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Multirate Perfect Tracking Control of Single-phase Inverter
with Inter Sampling for Arbitrary Waveform
Hironori Abe and Hiroshi Fujimoto
Department of Electrical and Computer EngineeringYokohama National University
79-5 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501 Japan
Phone: +81-45-339-4107, Fax: +81-45-338-1157
E-mail: [email protected], [email protected]
Abstract In this paper, multirate perfect tracking controlis proposed for a single-phase inverter. Although conventionalsingle-rate deadbeat control could not guarantee zero trackingerror for arbitrary reference signals, the proposed multiratecontrol can achieve perfect tracking at every sampling point.Feedback characteristic is enhanced by inter sampling tech-
nique. Simulations and experiments are carried out to comparethe proposed method with a single-rate deadbeat controllerfor non-sinusoidal reference waveform. Finally, the proposedmethod is applied to a nonlinear rectifier load.
Index Terms inter sampling, single-phase inverter, multi-rate control, deadbeat control, arbitrary waveform
I. INTRODUCTION
The deadbeat control is well known and widely used
technique in high speed and high precision control. Several
papers have tried to apply the deadbeat control to inverter
systems [1][2][3]. However, it is proven theoretically that the
conventional single-rate deadbeat control could not guarantee
zero tracking error for arbitrary reference signals [4][5].We have applied multirate perfect tracking control [4] to a
single-phase inverter [5][6][7]. The terminology of perfect
tracking control(PTC) is originally defined in [8], which
means the plant output perfectly tracks the desired trajectory
with zero tracking error at every sampling point. In the
perfect tracking control, the tracking error of plant state be-
comes completely zero at every sampling period of reference
input for a nominal plant without disturbance. Moreover,
by combining the proposed feedforward controller with a
robust feedback controller such as disturbance observer or
H controller, high tracking performance is preserved even
if the plant has modeling error and disturbance.
In [6], there was a possibility that the feedback characteris-tic of PTC worsens more than the single-rate control because
the output period became longer than the input period, by
synchronizing the sampling period of reference signal for the
simplification. In [7], the feedback characteristic of PTC is
equal to single-rate control by synchronizing the sampling
period of the output with the career period. In both [6] and
[7], the output samples only at the bottom vertices of the
triangle career. However, there are development environment
that the output samples both at the top and bottom vertices of
the career recently. On the other hand, it was proposed multi-
sampling method with FPGA based on hardware controller
in [9].
In this paper, we proposed the error suppression technique
by increasing and decreasing the pulse width every half
cycle of the career. Then, the PWM pulse can become an
asymmetry type when the tracking error is caused by themodeling error and disturbance. Thus, it is possible that
the bandwidth of feedback system becomes twice, because
sampling period of the feedback loop is half. The proposed
method is applied to arbitrary AC power supply in the
simulations and experiments. Also we apply the proposed
method to a nonlinear load.
II. SINGLE-RATE CONTROL
A. Plant Model of Single-phase Inverter
In this section, the plant model is introduced in order
to apply the PTC to the voltage control of a single-phase
inverter. Tracking performance is very important not only in
motor drive but also in active filter and UPS. Fig. 1 shows
the controlled plant. As shown in Fig. 2, the PWM inverter
bridge can generate output voltage ofVDC or 0 as vinv(t).The load of vinv(t) is considered as plant P(s), which ismodeled as
x(t) = Apx(t) + bpvinv(t), y(t) = cpx(t). (1)
It is assumed that this plant is n-th order linear load, i.e.,
the number of inductance L and capacitance C is n. In the
example of Fig. 1, the plant system has a LC filter and a
resistive load R. Hence n = 2. The plant coefficients arerepresented as
Ap =
0 1
1LC
1RC
, bp =
01LC
cp =
1 0
, x(t) =
vcvc
. (2)
In order to design the digital controller in discrete-time
domain, we need to discretize the continuous-time plant
model (1). The discrete-time state space model is formulated
with the period Tu as
x[k + 1] = Asx[k] + bsu[k], y[k] = csx[k], (3)
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L=1mH
C=10F
R=
23.8
vcv invVDC=60V
DSP
Gate Signalvc
Fig. 1. Inverter system.
u[k]
kTu (k+1)Tu
Tu
r[i]
r[i+1]
VDC
T[k]
Fig. 2. Single-rate PWM control.
where x[k] = x(kTu) and the pulse width is regarded ascontrol input u[k] = T[k]. In this paper, Tu =50s with20kHz carrier frequency. [1] proposed more precise model
which can evaluate the instantaneous value than the zero-
order hold. When the pulse is allocated is the center of
control period Tu, the PWM holder can be modeled as
follows.
