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Control Method for IPMSM Based on PTC and PWM Hold Modelin Overmodulation Range
-Study on Robustness and Comparison with Anti-Windup Control-
Takayuki MiyajimaStudent Member, IEEE
Yokohama National University79-5 Tokiwadai, Hodogaya-kuYokohama, 240-8501 Japan
Hiroshi FujimotoMember, IEEE
The University of Tokyo5-1-5 Kashiwanoha,
Kashiwa, Chiba, 277-0804 [email protected]
Masami FujitsunaNon Member
DENSO CORPORATION1-1 Syouwacho,
Kariya, Aichi, 448-8661 JapanMASAMI [email protected]
Abstract— Interior permanent magnet synchronous motors(IPMSMs) are employed for drive motors of electric and hybridvehicles. Therefore, the control method in overmodulationrange of inverter is important to achieve fast torque responseand expand the operating range of IPMSMs. The authorsproposed the control method over the full operating rangebased on perfect tracking control (PTC) with PWM hold model.This method takes into consideration the voltage limit ellipseof IPMSM by feedforward (FF) controller. However, whenmodeling error exists, FF controller limits the operating rangeor cannot prevent the windup phenomenon. In this paper,a method which can improve the robustness is developed.Proposed method considers the voltage limit ellipse by FFcontroller. But, in general, the voltage limit ellipse is takeninto consideration by anti-windup control. Therefore, in thispaper, proposed method is compared with a control methodusing anti-windup control.
Index Terms—IPMSM, overmodulation range, PWM holdmodel, perfect tracking control
I. INTRODUCTION
Permanent magnet synchronous motors (PMSMs) arewidely employed in many industries. Especially, IPMSMsare used as drive motors of electric and hybrid vehicles. So,it is necessary that IPMSM drive systems achieve fast torqueresponse and wide operating range.
For realization of these requests, high output voltage ofinverter is necessary. Therefore, the control in the over-modulation range of inverter is important in addition to thelinear range. One of the control methods of IPMSMs in theovermodulation range is presented in [1] which is composedof current vector control and voltage phase control. How-ever, the mode of the controller should be switched fromcurrent vector control to voltage phase control. This switchdeteriorates torque response, so it is desirable to control theIPMSMs using only one mode of controller over the fulloperating range.
When output voltage is saturated, flux-weakening andanti-windup controls are applied. In [2], a method whichachieves flux-weakening and anti-windup controls simul-taneously is proposed with feedback (FB) controllers andreference tables. In general, a control system consists of only
FB controller like [2]. However, FF controller is necessaryfor fast torque response. [3] proposed a control system usingmodel predictive control (MPC). Although MPC controllerhas no integrator, MPC needs high computational cost andonly one voltage vector is output during one control period.Moreover, harmonic current increases in the overmodulationrange. [4] and [5] proposed methods based on estimatingharmonic current to suppress the voltage saturation whichis caused by harmonic current. However, the discrete-timemodel was not discussed.
In [6], the authors proposed a control method over the fulloperating range based on perfect tracking control (PTC) [7]with PWM hold model [8]. This method takes into consider-ation the voltage limit ellipse of IPMSM by FF controller, soit can simultaneously compensate the windup phenomenonand harmonic current. However, when modeling error exists,FF controller limits the operating range or cannot compensatethe windup phenomenon.
In this paper, a method which can improve the robust-ness is developed. The robustness is verified when there islarge inductance error or parameter estimation scheme [9]is applied to suppress inductance error. Proposed methodconsiders the voltage limit ellipse by FF controller. But, ingeneral, the windup phenomenon is compensated by anti-windup control. Therefore, proposed method is comparedwith a control method using anti-windup control in thispaper.
