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Additional Adaptive Controller for Mutual Torque Ripple Minimization in PMSM Drive Systems
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Additional Adaptive Controller for Mutual Torque Ripple Minimization in PMSM Drive Systems ⋆ Marián Tárník ∗ Ján Murgaš ∗ ∗
Institute of Control and Industrial Informatics, Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Ilkovičova 3, 812 19 Bratislava, Slovakia (e-mail: [email protected], [email protected]).
Abstract: This paper presents an additional adaptive current controller for mutual torque ripple minimization in PMSM. It is assumed that the basis of the control system is a conventional current control loop commonly used in vector control of the PMSM. No change in the structure of current loop is required, only access to the torque reference input is needed. Parameters of the additional controller are adapted directly by the derived adaptation law. Finally, the proposed control scheme is verified by means of simulations. Keywords: Permanent magnet synchronous motor, Torque ripple, Adaptive control. 1. INTRODUCTION
PMSM with the isotropic rotor and with the negligible reluctance torque.
Permanent magnet synchronous motors (PMSM) are used in drives for their features such as high efficiency, low moment of inertia and high torque to motor size ratio. Drives with PMSM are also used in machine tools and robots. In these applications an undesirable feature of the motor appears – the torque ripple. There are several reasons of the torque ripple, and these can be divided by type of torque produced by the motor into three categories, see Petrović et al. (2000):
The common control strategy for PMSM in servodrives is vector control oriented in rotor dq reference frame. In that case, the motor control system has a certain standard structure. The control system of servodrive can be divided by controlled variable into several parts: current control loop, speed control loop and position control loop. The basis is always the current control loop, because the torque is given by current, especially when the motor magnetic flux is constant, provided by the permanent magnets. Usually the current control subsystem consists of PI controllers of d-axis and q-axis currents, decoupling block and the block which we will call a Current Factor (CF). The Current Factor is used to convert the torque reference value to current reference value. In standard case, the decoupling block and the Current Factor are derived from the motor model, which describes only the basic dynamics of the motor. The higher harmonics in the signals are neglected. This means, that standard current loop does not count with higher harmonics in the motor electromagnetic subsystem. Therefore, the higher harmonics become a part of disturbance signal. PI controllers do not remove these disturbances suitably which results in the torque ripple.
• Mutual torque – Higher harmonics in mutual torque are caused by nonsinusoidal distribution of stator windings or rotor magnets. Smooth mutual torque is produced only if mutual flux through stator windings is purely sinusoidal. • Reluctance torque – It is nonzero only if stator inductances depend on rotor position. If stator windings are sinusoidally distributed then the reluctance torque has only DC component. • Cogging torque – It depends only on geometry and number of stator slots. This torque component contributes only to torque ripple, it has no DC component. The cogging torque is usually reduced in machine design procedure. A large number of PMSMs is manufactured with an isotropic rotor. In this case the reluctance torque is negligible, and therefore it does not contribute to torque ripple. In applications where there is a reason to deal with high quality of produced torque it is also reasonable to consider that the motor has reduced cogging torque by the machine design procedure. Motor with this construction produces only mutual torque with large DC component and with higher harmonics. This paper deals with the control of ⋆ This work has been supported by Slovak scientific grant agency through grant VEGA-1/0592/10.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Many techniques for PMSM torque ripple minimization have been proposed in the literature, see Petrović et al. (2000). There are two classes of these techniques. First, techniques that focus on the motor design so that PMSM closely approaches its ideal characteristics. This brings more complicated manufacturing process and increases the cost of the motor. The second class of techniques includes control algorithms which are developed taking into account the nonideal characteristics of the motor. The control scheme proposed in this paper belongs to the second class. The oldest approach in this class uses the preprogrammed current waveform to cancel higher
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harmonics effects in the motor electromagnetic subsystem. This approach is usually based on off-line calculations, but on-line identification of necessary parameters is also used, for example see Holtz (1996). In this class, the use of adaptive control based on identification in design of control algorithm for PMSM torque ripple minimization is represented by several algorithms. Their common feature is the use of motor model which sufficiently describes the motor properties, while it remains compact and suitable for control design. The most significant higher harmonic components of magnetic flux are preserved in the model, see Petrović et al. (2001); Wallmark (2001), and only those with too small amplitude are truncated. In conventional model only fundamental components are considered, see Ohm (2000); Dolinar (2007); Krause et al. (2002). Petrović et al. (2000) proposed the adaptive current controller, which is theoretically able to completely cancel the most significant components of mutual torque ripple which come from the electromagnetic subsystem of motor. In the paper they also deal with the implementation of the proposed controller. In Qian (2005) so-called Compensating Current Factor (CCF) is proposed by which current reference