International ournal o Systems Control (Vol.1-010/Iss.1) Merzoug an Benalla / Nonlinear Backstepping Control o ermanent... / pp. 30-34 Nonlinear Backstepping Control o ermanent Magnet Synchronous Motor (MSM) M.S. Merzoug *, H. Benalla ** * Department o Department o Electrical Engineering, University o Mentouri Constantine, B.. 35, avenue Ain El Bey, Constantine 5017, Algérie tel/ax: (13)31.81.87.11 e-mail: merzougmohamesalah@yahoo.r ** Department o Department o Electrical Engineering, University o Mentouri Constantine, B.. 35, avenue Ain El Bey, Constantine 5017, Algérie tel/ax: (13)31.81.87.11 e-mail: Hbenalla@yahoo.z Submitte: 08/01/010 Accepte: 11/01/010 Appeare: 16/01/010 HyperSciences.ublisher Abstract This paper presents a novel spee control techniue or an permanent magnets synchronous (MSA) rive base on newly Nonlinear backstepping techniue. The most appealing point o it is to use the virtual control variable to make the high-orer system simple, an thus the inal control outputs can be erive step by step through appropriate Lyapunov unctions Backstepping control approach is aapte to erive the control scheme, which is robust to parameter uncertainties an external loa isturbance. Simulation results clearly show that the propose controller can track the spee reerence signal successully uner parameter uncertainties an loa torue isturbance rejection. Keywors: MSM, Backstepping, Lyapunov, Stability, Nonlinear V,V I,I L,L ϕ S 1. NOMENCLATUE Direct-an uarature-axis stator voltages Direct-an uarature-axis stator voltages Direct -an uarature-axis inuctance Number o poles Stator resistance rotor magnet lux linkage ϕ, ϕ Flux in ( ) ωr Ω θ T T e Tr F, reerence rame rotor spee in electrical Mechanical rotor spee Electrical rotor position Electromagnetic torue Friction torue Loa torue Inertia Damping coeicient. INTODUCTION ermanent magnet (M) synchronous motors have attracte increasing interest in recent years or inustrial rive application. The high eiciency, high steay state torue ensity an simple controller o the M motor rives compare with the inuction motor rives make them a goo alternative in certain applications. Moreover, the availability o low-cost power electronic evices an the improvement o M characteristics enable the use o M motors even in some more emaning applications [4]. Avantages o MSM inclue low inertia, high eiciency, high power ensity an reliability. Because o these avantages, MSM are inee excellent or use in highperormance servo rives where a ast an accurate torue response is reuire. Usually, high-perormance motor rives reuire ast an ac- curate response, uick recovery rom any isturbances an insensitivity to parameter variations. The ynamic behavior o an ac motor can be signiicantly improve using vector control theory where motor variables are transorme into an orthogonal set o axes such that spee an torue can be controlle separately. This gives the IMSM machine the highly esirable ynamic perormance capabilities o the separately excite c machine, while retaining the general avantages o the ac over c motors. Originally, vector control was applie to the inuction motor an a vast amount o research work has been evote to this area. The vector control metho is relevant to the IMSM rive as the control is completely carrie out through the stator, as the rotor excitation control is not possible. [6] The two major classes o controllers which are capable o ealing with nonlinear uncertain systems are aaptive an robust controllers. Backstepping control is an approach to Copyright 010 HyperSciences_ublisher. All rights reserve 30 www.hypersciences.org
International ournal o Systems Control (Vol.1-010/Iss.1) Merzoug an Benalla / Nonlinear Backstepping Control o ermanent... / pp. 30-34 nonlinear control esign which has attracte a great eal o research interest in recent years. It is mainly applicable to systems having a cascae or triangular structure. The central iea o the approach is to recursively esign controllers or motor torue constant uncertainty subsystems in the structure an step back the eeback signals towars the control input. This iers rom the conventional eeback linearization in that it can avoi cancellation o useul nonlinearities in pursuing the objectives o stabilization an tracking. In aition, by utilizing the control Lyapunov unction, it also has the lexibility in introucing appropriate ynamics to make the system behave in a esire manner. The presentations o backstepping control in the literature are mostly in pure mathematical settings. [] The Backstepping control is a systematic an recursive esign methoology or nonlinear eeback control. Appling those esign methos, control objectives such as position, velocity can be achieve. [1] A nonlinear backstepping control esign scheme is evelope or the spee tracking control o MSM that has exact moel knowlege. The asymptotic stability o the resulting closeloop system is guarantee accoring to Lyapunov stability theorem. 