II. MODEL AND DISCRETIZATION
A. dq Model of IPMSM
A continuous-time state equation of a plant is given by
x(t) = Ac(ωe)x(t)+Bc
{u(t)− [ 0 ωeKe
]T}, (1)
y(t) = Ccx(t), (2)
[Ac(ωe) Bc
Cc 0
]:=
⎡⎢⎣ − R
Ldωe
Lq
Ld
−ωeLd
Lq− R
Lq
1Ld
0
0 1Lq
I 0
⎤⎥⎦ , (3)
where u = [vd vq]T , vd, q are d-axis and q-axis voltage,
R is stator winding resistance, Ld, q are d-axis and q-axis
irefd [k]
irefq [k]
vrefd [k]
vrefq [k]
vrefu [k] iu[k]
iw[k]θe[k]
uw
dq iq[k]
id[k]vrefv [k]
vrefw [k]uvw
dqCd[z]
Cq[z]+
+
−
−
+
+
+
+
Decoupling Control
IPMSM+
INV
Fig. 1. Block diagram of current vector control.
inductance (Ld �= Lq), ωe is electric angular velocity, y =x = [id iq]
T , id, q are d-axis and q-axis current, and Ke isback EMF constant.
The motor torque T is given by
T = Kmtiq +Krtidiq, (4)
where Kmt = PKe, Krt = P (Ld − Lq), and P is numberof pole pairs. In this paper, 2-phase/3-phase transform isrepresented as absolute transformation.
B. PWM hold model of IPMSM
(1) is discretized based on PWM hold [8]. Here, it isassumed that the speed variation during one control periodcan be neglected. ωeKe can be considered to be a pulse ofmagnitude ωeKe and width Tu. Therefore, the PWM holdmodel of IPMSM is designed as follows:
x[k + 1] = As(ωe)x[k]
+Bs(ωe)
{ΔT [k]−
[0 ωeKeTu
Vdc
]T}, (5)
y[k] = Csx[k] = Ccx[k], (6)
As(ωe) = eAc(ωe)Tu , Bs(ωe) = eAc(ωe)Tu2 BcVdc, (7)
where Vdc is dc-bus voltage of three-phase inverter, ΔT =[ΔTd ΔTq]
T , and ΔTd, q are d-axis and q-axis ON time.As(ωe) and Bs(ωe) are functions of ωe. Therefore, thesefunctions are calculated again when speed is changed.
III. CONTROL SYSTEM DESIGN
A. Conventional Method
The block diagram of current vector control is shown inFig. 1. The decoupling control which is shown by (8) and(9) is applied to the plant (1) as follows:
vrefd [k] = v′dref
[k]− ωe[k]Lqiq[k], (8)
vrefq [k] = v′qref
[k] + ωe[k](Ldid[k] +Ke). (9)
In addition, the d-axis and q-axis current controllers Cd, q(s)are designed as
Cd, q(s) =Ld, qs+R
τs, (10)
where τ = 10Tu. By discretizing (10), Cd, q[z] are obtainedby Tustin transformation with control period Tu.
In the rectangular wave operation, the fundamental com-ponent of output voltage is constant. Therefore, the motortorque T is controlled by the voltage phase δ. The blockdiagram of voltage phase control [1] is shown in Fig. 2. ThePI controller Cδ[z] calculates δ which reduces torque error.
T ref [k]
iu[k]iw[k]θe[k]
IPMSM+
INV
vrefu [k]
vrefv [k]
vrefw [k]Calculator
Voltageδ[k]
uw
dq
Torque
Estimator iq[k]
id[k]T [k]
Cδ[z]+−
Fig. 2. Block diagram of voltage phase control.
C2[z]
+
++
u0[k]
y0[k] y[k]
u[k]
e[k]S
(Tu)−
P c(s)y(t)
B−1(I − z−1A)S
D
z−1C
r(t)
r[k] = xd[k + 1]
C1[z]
+
+
(Tu)
PLANT
Fig. 3. Singlerate perfect tracking control system of multi-input multi-output system.
Next, the rectangular wave voltage is generated based onδ. However, the relationship between T and δ is nonlinear,so Cδ[z] cannot be designed by model-based design. In thispaper, the proportional gain and the integral gain of Cδ(s)were set to be 0.15 and 30 by try and error. By discretizingCδ(s), Cδ[z] is obtained by Tustin transformation with Tu.