As = eApTu , bs = e
Ap Tu2 bpVDC, cs = cp (4)
B. Single-rate Deadbeat Control
In this section, the conventional deadbeat controller is
designed to see the problem of single-rate method. The
single-rate feedback deadbeat (SR-FBDB) control law is
given byu[k] = f x[k] + gr[k], (5)
where f is the feedback gain, g is feedforward gain, and
r[k] is the reference signal.1) Single-rate Feedback Deadbeat Controller 1: [6]
In this section, first single-rate feedback deadbeat controller
(SR-FBDB-1) is designed. The feedback gain f places
the closed-loop poles to origin of z-plane for the discrete-
time plant (3) with period Tu. The feedforward gain g is
determined to make the DC gain from r[k] to y[k] unity.Then, this controller can track the step-type reference signal
with 2 steps delay since the plant is second order system.2) Single-rate Feedback Deadbeat Controller 2: [1][2]
Next, second single-rate feedback deadbeat controller (SR-
FBDB-2) is designed. In [1] and [2], it is proposed the
control law is given by
u[k] = f x[k] + gr[k] (6)
f = (As11
bs1
As12
bs1), g =
1
bs1,
from first row of (3). This law has deadbeat characteristic to
track with 1 step as vc[k +1] = r[k]. Then, transfer functionfrom r[k] to the control input u[k] corresponds to the inverseof the discrete-time plant model P[z]. Hence, this controller
must be able to assure perfect tracking.However, P[z] discretized by zero-order hold has unstablezeros when the relative degree of P(s) is greater than 2 [10].Thus, feedforward controller becomes unstable because the
closed-loop system has the unstable zeros.
In the case of example Fig. 1, the transfer function of (4)
can be calculated as
P[z] = cs(zI As)1bs =
4.77 105(z + 0.95)
z2 1.45z + 0.90. (7)
Even though the relative degree of (2) is 2, P[z] has zero atz = 0.95. Although it is in the stable region (| z |< 1), this
C=10F
R=
23.8vv invVDC=
60V
L1=1mH L2=2mH
Fig. 3. Inverter system with 3rd order load.
-150
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-50
0
50
100
150
0 5 10 15 20
voltage[V]
time[ms]
Vc(t)r[k]
Fig. 4. Simulation result for 3rd order plant(SR-FBDB-2).
zero is very oscillatory as it is closed to z = 1. Thus, theoutput signal of feedforward controller oscillates with high
frequency as the sign of u[k] alternates at every Tu.Moreover, as described above, the discrete-time plant with
zero-order hold has unstable zeros when relative degree ofplant is greater than 2. Here, this method is applied to the
inverter system of Fig. 3. The transfer function where from
vinv to v can be calculated as
V
Vinv=
R
L1L2Cs3 + L1CRs2 + (L1 + L2)s + R. (8)
Fig. 4 shows the simulation result by SR-FBDB-2. The
output becomes unstable because the discrete-time plant
model of (8) that discretized of (4) has zero at z = 2.92.Therefore, in conventional single-rate control systems, per-
fect tracking control is generally impossible.
III. MULTIRATE CONTROL
A. PTC by multirate control
A digital tracking control system generally has two sam-
plers S for the reference signal r(t) and the output y(t), andone holder H on the input u(t). Therefore, there exist threetime periods Tr , Ty , and Tu which represent the periods
of r(t), y(t), and u(t), respectively. The input period Tuis generally decided by the speed of the actuator, the D/A
converter, or the calculations on the CPU. In case of inverter
systems, Tu is determined by career frequency. On the other
hand, the output period Ty is determined by the speed of
the sensor or the A/D converter. In conventional single-rate
systems, these three periods are set to equal value for the
simplification both in controller design and implementation.In the multirate feedforward control [4], the control input
u(t) is changed n times during one sampling period (Tr) ofreference input r(t). In other words, the reference samplingperiod Tr is set to n times longer than the control period Tu.
