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Feedforward control for SPMSM with Final-State ControlBased on Voltage Limit Circle with Transient Term
Takayuki MiyajimaThe University of TokyoKashiwa, Chiba, Japan
Hiroshi FujimotoThe University of TokyoKashiwa, Chiba, Japan
Masami FujitsunaDENSO CORPORATION
Kariya, Aichi, JapanMASAMI [email protected]
Abstract— SPMSMs (Surface Permanent Magnet Syn-chronous Motors) are employed for many industrial appli-cations. SPMSM drive systems should achieve quick torqueresponse and wide operating range. For quick current response,authors proposed final-state control considering voltage limit.Final-state control drives an initial state to a final state withfeedforward input during finite time. The proposed methodachieved quick current control in field-weakening region but in-verse response of q-axis current was caused. In this paper, a newfeedforward control method which improves current responseis proposed. Simulations and experiments are performed tocompare two types of feedforward control method. In addition,current response under voltage limit is analyzed by the voltagelimit circle of SPMSM with transient term.
I. INTRODUCTION
Permanent magnet synchronous motors (PMSMs) arewidely employed in many industries because of high effi-ciency, high power density. Surface PMSMs (SPMSMs) areused for machine tool and electric power steering (EPS).In these applications, SPMSM drive systems are demandedhigh-speed operation and quick torque response but outputvoltage is limited. Especially, in the EPS system, the dc volt-age supply is battery. Therefore, to achieve high performanceis difficult due to low output voltage. In addition, from viewpoint of torque ripple, the overmodulation range of inverteris not unavailable.
In the SPMSM drive systems, field-weakening control isapplied to expand the operating range. However, in field-weakening region, voltage amplitude is reached the voltagelimit. Therefore, the manipulated variable is voltage phaseonly. General methods determine d-axis and q-axis voltageindependently but it cannot operate voltage phase directly.Thus, quick current response cannot be achieved and controlmethod which can operate voltage phase directly is required.
Many methods for qucik torque response under the voltagelimit had been studied. Methods which operate voltage phasewith the voltage limiter [1][2], control torque and modulationindex by torque loop [3][4], and operate voltage phasedirectly by feedback (FB) controller [5][6] were proposed.However, for quicker response, feedforward (FF) controllershould be used.
For the quickest current response under voltage limit,authors proposed a final-state control [7] for SPMSM con-sidering voltage limit [8]. Conventional final-state controlsfeatured single-input single-output system. On the other
hand, this proposed method controls SPMSM of two-inputtwo-output system. SPMSM control under voltage limit canbe described as the programming problem whose evolutionfunction and constraint function are quadratic function andquadratic inequality in form of linear matrix inequality(LMI), respectively. The quickest current response was re-alized by FF input (FSC input) which is obtained by solvingthese LMIs. In this paper, for improving current response, anew FSC input is proposed. In addition, this paper analyzesFSC input by voltage limit circle with transient term.
II. MODEL AND DISCRETIZATION
A. dq Model of SPMSM
The voltage equation of SPMSM in dq-axis is representedin the form of state equation as y = x = [id iq]
T andu = [vd vq]
T 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
L ωe
−ωe −RL
1L 00 1
L
I 0
⎤⎦ , (3)
where vd, q are the d-axis and q-axis voltage, R is the statorwinding resistance, L is the inductance, ωe is the electricangular velocity, id, q are the d-axis and q-axis current, andKe is the back EMF constant.
The motor torque T is given by
T = Kmtiq, (4)
where Kmt := PKe and P is the number of pole pairs. Inthis paper, 2-phase/3-phase transform is absolute transforma-tion.
B. Discretization Based on PWM Hold
In order to discretize a plant, a zero-order hold is generallyapplied. However, in the case of single-phase inverter, itcannot output arbitrary voltage but only 0 or ±E (E is thedc-bus voltage of single-phase inverter). Therefore, in orderto control instantaneous values precisely, the zero-order holdis not suitable. When the pulse is allocated in the center ofcontrol period Tu, the plant in inverter drive system can bediscretized as follows [11].