When the absolute value of dq axis voltage is√
32Vdc
2 ×1.15 or more, the mode of controller is changed from thecurrent vector control to the voltage phase control. Here,coefficient
√3/2 is the coefficient to transform two-phase
into three-phase. Then, the initial state variable of Cδ iscompensated with the voltage phase at this point.
On the other hand, when the operating range is assumedthat the absolute value of dq axis voltage in the case of
current vector control is√
32Vdc
2 or less, the mode is changedto the current vector control.
B. Perfect Tracking Control
Perfect tracking control (PTC) system consists of the 2-DOF control system as show in Fig. 3. The feedforwardcontroller C1[z] is a stable inverse system of the plant asthe reference is state variable xd[k + 1]. Therefore, theperfect tracking is assured at the sample time. Moreover, thefeedback controller C2[z] suppresses the error between theoutput y[k] and the nominal output y0[k] to assure robustnessonly when disturbances or plant variations exist.
Here, a discrete-time state equation is given by
x[k + 1] = Ax[k] +Bu[k], y[k] = Cx[k]. (11)
Therefore, a stable inverse system of the plant is obtained as(12) and the nominal output is obtained as (13).
u0[k] = B−1(I − z−1A)xd[k + 1] (12)
y0[k] = z−1Cxd[k + 1] (13)
z−1
iref [k] =
C1[z]
C2[z]
+
+
+
i[k]e[k]−
ΔT ff [k]
i∗0[k]
ΔT [k]
i0[k]
xd[k + 1]
+
+
uw
dq S(Tu)
IPMSM
INV+
(SVM)HPWM
θe + Δθe
θe
NOE
Tu
VdcI ΔT ′[k]
B−1(ωe)(I − z−1A(ωe))
[0 ωeKeTu/Vdc]T
Anti windupcontrol
-
(Proposed 1)
Fig. 4. Block diagram of proposed method 1.
ΔT [k]
+−
ΔTh[k] i∗0[k]
ΔTo[k]
+
+
= ΔTff [k]
PWMHoldModel
HPFΔT ′[k]
ΔT [k]
Fig. 5. NOE of proposed method 1.
C. Proposed Method 1
Proposed method 1 is shown in Fig. 4. The PTC systemis composed of the inverse system of (5). The saturationof fundamental component voltage causes the windup phe-nomenon. The harmonic current which is generated in theovermodulation range deteriorates the torque response. Thesaturation of fundamental component voltage is compensatedby anti-windup control and harmonic current is compensatedby estimating the nominal output considering harmonic cur-rent.
In this paper, error correction anti-windup control [10] isapplied. When state variables are updated, error correctionanti-windup control is calculated as follows:
xfbd[k + 1] = xfbd[k] +Bfbdeid[k]
− BfbdVdc
DfbdTu(ΔTd[k]−ΔTd[k]), (14)
xfbq[k + 1] = xfbq[k] +Bfbqeiq[k]
− BfbqVdc
DfbqTu(ΔTq[k]−ΔTq[k]), (15)
where Bfbd, fbq and Dfbd, fbq are B and D matricies of d-axis and q-axis current FB controllers. xfbd, fbq are statevariables of FB controllers. eid, iq are d-axis and q-axiscurrent error. ΔTd, q are limited value of ΔT . ΔT =[ΔTd ΔTq] is calculated as follows:
ΔT =
{ΔT|ΔT |
√322πTu if |ΔT | >
√322πTu
ΔT otherwise(16)
The other limiters of Fig. 4 calculate in the same way as(16).
The nominal output estimator (NOE) of proposed method1 is shown in Fig. 5. The NOE estimates i∗0 = [i∗d0 i∗q0]
T
which is the nominal output considering harmonic current.The overmodulation is realized by containing harmonic com-ponent ΔT h into the reference. Therefore, ΔT h is calculatedas the difference between ΔT ′ which is the reference beforethe overmodulation and ΔT which is the reference after the
z−1
iref [k] =
C1[z]
C2[z]
+
+
+
i[k]e[k]−
ΔT ff [k]
i∗0[k]
ΔT [k]
i0[k]
xd[k + 1]
+
+
uw
dq S(Tu)
IPMSM
INV+
(SVM)HPWM
θe + Δθe
θe
Tu
VdcI ΔT ′[k]
B−1(ωe)(I − z−1A(ωe))
[0 ωeKeTu/Vdc]T
NOE(Proposed 2)
Fig. 6. Block diagram of proposed method 2.
