# Disturbance Observer-Based Simple Nonlinearity Compensation for Matrix Converter Drives

Authors: Kyo-Beum Lee; Frede Blaabjerg
Journal: Research Letters in Electronics
 (2009)

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## Abstract

This paper presents a new method to compensate the nonlinearity for matrix converter drives using disturbance observer. The nonlinearity of matrix converter drives such as commutation delay, turn-on and turn-off time of switching device, and on-state switching device voltage drop is modeled by disturbance observer and compensated. The proposed method does not need any additional hardware and offline experimental measurements. The proposed compensation method is applied for high-performance induction motor drives using a 3 kW matrix converter system without a speed sensor. Simulation and experimental results show that the proposed method using disturbance observer provides good compensating characteristics.


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## Body

## 1. Introduction

Induction motor drives fed by matrix converter have been developed for the last
two decades [1]. The matrix converter drive has recently attracted the industry
application, and the technical development has been further accelerated because
of the increasing importance of power quality and energy efficiency issues [2].Since the fundamental output voltage cannot be
detected directly from the matrix converter output terminal in a matrix
converter drive, the command voltage is usually used instead of the actual one.
However, the commanded voltage does not agree with the actual fundamental
output voltage due to nonlinear characteristic of the matrix converter, such as
commutation delay, turn-on and turn-off time of switching device, and on-state
switching device voltage drop. In order to compensate this problem, the authors
have made an attempt to compensate the nonlinear matrix converter effects with
current sign and offline manners [3]. However, it is difficult to
determine the current sign when the phase currents are closed to zero. If the
current sign is misjudged, the nonlinearity model of a matrix converter
operates improperly. It is also difficult to compensate the nonlinearity
effects perfectly by offline manners because the switching times and voltage
drops of the power devices are varied with the operating conditions [4].In this paper, a new online nonlinearity compensation
method for matrix converter drives using a disturbance observer is proposed. Figure1 shows the whole control block diagram of a sensorless vector-controlled
matrix converter drive with disturbance observer-based nonlinearity
compensation. The proposed method does not need any additional hardware and
offline experimental measurements. The paper calculates the nonlinearity of
matrix converter drives using disturbance observer, and the simple feed-forward
control method applies to make the compensation voltages in order to compensate
the nonlinear model of matrix converter drives (see Figure
1).Figure 1
The proposed sensorless vector control for matrix converter drives.


## 2. Disturbance Observer-Based  Nonlinearity Compensation

### 2.1. Back-EMF Voltage Estimation

The back-EMF voltage equation for the stator of an induction motor in the stationary reference frame
can be expressed as(1)e=vs−Llsddtis.Ifvs* is the desired applied voltage, (1) can be restated as(2)e^=vs*−Llsddtis, where e^ is the
estimated back-EMF voltage.Because of the
nonlinearity of a matrix converter drive, the command voltage from the
controllervs* is not the real applied
voltage vs that appears on the motor terminals.From (1) and (2), the error equation for
the back-EMF voltage can be written as(3)e−e^=vs−vs*, where Δv is the voltage error vs−vs* by the nonlinearity of a matrix converter drive [5].Using (1)–(3) can be rewritten as(4)e=vs*−Llsddtis+Δv.In order to estimate the
voltage error vectorΔv, especially important at low speed, this paper presents a
method to compensate for the nonlinear matrix converter effects with a
disturbance observer.


### 2.2. Disturbance Observer-Based Nonlinearity Compensation

The voltages due to the
nonlinearity of matrix converter drives cause the matrix converter output
voltage distortion, results in the phase current distortion and torque ripple.
The disturbance voltages are a function of a commutation delay, turn-on and
turn-off times of the switching devices, on-state switching device voltage drop,
and a current polarity. However, the commutation delay, on-state voltage drop
of the switching devices, and a current polarity are varying with the operating
conditions. Since it is very difficult to measure the commutation delays and
the on-state voltage drops, as well as to
determine the current sign when the phase currents are closed to zero, it is
not easy to compensate the nonlinearity of matrix converter drives in an
offline manner. To alleviate these problems, a new online nonlinearity
compensation method using disturbance observer is proposed. The proposed scheme
is shown in Figure2, which consists of an inverse machine model and a simple
first-order low-pass filter.Figure 2
Block
diagram of nonlinearity observer.The estimatedd- and q-axis voltages can be written as(5)v^dse=Rsidse−ω^e(λqme+Llsiqse),vqss=Rsiqss+ω^e(λdme+Llsidse).The disturbance observer
estimates the nonlinearity distorted voltagevdis by
using the inverse motor model Pn−1(s) as follows:(6)δ=u−Pn−1(s)(y−γ)=(P−1(s)−Pn−1(s))y+vdis+Pn−1(s)γ≜ΔP(s)y+vdis+Pn−1(s)γ, where the outputs  y are currents, P(s) is the real motor model, Pn(s)
is the nominal motor model, vdis is the total nonlinearity of
a matrix converter drive, γ is the
measurement noise (currents and motor speed), and Δp(s) in the first term is the
perturbation of the real model from the nominal model. Therefore, the
nonlinearity observer estimates not only the nonlinearity of a matrix converter,
but also the perturbation of the real model from the nominal model.Since the inverter nominal
plantPn−1(s) includes some derivatives,
the disturbance observer estimates the disturbance Δthrough a low-pass
filter Q(s) as follows:(7)Δ=Q(s)δ.If the estimated
disturbanceδ is fed back as
shown in Figure 2, the output signal can be described by the disturbance, the
measurement noise, and the control input signal as follows:(8)y=u−(1−Q(s))d+Q(s)Pn−1(s)γQ(s)Pn−1(s)+(1−Q(s))P−1(s)=Pn(s){u−(1−Q(s))d}+Q(s)γ1+(1−Q(s))ΔP(s)Pn(s).If conditions(1−Q(s))vdis≈0, (1−Q(s))ΔP(s)Pn(s)≈0, and Q(s)γ≈0 are satisfied, the output signal y can be obtained as in (9), in which it has no relation with the disturbance. As
a result, the estimated disturbance is nearly close to the practical
disturbance:(9)y=Pn(s)u.Q(s)
must be determined according to the characteristics of the disturbance d and the plant perturbation ΔP(s),
so that (1−Q(s))vdis and (1−Q(s))ΔP(s)Pn(s) fulfil a specification. Since plant perturbation ΔP(s)
affects the stability of control system, it is desirable to make ΔP(s)
small, which is realized by determining the nominal plant Pn(s)
close to the plant P(s). On the other hand, since Q(s)
is the sensitivity to the sensor noise, Q(s) must be determined,
so that influence of the sensor noise does not become excessively large.
Therefore, the bandwidth of the low-pass filter Q(s) is limited
by the bandwidth of the sensor noise γ.