value is shaped in order to suppress the torque ripple. To obtain the parameters of Compensating Current Factor a MRAS-based flux linkage estimator and a LS-based harmonic selective filter is used. The proposed algorithm does not solve the whole problem. Some of the problem parts are not addressed, since higher harmonics in the interactions of d and q axis remain uncompensated. The current control scheme with an adaptive internal model is proposed in Mohamed (2008). The paper takes aim at compensation of all motor parameter uncertainties influence. By estimating the total voltage error a minimization of torque ripple is also achieved. The remainder of this paper is organized as follows: In Section 2 we introduce two types of PMSM models. The standard current control loop is described in Section 3. In Section 4 the current control loop for the mutual torque ripple minimization is presented and its modification for use with the standard current control loop is developed. In Section 5 an adaptation mechanism for additional controller parameters introduced in Section 4 is derived. Section 6 shows the simulation results. 2. MODELING OF THE PMSM IN DQ REFERENCE FRAME Commonly used model of PMSM with an isotropic rotor, i.e., its self and mutual inductances of stator windings do not depend on rotor position, has in dq reference frame the form dψd ud = Rid + − ωψq (1) dt dψq uq = Riq + + ωψd (2) dt ψd = Ld id + ψd rot (3) ψq = Lq iq + ψq rot (4) 3 Pmech (5) Mm = p 2 ω dω = Mm − Mz − Bf ω (6) J dt
dϑ =ω (7) dt where Ld and Lq are the stator inductances in dq reference frame, and Ld = Lq , R is the stator winding resistance, J is the moment of inertia, Bf is the friction constant and p is number of pole-pairs. ud and uq are the d-axis and q-axis stator voltages, id and iq are d-axis and qaxis stator currents, Mm is the torque produced by the motor, Pmech is part of electrical power which is converted to mechanical power, Mz is the load torque, ω is the rotational speed of the rotor in electrical rad/sec and ϑ is the position measured in electrical radians. ψd a ψq are the total flux linkages and ψd rot and ψq rot are the flux linkages established by the permanent magnets. In the conventional model it is considered, see Ohm (2000); Krause et al. (2002) ψd rot = ψPM (8) ψq rot = 0 (9) where ψP M is the amplitude of permanent magnet flux linkage. In this case, the rotor magnetic field distribution is considered to be sinusoidal. When the rotor and stator magnetic field distributions are not sinusoidal, the permanent magnet flux linkages can be viewed as the sum of a fundamental component and the series of higher harmonics, see Petrović et al. (2001); Wallmark (2001); Qian (2005) ψd rot = ψd0 + ψd6 cos(6ϑ) + ψd12 cos(12ϑ) + . . . (10) ψq rot = ψq6 sin(6ϑ) + ψq12 sin(12ϑ) + . . . (11) For good approximation of the permanent magnet flux linkages it is sufficient to keep only series members up to 12th harmonics. To determine the decoupling block and the Current Factor, it is convenient to write equations in the following form. For the case when the conventional model is considered, i.e. using (8) and (9) did Ld = −Rid + ω (Lq iq ) + ud (12) dt diq Lq = −Riq − ω (Ld id + ψPM ) + uq (13) dt 3 (14) Mm = p (ψPM iq + (Ld − Lq ) id iq ) 2 dω J = −Bf ω + Mm − Mz (15) dt For the case when the nonsinusoidal distributions are considered (equations (10) and (11) are used) did = ud − Rid + ω (Lq iq + Ld (16) dt + Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) diq Lq = uq − Riq − ω (Ld id + (17) dt + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) 3 Mm = p ((Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) iq − 2 − (Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) id + + (Ld − Lq ) id iq ) (18) dω = −Bf ω + Mm − Mz (19) J dt
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where Ψd0 = ψd0 (20) Ψd6 = 6ψq6 + ψd6 (21) Ψd12 = 12ψq12 + ψd12 (22) Ψq6 = 6ψd6 + ψq6 (23) Ψq12 = 12ψd12 + ψq12 (24) are parameters having physical interpretation, see Petrović et al. (2000). For the rest of paper, only the model of PMSM considering nonsinusoidal distributions of magnetic field will be used in simulations. Conventional model will be used only to derive the standard decoupling block and the Current Factor. 3. STANDARD CURRENT CONTROL LOOP In the standard current control loop, the controlled system is given by equations (12) – (14). It is clear from equation (14) that the torque Mm is proportional to current iq . The current id has no contribution to the torque production because Ld = Lq , therefore zero reference value of id is chosen idW = 0. The reference value iq W is governed by the torque reference value according to 1 (25) Mm W iq W = 3 pψ PM 2 which we call Current Factor CF =
1 . 3 2 pψPM
In equation
(25) is term KT = 32 pψPM called motor torque constant. PI controllers are commonly used for control of both current components. Equations (12) and (13) are coupled. Decoupling is obtained using the voltage feed-forward signals which are added to both outputs of PI controllers. Then the voltages in equations (12) and (13) are ud = uPId + dd (26) uq = uPIq + dq (27) where dd = −ω (Lq iq ) (28) (29) dq = ω (Ld id + ψPM ) are the voltage feed-forward signals and uPId , uPIq are the outputs of PI controllers. The parameters of decoupling block and the parameters of Current Factor are given by model of PMSM. The parameters of current PI controllers can be determined using any method. For example, in Dolinar (2007) socalled compensation method is applied. Block diagram of the current control loop is in Fig. 1. Outputs of the PI controllers ud , uq are transformed to the three phase abc stator reference frame and modulated using pulse-width modulation in voltage source inverter. In industrial drives all components of standard current control loop are contained in a single device connected to the motor. In such the case, only one external input is available – the torque reference. Moreover, the components can be changed only by the parameters. The standard current control loop is not designed to minimize mutual torque ripple so the standard industrial drive is not capable to cope with the torque ripple.