ΔT [k]
+−
ΔTh[k] i∗0[k]
ΔTo[k]
+
+
= ΔTff [k]
PWMHoldModel
HPFΔT ′[k]
ΔT [k] ΔTo[k]
Fig. 7. NOE of proposed method 2.
overmodulation. Here, the overmodulation is achieved as thefundamental component of ΔT ′ is equal to ΔT .
If ΔT is oscillated by estimation error of harmonic com-ponent, the overmodulation cannot be achieved completely.In this case, ΔT h has dc component and i
∗0 are offset.
Therefore, real current i cannot track target trajectory iref .Therefore, dc component of ΔT h is cut by high pass filter(HPF). In this paper, the cutoff frequency of HPF is 60[rad/s]. In addition, state variables of PWM hold modelare changed into target trajectory to compensate attenuationof harmonic current with FB controller when the operatingrange shifts from the overmodulation range to the linearrange.
Here, Δθe(= 0.5ωeTu) is added to θe of the space vectormodulation for sampling error compensation [11]. Δθe is apole position variation between the sampling point and thecenter of the PWM pattern output point.
D. Proposed Method 2
Fig. 6 shows the block diagram of proposed method 2. Thenominal output estimator of proposed method 2 is shown inFig. 7.
Anti-windup control deteriorates torque response becauseanti-windup control operates state variable of FB controller.Therefore, anti-windup control is not applied in proposedmethod 2. In proposed method 2, the nominal output es-timator takes into consideration the voltage limit ellipseof IPMSM by the limiter of ΔT o. Therefore, IPMSM iscontrolled inside the voltage limit ellipse by the PTC systemand windup phenomenon is not caused.
E. Proposed Method 3
In proposed method 2, if error of voltage limit ellipseexists, the PTC system limits the operating range or cannotcompensate windup phenomenon. However, operating pointcontinually is changed in the drive systems of electric andhybrid vehicles, so it is difficult to correct all parameter ofIPMSM. Therefore, in this paper, the dc-bus voltage of PTCsystem is focused to correct the error of voltage limit ellipse.
B−1(ωe)(I − z−1A(ωe))
z−1
iref [k] =
C1[z]
C2[z]
+
++
i[k]e[k]−
ΔT ff [k]
Tu
VdcI
i∗0[k]
ΔT [k]
i0[k]
[0 ωeKeTu/Vdc]T
xd[k + 1]
+
+
uw
dq S(Tu)
IPMSM
INV+
(SVM)HPWM
θe + Δθe
θe
ΔT ′[k]
+
−CΔV dc[z]
ΔVdc[k]
ABS
Vdc
Tu
ABS
NOE(Proposed 2)
Fig. 8. Block diagram of proposed method 3.
The nominal voltage limit ellipse and actual voltage ampli-tude which is required to track the nominal output on steadystate are represented by
V 2an = (Riqn + ωeKe + ωeLdnidn)
2
+ (Ridn − ωeLqniqn)2 ≤
(√3
2
2
πVdcn
)2
, (17)
V 2a = (Riqn + ωeKe + ωeLdidn)
2+ (Ridn − ωeLqiqn)
2,
(18)
where Van is nominal voltage amplitude, Va is actual voltageamplitude, Vdcn is nominal dc-bus voltage which is equal toVdc in this equation, idn, qn are fundamental component of d-axis and q-axis nominal output, and Ldn, qn are d-axis and q-
axis nominal inductance. If V 2an =
(√322πVdcn
)2and V 2
an >
V 2a , the operating range is limited. On the other hand, if
V 2a >
(√322πVdc
)2and V 2
an < V 2a , windup phenomenon is
caused. Here, the nominal voltage limit ellipse is corrected by
changing Vdcn and V 2a can be limited within
(√322πVdc
)2.