Again, n is the plant order. The advantage of this method is
that the control laws to generate u1[i], , un[i] can be setindependently. By using these extra degrees of freedom, this
method can guarantee perfect tracking at every Tr . Note that
the single-rate systems with Ty = Tu cannot achieve perfect
tracking even at Tr = nTu because it has one control law.
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0
5
10
15
20
25
0 5 10 15 20
voltage[V]
time[ms]
Vc(t)r[k]
Fig. 5. Simulation result for 3rd order plant(PTC).
B. Plant Formulation with Multirate PWM
The case of Ty = Tu is considered. We need to obtainthe discrete-time plant formulation with multirate input. The
state transition from kTu to (k + 1)Tu is given by (3) and(4) for single-rate PWM. Thus, the state-space model from
iTr = kTu to (i + 1)Tr = (k + n)Tu is represented as
x[i + 1] = Ax[i] + Bu[i] (9)
y[i] = Cx[i] + Du[i] (10)
where x[i] = x(iTr), and multirate input and output vectorsu, y are defined as
u[i] = [u1[i], , un[i]]T
= [T[k], , T[k + n 1]]T
, (11)
y[i] = [y1[i], , yn[i]]T
= [y(kTy), , y((k + n 1)Ty)]T
. (12)
The coefficient matrices are given byA B
C D
=
Ans An1s bs A
n2s bs Asbs bs
cs 0 0 0 0csAs csbs 0 0 0
......
...
csAn1s csA
n2s bs csA
n3s bs csbs 0
. (13)
C. Design of Perfect Tracking Controller
In the ideal tracking control system, the transfer charac-
teristic from the command yd to the output y is unity. In
this section, the feedforward controller is designed so that
the transfer characteristic from the desired state xd[i] to theplant state x[i] can be identity matrix I.
From (9) and (10), the transfer function from x[i + 1] tou[i] and y[i] is described by
u[i] = B1(I z1A)x[i + 1]
=
0 I
B1A B1
x[i + 1] (14)
y[i] = z1Cx[i + 1] + Du[i], (15)
where z = esTr . In (14), the nonsingularity of matrix B is
assured for controllable plant, because B in (12) coincides
with the controllability matrix. Because all poles of the
transfer function (14) are zero, it is found that (14) is a
stable inverse system. Thus, if the control input is calculated
B (I - z A)-1 -1+
- +
+
yo
uff u y(t)r(t)(Tr)
(Ty)Pn(s)
Cfb
ufb
P(s)
(Ty)
(PWM)(Tu)
(PWM)(Tu)
Fig. 6. Block diagram of PTC with inter sampling.
VDC
T[k] T[k+1]
kTu (k+1)Tu (k+2)Tu
iTr (i+1)Tr Tu Tu
Tr
(j+1)Ty
TyTyTyTy
jTy (j+2)Ty (j+3)Ty (j+4)Ty
y[j+2]
y[j]r[i]
y[j+4]r[i+1]
y[j+1]
y[j+3]
Fig. 7. Multirate PWM control with inter sampling.
by (16), perfect tracking is guaranteed since (16) is an exact
inverse plant.
u0[i] = B1
(I z1
A)xd[i + 1] (16)Here, xd[i + 1] is previewed desired trajectory of plant state.The output of the nominal plant model can be calculated by
y0[i] = z1Cxd[i + 1] + Du0[i], (17)
When the tracking error e[k] = y[k] y0[k] is caused bymodeling error or disturbance, it can be eliminated by the
robust feedback controller as
u[k] = u0[k] + C2[zy]e[k]. (18)
The inverter system of Fig. 3 is applied to PTC as in the
case of SR-FBDB-2. Fig. 5 shows the simulation result. Fig.
5 shows that perfect tracking is achieved even though therelative degree of plant is greater than 2 as (8).