3. MATHEMATICAL MODEL OF THE MSM The electrical an mechanical euations o the MSM in the rotor reerence (- ) rame are as ollows [5]: V = s I + ϕ ωrϕ t (1) V = s I + ϕ + ωrϕ t ϕ = L I + ϕ () ϕ = L I (3) An the electromagnetic torue T e is given by: 3 Te = [( L )I I + I ϕ )] (4) The euation or the motor ynamics, on the other han, is Ω T T T = (5) t e r principle, the irect axis current i is always orce to be zero in orer to orient all the linkage lux in the - axis an achieve maximum torue per ampere. With the state assignments, the ynamic x 1 = I, x = ωr an x3 = I. Moel o MSM can be rewritten as ollows: L s 1 xɺ 1 = x1 + xx3 + V 1 xɺ = x3 x Tr s 1 xɺ 3 = x3 xx1 ϕ x + V L L L L 4. NONLINEA BACKSTEING DESIGN The schematic iagram o the spee control system uner stuy is shown in Fig.. The parameters o the synchronous machine are given in the Appenix. In this section, we employ the nonlinear backstepping schemes to esign the controllers or MSM systems with angular velocity measurement. Fig.. System coniguration o Backstepping control With the choice o appropriate regulate variables, the backstepping esign proceure consists o the ollowing three steps : [7] Step 1: First o all, since the irect axis current i must be orce to be zero, the irst regulate variable is introuce by z = x (7) 1 1 (6) The erivative o (7) is compute as zɺ =ɺ x 1 1 L 1 = x + x x + V (8) L L L s 1 3 Fig. 1. MSM euivalent circuit rom ynamic euations From (1), it is obvious that the ynamic moel o MSM is highly nonlinear because o the coupling between the spee an the electrical currents. Accoring to the vector control The irst Lyapunov caniate V 1 is chosen as: So the erivative o (9) is compute as: 1 V1 = z1 (9) 31
International ournal o Systems Control (Vol.1-010/Iss.1) Merzoug an Benalla / Nonlinear Backstepping Control o ermanent... / pp. 30-34 V ɺ = z z ɺ 1 1 1 L 1 s = z1 z1 + xx3 + V L At this point, the irect axis voltage control input V can be selecte by L V = c1 z1 + xx3 L Where c 1 is a positive esign constant, so (8) becomes: 1 1 1 (10) (11) zɺ = ( c + ) z (1) Thereore, (10) can be rewritten as: Vɺ c z (13) 1 = ( 1 + ) 1 Step : The purpose o this control esign is to achieve the reerence spee tracking, so the secon regulate variable is chosen as z = x Ω (14) Where Ωre (14) is calculate as: re is the spee reerence, hence the erivative o 1 zɺ = x3 x Tr Ω ɺ re (15) By eining the error variable z3 = x3 α where α is the stabilizing unction chosen as ollows: 1 α = re Tr re Ω + + Ω ɺ ϕ (16) (15) can be rewritten as zɺ = z + z3 (17) With the choice o the secon Lyapunov caniate V 1 = (18) cz L zɺ = x x x ϕ x s 3 3 1 L L L 1 + V re re L Ω ɺ + Ω ɺɺ (0) With the selection o the complete Lyapunov unction 1 V = V1 + V + z3 (1) From (13), (19) an (0) an the erivative o (1) is compute as ollows: V = Vɺ + Vɺ + z zɺ 1 3 3 = ( c + ) z c z 1 1 L + z c z x x x ϕ x s 3[ 3 1 L L L 1 + V ( Ω ɺ re + Ωɺɺ re )] L () At last, in orer to make the erivative o the complete Lyapunov unction (1) be negative einite, the -axis voltage control input is chosen as ollows: s V = L[ c z + x3 + xx1 + ϕ x L L L (3) + ( Ω ɺ re + Ωɺɺ re ) c3 z3] Thereore, substituting (3) into (), we are able to obtain Vɺ = ( c + ) z c z c z (4) 1 1 3 3 Clearly, in (4) is negative einite, so it implies that the resulting close-loop system is asymptotically stable an, hence, all the error variables z 1, z an z 3 will converge to zero asymptotically. 5. VOLTAGE SOUCE WM INVETE A typical voltage-source WM converter perorms the ac to ac conversion in two stages: ac to c an c to variable reuency ac. The basic converter esign is shown in Fig.3. where c is a positive esign constant, the erivative o (18) is compute as : Vɺ = c z zɺ = c z + c z z 3 (19) Step 3: The erivative o the given error variable z 3 is compute as Fig. 3. Basic three-phase voltage-source converter circuit 3