idW +
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− iq Fig. 1. The current control loop 4. CURRENT CONTROL LOOP FOR MUTUAL TORQUE RIPPLE MINIMIZATION Mutual torque ripple is induced by the higher harmonics in permanent magnet flux linkages. This phenomenon can be modeled by equations (16)–(18). Using these equations components of the current control loop can be designed. Higher harmonics in the back EMFs are then compensated and thus the disturbances added to PI controller output are reduced. Conversion of torque reference value to current reference value is realized using the extended current factor. The torque constant is replaced by a term which contains harmonic functions. The current waveform is then shaped so that the produced mutual torque is smooth and its ripple is suppressed. For decoupling of equations (16) and (17) the signals dd and dq have form dd = −ω (Lq iq + Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) (30) dq = ω (Ld id + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) (31) From the torque equation (18) it follows, that the nonzero id contributes only to torque ripple, because it is multiplied only by higher harmonics. Thus it is desirable to set the reference value of id to zero. The current iq contributes to DC component but also to higher harmonics. In this case the Current Factor must be 1 CF = 3 (32) p (Ψ + Ψ cos(6ϑ) + Ψd12 cos(12ϑ)) d0 d6 2 As in previous case, the PI controllers are used for current control so these components of current control loop remain unchanged. However, it is necessary to reflect new requirements in their design. In q-axis the reference value signal iqW is much more rich in frequencies, which requires wider bandwidth of the closed loop. Generally speaking, the standard current control loop can be complemented by parts which ensure the minimization of torque ripple. This complementation can not be made directly if the whole current control loop is contained in common industrial voltage source inverter. Then the structure of control loop can not be changed and another signal can not be added to signals (eg, the output of PI controller). The only available input is the torque reference. Let’s assume the standard current control loop as described in section 3 which is implemented in a single device. The decoupling of equation (16) can not be complete because standard signal dd does not compensate the higher harmonics. Theoretically, it is possible to amend dd , but practically it is not feasible because the signal uPI d is not available for adding. So the higher harmonics of
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dq A Fig. 2. The additional decoupling of q-axis, location of TF disturbance added to d-axis PI controller output remain uncompensated. In q-axis the situation is different. Instead of adding the missing part of signal dq to signal uPI q it can be added to the torque reference signal MmW . First, let us determine the missing part of the signal. For decoupling of equation (17) it is required: d⋆q = ω (Ld id + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) = = ω (Ld id + γΘ) (33) T
where γ = [1 cos(6ϑ) cos(12ϑ)], Θ = [Ψd0 Ψd6 Ψd12 ] . In standard current control loop dq = ω (Ld id + ψPM ) (34) to satisfy d⋆q = dq + dq A (35) dq A must be in the form dq A = ω ((Ψd0 − ψPM ) + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) (36) Before adding the signal dq A to Mm W , it is necessary to adjuste it. The adjustment consists in transition of dq A through the inverted PI controller and subsequent multiplication by the inverted value of standard current factor. This yields to additional decoupling for complete decoupling of equation (17). The additional decoupling is shown in Fig. 2. Next, it is necessary to add a block of the torque reference shaping so that the current reference value has the same properties as in the case of current factor (32). We require 1 (37) Mm W iq W = 3 2 pγΘ which can be written in the form ψPM 1 · iq W = 3 MmW (38) γΘ 2 pψPM | {z } | {z } standard CF
TF
Waveform of torque reference is obtained using the Torque Factor (TF). To torque reference waveform a signal of additional decoupling is added and the resulting signal serves as a torque reference for the standard current control loop. Location of Torque Factor is shown in Fig. 2. Assuming that the control of PMSM is based on the standard current control loop allows to use the additional decoupling and torque factor which improves characteristics of the current control loop so that the resulting torque ripple is minimized.
Fig. 3. The estimation error e 5. ADAPTATION The Torque Factor and the additional decoupling block depend on the vector of parameters Θ. To obtain the best possible function of the additive current control loop components it is necessary to know their exact values. The values of these parameters are not normally listed in the motor data sheet and are therefore unknown. They can be obtained from measurements of induced voltage waveform, when motor works as a generator. However, it is the offline measurement and these parameters are sensitive to changes in the operating conditions. Therefore, we have developed an adaptive control scheme which ensures the suppression of disturbances in the q-axis control loop. This scheme can be also modified to reduce the effect of unmodelled dynamics and external disturbances, see Miklovičová et al. (2005); Poliačik et al. (2010); Veselý et al. (1993). Recall that we consider PMSM model in the form (16)– (18). The closed loop dynamics of q-axis is expressed by the equation diq = −Riq + uPIq + Lq (39) dt + ω (Ld id + γΘ) − ω (Ld id + γΘ⋆ ) where Θ⋆ is the vector of unknown parameters. The ideal situation is when the last two right-hand side terms cancel each other out. Then di⋆q = −Ri⋆q + uPIq (40) Lq dt ⋆ Thus, zero value of the error e = iq − iq is required. The situation is shown in Fig. 3. After subtracting (40) from (39) we obtain the following equation governing estimation error dynamics de R 1 = − e + ω γθ (41) dt Lq Lq where θ = Θ − Θ⋆ represents parameters deviation from their desired (true) values. Equation (41) relates the parameter error θ to the estimation error e and motivates the use of the Lyapunov-like function 1 1 V = α e2 + θ T θ (42) 2 2 where α > 0 is a design parameter. The time derivative of this function along the trajectories of error e is 1 R (43) V˙ = −α e2 + αe ωγθ + θ T θ˙ Lq Lq The term −α LRq e2 is negative semidefinite.
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In an attempt to make V˙ negative semidefinite, we choose the adaptation law so that the second term of V˙ is canceled ˙ ≈ θ˙ = −αe 1 ωγ T Θ (44) Lq By changes in design parameter α we can control the estimated parameters convergence rate.
The initial values of the adapted parameters have been set T to Θinit = [0.2 0 0] . Ψd0 can be considered to be equal to the value of ψPM which is often referred in the motor datasheet. Nonzero and sufficiently accurate initial value of Ψd0 ensures that the division by zero in the Torque Factor is avoided.
6. SIMULATION EXPERIMENTS
Several simulations have been performed to find the appropriate value for the parameter α = 0.1. Convergence of the parameters is then fast enough with low oscillations.