D. Perfect Tracking Control with Inter Sampling
In this section, the output samples both at the top and
bottom vertices of the triangle career, although in foregoing
section it samples only at the bottom vertices. We proposed
the error suppression technique by increasing and decreasing
the pulse width every half cycle of the career. Fig. 6
shows block diagram of PTC with inter sampling, where
H(PWM) represents PWM holder. The nominal output y0[j]is calculated every half cycle of the career in real-time. Thus,
the sampling period of PTC with inter sampling becomes Tu
=
Tr
2 = 2Ty. The solution x(t) of (1) is represented asx(t) = eAp(tt0)x(t0) +
tt0
eAp(t)bpu()d, (19)
by giving initial value x(t0) and piecewise continuous inputu(t). From (19), the nominal plant output y0[j](= x0[j]) canbe calculated
x0[j + 1] = A0x0[j] + b0(Ton, Toff) (20)
A0 = eApTy
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0
2
4
6
8
2 2.2 2.4 2.6 2.8 3
voltage[V]
time[ms]
Vc(t)r[k]
Vc*(t)
Fig. 8. Simulation result for sinusoidal wave with 3rd harmonics (SR-FBDB-1).
0
2
4
6
8
2 2.2 2.4 2.6 2.8 3
voltage[V]
time[ms]
Vc(t)r[i]
Vc*(t)
Fig. 9. Simulation result for sinusoidal wave with 3rd harmonics (PTC).
101
102
103
104
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-30
-20
-10
0
10
Frequency [Hz]
G
ain[dB]
S1[z]S2[z]
Fig. 10. Sensitivity function.
b0 =
TyTon(k)0
eAptbp(E)dt (of f on)
TyTyToff(k)
eAptbp(E)dt (on of f)
,
where Ton(k) is the time PWM pulse becomes on from of f,Toff(k) is the time PWM pulse becomes of f from on.
The PWM pulse is increased and decreased by feedbackcontroller Cfb when the tracking error is caused by modeling
error or disturbance. The nominal plant output y0 from (20)
and detection value y are compared every sampling period
of the output Ty. Therefore, as Fig. 7, the PWM pulse can
become an asymmetry type when the tracking error is caused
by the modeling error and disturbance.
Analytical solution of (20) can be obtained for low degree
of the plant like this paper. However, it is necessary to
calculate (20) numerically for higher-order plant by using
the Pade approximation etc. Moreover, if the reference signal
is already known, the computation cost can be reduced by
computing (20) off-line.
IV. SIMULATION RESULTS
A. Output for Arbitrary Waveform
As [11], the reference signal is given
Vc = 10sin(2500)t + 2sin(21500)t. (21)
The three periods of SR-FBDB-1 are set to Ty = Tu =Tr =50s. The career frequencies of SR-FBDB-1 and PTCare selected to be equal with Tu =50s for fair comparison.Thus, the three periods of PTC are set to Ty = Tu =50sTr =100s. Fig. 8 shows that SR-FBDB-1 has tracking error
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25 30 35 40 45 50
voltage[V]
time[ms]
Inter Samplingwith Inter Sampling
(A)(B)
without
Fig. 11. Simulation result(Error comparison).
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-10
0
10
20
30
40
0 2 4 6 8 1 0
voltage[V]
time[ms]
Vc(t)Vc*(t)
(A)without Inter Sampling
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-10
0
10
20
30
40
0 2 4 6 8 1 0
voltage[V]
time[ms]
Vc(t)Vc*(t)
(B)with Inter Sampling
Fig. 12. Simulation results with plant variation.
0 500 1000 1500 2000 2500 30000
0.5
1
1.5
2
2.5
3
3.5
frequency[Hz]
voltage[V]
(A)without inter sampling(B)with inter sampling
Fig. 13. Simulation results with plant variation(FFT).
even at the sampling points. On the other hand, Fig. 9 shows
that PTC has zero tracking error at every sampling period.
Thus, we find that the multirate control has better tracking
performance than the single-rate system even though thesetwo methods have same Tu.
B. Effectiveness of Inter Sampling
In this section, we design the feedback controller Cfb and
consider the stability of the closed-loop system about each of
(A) conventional PTC (Ty =50s) and (B) PTC with intersampling (Ty =25s). We use the lead-lag controller as Cfb.We determined that the cutoff frequency of the closed-loop
transfer function is equal to 1100 of the nyquist frequency for
(A) and (B). Then, it becomes possible to achieve higherbandwidth because (A) is discretized by sampling frequencyTy =50s, but (B) is discretized by Ty =25s. In Fig. 10,
S1(z) and S2(z) are the sensitivity function of the closed-loop system of (A) and (B), respectively. From Fig. 10, wefind that it makes bandwidth twice by using inter sampling.
