Pulsating Torque Reduction for Permanent Magnet

Pulsating Torque Reduction for Permanent Magnet AC Motors IEEE Conf. of Control Application 2001 ˇ Bojan Grˇcar, Peter Cafuta, Gorazd Stumberger University of Maribor, Faculty of Electrical Engineering and Computer Science Maribor, Slovenia

Aleksandar M. Stankovi´c Northeastern University, Dept. Electrical and Computer Engineering Boston,MA

Abstract— Control methods for pulsation of torque reduction for the surface-mounted permanent magnet motors are discussed in the paper. The pulsation torque is a consequence of the non-sinusoidal flux distribution and due to interaction of the rotor’s permanent magnets with the changing stator reluctance. Proposed control method is estimator based. To assure parameter convergence Lyapunov’s direct method is used in estimator design for the flux Fourier’s coefficients. Novel nonlinear torque controller based on flux/torque estimate is introduced to reduce the influence of the flux harmonics. The influence of the cogging torque is considerably reduced at lower motor speed using internal model principle and adaptive feedforward compensation technique. Overall control scheme and experimental results are also presented.

I. I NTRODUCTION The PMAC motors are often used in various high performance drives where high efficiency, torque smoothness, and high torque to inertia ratio are required. Although the basic control structure is relatively simple, the problem of pulsation in the generated electrical torque requires additional attention [1]. For the surface mounted PM motor-type with non-skewed windings besides nominal torque t 0 (i) at least two additional torque components should be considered. The first component is a consequence of the non-sinusoidal flux distribution (ripple torque), while the second is due to interaction of the rotor’s permanent magnets with changing stator reluctance (cogging torque). Ripple torque tr (i; θe ) (θe is the rotor angle in electrical degrees; subscript e is used through the text to denote electrical variables while subscript m stands for mechanical variables) is produced by the interaction of the stator current MMF and the rotor MMF. The predominant ripple frequencies are six and twelve times the stator frequency [4], and ripple torque can reach typically 2 4 % of the nominal torque. Cogging torque t c (θm ), which is current independent, can reach up to 3 % of the nominal torque with pulsation frequency f c = ωm Ns= p , where ωm is mechanical speed, and Ns= p is the number of stator slots per pole pair. Pulsating torque represents the sum of the ripple and cogging torque with zero mean value and produces only vibration and acoustic noise which may be even amplified in variable speed drives when the torque frequency coincides with a resonant frequency of the mechanical system. Total electrical torque for the assumed PMAC machine type is given by the sum of its components: te (i; θ ) = t0 (i) + tr (i; θe ) + tc (θm )

(1)

The torque pulsation reduction is equally challenging for the motor designer as for the control engineer. Methods based on the motor design (skewing, shifting the magnet position, special

windings, etc.) [5], [6] are in general more efficient than control based methods. The main drawbacks of the quoted methods are the reduced average (nominal) torque and the increased complexity of motor construction, resulting in higher overall costs. On other hand, control based methods [2], [4], [10] without influencing the drive hardware can be easily and cheaply implemented; however the obtained torque is never entirely smooth. The torque pulsation under 1 2 % is typically considered as the desired objective. Control-based methods in the reduction of the ripple torque can be subdivided into two basic groups. The first one is based on a-priori knowledge of the torque ripple, and is structured as open-loop control (harmonic cancellation technique based on programmed current waveform). The resulting control structure is highly sensitive due to the imperfect pulsating torque model, and due to parameter variations of the motor. More efficient are methods in the second group structured as a closed-loop including flux and/or torque estimators. Two major classes of estimators can be introduced in this framework: observers for total flux/torque and estimators for the determination of Fourier’s coefficients. The first class is simpler for on-line implementation, but the phase lag reduces the accuracy at higher motor speed. Methods in the second class are more accurate, but they require adaptive updating of Fourier’s coefficients. Its on-line implementation is consequently more time consuming. The compensation of high frequency torque ripple requires extended bandwidth of the current controllers as torque harmonics increase proportionally to the motor speed. The compensation of the cogging torque is in some respect more difficult. The main reason is that adequate, simple models with lumped parameters do not exist. Most often the problem of predicting cogging torque is addressed in motor design, and the solution is based on demanding field calculations [3]. The number of stator slots per pole pair and geometry of the stator slots and permanent magnets define the waveform and harmonic contents of the generated cogging torque. Additional difficulty is due to the empirical fact that the characteristic waveform of the cogging torque varies with the operation conditions (temperature) and may substantially differ for the nominally identical machines of the same manufacturer. From the control theory viewpoint the cogging torque represents self-generated unmatched periodical disturbance. Satisfactory rejection of the periodical oscillations at low speed requires on-line estimation of the significant cogging torque components, and their feedforward compensation [7] and [8]. The organization of the paper is as follows. First the adaptive flux/torque estimator based on Lyapunov’s second method

