Nonlinear Control of Interior PMSM Using Control Lyapunov Functions

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1 Journal of Power an Energy Engineering, 24, 2, 7-26 Publihe Online January 24 ( Nonlinear Control of Interior PMSM Uing Control yapunov Function Maha Sabra, Bahar Khaawneh, Mohame A. Zohy Department of Electrical an Computer Engineering, Oaklan Univerity, Rocheter, USA. Receive Oct. 3 t, 23; revie Dec. 5 th, 23; accepte Dec. 23 r, 23 Copyright 24 Maha Sabra. Thi i an open acce article itribute uner the Creative Common Attribution icene, which permit unretricte ue, itribution, an reprouction in any meium, provie the original work i properly cite. In accorance of the Creative Common Attribution icene all Copyright 24 are reerve for SCIRP an the owner of the intellectual property Maha Sabra. All Copyright 24 are guare by law an by SCIRP a a guarian. ABSTRACT In thi paper, we introuce a non-linear torue control for an interior permanent-magnet ynchronou motor (IPMSM). The nonlinear control i bae on a Control yapunov Function (CF) techniue. The propoe tabilizing feeback law for the IPMSM rive i a amping control metho an i hown to be globally aymptotically table. The CF metho take the ytem nonlinearitie into account in the control ytem eign tage. Such nonlinearitie are ue to the cro coupling between the an the current in aition to the ytem parameter like the inuctance an the flux linkage. The complete IPMSM rive incorporating the propoe CF ha been uccefully imulate in a plant moel for both motor an inverter. The performance of the propoe rive i invetigate in imulation at ifferent operating conition. It i foun that the propoe control techniue provie a goo torue control performance for the IPMSM rive enuring the global tability. In later work, we are planning to invetigate other phenomena uch a magnetic aturation, nonlinear loa, mechanical friction an flexibilitie. KEYWORDS Nonlinear Control; Interior Permanent Magnet Motor (IPMSM); Control yapunov Function (CF); IPMSM Experimental Parameter Determination. Introuction The permanent-magnet ynchronou motor (PMSM) ha been gaining popularity epecially in the automotive inutry mainly ue to it relatively high efficiency, mall ize an robutne. All thee avantage reult from the high-energy rare-earth alloy. Depite the high cot of thee magnet, the PMSM i till the AC rive of choice in the automotive inutry. The interior permanent-magnet ynchronou motor (IPMSM) referre to a embee PMSM alo ha the magnet burie in the rotor core, a oppoe to the urface-mount PMSM where the magnet are mounte on the outer urface of the rotor. In aition to a mechanically robut rotor, the IPMSM enable an accurate etimation of the initial rotor poition by inuctance variation ue to motor aliencie. The avance in both power electronic an control theory ha mae the high-performance reuirement poible by enabling the IPMSM to be operate in highpee moe in the contant power region. The ynamic behavior of the IPMSM i ignificantly improve by uing vector control theory. The motor variable are tranforme from the fixe tator reference frame to the rotor reference frame. Precie control of an IPMSM rive ha it challenge ue to nonlinear coupling of current an the rotor pee, a well a the nonlinearity preent in the torue euation. To achieve precie control of PMSM, many nonlinear metho have been tuie. The PMSM control propoe in [] ue the backtepping metho which i a nonlinear control metho, but the author aume that i = which cancel many nonlinearitie an limit the operating range of the electric machine. Feeback inearization i alo another nonlinear control metho ue in [2], an the author are uing a urface mount PMSM where = =, an the torue i a function of i. But the mot

2 8 Nonlinear Control of Interior PMSM Uing Control yapunov Function wiely ue techniue in inutry i a PI controller [3-7]. There are two PI controller ue for the inner loop torue control, one for the -current an the other for the - current control. Another PI controller i ue for the outer pee control loop. The PI controller eign aume that the ytem i linear below the bae pee an a an anti-winup compenation to account for the highly nonlinear behavior in the flux weakening region. Different variation of the anti-winup techniue are preente in [4,5]. Tuning of the PI controller parameter ha alo prove to be challenging ince the ame fixe parameter cannot be ue acro the whole operation range. Thi reulte