J J J 0 0 0 C sl = an D sl = 0 0 0 p - J (7) III. ROBUS MONOVARIABLE H DESIGN CONROLLER As mentione above, the control law is esigne for the purpose of regulating the spee w r an the irect current I. But by referring to equation (5), we see a strong coupling between the spee an the axis current. In this regar, to obtain totally ecouple variables we aopte the following representation where the nominal control moel inclue the following transfer matrix of variable we want to control that is to say H inom(s) an H wrnom(s). H inom(s) 0 H sl(s)= 0 H wrnom(s) (9) where: H inom(s)= R+L s p 2 Φ H wrnom(s)= JL 2 2 2 qs +(Lq f c+ JR)s+ Rf c+ p Φ As mentione in [9], parameters most probable to change, uner several conitions, are the electrical an mechanical time constants ( τ elc an τ mec ). Inee R influence by temperature can vary, J increase an Φ ecrease after an extene utilization. aking in consieration these variations pushes us to aopt a variation of ± 0% of all machine parameters from its nominal values. For the selection of weighting functions W, W2 an W 3, is base on the choice of templates. Which being impose by esire specifications. It is note that the introuction of the weights on the ifferent signals take the form of filters that allowing to the signal to which they apply, to focus a particular area of frequencies. We can summarize these esire specifications by imposing: Frequency template emporal template (0) ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 556

In our application ks+α W(s)= s Where k is taken equal to 400 for a moule margin less than 6b, α = 398 which settle the rise time. A way that α is greater, the rise time is shorter, an the close loop response is fast. In our case, the rise time is equal to 0.5s an the overshoot of the close loop system is almost null. W(s), u which acts explicitly on the control signal u, is selecte in orer to limit its amplitue an reuce the orer of the controller K. W(s) u is chosen a positive scalar W(s)=0.0 u for o stanar H of () less then γ = 0.056 he weight W(s) is chosen base on uncertainties norm that the close-loop system must accept [5]. as+b 0.s+ W(s)= = -5 cs+ 78e s+0. he steps of calculating a robust controller by H synthesis can be summarize in the following flow chart: Setting G nom(s)uner state form Choice of W, W, 2 W 3 Writting the increase system P(s) Ajust γ Synthesis of the H controller W WS s < If no If ok Performance evaluation Satisfying performance If no If ok Controller restraint En ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 557

Notice: Control of the -axis current I can be realize by a classic PI corrector. his kin of control allows the PMSM to operate in oriente flow while ensuring at machine moel, perfect ecoupling. In our stuy I =0.because as we have a smooth poles machine, the best selection for its operation is obtaine for a value where the internal angle of the machine is equal to π 2. Source 380V 50Hz P W M PMSM Encoer v abc I ref - PI v ref Park 2 3 w ref - H v qref I w r Park 3 2 Fig 4 Overall scheme of the monovariable H robust control of the PMSM he robust controller K euce from the algorithm escribe above an satisfying () takes the form: 5 3 7 2 9 9 2.3e s + 6.79e s +.27e s + 3.06e K(s)= s +.72e s +.48e s + 3.9e s + 0.54e 4 4 3 8 2 8-3 IV. EXPERIMENAL RESULS AND INERPREAIONS he photo given in Fig.5 present the test bench use Fig 5 Photo of the test bench ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 558

he experimental setup shown in Fig. 5 is constructe to inspect the effectiveness of the propose controller. wo ientical permanent magnet synchronous machines PMSM are use, one as a motor an the other as a loa. heir parameters that were use are given in table below Machine power ABLE I Parameter of the PMSM KW Rate current 6.5A Pole pair number (p) 2 -axis inuctance L q-axis inuctance Lq Stator resistance R 4.5mH 4mH 0.56 Ω Machine inertia J 2.08.0-3 Kg.m 2 Friction coefficient fc 3.9.0-3 Nm.s.ra - Magnet flux constant Φ 0.064wb he experimental set-up assemble in the LAII laboratory is mae of a Dspace 04 controller car with its interface, a voltage inverter using the space vector PWM (SVPWM) technique an a permanent magnet synchronous servo-motor (PMSM) equippe with 024 pulse/revolution incremental encoer. he current measurements are obtaine via hall effect sensors. he control algorithm was compile by the software Matlab / SIMULINK package, an implemente to machine language via software control esk. he igital sampling perio was taken equal to 0.ms. In orer to verify the effectiveness of the esigne controller, a series of measurements has been accomplishe. he results are the machine axis current an the rotor spee in which we observe the performance specifications of the synthesize robust controller to the PMSM. Inee, the measure spee an -axis current track well the trajectory of reference one with almost no static error an this over the whole spee range 0.4 measure current reference current -axis current an its reference in A 0.2 0-0.2-0.4-0.6-0.8 0 2 4 6 8 0 2 4 time in s Fig 6 -axis current an its reference ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 559