is presented. Additionally a new non-linear torque controller is introduced which enables tracking of the torque command in high frequency range. In the second section adaptive compensation technique of significant cogging torque components is introduced. The efficiency of the proposed techniques in reducing the pulsation torque was verified with experiments. Some illustrative results presenting the possibilities and limitations of the proposed control methods are included in the third section. In the appendix-A estimator’s convergence proofs and in the appendix-B nominal PMAC machine parameters are given. II. N ON - LINEAR TORQUE CONTROL BASED ON FLUX ESTIMATE

The a; b; c motor model is transformed into a synchronous rotating d q reference frame by standard 3 phase to orthogonal d q coordinate transformation (Blondel-Park). The resulting model under the assumption of the non-sinusoidal flux distribution and using standard notation is of the following form: Ld R Ld + Ld ωe iq

d dt id =

Lq ωe Ψq (θe ) R 1 + L uq Lq Ld ωe id Lq q 3p d ω = ( Ψ ( θ ) i + Ψ ( θ ) i + ( L m e q q e q d d dt 2J tc (θm ) Bω m 1 t J J J L

(2)

Lq )id iq )

where Ψ(θe ) is flux linkage of the permanent magnet and t c (θm ) is the cogging torque. Since only symmetrical machines are considered, the inductances in the d- and q- direction are equal, so the model above is partially simplified. Additionally the reluctance torque does not appear in the expression for the mechanical speed. After introducing the concept of field orientation, it can be further assumed that the d-axis current is controlled to zero (outside the regime of field weakening). The generated machine torque is maximized, and can be described as: te = cΨq (θe )iq

(3)

The flux linkages in the d- and q- axes are modeled as:



 =

e ωe F )ei + DΨ

Kei = (A



Ψd6 sin(6θe ) + Ψd12 sin(12θe ) + : : : Ψq0 + Ψq6 cos(6θe ) + Ψq12 cos(12θe ) + : : :

(4)

db b (θe ) + L 1 u + Kei i = Abi ωe Fbi ωe L 1 Ψ dt

(5)

where symbol b: refers to estimated quantities and symbol e: refers to the error between actual and model variables. The introduced matrices A; F; L 1 and the gain matrix K are defined as: A = diag( LR ; LR ) F = antidiag( 1; 1) (6) L 1 = diag( L1 ; L1 ) K = diag(kd ; kq )

e K )ei + DΨ

(7)

where P is a symmetrical positive definite solution of the Lyapunov’s matrix equation and δ is a positive constant. If we assume that total motor flux changes slowly compared with the estimator dynamic (dΨ(θ e )=dt  0 ), then the stability condition dV =dt  0 leads to the adaptation law: 1

DT Pei

(9)

The introduced assumption d =dtΨ(θ e )  0 implies that estimator gains must be selected high enough to achieve a fast response compared to the time variations of the flux. High gains can lead to numerical instability, or cause noise amplification with significant influence for the torque controller performance at a higher motor speed. For the modified estimator only the Fourier’s coefficients of the flux need to be updated. This estimator performs equally well at all speeds, but is computationally more demanding. The design differs only slightly from the previous procedure, so only the main differences will be pointed out. Considering (4) the flux model is written in the following form:



Ψd (θe ) Ψq (θe )





sin(6θe ) sin(12θe ) 0 0 0 0 0 1 cos(6θe ) cos(12θe ) = ψ (θe ) pΨ =

 pΨ

(10) where the parameter vector is defined as p TΨ = [ pd6 ; pd12 ; pq0 ; pq6 ; pq12 ]. For the new Lyapunov function candidate: V

The control scheme based on adaptive flux/torque estimate is introduced to reduce the torque ripple. An estimator of the total motor flux is obtained using the error between the actual motor variables (reference model) and adjustable model. To derive the adaptation law (see appendix for more detailed derivation), the current model is introduced in the following form:

ωe F

where D = diag( ωe =L; ωe =L). To achieve asymptotic convergence of the estimated and actual quantities, the error vector should tend to zero after an initial perturbation. The following Lyapunov function candidate is used in the determination of a stable solution: eTΨ e (8) V = eiT Pei + δ Ψ

d b Ψ(θe ) = δ dt

d dt θm = ωm

Ψd (θe ) Ψq (θe )

de i = (A dt

ωe Ψd (θe ) 1 + L ud Ld d

d dt iq =



i and the flux error For the introduced current error ei = i b e = Ψ(θe ) Ψ b (θe ) the differential equation of the error vecΨ tor dynamics is obtained as:

where peΨ = pΨ

=e iT Pei + δ peTΨ peΨ

(11)

pbΨ , the adaptation law is derived as: d pb = δ ψ (θe )DT Pei dt Ψ

(12)

Using the flux and current estimate generated machine torque is calculated in accordance with (3):

b q (θe )biq te = cΨ

(13)

As the model current bi converges to the actual current i very quickly and considering the uniqueness property of the Fourier b converge to some value this series expansion (meaning that if Ψ te will represent a value is also actual one), the estimated torque b good approximation of the actual machine torque.

A nonlinear output tracking controller [9] based on the presented adaptive flux/torque estimators is introduced next to reduce torque ripple due to flux harmonics. The feedback linearization approach is used in the design of the tracking controller to accommodate the time-varying (periodical) signals entering the control loop. The model of the controlled plant obtained under the assumption of complete decoupling between dand q-axes (the term Lω e id is neglected) is described as: Rb iq L

db iq = dt

ωe b Ψ(θe ) + kq(iq L

biq ) + 1 uq L

(14)

Considering the expression for the torque estimate (13) (with c = 3p=2) we obtain:

b (θ ) dΨ dbi b d bte = c q e biq + c q Ψ q (θe ) dt dt dt

(15)

Further it follows: b q (θe ) dΨ db dt te = ( dt

Rb b L Ψq (θe ))ciq

cωe b 2 L Ψq (θe )

b q (θe )kqeiq + c Ψ b +cΨ L q (θe )uq

(16)

where eiq = iq biq . Following the input-output linearization procedure we choose the control voltage u q as: uq = b L (( cΦq (θe )

b q (θe ) dΦ b b cωe b 2 +R dt L Ψq (θe ))ciq + L Ψq (θe )

b q (θe )kqeiq + vq) cΨ

(17)

The new control input v q is designed as: vq = k pqe te + kdq

de te dt

(18)

where e te = tre f b te is the torque error between the reference and estimated machine torque and, k pq ; kdq are design parameters.

rejection, a model of the disturbance must be included in the feedback system. For periodic disturbances this implies that the feedback system must have a pole pair(s) at the imaginary axis at the particular frequency(s) of the disturbance. Adaptive feedforward compensation represents a realization of this principle in the time domain. The disturbance is simply compensated at the input of the system. Since the magnitudes and the phases are usually not known, they are estimated. Arbitrary harmonic orders can be considered in the adaptive compensation, but the number should be limited only to the relevant ones, due to limitations in practical implementation. The simplified model for the cogging torque is introduced: tc (θm ) = Σk (ak cos(kθm ) + bk sin(kθm ))