in other PI controller eign that were elf-tuning a in [6,7]. Thoe PI controller have ifferent controller gain in ifferent operating region. Thee elf-tuning controller cannot be ue in high performance application like the automotive application with a very wie operating range. In thi paper, CF metho i ue a the nonlinear control metho that enure the tability of the ytem. No ecoupling or implification are aume, an the motor parameter are erive experimentally. The metho i proven to be veratile in all operating region, eaier to tune, an uperior to the repone of both the traitionally ue PI controller with anti-winup an the feeback linearization metho. The paper i arrange a follow: Section 2 ecribe the nonlinear IPMSM rive moel in the reference frame. Section 3 ecribe the experimental metho ue to obtain the motor parameter uing ynamometer meaurement, an the nonlinearitie are ientifie an ummarize. In Section 4, the theory of Control yapunov Function i preente an choen a the control metho. The CF metho propoe effectively compenate for the IPMSM nonlinearitie. The reult are icue in Section 5 an finally the concluion i given in Section Internal Permanent Magnet Synchronou Motor (IPMSM) The euivalent circuit moel of an embee permanent magnet ynchronou machine in the rotor - reference frame [8] i hown in Figure. The correponing mathematical moel in () how the irect v an uarature v voltage euation. Note that the motor parameter are not contant but are function of i an i. λ v = ir + ωλ e t () λ v = ir + + ωλ e t where, λ = i (2) λ = i + λ m Figure. Euivalent circuit of an interior PMSM. -axi circuit; -axi circuit. By ubtituting (2) into (), we get (3): i Ri + ωei v t v = (3) i ωei + Ri + + ωλ e m t The torue euation i hown in (4) 3 Te = n p λmi ( ) ii 2 + (4) ωr Te = Tl + kfωr + J t where: i an i are the irect an uarature current [A] v an v are the irect an uarature voltage [V] R i the tator reitance [Ω] an are the irect an uarature inuctance [H] T l i the loa torue [Nm] T e i the electrical torue [Nm] J i the moment of inertia of the motor an loa k f i the friction coefficient of the motor ω r i the rotor angular pee [ra/] n p i the number of pole pair of the motor ωe = npωr i the electric angular pee [ra/] λ m i the permanent magnet flux [Wb] λ an λ are the uarature an irect flux linkage 3. Experimental Determination of Motor Parameter A the pee increae, the nonlinearity of the voltage (3)

3 Nonlinear Control of Interior PMSM Uing Control yapunov Function 9 an torue (4) euation increae. The motor parameter a to the nonlinearity of the ytem. Therefore, the accuracy of the motor parameter i the firt tep to improving the control performance of the interior PMSM rive. The parameter are etermine to take into account the aturation effect. The effect of temperature however i not taken into account. All meaurement are taken at a contant motor coolant temperature. The IPMSM ue i rate at 2 Nm peak torue an 5 rpm maximum pee with a tator reitance of mohm an 4 pole pair. The inverter i rate at kw. 3.. Flux inkage λ an λ To etermine λ an λ a tet i performe by running the PMSM at contant pee ω e an weeping through the motoring an generating uarant, an by commaning all the poible i an i combination. At teay tate, () become ( v ir) λ = + e ( v ir) λ = ω ω e By proceing the meaure phae current an voltage waveform, every I an I combination i matche with the correponing v an v. Flux_ ( λ ) an Flux_ ( λ ) are then etermine from (5) an hown in Figure 2. The fitting of the flux ata a hown alo in Figure 2 i ue to etermine the type of nonlinearity preent in the ytem parameter Permanent Magnet Flux inkage λ m The permanent magnet flux can be etermine by etting the tator current to zero ( i = i = ). Euation () an (2) above become: v = (6) v = ωλ e m (5) Figure 2. Flux an Flux a a function of I (x-axi) an I (y-axi) (blue: experimental meaurement; color map: fitte ata). -axi Flux inkage; -axi Flux inkage. v = v + v = ωλ (7) 2 2 e m Therefore, λ m can be etimate by iviing the funamental voltage by the electrical angular freuency. Alternatively, to etermine λ m a a function of i, the flux λ ( i, i) reult ha be ue. The permanent flux i etermine by etting i =. The Permanent flux i therefore λ m = λ (, i ). The permanent flux i hown a a function of i in Figure Inuctance an an are calculate uing (2). an are function of I an I a hown in Figure 4 an 4 repectivly. Figure 3. Permanent magnet flux linkage.

