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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: abe@hfl.dnj.ynu.ac.jp, hfuji@ynu.ac.jp

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

-100

-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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-15

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-5

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

-40

-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).

-30

-20

-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

-30

-20

-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

-40

-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25

voltage[V]

time[ms]

Vc(t)Vc*(t)

Id

(a)without disturbance compensation.

-40

-30

-20

-10

0

10

20

30

40

-25 -20 -15 -10 -5 0 5 10 15 20 25

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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