(19)

where ak and bk are unknown coefficients and k = (N s = p)ν ; ν = 1; 2 : : :. Instead of the actual values their estimates abk and b bk are introduced in the calculation of the compensation torque: tc (θm ) = ∆tc = b

∑(abk cos(kθm ) + bbk sin(kθm ))

(20)

k

The disturbance is exactly compensated when parameter estimates converge to their actual values. For the feedback system we get: d (21) J ωm = Bωm + wT pe dt

where wT is the regresor vector defined as w T = [cos(θm ); sin(θm ); : : : ; cos(kθm and pe = p pb is the error vector between actual and estimated parameters. The control goal is to estimate the parameter vector pb, so that the system output (oscillations in the rotor speed) is forced to remain bounded and converge to zero. In practice, however, only some of the cogging torque harmonic components are employed, so that rotor speed actually will remain bounded and will converge to some lower (finite) value. Lyapunov’s direct method is used again in the parameter vector estimation. Based on the Lyapunov’s function candidate:

III. C OGGING TORQUE COMPENSATION Two assumptions are adopted to enable a partial compensation of the machine cogging torque. The first is a standard assumption concerning the time-scale separation of the mechanical and the electrical dynamics. Using the singular perturbation approach the electrical part of the machine can be represented only by a constant describing the relation between the control voltage and the produced machine torque. The second assumption deals with the cogging torque itself. It is assumed that the dominant frequencies are known (say from previous measurements on the particular machine), but the corresponding magnitudes are to be determined on-line. An adaptive feedforward compensation technique is used to reduce the cogging torque. The proposed method is a generalization of the internal model principle that is known from the control design in the frequency domain. The cogging torque is treated as a self-generated periodic disturbance with known harmonic content, but with unknown amplitudes and phases. Because of the low-pass properties of the mechanical system, only a handful of harmonics actually needs to be considered in the practical implementations. The internal model principle states that when dealing with disturbance

V

=

1 2 1 T J ω + γ pe pe 2 m 2

(22)

and its derivative the parameter update law is obtained as: d pb = γ dt

1

ωm w

(23)

The adaptive cogging torque compensation signal is simply added to the torque controller input. As the actual motor speed increases the output of the adaptive compensator can be additc = b tc exp f (jωm j) . To avoid noise gentionally weighted as ∆b eration above mechanical cut-off frequency the weighting function f (jωm j) should be properly selected in accordance with the actual drive inertia and friction. IV. E XPERIMENTAL RESULTS In Fig. 1 the overall q-axis feedback structure including flux harmonics and cogging torque compensation is presented. To simulate the effect of the flux harmonics on a perfectly sinusoidal laboratory PMAC machine additional voltage ∆u q , based on mea-

Variable load and cogging torque genarated with torque controlled DC motor

∆ud Reference speed trajectorie

ωre f

idre f

Speed control

=0 id

tre f

ud

estimation

Rb uq

Nonlinear torque control

+

∆tc

ωm

id control &R

+

tL

+

tc (19)

Nominal electrical part

Nominal mechanical ωm part

te

(17),(18)

θm

bte

∆uq

iq ; id

Torque\

Flux harmonics model (24)

b q flux Ψ

estimation

(12),(13)

Experimental PMAC-DC Drive

iq

Cogging torque compensation (20)