4 2 Nonlinear Control of Interior PMSM Uing Control yapunov Function The nonlinearitie of the IPMSM rive can be ummarize in Table. Figure 4. an. -axi Inuctance; -axi inuctance Stator Reitance R The tator reitance i etermine by meauring the phae-to-phae reitance an calculating the reitance epening on the tator wining having a wye or a elta connection. The tator reitance i a linear function of the tator temperature an will be aume contant Summary of the Nonlinearitie in IPMSM Control The nonlinearitie preent in the IPMSM ytem are ue to both nonlinear parameter an the ytem euation. The -axi an -axi inuctance were etermine to be function of both I an I an are ue a uch in the moel. But to practical purpoe, we can afely aume that an can be fitte into 3 polynomial a function of I an I repectively without too much impact on the control. The fitting hown in Figure 2 how that the flux an flux are inuoial function of I an I. Other nonlinearitie are preent ue to the multiplication of the tate a hown in (9) an alo in the torue euation 4. Nonlinear Control of IPMSM The IPMSM rive ytem ue in the imulation i hown in Figure 5. The moel conit of the voltage ource inverter, the PMSM an two control loop. The inner loop i the current control loop an the outer loop i the pee control loop. Simplifie verion of thi control cheme ue the linear ecouple voltage euation for the IPMSM with PI controller for each of the an term by auming that the pee an the motor parameter are contant. For pee control an aitional PI controller i ue for the outer loop control. The motor parameter cannot be aume contant otherwie the inner loop will be a linear moel a in (8). i R w e t i v = i i + v (8) R w e t Thi will lea to an unatifactory performance of the IPMSM ue to the over implification. Thi i particularly true uring operation in the flux weakening region. By etting [ xxx 2 3] = iiω r, (3) an (4) can be combine into the nonlinear tate Euation (9) to be ue R ( x2 ) x+ np xx 2 3 ( x) ( x) x R ( x ) λ m x 2 x2 np xx 3 np x = 3 ( x2) ( x2) ( x) x 3 k f T x3 J J (9) ( x ) + u + u 2 + u3 ( x2 ) J Table. Nonlinearitie coniere in the IPMSM rive. Nonlineartie Parameter Nonlinearitie Polynomial(I ) Polynomial(I ) λ Sinuoial(I, I ) λ Sinuoial(I, I ) λ m Sinuoial(I ) Sytem T e Coupling between I an λ State Coupling between I an w r

5 Nonlinear Control of Interior PMSM Uing Control yapunov Function 2 Figure 5. Block iagram of current controlle IPMSM rive ytem. where: (.) operator i the erivative ()/t x, x 2, an x 3 are the ytem tate u, u 2, an u 3 are the control input To achieve a high performance control for the IPMSM, Control yapunov Function (CF) metho i ue. The iea behin thi metho i to ue the ytem in (9), eign a control yapunov function, an a control input that minimize that function. Thi metho oe not implify the nonlinearitie that are inherent to the motor moel. The propoe CF metho oe not aume any implification or ecoupling in the moel. In thi ection alo, a PI control metho an a feeback linearization metho will be briefly preente to be later compare to the propoe metho. 4.. Control Uing CF 4... Backgroun on CF Conier the non-linear ytem repreente by the tate pace euation ( ) ( ( )) x = F x = f xu, x ; x R n ; u R k () Where xx i the tate vector an u i the control vector. Accoring to [9], the ytem i ai to be affine with repect to the input when it ha the form k = + i i i= ( ) ( ) x f x uf x () where, f, f, f2,, fm are continuou vector fiel in R n. f ( x ) i a table unforce ytem, u i i the eignate control, an fi ( x ) i a mooth vector fiel in R n. When ealing with affine ytem, (2) i reuire. ( ) ( ( )) x = f, u = (2) Choice of a Suitable CF There i no ytematic approach for fining yapunov function; in ome cae they are natural energy function for mechanical or electrical ytem, in other cae it i jut a matter of trial an error []. The convere yapunov theorem prove that the exitence of a yapunov function