i∗d[k]
i∗q[k]
ΔTd[k]
ΔTq[k]
iu[k]
iw[k]
θe[k]
uw
dq iq[k]
id[k]
SVMC[z]
C[z]
−
Decoupling Control
SPMSM+
INVT ∗[k]
Tu
Vdc
Tu
Vdc
CurrentReferenceGenerator
−
+
++
+++
θe[k]+Δθe[k]
Fig. 1. Conventional method
d−axis
q−axis
T ∗
Kmt
iq max
iq min
voltage limit circle
Fig. 2. Intersection point of voltage limitcircle with constant torque line
A continuous-time state equation of a plant is given by
x(t) = Acx(t) +Bcu(t), y(t) = Ccx(t). (5)
The precise discrete model in which the input is the ON timeΔT [k] is obtained as
x[k + 1] = Asx[k] +BsΔT [k], y[k] = Csx[k], (6)
As := eAcTu , Bs := eAcTu/2BcE, Cs := Cc. (7)
In derivation of (6) and (7), the voltage amplitude is consid-ered +E only. Therefore, if ΔT [k] is negative, the voltageamplitude is −E.
C. PWM Hold Model of SPMSM
(1) is discretized based on PWM hold. Here, it is assumedthat the speed variation during one control period can beneglected. Hence, the back EMF ωeKe can be presumedas the zero-order hold. Therefore, the PWM hold model ofSPMSM is designed as E = Vdc in (5) as follows:
x[k + 1] = As(ωe)x[k] +Bs(ωe)ΔT [k]
−Bs2(ωe)[0 ωeKe
]T, (8)
y[k] = Csx[k], (9)
As(ωe) := eAc(ωe)Tu , Bs(ωe) := eAc(ωe)Tu2 BcVdc,
Bs2(ωe) := A−1c (ωe)
(eAc(ωe)Tu − I
)Bc, (10)
where Vdc is the dc-bus voltage of three-phase inverter,ΔT = [ΔTd ΔTq]
T , and ΔTd, q are the d-axis and q-axisON time. As(ωe), Bs(ωe), and Bs2(ωe) are functions ofωe. Thus, these functions should be calculated again whenωe is changed.
Here, input of three-phase can be generated based on spacevector modulation (SVM) [12].
III. CONTROL SYSTEM DESIGN
A. Conventional method
Fig. 1 shows the block diagram of the conventionalmethod. The decoupling control which is given by (11), (12)is applied.
vrefd [k] = v′dref
[k]− ωe[k]Liq[k] (11)
vrefq [k] = v′qref
[k] + ωe[k](Lid[k] +Ke) (12)
The current PI controller is designed with pole-zero cancel-lation as follows:
C(s) =Ls+R
τs, τ = 10Tu (13)
By discretizing (13), C[z] is obtained by Tustin transforma-
tion with control period Tu. Here, if Va :=√
v2d + v2q >
Vmax (Va is the voltage amplitude and Vmax :=√
32Vdc√
3is
the maximum value of voltage amplitude.), the integrator ofC[z] is stopped.
The current reference generator makes d-axis and q-axiscurrent references i∗d and i∗q from torque reference T ∗. Itselects the intersection point of the voltage limit circle withthe constant torque line as Fig. 2. The generation of currentreferences is represented by
i∗d[k] =−ω2
eKeL
R2 + ω2eL
2+√
(iqmax − i∗q)(i∗q−iqmin), (14)
i∗q [k] =
⎧⎨⎩
iqmax if T ∗[k] > Kmtiqmax
iqmin elseif T ∗[k] < KmtiqminT∗[k]Kmt
otherwise, (15)
iqmax := −ωeKeR− Va
√R2 + ω2
eL2
R2 + ω2eL
2, (16)
iqmin := −ωeKeR+ Va
√R2 + ω2
eL2
R2 + ω2eL
2. (17)
If d-axis current reference is selected to strengthen field flux,i∗d[k] = 0.
Δθe(= 0.5ωeTu) is added to θe in the dq/2-phase trans-form for sampling error compensation [13]. The ON timelimit in SVM is described by
ΔT [k] =
{ΔT [k]
|ΔT [k]|ΔTmax if |ΔT [k]| > ΔTmax
ΔT [k] otherwise, (18)
where ΔTmax
(:=
√32Tu√3
)is the maximum value of dq-axis
ON time vector amplitude and ΔT [k] is the limiter output.In this paper, all limiters are calculated in the same way as(18).