Fig. 1. Overall feedback structure

cΨk sin(kθe )ωe + RL Ψk cos(kθe ))iq

diq +Ψk cos(kθe ) dt

Ψk cos(kθe )ωe ) ; k = 6; 12; : : : (24) The effect of the cogging torque was simulated with the torque controlled DC motor according to (19) where k = N s= p ν ; ν = 12; 24 was considered. Compared with the classical control structure additional blocks for torque estimation and cogging torque compensation are included in the feedback system. The torque controller itself was also modified in accordance to (17) and (18). Accurate measurements of the state variables (currents, rotor speed and position) are required in practical implementation. The proposed nonlinear torque control is estimator-based, so a good knowledge of the machine parameters is needed. Since motor parameters can considerably vary under a wide range of operating conditions (temperature, saturation, load variations), a sensitivity analysis must be included as a part of the design for every particular drive. The stator resistance and inductance are important parameters in the estimation of the motor flux; the non-linear torque controller based on feedback linearization technique is also sensitive to large parameter changes. It can be shown that a bounded variation of the motor inductance (decrease) influences the flux estimation during the transient state, and does not affect the steady-state performance. The torque ripple component is higher than in nominal conditions; but very large variations during saturation can even cause instability. Inductance variations as functions of motor currents L(i) should be included in the flux estimation in these cases. Resistance variations are more critical during operation at lower speed with heavy loads, because the corresponding voltage drop is dominant under these conditions. Increased motor resistance actually reduces the torque ripple, but on the other hand introduces steady-state error in the flux estimate. The voltage drops on transistors and on supply cables are also not negligible at lower speed when back-EMF is small. In our experiments it was therefore necessary to consider ohmic resistance

10

With compensation

5

Without comp.

0

Speed tracking −5

Load step change −10

0

2

4

6

Compensation torque (Nm)

((

0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

0

2

Time (s) 8

6

4

6

10

6 4 2 0 −2

4

Time (s)

Voltage in q−axis (V)

0

Omega (rad/s)

∆uq = ∑k ΨL

also as an adjustable parameter which was estimated on line. Voltage and current in d-axis were used for this estimation (appendix A). In Fig. 2 the cogging torque compensation during speed tracking is presented. The machine was loaded with 20% of the nominal torque. The oscillation in the mechanical speed

Current in q−axis (A)

sured variables iq ; ωe and θe , was simultaneously injected. Using dynamic inversion ∆u q was calculated as:

0

2

4

Time (s)

6

8 6 4 2 0 −2

0

2

Time [s]

Fig. 2. Cogging torque compensation during speed tracking.

is substantially reduced during acceleration and in steady state. With increased speed the compensation effect is more and more reduced, the oscillations are filtered out merely because of drive inertia and friction. In Fig. 3 and Fig. 4 experiments referring to the compensation of the ripple torque are presented. First the open-loop estimation of the total machine flux and its sixth and twelfth harmonic components are shown for steady state operation in Fig. 3. The nominal value of Ψ 0 according to machine b was data is 0:0957 Vs, while the mean estimated value of Ψ 0 0:1157 Vs; a similarly high accuracy was achieved also in the b and Ψ b . According to (24) the magnitude of Ψ estimated Ψ 12 6 6 was set to 0:004 Vs and Ψ 12 to 0:002 Vs. In Fig. 4 the results of the classical and proposed control scheme for the step change in the reference speed are presented. Parameters of PI speed con-

−3

0

4 0

−5

o −10

0

0.2

0.4

0.6

0.8

−5

1

0

0.2

Time (s)

0.4

0.6

0.8

Estimated torque (Nm)

Total flux (Vs)

Flux−12th harm.(Vs)

0.115

0 −1

0.11 0.105

−2

0.1

−3

0.095

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

a)

a)

5

8

Omega (rad/s)

6 0

−5

4 2 0

2

3

−2

4

0

1

2

3

4

Time (s)

Time (s) b)

b)

5

8

Omega (rad/s)

6 0

−5

4 2 0

0

1

2

2

2.2

2.4

2.6

2.8

3

Time (s) 2

10

−1

10

−2

10

0

10

−2

10

−4

0

10

20

30

10

0

Frequency (Hz)

5

10

15

Frequency (Hz)

Fig. 5. Discrete spectra of b te and ωm during steady-state operation; o-with compensation, *-without compensation

troller are the same in both cases. Significant compensation of the flux harmonics can be seen both from the estimated electrical torque as well as from the motor speed. In Fig. 5 the dis-

1

*

Time (s)

Fig. 3. Estimated total flux and components Ψ0 ; Ψ6 and Ψ12 .