Proposed method 3 is shown in Fig. 8. CΔVdc[z] calculates
ΔVdc from the error between |ΔT ff | and |ΔT |. Vdcn isupdated as Vdcn = Vdc + ΔVdc, so the nominal voltagelimit ellipse is corrected. In this paper, the proportional gainand the integral gain of CΔV dc(s) were set to be 0.001 and2000 by try and error. By discretizing CΔV dc(s), CΔV dc[z]is obtained by Tustin transformation with Tu.
F. Parameter estimation[9]
In this paper, parameter estimation scheme is based onrecursive least-squares. The algorithm is represented as
θ[k] = θ[k − 1]
+ Projˆθ
{K[k](η[k]−ϕ[k]T θ[k − 1])
}, (19)
K[k] =P [k − 1]ϕ[k]
λ+ϕT [k]P [k − 1]ϕ[k], (20)
P [k] = (I −K[k]ϕ[k]T )P [k − 1]/λ, (21)
Projθ(·j) :=⎧⎨⎩
0, if θj ≥ θmax&·j > 0
0, if θj ≤ θmin&·j < 0·j , otherwise
(22)
where λ is the forgetting factor. When persistent excitationof regressor ϕ is not satisfied, θ and P are not update.
TABLE I
PARAMETERS OF IPMSM.
d-axis Inductance Ld 0.664 [mH]q-axis Inductance Lq 2.08 [mH]
Resistance R 138.2 [mΩ]Pairs of poles P 3
Induced voltage constant Ke 43.23 [mV/(rad/s)]
In this paper, θq = Ld and θd = Lq are estimated byvoltage equation on steady state as follows:
ϕd = F (s)(vd −Rid), ηd = F (s)(−ωeiq), (23)
ϕq = F (s)(vq −Riq − ωeKe), ηq = F (s)ωeid. (24)
F (s) = ωc/(s+ ωc) is discretized by Tustin transformationwith Tu. λ and ωc were set to be 0.99 and 100 [rad/s].
IV. SIMULATION
The parameters of IPMSM for the simulation are shown inTable I. The dc-bus voltage of the three-phase inverter Vdc is36 [V]. The control period Tu is 0.1 [ms]. In all simulations,the motor speed is 2000 [rpm].
Simulation results in the case of Ldn, qn = 0.7Ld, q areshown in Fig. 9. The nominal d-axis and q-axis inductanceLdn, qn of FF controller is Ldn, qn = 0.7Ld, q until timet = 0.25 [s]. The parameter estimation scheme estimates d-axis and q-axis inductance Ld, q until t = 0.18 [s] (estimationinterval). After t = 0.25 [s], the FF controller is composedwith Ld, q . Fig. 9(f), 9(g), and 9(h) are close-up view oftransient response when the FF controller is composed withestimated value.
The “flag” in the conventional method is a signal ofswitching controllers. When the flag is “high”, the voltagephase control is applied. On the other hand, the current vectorcontrol is applied when the flag is “low”. T , T0, and T ∗
0 arethe torque estimated from i, i0, and i∗0 by (2) respectively.The overmodulation range of proposed method is representedas the range between two dotted lines like Fig. 9(i) (From0.0707 [ms] to 0.0780 [ms]).
The conventional method cannot achieve fast torque re-sponse because it is composed with only FB controllers.In the estimation interval, windup phenomenon happens inproposed method 2. Moreover, in proposed method 1, theanti-windup control deteriorates torque response. On theother hand, in proposed method 3, windup phenomenonis suppressed by changing Vdcn. When the FF controlleris recomposed with Ld, q , anti-windup control of proposedmethod 1 deteriorates torque response. In proposed method3, state variable of FB controller is not operated, so fasttorque response is achieved.
Simulation results in the case of Ldn, qn = 1.3Ld, q areshown in Fig. 10.
In the estimation interval, The PTC system of proposedmethod 2 excessively limits the operating range, so it cannottrack the reference. On the other hand, in proposed method3, the voltage limit ellipse is corrected by ΔVdc and it canachieve the tracking.