B. Proposed method
Fig. 3 and 4 show the block diagrams of the proposedmethod and FF controller C1[z], respectively. The proposedmethod consists of the 2-DOF control system and has twotypes of FF controllers.
i∗[k]
C2[z]
+
+
+
i[k]e[k]−
ΔT ff [k]
x[k]
uw
dq S(Tu)
(SVM)HPWM
θe[k]Tu
VdcI +Δθe[k]
θe[k]Pn[z]
CurrentReferenceGenerator
T ∗[k]
ΔT ff [k]
SPMSM
INV+
ΔT [k]C1[z]
Fig. 3. Proposed method
i∗[k]
FSC Input Generator
ΔT ff [k]
C1[z]
−B−1s (ωe)As(ωe)x[k] + B−1
s (ωe)i∗[k]
Inverse system
+B−1s (ωe)Bs2(ωe) [0 ωeKe]
T
Fig. 4. Feedforward controller of proposed method
The inverse system of (8) is derived as follows:
ΔT ff [k] =−B−1s (ωe)As(ωe)x[k]+B−1
s (ωe)xd[k + 1]
+B−1s (ωe)Bs2(ωe)
[0 ωeKe
]T(19)
The inverse system used as a feedforward controller assuresperfect tracking if the plant is nominal and the feedforwordinput is not limited. x[k] := [id[k] iq[k]]
T is the nominalouput which takes the input limit into consideration.
If the feedforward input from the inverse system is not lim-ited, the inverse system is used as a feedforward controller.On the other hand, if the feedforward input from the inversesystem is limited, the feedforward input is recalculated withFSC input generator to achieve quick current response.Here, the proposed method switches the type of feedforwardcontroller but the inverse system is equal to final-state controlwhose prescribe time interval is one. Therefore, the switchdoes not make unstable.
The FB controller C2[z] suppresses the current error e[k]between x[k] and x[k]. Here, C2[z] is the same as the FBcontroller of conventional method. However, in the proposedmethod, anti-windup control is not necessary because x[k]considers the input limit.
1) Proposed method 1[8]: Final-state control drives aninitial state to a final state by prescribed time interval. Here,final-state control considering voltage limit is expressedas a programming program whose evaluation function andconstraint functions are quadratic function and quadraticinequality in the form of LMI, respectively. The FF inputcalled FSC input is generated by solving these LMIs.
From PWM hold model of SPMSM, a final state x[N ] is
represented with an initial state x[0] as follows:
Y = ΣU (20)
Y := x[N ]−ANs x[0]−Σ2UEMF (21)
Σ :=[AN−1
s Bs AN−2s Bs · · · Bs
](22)
Σ2 :=[AN−1
s Bs2 AN−2s Bs2 · · · Bs2
](23)
U :=[ΔT T [0] ΔT T [1] · · ·ΔT T [N − 1]
]T(24)
UEMF := −ωeKe
[0 1 · · · 0 1
]T(25)
Here, it is assumed that ωe is constant in this derivation.The FF input series U which satisfies (20) is not deter-
mined uniquely. Thereupon, the square sum of the currenterror is minimized. The evaluation function J is representedby
J = ETQE, Q > 0, (26)
E := [e[1]T e[2]T · · · e[N ]T ]T , (27)
e[k] := i∗ − x[k] =[i∗d i∗q
]T − x[k], (28)
E := I∗ −Ax[0]−BU −B2UEMF , (29)
A :=[As A2
s · · ·ANs
]T, (30)
B :=
⎡⎢⎢⎢⎣
Bs 0 · · · 0AsBs Bs · · · 0
......
. . ....
AN−1s Bs AN−2
d Bs · · · Bs
⎤⎥⎥⎥⎦ , (31)
B2 :=
⎡⎢⎢⎢⎣
Bs2 0 · · · 0AsBs2 Bs2 · · · 0
......
. . ....
AN−1s Bs2 AN−2
s Bs2 · · · Bs2
⎤⎥⎥⎥⎦ , (32)
I∗ :=[(i∗)T (i∗)T · · · (i∗)T
]T. (33)
Moreover, (8) is controllable. Thus, Σ is full row rank.There, Σ⊥ ∈ R
2N×(2N−2) and Σ† ∈ R2N×2 which fulfill
ΣΣ⊥ = 0 and ΣΣ† = I respectively are defined. Inaddition, U is obtained by
U := [Σ† Σ⊥]U . (34)
By assigning (34) to (20), Y = [I 0]U is given. Therefore,U is represented by
U =[Y q
]T, (35)
where q ∈ R(2N−2)×1 is a free parameter.