0

o

0

−2

3

−3

Time (s)

Estmated Torque(Nm)

2.8

0

0.12

10

Estmated Torque(Nm)

2.6

1

−1

10

x 10

1

−10

2.4

2

Time (s)

2

−10

2.2

3

Time (s)

−3

3

2

1

*

Omega (rad/s)

0.05

5

Omega (rad/s)

0.1

5

x 10

Estimated torque (Nm)

5

Flux−6th harm.(Vs)

Nominal flux (Vs)

0.15

3

4

−2

0

Time (s)

1

2

3

4

Time (s)

Fig. 4. Speed control without a) and with compensation of flux harmonics b).

crete spectra of the estimated torque b te and measured ω m during steady-state operation are presented. DC components in torque and speed are not affected but, practically all other frequencies are substantially suppressed and none of them is amplified by the proposed control scheme. V. C ONCLUSION In the paper the techniques for reducing the pulsating torque in PMAC motor drives are presented. Ripple torque and cogging torque are considered. The adaptive estimation technique is used in the determination of the Fourier’s coefficients while the frequencies of the torque oscillations are assumed to be known. Lyapunov’s direct method is used to assure convergence be-

tween actual and estimated parameters. According to experimental results the proposed feedback technique enables substantial reduction of the pulsating torque and assures a smoother shape of the produced machine torque particularly, at lower speed. Practical implementation in demanding industrial drives requires some additional work concerning the overall feedback robustness due to parameter perturbations, measurement noise, delays and because of additional harmonics generated by the voltage inverter. With some minor modifications the proposed technique can also be used for other types of the machines. R EFERENCES [1] T. M. Jahns and W.L.Soong, “Pulsating Torque Minimization Techniques for Permanent Magnet AC Motor Drives-A Review,” IEEE Trans. on Industrial Electronics, vol.43, no. 2, pp.:321–329,1996. [2] J. Holtz and L. Springob, “Identification and Compensation of Torque Ripple in High Precision Permanent Magnet Motor Drive,” IEEE Trans. on Industrial Electronics, vol.43, no. 2, pp.:309–320,1996. [3] T. Sebastian and V. Gangla, “Analysis of Induced EMF Waveforms and Torque Ripple in a Brushless Permanent Magnet Machine,” IEEE Trans. on Industrial Applications, vol.32, no. 1, pp.:195–200,1996. [4] Chung, H. Kim, C. Kim, and M. Yung, “A New Instantaneous Torque Control of PM Synchronous Motor for High-performance Direct-Drive Applications,” IEEE Trans. on Power Electronics, vol.13, no. 3, pp.:388– 400,1998. [5] E. Favre, L. Cardoletti, and M. Jufer, “Permanent magnet Synchronous Motors: A Comprehensive Approach to Cogging Torque Suppression,” IEEE Trans. on Industrial Applications, vol.29, no. 6, pp.:1141–1148,1993. [6] T. Ishikawa and G. Slemon, “A Method of Reducing Ripple Torque in Permanent Magnet Motors without Skewing,” IEEE Trans. on Magnetics, vol.29, no. 2, pp.:2028–2031,1988. [7] K.B. Ariyur and M. Krstic, “Feedback Attenuation and Adaptive Cancellation of Blade Vortex on a Helicopter Blade Element,” IEEE Trans. on Control Systems Technology, vol.7, no. 5, pp.:596–605,1998. [8] A. Sacks, M. Bodson, and P. Khosla, “Experimental Results of Adaptive Periodic Disturbance Cancellation in a High Performance Magnetic Disk Drive,” Journal of Dynamic Systems, Measurement, and Control, vol.118, pp.:416–424,1996. ˇ [9] B. Grˇcar, P. Cafuta, M. Znidariˇ c, and F. Gausch, “Nonlinear Control of Synchronous Servo-Drive,” IEEE Trans. on Control Systems Technology, vol.4, no. 2, pp.:177–184,1996. ´ c, and I. Agirman, “On Torque Rip[10] A.M. Stankovi´c, G. Tadmor, Z.J. Cori´ ple Reduction in Current-Fed Switched Reluctance Motors,” IEEE Trans. on Industrial Electronics, vol.46, no. 1, pp.:177–183,1999.