250 200 measure spee reference spee 50 Spee an its reference in r/s 00 50 0-50 -00-50 -200-250 0 2 4 6 8 0 2 4 time in s Fig 7 Spee an its reference V. CONCLUSION his paper evelops a methoology to esign a robust monovariable H controller in orer to ensure goo performance robustness uner uncertainty perturbations. An efficient algorithm base on iterative technique is propose to minimize the H norm of an augmente plant, inclues nominal moel of the PMSM, the controller K an weighting functions, in a way it is less than a positive constant γ he controller has been implemente with success on a permanent magnet synchronous machine (PMSM). ACKNOWLEDGMEN he authors wish to thank all the team of the Mixe Committee for the Acaemic Cooperation CMCU project an Research Unit of Inustrial Processes Control UCPI in the National Engineering School of Sfax. REFERENCES [] B.K. Bose, echnology trens in microcomputer control of electrical machines. IEEE ransactions on Inustrial Electronics, vol 35 (), pp 60-76, 988. [2] P.C.Sen, Electric motor rives an control-past, present an future. IEEE ransactions on Inustrial Electronics, vol 37 (), pp 562-575, 990. [3] P.M.Pelecezewski., W.Oberschelp., & U.K.Kunz, Optimal moel-following control of a positioning rive system with a permanentmagnet synchronous motor. IEEE Proceeing Control heory an Applications, Part-D, vol 38 (3), pp 267-273, 99. [4] N.Matsui., & H.Ohashi, DSP-base aaptative control of a brushless motor. IEEE ransactions on Inustry Applications, vol 28 (2), pp 448-454, 992. [5] E.Cerruto, A.Consoli, A.Racitti, & A.esta, A robust aaptative controller of PM motor rives in robotic applications. IEEE ransactions on Power Electronics, vol 0 (), pp 62-7, 992. [6] N.Hemati, J.S.horp & M.C.Leu, Robust non linear control of brushless DC motors for irect-rive robotic applications; IEEE ransactions on Inustrial Electronics, vol 37 (6), pp 460-468, 990 [7].H.Liu & C.P.Cheng, Controller esign for a sensorless permanent magnet synchronous rives system. IEEE. Proceeings Electrical Power Applications Part B, vol.40 (6), pp 368-378 [8] J.Kheri, A Fuzzy Moel racking observer-base Control Design for a Permanant Magnet Synchronous Machine. Eighth Multiconference International System, Signals & Devices SSD, PES 569397283, 22-25 March 20, Sousse-unisia [9] Zhou, K.Doyle, J.C an Glover, K, Robust an optimal control, Prentice Hall, Upper Sale River, NJ (995). [0] Doyle, J., Glover, K., Khargonker, P. an Francis, B, State space solution to stanar H2 an H control problems, IEEE rans. Automat.Contr.AC-34N 8 August 989, pp 83-84 [] Duc, G., Font, S, Commane H infini et -analyse es outils pour la robustesse, Hermes Sciences Publications, Paris 999. [2] Glover, K., Doyle, J.C, State space formulae for all stabilizing controllers that satisfy an H norm boun an relations to risk sensitivity, Systems an Control Letters 988. [3] Petkov, D.W., Gu, P.H., Konstantinov, M.M, Robust control esign with Matlab, Springer Verlag, Lonon 2005 [4] Bose, B. K., Power electronics an AC rive, New york, Prince Hall, first Eition, chapiter 2, page 95. (986) [5] Sturtzer, G. an Sanigiel, E., Moeling an Control of three-phase motors. [6] Peaucelle, D. an Arzelier, D., An efficient numerical solution for H2 static output feeback synthesis., European Control Conference, 200, pp.3800-3805 [7] Giriharan.K an al, Vector Control of a Permanent Magnet Synchronous Motor using high performance Hexagram Inverter. International Journal of Engineering an echnology IJE Vol 4 n 6 pp486-493 December 202-January 203 ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 560

[8] A.Kaouri, S.Blais, M.Ghribi an O.Akhrif, New Software eicate to the esign of Aaptative Nonlinear Controllers Base on feeback Linearization, echniques International ransactions on computer Science an Engineering vol 8, October 2005 [9] Azaiz, A., Ramani, Y., Meroufel, A. an Belabbes, B, H esign of controllers ensuring the regulation of currents of the ecouple fiel orientation control applie to a PMS Motor, Acta Electrotechnica an Informatica, Vol 8. N, pp 5-59, (2008) AUHOR PROFILE Jamel KHEDRI receive the Ph. D egree from National School of Engineers of Sfax (ENIS), University of Sfax, unisia, in 200. His currents research interests robust control, fuzzy logic control, D-stability analysis an applications of these techniques to inuction motors. ISSN : 0975-4024 Vol 7 No 2 Apr-May 205 56