i euivalent to aymptotic tability [, 2]. V ( x ) i ai to be a yapunov function of () if there exit a region in the neighbourhoo of the origin uch that: ( ) ( ) ( ) ( ) V x >, V x <, an V x V x < (3) Thi region i calle the region of attraction [] n V V ( x) = V ( x) F( x) = fi ( x) (4) x where: ( ) V x i= = V ( x), V ( x), V ( x),, V( x) x x2 x3 xn i (5) The yapunov tability theorem tate that if () ha the origin a an euilibrium point an a uitable yapunov function V uch that ( V ( ) = an V ( x ) > for all x ) an V ( x) for all x, then the origin i table. In aition if V ( x ) < for all x, then the origin i globally aymptotically table []. The election of CF i uch that V(x) continuouly ifferentiable an poitive efinite. Stability i guarantee if V ( x ) i poitive efinite an V ( x) i negative efinite for all n x R The CF for IPMSM Control The control objective i torue control. Therefore, the i an i error nee to be reuce to zero a well a the

6 22 Nonlinear Control of Interior PMSM Uing Control yapunov Function pee error. ooking at the mall cale ynamic moel, the loa torue term in (9) i cancele. Different application have ue ifferent type of yapunov function; the author in [3] have ue the natural logarithm an in [4] they ue an energy function in their yapunov function eign. For thi application, the electe poitive efinite control yapunov function i efine a in (6) V ( x) = px + x2 + rx3 (6) where p >, >, an r > are the control parameter V ( x) = V ( x), V ( x), V( x) x x2 x3 = [ px, x, rx ] 2 3 The control input are choen a in (8) p u x = V x f x = x ( ) ( ) ( ) u x = V x f x = x ( ) ( ) ( ) (7) (8) r u3( x) = V ( x) f3( x) = x3 J Uing the control input in (8) an (), (4) become V ( x) = V ( x) f( x) 3 2 (9) V x f x For V ( x) V ( x) f ( x) V ( x) f ( x) ( ( ) i( i) ) i= to be negative efinite, the firt term ha to be negative efinite. R R k = p x x r x + n p x x x J 2 2 f p 2 3 λ n p xxx + n xx R 2 R k 2 f 2 = p x x2 r x3 J + n p xx n xx m 2 3 p 2 3 p λ p λ 2 3 (2) To enure the tability of the propoe control, the control parameter are electe a in (2) to guarantee V x for all x. that ( ) p = (2) 2 2 The control variable are now reuce to changing one variable a given by (2) Control Uing a PI with Anti-Winup The proportional-integral (PI) controller i till wiely ue in the control of PMSM. However; to account for the nonlinearitie ue to magnetic aturation, ifferent anti-winup trategie for the integral term have been ue. The author in [4] compare the performance of four ifferent anti-winup eign: Dea zone, tracking, tracking with gain, an conitione. For comparion purpoe, a PI controller with conitione anti-winup i ue. Figure 6 how the block iagram of the PI controller ue where the integrator hol it lat value by etting the output of the anti-winup block to zero when the aturation flag goe high. The parameter ue for the PI controller are K p = 3 an K i = Control Uing Feeback inearization The main iea of the feeback linearization control i to tranform nonlinear ytem ynamic into linear ynamic by canceling the nonlinearitie. A in [2], exact feeback linearization i applie to the inner loop by utilizing the auxiliary input in (22). For comparion purpoe, feeback linearization i alo implemente an it performance i compare to the propoe CF metho. ( x ) ( x2 ) ( ) ( x ) ( ) R u = u x+ n x x x xx R u = u x n x x x xx λm np x3 5. Reult p 2 3 ( ) ( ) 2 2 p 3 ( 2) ( 2) 2 (22) For a fixe torue comman, many current vector can be commane. For optimal current vector etermination, the operation of inverter fe PMSM rive can be categorize into 2 moe: The contant torue moe which i below the bae pee an the contant power moe which i above the bae pee. The bae pee varie with voltage. The contant torue operation i hown in Figure 7 a Region I. Thi moe of operation i along the maximum torue per current (MTPC), alo known a maximum torue per ampere or MTPA. A the pee increae from bae to maximum pee, the PMSM enter the contant power moe hown in Figure 7 a Region II an III. Thi i alo known a the contant voltage moe. The high pee contant vol-