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
Tref
flag
(a) Torque (Conventional)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(b) Torque (Proposed 1)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(c) Torque (Proposed 2)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(d) Torque (Proposed 3)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
Time [s]
Tor
que
[Nm
]
T^
Tref
(e) Torque (Conventional)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
Time [s]
Tor
que
[Nm
]
T^
T0
(f) Torque (Proposed 1)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
Time [s]
Tor
que
[Nm
]
T^
T0
(g) Torque (Proposed 2)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
Time [s]
Tor
que
[Nm
]
T^
T0
(h) Torque (Proposed 3)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(i) |ΔT | (Proposed 1)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(j) |ΔT | (Proposed 2)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(k) |ΔT | (Proposed 3)
Fig. 9. Ldn = 0.7Ld, Lqn = 0.7Lq (Simulations).
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(a) Torque (Proposed 1)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(b) Torque (Proposed 2)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(c) Torque (Proposed 3)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(d) |ΔT | (Proposed 1)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(e) |ΔT | (Proposed 2)
0 0.1 0.2 0.3 0.4 0.50.04
0.06
0.08
0.1
0.12
Time [s]
Inpu
t [m
s]
(f) |ΔT | (Proposed 3)
Fig. 10. Ldn = 1.3Ld, Lqn = 1.3Lq (Simulations).
V. EXPERIMENT
The encoder resolution is 2000 [pulse/r]. In order to reduceoscillation of input which is caused by ωe, Synchronous-measurement method [12] is applied. Δθ in experiments is0.9ωeTu to consider also the computational time 0.4Tu.
Experimental results in the case of Ldn, qn = 0.7Ld, q isshown in Fig. 11.
The conventional method cannot achieve fast torque re-sponse. In proposed method 1, the anti-windup control dete-riorates torque response. In the estimation interval, windup
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
Tref
flag
(a) Torque (Conventional)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(b) Torque (Proposed 1)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(c) Torque (Proposed 2)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(d) Torque (Proposed 3)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
Tref
(e) Torque (Conventional)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(f) Torque (Proposed 1)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(g) Torque (Proposed 2)
0.31 0.32 0.33 0.34 0.35 0.36
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(h) Torque (Proposed 3)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(i) |ΔT | (Proposed 1)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(j) |ΔT | (Proposed 2)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(k) |ΔT | (Proposed 3)
Fig. 11. Ldn, qn = 0.7Ld, q (Experiments).
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(a) Torque (Proposed 1)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(b) Torque (Proposed 2)
0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
5
Time [s]
Tor
que
[Nm
]
T^
T0
(c) Torque (Proposed 3)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(d) |ΔT | (Proposed 1)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(e) |ΔT | (Proposed 2)
0 0.1 0.2 0.3 0.4 0.50.02
0.04
0.06
0.08
0.1
0.12
0.14
Time [s]
Inpu
t [m
s]
(f) |ΔT | (Proposed 3)
Fig. 12. Ldn, qn = 1.3Ld, q (Experiments).
phenomenon happens in proposed method 2. Moreover, pro-posed method 2 cannot prevent windup phenomenon becauseminute modeling error exists. On the other hand, in proposedmethod 3, windup phenomenon is suppressed by changingVdcn and fast torque response is achieved.
Experimental results in the case of Ldn, qn = 1.3Ld, q areshown in Fig. 12.
In the estimation interval, proposed method 2 cannot trackthe reference because of excessive voltage limit. On the otherhand, in proposed method 3, tracking is achieved by ΔVdc.
VI. CONCLUSION
In this paper, the method which improves robustness isdeveloped. Simulations and experiments are performed toshow the robustness of proposed method 3. In addition,proposed method 1 which uses anti-windup control cannotachieve fast torque response. Proposed method 3 whichcompensates windup phenomenon by considering the voltagelimit ellipse achieves fast torque response.
The target trajectory was not changed during the saturationof input amplitude. Therefore, in proposed method 3, thetorque response was oscillated by free response of theplant. The future work is improvement of torque responsewith voltage phase operation during the saturation of inputamplitude.
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