By substituting (29), (34), and (35) to (26), the evaluationfunction J is transformed as follows:
J = R(q) + ST (q)QS(q), (36)
Z := I∗ −Ax[0]−BΣ†Y −B2UEMF , (37)
R(q) := ZTQZ − 2ZTQS(q), (38)
S(q) := BΣ⊥q. (39)
With LMI, the condition which satisfies J < γ for providedγ is given by (40) [9].[
γ −R(q) S(q)T
S(q) Q−1
]> 0. (40)
Next, the voltage limit is described with LMI. The voltagelimits at each sampling point are described as
ΔT T [k]ΔT [k] = ΔT 2d [k] + ΔT 2
q [k]
≤ ΔT 2max (k = 0, 1, · · · , N − 1). (41)
Here, g(i) ∈ R2×2N (i := 2k + 1) of which (1, i) th and
(2, i+ 1) th entries are 1 and other entries are 0 is defined.g(i) separates input vector ΔT [k] as follows:
ΔT [k] = g(i)U(q). (42)
With g(i), (41) is transformed as
{g(i)U(q)}T {g(i)U(q)} ≤ ΔT 2max. (43)
(43) is expressed with N LMIs by[ΔT 2
max U(q)Tg(i)T
g(i)U(q) I
]≥ 0, (44)
where i = 2k + 1, k = 0, 1, · · · , N − 1.By minimizing γ under (40) and (44), the FF input, which
drives an initial state x[0] to a final state x[N ] by theprescribed time interval N for the PWM hold model ofSPMSM and minimizes the square sum of current error tofulfill the voltage limit, is obtained.
2) Propsoed method 2: Proposed method 1 can be causedinverse response of q-axis current because q-axis currentresponse is untrammeled. Generally, inverse response of q-axis current is not desirable, so constraint functions of q-axiscurrent response are added in proposed method 2.f(i) ∈ R
1×2N (i := 2k + 1) of which (1, i+ 1) th entryis 1 and other entries are 0 is defined. iq[k+1] is describedwith f(i) as follows:
iq[k + 1] = f(i) (Ax[0] +BU +B2UEMF ) . (45)
Therefore, the constraint function which wraps inverse re-sponse of q-axis current is represented by
f(i) (Ax[0] +BU +B2UEMF )− iq[0] ≥ 0. (46)
By minimizing γ under (40), (44), and (46), the FSC inputwhich drives an initial state x[0] to a final state x[N ] bythe prescribed time interval N for the PWM hold modelof SPMSM, minimizes the square sum of current error tofulfill the voltage limit, and prevent inverse response of q-axis current is obtained.
Here, the FSC input exists in proposed method 2 becauseit is assumed that ωe is constant in this paper. However, ifspeed increases during prescribed time interval, FSC inputof proposed method 2 could not exist.
TABLE I
PMSM PARAMETERS
Inductance L 1.80 [mH]Resistance R 0.157 [Ω]
Pairs of poles P 4Back EMF constant Ke 74.6 [mV/(rad/s)]
IV. SIMULATION
The parameters of SPMSM for the simulation are shownin Table I. The dc-bus voltage of the three-phase inverter Vdc
is 36 [V]. The control period Tu is 0.1 [ms].Fig. 5 shows simulation results when step torque reference
was given and the rotor velocity was 1000 [rpm]. Here, thevoltage amplitude of current reference generator was set tobe 0.98Vmax. The dq-axis ON time vector amplitude ΔTa
and the dq-axis ON time vector phase (voltage phase) δ arerepresented by
ΔTa =√
ΔT 2d +ΔT 2
q , (47)
δ = tan−1 −ΔTd
ΔTq. (48)
The dotted lines in Fig. 5(c), 5(g), and 5(k) are describedΔTmax.
Conventional method cannot operate the voltage phasebecause it consists of current vector control. Therefore, thevoltage phase variation was slow and quick response was notachieved. In addition, when the voltage amplitude was notlimited, current response still is slow because anti-windupcontrol made closed-loop discontinuous and conventionalmethod is only composed with FB controllers.
In the proposed method 1 and 2, the FSC inputs weregenerated as Q = I . The prescribed time interval N waschosen to be the quickest response (proposed method 1:N = 40, proposed method 2: N = 49). During the beginningof transient response, inverse reponse of q-axis current hap-pened but d-axis current variation is large. This made larged-axis current quickly. Finally, proposed method 1 achievedthe quickest response. On the other hand, proposed method2 suppressed inverse respose of q-axis current and maximumovershoot. However, the settling time is larger than one ofproposed method 2.