All simulations have been performed using Matlab & Simulink software. To model the motor the equations (16) – (19) have been used. Motor parameter values are given in Table 1. The benefit of an additional controller is demonstrated by comparing the simulation results without additional controller with the simulation results when additional adaptive controller is used. In order to see the ripple and fundamental dynamics at the same time scale, the desired value of the mechanical angular velocity has been chosen relatively low: ωm W = 25rad/s. The electrical angular velocity is two times higher, because motor has two pole-pairs, so the first harmonic of electrical quantities is fe = 7.96Hz. The ripple of electrical quantities appears at the frequency 6fe = 47.7Hz and 12fe = 95.49Hz. The ripple at these frequencies then appears in mutual torque of the motor. These frequencies are not filtered enough by the motor mechanical subsystem mainly due to a small moment of inertia and thus ripple appears in angular velocity, too. Let’s design the current closed loop dynamics to be one order faster than the fastest dynamics expected in the motor electromagnetic subsystem. 12fe corresponds to the time constant of approximately 10ms. By compensation of the open-loop transfer function pole by the zero of the PI controller with the integral time constant TIi = Lq /R and by choosing the proportional gain Pi = R, it is achieved that the closed loop transfer function is the first order system with the time constant TIi = 0.85ms and with the unity gain, see Dolinar (2007). The standard time separation principle is used in the speed controller design. The speed loop controller is designed assuming that the electrical dynamics is infinitely fast. When the speed controller is PI-type with transfer function P Iω (s) = Pω (1 + 1/TIω s), the closed loop transfer function has the following characteristic polynomial ch(s) = s2 + 2bω0 s + ω02 (45) Pω while Pω = 2bω0 J − Bf and TIω = Jω 2 . The PI controller 0 parameters depend on the bandwidth ω0 and damping b. We have chosen ω0 = 50rad/s and b = 1.
In simulations the step increase of a load torque from zero to 0.03N m is simulated at the time 0.2s. Table 1. Parameters of the simulated motor Parameter p R Ld Lq J Bf
Value [Unit] 2 33.6 [Ω] 0.0284 [H] 0.0284 [H] 0.000016 [kg/m2 ] 0.0000082 [kg m2 /s]
Parameter ψPM Ψd0 Ψd6 Ψd12 Ψq6 Ψq12
Value [Wb] 0.303 0.303 0.0181 0.0024 0.0036 0.0022
The results of the simulation without the additional adaptive controller are shown in Fig. 4. The results of the simulation with the adaptive controller are shown in Fig. 5. 7. CONCLUSION The additional adaptive controller supplements the standard control scheme of PMSM for achieving minimization of the mutual torque ripple. Motivation for its development has been an effort to employ the motor model which takes into account the higher harmonics while the control system is based on the standard current control loop with PI controllers. The additional controller ensures that not only DC component of the d-axis to q-axis interaction but also higher harmonics of this interaction are compensated. Decoupling in d-axis remains incomplete because common realization of the current control loop does not allow its addition. The id current ripple occurs only when the load torque is not zero. At the load torque 0.03N m, the steady state torque ripple was in the range 0.03 ± 0.0007N m. When 100% load torque was simulated, the steady state torque ripple was in the range 0.5 ± 0.01N m. Thus the id current ripple was found to be negligible. The additional adaptive controller represents a compromise between maximally possible suppression of the mutual torque ripple and implementation simplicity of control scheme for the torque ripple minimization. Its addition to the standard servosystem requires access only to the speed control loop, which is often more accessible than the current control loop. This is the main advantage of the proposed control scheme. REFERENCES Dolinar D.,Štumberg G. (2007). Modeling and Control of Electrical Machines. University of Maribor. Holtz J., Springob L. (1996). Identification and Compensation of Torque Ripple in High-Precision Permanent Magnet Motor Drives. IEEE Trans. on Industrial Electronics, vol.43, No5, 309-320. Krause P.C.,Wasynczuk O., Sudhoff S.D. (2002). Analysis of Electic Machinery and Drive Systems. IEEE Press. Miklovičová E., Murgaš J., Gonos M. (2005). Reducing the effect of unmodelled dynamics by MRAC control law modification. 16th IFAC World Congress, Czech Republic, s. Tu-M02-TP/11. Mohamed Y.A-R.I., El-Saadany E.F. (2008). A Current Control Scheme With an Adaptive Internal Model for Torque Ripple Minimization and Robust Current Regulation in PMSM Drive Systems. IEEE Trans. on Energy Conversion, vol.23, no.1 Ohm D.Y. (2000). Dynamic model of PM synchronous motors. Drivetech, Inc., Virginia.
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Fig. 4. Simulation of standard PMSM control without additional controller Petrović V., Ortega R., Stanković A.M. and Tadmor G. (2000). Design and implementation of an adaptive controller for torque ripple minimization in PM synchronous motors. IEEE Trans. on Power Electronics, vol.15, No.5, 871-880. Petrović V., Stanković A.M. (2001). Modeling of PM synchronous motors for control and estimation tasks. IEEE Proc. on Decision and Control, vol.3, 2229-2234. Poliačik M., Murgaš J., Farkas Ľ., Blaho M. (2010). A robust MRAC Modification and Performance Improvement in the Presence of Uncertainties. IFAC International Workshops on Adaptation and Learning in Control and Signal Processing, Turkey. Qian W., Nondhal A.T. (2005). Mutual Torque Suppression of Surface-mounted Permanent Magnet Synchronous Motor. Proc. of the Eighth International Conference on Electrical Machines and Systems, vol.1. Veselý V., Murgaš J. (1993). Decentralized adaptive stabilization of nonlinear systems. Internat. Journal of Control, Vol. 58, No.6, 1445-1460.
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Fig. 5. Simulation of PMSM control with the additional adaptive controller Wallmark O. (2001). Control of a Permanent Magnet Synchronous Motor with Non-Sinusoidal Flux Density Distribution. Dept. of Electric Power Engineering Chalmers University of Technology, Göteborg.