7 Nonlinear Control of Interior PMSM Uing Control yapunov Function 23 Figure 6. PI controller with anti-winup. Figure 7. Operation region of IPMSM rive. In the i /i plane; In the torue-pee plane. tage operation i realize by reucing the tator flux linkage. Region II an III therefore repreent the fluxweakening operation. Region II repreent the current limit ue to the inverter harware limitation. Region III i given by the voltage-limit ellipe in the I -I plane a hown by the ahe line in Figure 7. The voltagelimit ellipe become maller a the pee increae an, a a reult, the current vector proucing maximum torue cannot atify the voltage contraint above the bae pee. In thi paper, the optimal flux control will be ue to achieve the highet PMSM efficiency an wier operating range. In [], by etting i =, the author eigne their control metho with a current vector eual to i. which i more applicable to urface mount PMSM but not very practical for IPMSM. The performance of the propoe metho i verifie in thi ection an compare to the repone of both the PI controller an the feeback linearization. Compare to both the PI control with anti-winup an the feeback linearization, the propoe CF metho how a very table repone in all region of operation with no ocillation or overhoot. Figure 8- each how a tep in torue in Region I, Region II, an Region III repectively where the commane an the actual current an torue are hown. The tep repone uing the CF approach i compare to that of a PI controller with anti-winup an the feeback linearization metho. The repone of the CF i uperior to thoe of the feeback linearization an PI controller. Uing the CF approach reulte in le overhoot an no ocillation. The feeback linearization metho ha lower overhoot in Region I but wore repone in Region II an Region III at higher pee. The CF metho wa eay to tune ince only one control parameter neee to be change. Tuning the feeback linearization metho i not traightforwar ince the repone got wore with pee no matter how fat or low the tuning i. Each PI controller ha two gain to be tune. Tuning the two couple PI controller i very ifficult epecially at high pee an i a topic for a

8 24 Nonlinear Control of Interior PMSM Uing Control yapunov Function (c) Figure 8. Torue an current repone (5 rpm). CF; Feeback linearization; (c) PI controller. ifferent tuy. Figure -(c) how the phae current an phae voltage when the CF metho, the feeback linearization, an the PI controller metho are ue repectively. (c) Figure 9. Torue an current repone (2 rpm). CF; Feeback linearization; (c) PI controller. The overhoot an ocillation in the tep repone will reult in over current an itortion of the inuoial current waveform.

9 Nonlinear Control of Interior PMSM Uing Control yapunov Function 25 3 Phae Current Ia, Ib, Ic 2 - Ia Ib Ic Phae Voltage va, vb, vc Va Vb Vc Phae Current 2 Ia, Ib, Ic -2 Ia Ib Ic Phae Voltage va, vb, vc Va Vb Vc Phae Current Ia, Ib, Ic 2 - Ia Ib Ic Phae Voltage (c) Figure. Torue an current repone (4 rpm). CF; Feeback linearization; (c) PI controller. 6. Concluion The propoe nonlinear control for an interior permanent-magnet ynchronou motor (IPMSM) for automo- va, vb, vc (c) Figure. Phae current an phae voltage. CF; Feeback linearization; (c) PI controller. Va Vb Vc