International ournal o Systems Control (Vol.1-010/Iss.1) Merzoug an Benalla / Nonlinear Backstepping Control o ermanent... / pp. 30-34 5.1. Sinusoial pulse with moulation Three-phase reerence voltages o variable amplitue an reuency are compare in three separate comparators with a common triangular carrier wave o ixe amplitue an reuency Fig. 4. Each comparator output orms the switching-state o the corresponing inverter leg. [3] 6. CONCLUSION A nonlinear backstepping control metho has been propose an use or the control o a permanent magnet synchronous machine. Simulation esults show goo perormances obtaine with propose control, with a goo choice o parameters o control. The spee control operates with enough stability. In this approach the components I an I is regulate using backstepping control, so that I is zero, the controller is esigne in the total system incluing switching evices. The simulation results show, Fast response without overshoot an robust perormance to parametric variation an isturbances in all the system Fig. 4. Sinusoial WM, current control an switching logic 6. SIMULATION ESULTS Extensive simulations have been perorme using Matlab/ Simulink Sotware to examine control algorithm o the nonlinear backstepping applie or MSM. In orer to valiate the control strategies as iscusse, igital simulation stuies were mae the system escribe in Fig.. The spee an currents loops o the rive were also esigne an simulate respectively with backstepping control. The eeback control algorithms were iterate until best simulation results were obtaine. The spee loop was close, an transient response was teste with both current controller an spee control simulation o the starting moe without loa is one ollowe by step in spee reerence Ω re = 100ra/s, The loa torue T r is applie in t = 0.1s. The valiity o the control is emonstrate through the results shown in Fig. (5, 6, 7, 8) the response or step in spee comman is chow in Fig. 5, the spee response is completely robust with perect rejection o loa isturbances. Fig. 6, show the response o the components i an i. The stator currents which are controlle by WM generate by backstepping control, high reuency harmonics that prouce high reuency torue ripple. EFEENCES Chaouch. S. an Naït-Saï, M. (006). Backstepping Control Design or osition an Spee Tracking o DC Motors, Asian ournal o Inormation Technology 5. Ghaarim A. S. (005). Hybri stepper motor backstepping control in micro-step operation. roceeings o ASME International Mechanical Engineering Congress an Exposition, Orlano, Floria USA. Labriue, F. Buyse, H. Seguier, G. an. Bausière,. (1998). Converters power electronics comman an ynamic behaviour. Dept. Mines, Sophia-Antipolis, aris Univ, France, vol. 5. Maemlis, C. Ageliis, V. G. (001). On Consiering Magnetic Saturation with Maximum Torue to Current Control in Interior ermanent Magnet Synchronous Motor Drives. IEEE Trans. Energy Conversion, Vol. 16, N o. 3. ragasan,. an. Krishnan,. (1988). Moeling o permanent magnet motor rives. IEEE Trans. Inustrial electronics, vol. 35, no.4. ahman, M. A. Vilathgamuwa, D. M. Uin, M. N. an Tseng, K.. (003). Nonlinear Control o Interior ermanent-magnetsynchronous Motor. IEEE Trans. Inustry applications, vol. 39, no.. Shun-Sheng, K. an ung-shan, L. (005). Sensorless Spee Tracking Control with Backstepping Design Scheme or ermanent Magnet Synchronous Motors. roceeings o IEEE Conerence on Control Applications Toronto, Canaa. High accuracy an strong robustness o the nonlinear backstepping are proviing by Fig. 7, when spee reversion aroun (+100 ra/s, -100 ra/s) are applie. Fig. 8, shows the robustness to parametric variations, where inertia moment is ouble. There is no overshoot, but the settling time is ouble compare with that o Fig. 6. 33
International ournal o Systems Control (Vol.1-010/Iss.1) Merzoug an Benalla / Nonlinear Backstepping Control o ermanent... / pp. 30-34 Appenix Three phases MSM parameters: ate output power 1500 W, ate phase voltage 0/380V, s=1.4 Ω, φ=0.154wb, =6.6mH, L =5.8mH F=0.00038N.m.s/ra, =0.00176, kg.m Fig. 7. Spee reversion (+100 ra/s, -100ra/s) Fig. 5. esponse uner loa isturbance Fig. 8. esponse uner inertia variation (=.n) Fig. 6. Stator current uner loa isturbance 34