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Additional Adaptive Controller for Mutual Torque Ripple Minimization in PMSM Drive Systems ⋆ Marián Tárník ∗ Ján Murgaš ∗ ∗
Institute of Control and Industrial Informatics, Slovak University of Technology in Bratislava, Faculty of Electrical Engineering and Information Technology, Ilkovičova 3, 812 19 Bratislava, Slovakia (e-mail: [email protected], [email protected]).
Abstract: This paper presents an additional adaptive current controller for mutual torque ripple minimization in PMSM. It is assumed that the basis of the control system is a conventional current control loop commonly used in vector control of the PMSM. No change in the structure of current loop is required, only access to the torque reference input is needed. Parameters of the additional controller are adapted directly by the derived adaptation law. Finally, the proposed control scheme is verified by means of simulations. Keywords: Permanent magnet synchronous motor, Torque ripple, Adaptive control. 1. INTRODUCTION
PMSM with the isotropic rotor and with the negligible reluctance torque.
Permanent magnet synchronous motors (PMSM) are used in drives for their features such as high efficiency, low moment of inertia and high torque to motor size ratio. Drives with PMSM are also used in machine tools and robots. In these applications an undesirable feature of the motor appears – the torque ripple. There are several reasons of the torque ripple, and these can be divided by type of torque produced by the motor into three categories, see Petrović et al. (2000):
The common control strategy for PMSM in servodrives is vector control oriented in rotor dq reference frame. In that case, the motor control system has a certain standard structure. The control system of servodrive can be divided by controlled variable into several parts: current control loop, speed control loop and position control loop. The basis is always the current control loop, because the torque is given by current, especially when the motor magnetic flux is constant, provided by the permanent magnets. Usually the current control subsystem consists of PI controllers of d-axis and q-axis currents, decoupling block and the block which we will call a Current Factor (CF). The Current Factor is used to convert the torque reference value to current reference value. In standard case, the decoupling block and the Current Factor are derived from the motor model, which describes only the basic dynamics of the motor. The higher harmonics in the signals are neglected. This means, that standard current loop does not count with higher harmonics in the motor electromagnetic subsystem. Therefore, the higher harmonics become a part of disturbance signal. PI controllers do not remove these disturbances suitably which results in the torque ripple.
• Mutual torque – Higher harmonics in mutual torque are caused by nonsinusoidal distribution of stator windings or rotor magnets. Smooth mutual torque is produced only if mutual flux through stator windings is purely sinusoidal. • Reluctance torque – It is nonzero only if stator inductances depend on rotor position. If stator windings are sinusoidally distributed then the reluctance torque has only DC component. • Cogging torque – It depends only on geometry and number of stator slots. This torque component contributes only to torque ripple, it has no DC component. The cogging torque is usually reduced in machine design procedure. A large number of PMSMs is manufactured with an isotropic rotor. In this case the reluctance torque is negligible, and therefore it does not contribute to torque ripple. In applications where there is a reason to deal with high quality of produced torque it is also reasonable to consider that the motor has reduced cogging torque by the machine design procedure. Motor with this construction produces only mutual torque with large DC component and with higher harmonics. This paper deals with the control of ⋆ This work has been supported by Slovak scientific grant agency through grant VEGA-1/0592/10.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Many techniques for PMSM torque ripple minimization have been proposed in the literature, see Petrović et al. (2000). There are two classes of these techniques. First, techniques that focus on the motor design so that PMSM closely approaches its ideal characteristics. This brings more complicated manufacturing process and increases the cost of the motor. The second class of techniques includes control algorithms which are developed taking into account the nonideal characteristics of the motor. The control scheme proposed in this paper belongs to the second class. The oldest approach in this class uses the preprogrammed current waveform to cancel higher
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harmonics effects in the motor electromagnetic subsystem. This approach is usually based on off-line calculations, but on-line identification of necessary parameters is also used, for example see Holtz (1996). In this class, the use of adaptive control based on identification in design of control algorithm for PMSM torque ripple minimization is represented by several algorithms. Their common feature is the use of motor model which sufficiently describes the motor properties, while it remains compact and suitable for control design. The most significant higher harmonic components of magnetic flux are preserved in the model, see Petrović et al. (2001); Wallmark (2001), and only those with too small amplitude are truncated. In conventional model only fundamental components are considered, see Ohm (2000); Dolinar (2007); Krause et al. (2002). Petrović et al. (2000) proposed the adaptive current controller, which is theoretically able to completely cancel the most significant components of mutual torque ripple which come from the electromagnetic subsystem of motor. In the paper they also deal with the implementation of the proposed controller. In Qian (2005) so-called Compensating Current Factor (CCF) is proposed by which current reference value is shaped in order to suppress the