A PPENDIX A-E STIMATORS :

must be satisfied. From equation (33) it follows immediately:

 Flux estimator; current model: db i = Abi ωe Fb i ωe L dt

1b

Ψ + L u + Kei 1

b = [Ψ b ;Ψ b q] . where bi = [bid ; biq ] and Ψ d A; F; L 1 and K are defined as:

(25)

Introduced matrices

A = diag( LR ; LR ) F = antidiag( 1; 1) T L 1 = diag( L1 ; L1 ) K = diag(kd ; kq )

d pbT dt

1

ωm wT

(34)

 Estimator for the ohmic resistance R; db dt id =

Rb b L id + ∆ud

d b e dt R = kR id

(26)

=γ

; R(0) = Rnominal ; eid

= id

A PPENDIX B:

The equation (25) is rewriten: de i = (A dt

e + Bu + Kei = Aei + DΨ e + Bu + Kei (27) ωeF )ei + DΨ

The equation of the error vctor dynamics is obtained as:



d dt i = Ai + DΨ + Bu db b b e dt i = Ai + DΨ + Bu + K i de e e e e e dt i = Ai + DΨ K i = Ae i + DΨ

)

(28)

e T Ψ, e Introducing a Lyapunov function cadidate V = eiT Pei + δ Ψ T where P is a symetric positive solution of A e P + PAe = Q with Q > 0 and δ > 0, the time derivative of V is given as: d deiT dt V = dt

eT

e

e

di Ψ e e T dΨ Pei + eiT P dt + δ dt Ψ +δΨ dt

e + 2δ Ψ e T ( dΨ e T DT Pei + eiT PDΨ =e iT (ATe P + PAe )ei + Ψ dt =

e dΨ dt )

e T DT Pei + 2δ Ψ e T ( dΨ eiT Qei + 2Ψ dt

Introducing the assumption

dΨ dt

(29)

 0 it follows:

e e T (DT Pei δ d Ψ ) eiT Pei + 2Ψ | {z dt }

d dt V =

=

d e 1 DT Pe i dt Ψ = δ

0

e dΨ dt )

(30)

)

The estimation of the Fourier’s coefficients ((12)) is derived with the similar precedure.  Cogging torque; mechanical model: J

d ωm = Bω + wT pe dt

(31)

where wT = [cos(θm ); sin(θm ); : : : ; cos(kθm ); sin(kθm )] and pe = p pb. For the Lyapunov function candidate V = 12 J ωm2 + 1 eT pe the time derivative of V is given as: 2γ p d dt V = J ωm (

=

B wT J ωm + J

d pb dt )

pe) + γ pet ( ddtp

Bωm2 + ωm wT pe + γ pe ddtp

γ ddtpb pe T

(32)

After introducing the assumption ddtp  0 in upper equation and considering that Bω m2  0 the condition: T (ωm w

γ

d pb )p e 0 dt

(33)

TABLE I N OMINAL MOTOR DATA Stator resistance Inductance PM flux magnitude Drive inertia Nominal electrical torque Number of pole pairs Number of stator slots

R = 0:34Ω L = 0:002H Ψ0 = 0:0957 Vs J = 0:0016 Kgm2 te = 9:5 Nm p=4 Ns = 24

bi

d

(35)