10 26 Nonlinear Control of Interior PMSM Uing Control yapunov Function tive application wa preente. The metho wa hown to be table an robut. Unlike the backtepping techniue ue in [], the Control yapunov metho prove effective in all operating region. The author in [] mae the aumption that i = which wa not realitic in an automotive application where it wa neceary to operate along the mot efficient trajectory in region I a hown in Figure 7. It i alo neceary to operate in the flux weakening. The propoe metho take the ytem nonlinearitie into account in the control ytem eign tage while operating in the mot efficient region even in the flux weakening area. The nonlinearitie that are taken into account are ue to the cro coupling between the an the current in aition to the ytem parameter. The motor parameter, which are a very important part in the control uch a the flux linkage an inuctance, are etermine experimentally. In aition, the complete IPMSM rive incorporating the propoe CF ha been uccefully imulate in a plant moel for both motor an inverter an compare to a feeback linearization controller an to a conventional PI controller. The performance of the propoe rive wa invetigate in imulation at ifferent operating conition. It i foun that the propoe control techniue provie a goo torue control performance for the IPMSM rive with no ocillation or overhoot while enuring global tability. Tuning of the propoe metho i alo traightforwar compare to the feeback linearization an the conventional PI control. In later work, we are planning to invetigate other phenomena uch a magnetic aturation, nonlinear loa, mechanical friction an flexibilitie. In aition, effort are being mae to incorporate iturbance, uantitative robutne an iagnoi of fault into the future of control cheme. REFERENCES [] M. A. Rahman, D. M. Vilathgamuwa, M. N. Uin an K.-J. Teng, Nonlinear Control of Interior Permanent- Magnet Synchronou Motor, IEEE Tranaction on Inutry Application, Vol. 39, No. 2, 23, pp [2] J. Solona, M. I. Valla an C. Muravchik, Nonlinear Control of a Permanent Magnet Synchronou Motor with Diturbance Torue Etimation, IEEE Tranaction on Energy Converion, Vol. 5, No. 2, 2, pp [3] G. S. akhmi, S. Kamakhaiah an T. R. Da, Cloe oop PI Control of PMSM for Hybri Electric Vehicle Uing Three evel Dioe Clampe Inverter for Optimal Efficiency, International Conference on Energy Efficient Technologie for Sutainability (ICEETS), Nagercoil, - 2 April 23, pp [4] J. Epina, A. Aria, J. Balcell an C. Ortega, Spee Anti-Winup PI Strategie Review for Fiel Oriente Control of Permanent Magnet Synchronou Machine, Compatibility an Power Electronic, Baajoz, 2-22 May 29, pp [5] P. March an M. C. Turner, Anti-Winup Compenator Deign for Nonalient Permanent-Magnet Synchronou Motor Spee Regulator, IEEE Tranaction on Inutry Application, Vol. 45, No. 5, 29, pp [6] J.. Gao an Y. F. Zhang, Reearch on Parameter Ientification an PI Self-Tuning of PMSM, 2n International Conference on Information Science an Engineering (ICISE), Hangzhou, 4-6 December 2, pp [7] R. G. Kanojiya an P. M. Mehram, Optimal Tuning of PI Controller for Spee Control of DC Motor Drive Uing Particle Swarm Optimization, International Conference on Avance in Power Converion an Energy Technologie (APCET), Mylavaram, 2-4 Augut 22, pp [8] P. Pillay an R. Krihnan, Moeling, Simulation, an Analyi of Permanent-Magnet Motor Drive. II. The Bruhle DC Motor Drive, IEEE Tranaction on Inutry Application, Vol. 25, No. 2, 989, pp [9] A. Bacciotti an. Roier, iapunov Function an Stability in Control Theory, Springer-Verlag, onon, 2, pp [] H. K. Khalil, Nonlinear Sytem, Prentice Hall, Upper Sale River, 22, pp. -6. [] E. Sontag, A yapunov-ike Characterization of Aymptotic Controllability, SIAM Journal on Control an Optimization, Vol. 2, No. 3, 983, pp [2] F. H. Clarke, Y. S. eyaev, E. D. Sontag an A. I. Subbotin, Aymptotic Controllability Implie Feeback Stabilization, IEEE Tranaction on Automatic Control, Vol. 42, No., 997, pp [3] F. A. Alazabi an M. A. Zohy, Nonlinear Uncertain HIV- Moel Controller by Uing Control yapunov Function, International Journal of Moern Nonlinear Theory an Application, Vol., No. 2, 22, pp [4] M. Pahlevaninezha, P. Da, J. Drobnik, G. Mochopoulo, P. K. Jain an A. Bakhhai, A Nonlinear Optimal Control Approach Bae on the Control-yapunov Function for an AC/DC Converter Ue in Electric Vehicle, IEEE Tranaction on Inutrial Informatic, Vol. 8, No. 3, 22, pp