V. EXPERIMENT
Experimental results are shown in Fig. 6. Experimentswere performed in the same condition as simulations. Theparameters of SPMSM and the control period were thesame too. The load motor which controled velocity was thesame with the controlled motor. Therefore, for quick currentcontrol of load motor, Vdc was set to be 80 [V]. In thecontroller calculation, the nominal three-phase inverter dc-bus voltage was Vdcn = 36 [V] and the input was multipliedby Vdcn/Vdc to compensate the difference between Vdc andVdcn in SVM. Δθ in experiments is 0.9ωeTu to consider thecomputational time 0.4Tu.
The variations of d-axis reference in all experimentalresults are due to speed variation. In conventional method,
0 10 20 30 40−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(a) id (Conventional)
0 10 20 30 400
2
4
6
8
10
12
14
Time [ms]
Cur
rent
[A]
iq
iq*
(b) iq (Conventional)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(c) ΔTa (Conventional)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Ang
le o
f Inp
ut [r
ad]
(d) δ (Conventional)
0 10 20 30 40−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(e) id (Proposed 1)
0 10 20 30 400
2
4
6
8
10
12
14
Time [ms]
Cur
rent
[A]
iq
iq*
(f) iq (Proposed 1)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(g) ΔTa (Proposed 1)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Ang
le o
f Inp
ut [r
ad]
(h) δ (Proposed 1)
0 10 20 30 40−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(i) id (Proposed 2)
0 10 20 30 400
2
4
6
8
10
12
14
Time [ms]
Cur
rent
[A]
iq
iq*
(j) iq (Proposed 2)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(k) ΔTa (Proposed 2)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Ang
le o
f Inp
ut [r
ad]
(l) δ (Proposed 2)
Fig. 5. Simulation result (Proposed 1: N = 40, Propsoed 2: N = 49)
the voltage phase variation was slow and anti-windup controldeteriorated current response.
In the proposed methods, the FSC inputs which are thesame as simulations were calculated by off-line. When torquereference was obtained, the FF inputs were output from timeseries table of FSC input. If the response is not settled duringthe prescribed time interval, the FF input was generated from(19). Proposed method 1 achieved the quickest response.current is not settled during prescribed time interval becauseof speed variation. The q-axis current response of proposedmethod 2 is slower than one of proposed method 1 butproposed method 2 improved undershoot of q-axis currentand maximum overshoot of d-axis current.
VI. ANALYSIS CURRENT RESPONSE USING VOLTAGE
LIMIT CIRCLE WITH TRANSIENT TERM
This section expresses the voltage limit circle with tran-sient term for SPMSM. In addition, the FSC inputs of bothproposed methods are analyzed by using it.
A. Voltage limit circle with transient term
The dq-axis voltage equation which is represented by (1)is described under voltage limit (v2d+v2q ≤ V 2
max) as follows:(Vmax
L
)2
≥ (id − Cd
)2+(iq − Cq
)2, (49)
(50)
id
iq
Cq
Cd
Fig. 7. Voltage limit circle with transient term (ωe ≥ 0, powering)
Cd = −R
Lid + ωeiq, (51)
Cq = −ωeid − R
Liq − ωeKe
L. (52)
This is the voltage limit circle with transient term. Here,Cd and Cq are the d-axis and q-axis center coordinates ofthe voltage limit circle with transient term, respectively. Ingeneral, the voltage limit circle of SPMSM represents d-axis and q-axis current under voltage limit. The voltage limitcircle with transient term expresses d-axis and q-axis currentvariation id and iq under voltage limit.
B. Current response of SPMSM under voltage limit
Cd ≥ 0 and Cq ≤ 0 when ωe ≥ 0 and the SPMSM ispowering. Therefore, the center of voltage limit circle withtransient term exists on fourth quadrant as shown in Fig. 7.