torque ripple. To obtain the parameters of Compensating Current Factor a MRAS-based flux linkage estimator and a LS-based harmonic selective filter is used. The proposed algorithm does not solve the whole problem. Some of the problem parts are not addressed, since higher harmonics in the interactions of d and q axis remain uncompensated. The current control scheme with an adaptive internal model is proposed in Mohamed (2008). The paper takes aim at compensation of all motor parameter uncertainties influence. By estimating the total voltage error a minimization of torque ripple is also achieved. The remainder of this paper is organized as follows: In Section 2 we introduce two types of PMSM models. The standard current control loop is described in Section 3. In Section 4 the current control loop for the mutual torque ripple minimization is presented and its modification for use with the standard current control loop is developed. In Section 5 an adaptation mechanism for additional controller parameters introduced in Section 4 is derived. Section 6 shows the simulation results. 2. MODELING OF THE PMSM IN DQ REFERENCE FRAME Commonly used model of PMSM with an isotropic rotor, i.e., its self and mutual inductances of stator windings do not depend on rotor position, has in dq reference frame the form dψd ud = Rid + − ωψq (1) dt dψq uq = Riq + + ωψd (2) dt ψd = Ld id + ψd rot (3) ψq = Lq iq + ψq rot (4) 3 Pmech (5) Mm = p 2 ω dω = Mm − Mz − Bf ω (6) J dt
dϑ =ω (7) dt where Ld and Lq are the stator inductances in dq reference frame, and Ld = Lq , R is the stator winding resistance, J is the moment of inertia, Bf is the friction constant and p is number of pole-pairs. ud and uq are the d-axis and q-axis stator voltages, id and iq are d-axis and qaxis stator currents, Mm is the torque produced by the motor, Pmech is part of electrical power which is converted to mechanical power, Mz is the load torque, ω is the rotational speed of the rotor in electrical rad/sec and ϑ is the position measured in electrical radians. ψd a ψq are the total flux linkages and ψd rot and ψq rot are the flux linkages established by the permanent magnets. In the conventional model it is considered, see Ohm (2000); Krause et al. (2002) ψd rot = ψPM (8) ψq rot = 0 (9) where ψP M is the amplitude of permanent magnet flux linkage. In this case, the rotor magnetic field distribution is considered to be sinusoidal. When the rotor and stator magnetic field distributions are not sinusoidal, the permanent magnet flux linkages can be viewed as the sum of a fundamental component and the series of higher harmonics, see Petrović et al. (2001); Wallmark (2001); Qian (2005) ψd rot = ψd0 + ψd6 cos(6ϑ) + ψd12 cos(12ϑ) + . . . (10) ψq rot = ψq6 sin(6ϑ) + ψq12 sin(12ϑ) + . . . (11) For good approximation of the permanent magnet flux linkages it is sufficient to keep only series members up to 12th harmonics. To determine the decoupling block and the Current Factor, it is convenient to write equations in the following form. For the case when the conventional model is considered, i.e. using (8) and (9) did Ld = −Rid + ω (Lq iq ) + ud (12) dt diq Lq = −Riq − ω (Ld id + ψPM ) + uq (13) dt 3 (14) Mm = p (ψPM iq + (Ld − Lq ) id iq ) 2 dω J = −Bf ω + Mm − Mz (15) dt For the case when the nonsinusoidal distributions are considered (equations (10) and (11) are used) did = ud − Rid + ω (Lq iq + Ld (16) dt + Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) diq Lq = uq − Riq − ω (Ld id + (17) dt + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) 3 Mm = p ((Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) iq − 2 − (Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) id + + (Ld − Lq ) id iq ) (18) dω = −Bf ω + Mm − Mz (19) J dt
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where Ψd0 = ψd0 (20) Ψd6 = 6ψq6 + ψd6 (21) Ψd12 = 12ψq12 + ψd12 (22) Ψq6 = 6ψd6 + ψq6 (23) Ψq12 = 12ψd12 + ψq12 (24) are parameters having physical interpretation, see Petrović et al. (2000). For the rest of paper, only the model of PMSM considering nonsinusoidal distributions of magnetic field will be used in simulations. Conventional model will be used only to derive the standard decoupling block and the Current Factor. 3. STANDARD CURRENT CONTROL LOOP In the standard current control loop, the controlled system is given by equations (12) – (14). It is clear from equation (14) that the torque Mm is proportional to current iq . The current id has no contribution to the torque production because Ld = Lq , therefore zero reference value of id is chosen idW = 0. The reference value iq W is governed by the torque reference value according to 1 (25) Mm W iq W = 3 pψ PM 2 which we call Current Factor CF =
1 . 3 2 pψPM
In equation
(25) is term KT = 32 pψPM called motor torque constant. PI controllers are commonly used for control of both current components. Equations (12) and (13) are coupled. Decoupling is obtained using the voltage feed-forward signals which are added to both outputs of PI controllers. Then the voltages in equations (12) and (13) are ud = uPId + dd (26) uq = uPIq + dq (27) where dd = −ω (Lq iq ) (28) (29) dq = ω (Ld id + ψPM ) are the voltage feed-forward signals and uPId , uPIq are the outputs of PI controllers. The parameters of decoupling block and the parameters of Current Factor are given by model of PMSM. The parameters of current PI controllers can be determined using any method. For example, in Dolinar (2007) socalled compensation method is applied. Block diagram of the current control loop is in Fig. 1. Outputs of the PI controllers ud , uq are transformed to the three phase abc stator reference frame and modulated using pulse-width modulation in voltage source inverter. In industrial drives all components of standard current control loop are contained in a single device connected to the motor. In such the case, only one external input is available – the torque reference. Moreover, the components can be changed only by the parameters. The standard current control loop is not designed to minimize mutual torque ripple so the standard industrial drive is not capable to cope with the torque ripple.