The gain of sensitivity can be attenuated about 6dB up to
300Hz.
We verify characteristic of disturbance rejection to com-
pare the feedback characteristic of (A) and (B). The sinu-soidal wave with amplitude 2V and frequency 100Hz is given
at the plant output as disturbance in Fig. 6 when the reference
r(t) is assumed to be 0. From Fig. 11, the characteristics ofdisturbance rejection of (B) is better than (A).
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0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
voltage[V]
time[ms]
Vc(t)Vc*[i]
-2
0
2
4
6
8
10
12
0 0.2 0.4 0.6 0.8 1
voltage[V]
time[ms]
Vc(t)Vc*[i]
(a) for sinusoidal wave with (b) for sinusoidal wave with
3rd harmoni cs(SR-FBDB-1) 3rd harmonics(PTC )
Fig. 14. Experimental result with nominal plant.
-1
-0.5
0
0.5
1
0 5 10 15 20 25 30 35 40
voltage[V]
time[ms]
Inter Samplingwith Inter Sampling
(A)(B)
without
Fig. 15. Experimental result(Error comparison).
0 2 4 6 8 10-30
-20
-10
0
10
20
30
40
t i me[ ms]
vo
tage[V]
Vc(t )
Vc*[i ]
0 2 4 6 8 10-30
-20
-10
0
10
20
30
40
t i me[ ms]
voltage[V]
Vc(t)
Vc*[i ]
HTGSWGPE[=*\?
XQNVCIG=8?
#YKVJQWV+PVGT5CORNKPI
$YKVJ+PVGT5CORNKPI
(a) without Inter Sampling (b) with Inter Sampling (c) FFT analysis
Fig. 16. Experimental result with plant variation.
Next, we examine that the plant has parameter varia-
tion from the nominal model. We consider the inductance
variation L = 0.4Ln where Ln is the nominal inductancevalue, and the reference signal is given the sinusoidal wave
with 3rd harmonics that consists of a combination first
harmonic with amplitude 30V and frequency 300Hz with 3rd
harmonics with amplitude 6V and frequency 900Hz with this
L(= 0.4Ln), the closed-loop system is still stable. Fig. 12shows the simulation results with plant variation, and Fig.
13 shows the FFT analysis of error both (A) and (B). From
Fig. 13, (B) can attenuate the error tracking than (A) in low-frequency region where the sensitivity improved in Fig. 10.
Thus, we find the effectiveness of inter sampling method.
V. EXPERIMENTAL RESULTS
Fig. 14 shows the experimental results. In the case of the
reference is sinusoidal wave with 3rd harmonics, SR-FBDB-
1 has tracking error even at the sampling points, as shown in
Fig. 14(a). On the other hand, Fig. 14(b) shows that perfect
tracking is achieved every sampling periods by PTC.
Fig. 15 shows the feedback characteristic of disturbance
rejection. As well as the simulation, the reference r(t) isassumed to be 0 and the sinusoidal wave of amplitude 2V,frequency 100Hz is given to the plant output as disturbance
in software. From Fig. 15, the disturbance has been sup-
pressed in (B) more than (A).
Fig. 16 shows the experimental results with plant variation.
As well as the simulation, L = 0.4Ln and the reference issinusoidal wave with 3rd harmonics. From Fig. 16(c), (B)can attenuate the tracking error than (A) in low-frequencyregion. Thus, we find the effectiveness of inter sampling
method even in experiments.
L=1mH
C=10F
vcv invVDC=60V
DSP
Gate Signalvc
LL=1.6mH
CL=220F
RL=
15.8
iL
iL
Fig. 17. Inverter system with rectifier load.
VI. APPLICATION TO NONLINEAR LOAD
In this section, the proposed method is applied to a
nonlinear load. Here, we think about the inverter system ofFig. 17. The plant is represented as
x(t) = Apx(t) + bp(vinv(t) L iL(t)), y(t) = cpx(t) (22)
Ap =
0 1
1LC
0
, bp =
01LC
, cp =
1 0
that is coincide with the model which has the resistance load
R = in (1). iL(t) is regarded as disturbance, which can besuppressed by feedback controller. The system of multirate
PWM control with inter sampling is designed by using the
discrete-time state space model of (22) with PWM holder.