0 10 20 30 40−30
−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(a) id (Conventional)
0 10 20 30 400
5
10
15
Time [ms]
Cur
rent
[A]
iq
iq*
(b) iq (Conventional)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(c) ΔTa(Conventional)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Pha
se o
f inp
ut [r
ad]
(d) δ (Conventional)
0 10 20 30 40−30
−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(e) id (Proposed 1)
0 10 20 30 400
5
10
15
Time [ms]
Cur
rent
[A]
iq
iq*
(f) iq (Proposed 1)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(g) ΔTa (Proposed 1)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Pha
se o
f inp
ut [r
ad]
(h) δ (Proposed 1)
0 10 20 30 40−30
−25
−20
−15
−10
−5
Time [ms]
Cur
rent
[A]
id
id*
(i) id (Proposed 2)
0 10 20 30 400
5
10
15
Time [ms]
Cur
rent
[A]
iq
iq*
(j) iq (Proposed 2)
0 10 20 30 400.06
0.065
0.07
0.075
0.08
Time [ms]
Am
plitu
de o
f inp
ut [m
s]
(k) ΔTa (Proposed 2)
0 10 20 30 40−0.4
0
0.4
0.8
1.2
Time [ms]
Pha
se o
f inp
ut [r
ad]
(l) δ (Proposed 2)
Fig. 6. Experimental result (Proposed 1: N = 40, Propsoed 2: N = 49)
Here, the area on fourth quadrant (id ≥ 0, iq ≤ 0) is large,so quick response as iq ≤ 0 can be achieved easily. On theother hand, quick response as iq ≥ 0 is not expected becausethe area on second quadrant (id ≤ 0, iq ≥ 0) is small. Thatis to say the operation which extends the area on secondquadrant is necessary.
Here, center coordinates Cd, Cq are partially differentiatedby −id, iq as follows:
∂Cd
∂(−id)=
R
L,
∂Cq
∂(−id)= ωe, (53)
∂Cd
∂iq= ωe,
∂Cq
∂iq= −R
L. (54)
This mentions that the center of voltage limit circle with tran-sient term is moved to fist coordinate and fourth coordinateby id < 0 and iq > 0, respectively. Especially, in high-speedoperation, the variations of Cd and Cq are subject to iq andid, respectively, because of ωe � R
L as shown in Fig. 8.Namely, large iq > 0 is undesirable at beginning of
transient response because it decreases the area on secondcoordinate. On the other hand, id < 0 increases the area onsecond coordinate. If d-axis current is preferentially flowed atbeginning of transient response, q-axis current response canbe improved. Moreover, if inverse response of q-axis currentis acceptable, a point on third coordinate (id ≤ 0, iq ≤ 0) canbe selected. The point of maximum d-axis current variationexists on third coordinate. In addition, if d-axis current is
id
iq
iq > 0
id < 0
Fig. 8. Transit of voltage limit circle with transient term on high-speedoperation (ωe ≥ 0, powering)
negatively larger than d-axis current reference, a point on firstcoordinate (id ≥ 0, iq ≥ 0) can be taken. Thus, quick q-axisresponse is prospected because there is a point of maximumiq . By the aforementioned process, quickest current responseis realized.
Fig. 9 shows the transit of voltage limit circle withtransient term in simulation. Conventional method selecteda point on second coordinate only. Therefore, the area onsecond coordinate decreased and q-axis current response isdeteriorated. In order to improve q-axis current variation,proposed method 1 moves the center of voltage limit circleto the second coordinate by q-axis current undershoot andd-axis current overshoot and achieve the quickest response.Proposed method 2 does not change q-axis current but d-axis
−20 −10 0 10 20 30−30
−20
−10
0
10
did/dt [A/ms]
diq/d
t [A
/ms]
Center ofCircle
(a) Conventional method
−20 −10 0 10 20 30−30
−20
−10
0
10
did/dt [A/ms]
diq/d
t [A
/ms]
CircleCenter of
(b) Proposed method 1
−20 −10 0 10 20 30−30
−20
−10
0
10
did/dt [A/ms]
diq/d
t [A
/ms]
CircleCenter of
(c) Proposed method 2
Fig. 9. Transit of voltage limit circle with transient term (simulation)
current at beginning of transient response. Thus, the centerof voltage limit circle is moved to the first coordinate andit increases the area on second coordinate. However, d-axiscurrent variation is negatively smaller than one of proposedmethod 2 because q-axis current undershoot does not happen.This mentions that changing q-axis current decreases thesettling time better than increasing d-axis current overshoot.Therefore, maximum d-axis current overshoot decreases.
The reason why proposed methods can act these operationsis because they use PWM hold model and consider voltagelimit.
VII. CONCLUSION
For quick current response under voltage limit, our pre-vious paper proposed final-state control considering voltagelimit but inverse response of q-axis current happened. Thispaper proposes a new FSC input which suppresses inverseresponse of q-axis current. Simulations and Experiments areperformed to show that proposed method 2 can suppressmaximum d-axis current overshoot and q-axis current un-dershoot. However, the settling time of proposed method 2is longer than one of proposed method 1. This reason isexplained by using voltage limit circle with transient term.In short, our proposed methods consider voltage limit circlewith transient term exactly.
In our future work, online calculation of proposed methodswill be realized and robust feedback controller for modelingerror is worked out.
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