idW +
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− iq Fig. 1. The current control loop 4. CURRENT CONTROL LOOP FOR MUTUAL TORQUE RIPPLE MINIMIZATION Mutual torque ripple is induced by the higher harmonics in permanent magnet flux linkages. This phenomenon can be modeled by equations (16)–(18). Using these equations components of the current control loop can be designed. Higher harmonics in the back EMFs are then compensated and thus the disturbances added to PI controller output are reduced. Conversion of torque reference value to current reference value is realized using the extended current factor. The torque constant is replaced by a term which contains harmonic functions. The current waveform is then shaped so that the produced mutual torque is smooth and its ripple is suppressed. For decoupling of equations (16) and (17) the signals dd and dq have form dd = −ω (Lq iq + Ψq6 sin(6ϑ) + Ψq12 sin(12ϑ)) (30) dq = ω (Ld id + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) (31) From the torque equation (18) it follows, that the nonzero id contributes only to torque ripple, because it is multiplied only by higher harmonics. Thus it is desirable to set the reference value of id to zero. The current iq contributes to DC component but also to higher harmonics. In this case the Current Factor must be 1 CF = 3 (32) p (Ψ + Ψ cos(6ϑ) + Ψd12 cos(12ϑ)) d0 d6 2 As in previous case, the PI controllers are used for current control so these components of current control loop remain unchanged. However, it is necessary to reflect new requirements in their design. In q-axis the reference value signal iqW is much more rich in frequencies, which requires wider bandwidth of the closed loop. Generally speaking, the standard current control loop can be complemented by parts which ensure the minimization of torque ripple. This complementation can not be made directly if the whole current control loop is contained in common industrial voltage source inverter. Then the structure of control loop can not be changed and another signal can not be added to signals (eg, the output of PI controller). The only available input is the torque reference. Let’s assume the standard current control loop as described in section 3 which is implemented in a single device. The decoupling of equation (16) can not be complete because standard signal dd does not compensate the higher harmonics. Theoretically, it is possible to amend dd , but practically it is not feasible because the signal uPI d is not available for adding. So the higher harmonics of
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dq A Fig. 2. The additional decoupling of q-axis, location of TF disturbance added to d-axis PI controller output remain uncompensated. In q-axis the situation is different. Instead of adding the missing part of signal dq to signal uPI q it can be added to the torque reference signal MmW . First, let us determine the missing part of the signal. For decoupling of equation (17) it is required: d⋆q = ω (Ld id + Ψd0 + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) = = ω (Ld id + γΘ) (33) T
where γ = [1 cos(6ϑ) cos(12ϑ)], Θ = [Ψd0 Ψd6 Ψd12 ] . In standard current control loop dq = ω (Ld id + ψPM ) (34) to satisfy d⋆q = dq + dq A (35) dq A must be in the form dq A = ω ((Ψd0 − ψPM ) + Ψd6 cos(6ϑ) + Ψd12 cos(12ϑ)) (36) Before adding the signal dq A to Mm W , it is necessary to adjuste it. The adjustment consists in transition of dq A through the inverted PI controller and subsequent multiplication by the inverted value of standard current factor. This yields to additional decoupling for complete decoupling of equation (17). The additional decoupling is shown in Fig. 2. Next, it is necessary to add a block of the torque reference shaping so that the current reference value has the same properties as in the case of current factor (32). We require 1 (37) Mm W iq W = 3 2 pγΘ which can be written in the form ψPM 1 · iq W = 3 MmW (38) γΘ 2 pψPM | {z } | {z } standard CF
TF
Waveform of torque reference is obtained using the Torque Factor (TF). To torque reference waveform a signal of additional decoupling is added and the resulting signal serves as a torque reference for the standard current control loop. Location of Torque Factor is shown in Fig. 2. Assuming that the control of PMSM is based on the standard current control loop allows to use the additional decoupling and torque factor which improves characteristics of the current control loop so that the resulting torque ripple is minimized.
Fig. 3. The estimation error e 5. ADAPTATION The Torque Factor and the additional decoupling block depend on the vector of parameters Θ. To obtain the best possible function of the additive current control loop components it is necessary to know their exact values. The values of these parameters are not normally listed in the motor data sheet and are therefore unknown. They can be obtained from measurements of induced voltage waveform, when motor works as a generator. However, it is the offline measurement and these parameters are sensitive to changes in the operating conditions. Therefore, we have developed an adaptive control scheme which ensures the suppression of disturbances in the q-axis control loop. This scheme can be also modified to reduce the effect of unmodelled dynamics and external disturbances, see Miklovičová et al. (2005); Poliačik et al. (2010); Veselý et al. (1993). Recall that we consider PMSM model in the form (16)– (18). The closed loop dynamics of q-axis is expressed by the equation diq = −Riq + uPIq + Lq (39) dt + ω (Ld id + γΘ) − ω (Ld id + γΘ⋆ ) where Θ⋆ is the vector of unknown parameters. The ideal situation is when the last two right-hand side terms cancel each other out. Then di⋆q = −Ri⋆q + uPIq (40) Lq dt ⋆ Thus, zero value of the error e = iq − iq is required. The situation is shown in Fig. 3. After subtracting (40) from (39) we obtain the following equation governing estimation error dynamics de R 1 = − e + ω γθ (41) dt Lq Lq where θ = Θ − Θ⋆ represents parameters deviation from their desired (true) values. Equation (41) relates the parameter error θ to the estimation error e and motivates the use of the Lyapunov-like function 1 1 V = α e2 + θ T θ (42) 2 2 where α > 0 is a design parameter. The time derivative of this function along the trajectories of error e is 1 R (43) V˙ = −α e2 + αe ωγθ + θ T θ˙ Lq Lq The term −α LRq e2 is negative semidefinite.
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In an attempt to make V˙ negative semidefinite, we choose the adaptation law so that the second term of V˙ is canceled ˙ ≈ θ˙ = −αe 1 ωγ T Θ (44) Lq By changes in design parameter α we can control the estimated parameters convergence rate.
The initial values of the adapted parameters have been set T to Θinit = [0.2 0 0] . Ψd0 can be considered to be equal to the value of ψPM which is often referred in the motor datasheet. Nonzero and sufficiently accurate initial value of Ψd0 ensures that the division by zero in the Torque Factor is avoided.
6. SIMULATION EXPERIMENTS
Several simulations have been performed to find the appropriate value for the parameter α = 0.1. Convergence of the parameters is then fast enough with low oscillations.