We apply the lead-lag controller as the feedback controller
Cfb , which is determined that the cutoff frequency of the
closed-loop transfer function is equal to 150 of the nyquistfrequency for (A) and (B). Fig. 18(a) shows the sensitivityfunction of the closed-loop system. The experiments are
carried out with the reference signal is given the sinusoidal
wave with amplitude 30V and frequency 50Hz. Fig. 18(b)
shows the FFT analysis of tracking error both (A) withoutinter sampling and (B) with inter sampling. From Fig.18(b), the tracking error of (B) is much smaller than thatof (A) in low frequency band around 50Hz. However, thesensitivity function is amplified in high frequency band over
the resonance frequency. The tracking error of (B) is larger
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101 102 103 104-40
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0
10
20
30
Frequency [Hz]
Gain[dB]
(A)without Inter Sampling(B)with Inter Sampling
(a)Sensitivity function
0 500 1000 1500 2000 2500 30000
0.2
0.4
0.6
0.8
1
frequency[Hz]
voltage[V]
(A)without inter sampling(B)with inter sampling
(b)FFT analysis
Fig. 18. Application to the rectifier load.
+-+Vinv u
d(t)
P(s)
d*[k]
Plant
(PWM)(Tu)
Tu
VDC
Fig. 19. Block diagram with disturbance compensator.
than that of (A) in high frequency band. When the nominalplant model is designed as the resistance load R = , it isdifficult to suppress the error in all frequency band only by
making closed-loop bandwidth higher with inter sampling.
Therefore we proposed the disturbance compensation asfeedforward manner with detection iL(t). Here, the rectifierload can be regarded as the current source in Fig. 17. Thus,
the plant model is represented as
Vc =1LC
s2 + 1LC
(Vinv LsIL) := P(s)(Vinv + d), (23)
which consists of LC filter and the current source sIL as
disturbance d. The disturbance compensation is proposed
as feedforward in Fig. 19. Because the control input u[k]is on-time T[s], the disturbance compensation d[k] isapproximately converted to Td[s] as
Td = TuVDC
d. (24)
Fig. 20(a) and (b) show the experiment results. Fig. 21
shows the FFT analysis of error both of Fig. 20(a) and (b).
From Fig. 21, the tracking error with disturbance compen-
sation is much smaller than that without disturbance com-
pensation. Thus, we find the effectiveness of the disturbance
compensation.
VII. CONCLUSION
A novel perfect tracking control (PTC) method was pro-
posed for inverter systems based on multirate PWM control.
The advantage of this method is that the feedforward con-
troller can be designed without considering the unstable zeroproblem. Moreover, by combining the proposed feedforward
controller with a robust feedback controller, robust tracking
performance is obtained.
In PTC, the output samples both at the top and bottom
vertices of the triangle career, although in conventional it
samples only at the top or bottom vertices. Therefore, it is
possible to achieve higher bandwidth of feedback system.
Then, the PWM pulse can become an asymmetry type when
the tracking error is caused by the modeling error and distur-
bance. The advantages of this approach were demonstrated
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0
10
20
30
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voltage[V]
time[ms]
Vc(t)Vc*(t)
Id
(a)without disturbance compensation.
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0
10
20
30
40
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voltage[V]
time[ms]
Vc(t)Vc*(t)
Id
(b)with disturbance compensation.
Fig. 20. Experimental Results with rectifier load.
0 500 1000 1500 2000 2500 30000
0.5
1
1.5
2
frequency[Hz]
voltage[V]
(a)without disturbance compensation
(b)with disturbance compensation
Fig. 21. FFT analysis with rectifier load.
through simulations and experiments on the voltage controlof single-phase inverter.
We also applied the proposed method to nonlinear load.
In the proposed method, the error is suppressed in the low
frequency region, but the error has increased in the high
frequency region. Therefore we proposed the disturbance
compensation as feedforward and confirmed the effectiveness
in experiments.
REFERENCES
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