All simulations have been performed using Matlab & Simulink software. To model the motor the equations (16) – (19) have been used. Motor parameter values are given in Table 1. The benefit of an additional controller is demonstrated by comparing the simulation results without additional controller with the simulation results when additional adaptive controller is used. In order to see the ripple and fundamental dynamics at the same time scale, the desired value of the mechanical angular velocity has been chosen relatively low: ωm W = 25rad/s. The electrical angular velocity is two times higher, because motor has two pole-pairs, so the first harmonic of electrical quantities is fe = 7.96Hz. The ripple of electrical quantities appears at the frequency 6fe = 47.7Hz and 12fe = 95.49Hz. The ripple at these frequencies then appears in mutual torque of the motor. These frequencies are not filtered enough by the motor mechanical subsystem mainly due to a small moment of inertia and thus ripple appears in angular velocity, too. Let’s design the current closed loop dynamics to be one order faster than the fastest dynamics expected in the motor electromagnetic subsystem. 12fe corresponds to the time constant of approximately 10ms. By compensation of the open-loop transfer function pole by the zero of the PI controller with the integral time constant TIi = Lq /R and by choosing the proportional gain Pi = R, it is achieved that the closed loop transfer function is the first order system with the time constant TIi = 0.85ms and with the unity gain, see Dolinar (2007). The standard time separation principle is used in the speed controller design. The speed loop controller is designed assuming that the electrical dynamics is infinitely fast. When the speed controller is PI-type with transfer function P Iω (s) = Pω (1 + 1/TIω s), the closed loop transfer function has the following characteristic polynomial ch(s) = s2 + 2bω0 s + ω02 (45) Pω while Pω = 2bω0 J − Bf and TIω = Jω 2 . The PI controller 0 parameters depend on the bandwidth ω0 and damping b. We have chosen ω0 = 50rad/s and b = 1.
In simulations the step increase of a load torque from zero to 0.03N m is simulated at the time 0.2s. Table 1. Parameters of the simulated motor Parameter p R Ld Lq J Bf
Value [Unit] 2 33.6 [Ω] 0.0284 [H] 0.0284 [H] 0.000016 [kg/m2 ] 0.0000082 [kg m2 /s]
Parameter ψPM Ψd0 Ψd6 Ψd12 Ψq6 Ψq12
Value [Wb] 0.303 0.303 0.0181 0.0024 0.0036 0.0022
The results of the simulation without the additional adaptive controller are shown in Fig. 4. The results of the simulation with the adaptive controller are shown in Fig. 5. 7. CONCLUSION The additional adaptive controller supplements the standard control scheme of PMSM for achieving minimization of the mutual torque ripple. Motivation for its development has been an effort to employ the motor model which takes into account the higher harmonics while the control system is based on the standard current control loop with PI controllers. The additional controller ensures that not only DC component of the d-axis to q-axis interaction but also higher harmonics of this interaction are compensated. Decoupling in d-axis remains incomplete because common realization of the current control loop does not allow its addition. The id current ripple occurs only when the load torque is not zero. At the load torque 0.03N m, the steady state torque ripple was in the range 0.03 ± 0.0007N m. When 100% load torque was simulated, the steady state torque ripple was in the range 0.5 ± 0.01N m. Thus the id current ripple was found to be negligible. The additional adaptive controller represents a compromise between maximally possible suppression of the mutual torque ripple and implementation simplicity of control scheme for the torque ripple minimization. Its addition to the standard servosystem requires access only to the speed control loop, which is often more accessible than the current control loop. This is the main advantage of the proposed control scheme. REFERENCES Dolinar D.,Štumberg G. (2007). Modeling and Control of Electrical Machines. University of Maribor. Holtz J., Springob L. (1996). Identification and Compensation of Torque Ripple in High-Precision Permanent Magnet Motor Drives. IEEE Trans. on Industrial Electronics, vol.43, No5, 309-320. Krause P.C.,Wasynczuk O., Sudhoff S.D. (2002). Analysis of Electic Machinery and Drive Systems. IEEE Press. Miklovičová E., Murgaš J., Gonos M. (2005). Reducing the effect of unmodelled dynamics by MRAC control law modification. 16th IFAC World Congress, Czech Republic, s. Tu-M02-TP/11. Mohamed Y.A-R.I., El-Saadany E.F. (2008). A Current Control Scheme With an Adaptive Internal Model for Torque Ripple Minimization and Robust Current Regulation in PMSM Drive Systems. IEEE Trans. on Energy Conversion, vol.23, no.1 Ohm D.Y. (2000). Dynamic model of PM synchronous motors. Drivetech, Inc., Virginia.
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Fig. 4. Simulation of standard PMSM control without additional controller Petrović V., Ortega R., Stanković A.M. and Tadmor G. (2000). Design and implementation of an adaptive controller for torque ripple minimization in PM synchronous motors. IEEE Trans. on Power Electronics, vol.15, No.5, 871-880. Petrović V., Stanković A.M. (2001). Modeling of PM synchronous motors for control and estimation tasks. IEEE Proc. on Decision and Control, vol.3, 2229-2234. Poliačik M., Murgaš J., Farkas Ľ., Blaho M. (2010). A robust MRAC Modification and Performance Improvement in the Presence of Uncertainties. IFAC International Workshops on Adaptation and Learning in Control and Signal Processing, Turkey. Qian W., Nondhal A.T. (2005). Mutual Torque Suppression of Surface-mounted Permanent Magnet Synchronous Motor. Proc. of the Eighth International Conference on Electrical Machines and Systems, vol.1. Veselý V., Murgaš J. (1993). Decentralized adaptive stabilization of nonlinear systems. Internat. Journal of Control, Vol. 58, No.6, 1445-1460.
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Fig. 5. Simulation of PMSM control with the additional adaptive controller Wallmark O. (2001). Control of a Permanent Magnet Synchronous Motor with Non-Sinusoidal Flux Density Distribution. Dept. of Electric Power Engineering Chalmers University of Technology, Göteborg.
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