## 2.1. Back-EMF Voltage Estimation

The back-EMF voltage equation for the stator of an induction motor in the stationary reference frame
can be expressed as(1)e=vs−Llsddtis.Ifvs* is the desired applied voltage, (1) can be restated as(2)e^=vs*−Llsddtis, where e^ is the
estimated back-EMF voltage.Because of the
nonlinearity of a matrix converter drive, the command voltage from the
controllervs* is not the real applied
voltage vs that appears on the motor terminals.From (1) and (2), the error equation for
the back-EMF voltage can be written as(3)e−e^=vs−vs*, where Δv is the voltage error vs−vs* by the nonlinearity of a matrix converter drive [5].Using (1)–(3) can be rewritten as(4)e=vs*−Llsddtis+Δv.In order to estimate the
voltage error vectorΔv, especially important at low speed, this paper presents a
method to compensate for the nonlinear matrix converter effects with a
disturbance observer.


## 2.2. Disturbance Observer-Based Nonlinearity Compensation

The voltages due to the
nonlinearity of matrix converter drives cause the matrix converter output
voltage distortion, results in the phase current distortion and torque ripple.
The disturbance voltages are a function of a commutation delay, turn-on and
turn-off times of the switching devices, on-state switching device voltage drop,
and a current polarity. However, the commutation delay, on-state voltage drop
of the switching devices, and a current polarity are varying with the operating
conditions. Since it is very difficult to measure the commutation delays and
the on-state voltage drops, as well as to
determine the current sign when the phase currents are closed to zero, it is
not easy to compensate the nonlinearity of matrix converter drives in an
offline manner. To alleviate these problems, a new online nonlinearity
compensation method using disturbance observer is proposed. The proposed scheme
is shown in Figure2, which consists of an inverse machine model and a simple
first-order low-pass filter.Figure 2
Block
diagram of nonlinearity observer.The estimatedd- and q-axis voltages can be written as(5)v^dse=Rsidse−ω^e(λqme+Llsiqse),vqss=Rsiqss+ω^e(λdme+Llsidse).The disturbance observer
estimates the nonlinearity distorted voltagevdis by
using the inverse motor model Pn−1(s) as follows:(6)δ=u−Pn−1(s)(y−γ)=(P−1(s)−Pn−1(s))y+vdis+Pn−1(s)γ≜ΔP(s)y+vdis+Pn−1(s)γ, where the outputs  y are currents, P(s) is the real motor model, Pn(s)
is the nominal motor model, vdis is the total nonlinearity of
a matrix converter drive, γ is the
measurement noise (currents and motor speed), and Δp(s) in the first term is the
perturbation of the real model from the nominal model. Therefore, the
nonlinearity observer estimates not only the nonlinearity of a matrix converter,
but also the perturbation of the real model from the nominal model.Since the inverter nominal
plantPn−1(s) includes some derivatives,
the disturbance observer estimates the disturbance Δthrough a low-pass
filter Q(s) as follows:(7)Δ=Q(s)δ.If the estimated
disturbanceδ is fed back as
shown in Figure 2, the output signal can be described by the disturbance, the
measurement noise, and the control input signal as follows:(8)y=u−(1−Q(s))d+Q(s)Pn−1(s)γQ(s)Pn−1(s)+(1−Q(s))P−1(s)=Pn(s){u−(1−Q(s))d}+Q(s)γ1+(1−Q(s))ΔP(s)Pn(s).If conditions(1−Q(s))vdis≈0, (1−Q(s))ΔP(s)Pn(s)≈0, and Q(s)γ≈0 are satisfied, the output signal y can be obtained as in (9), in which it has no relation with the disturbance. As
a result, the estimated disturbance is nearly close to the practical
disturbance:(9)y=Pn(s)u.Q(s)
must be determined according to the characteristics of the disturbance d and the plant perturbation ΔP(s),
so that (1−Q(s))vdis and (1−Q(s))ΔP(s)Pn(s) fulfil a specification. Since plant perturbation ΔP(s)
affects the stability of control system, it is desirable to make ΔP(s)
small, which is realized by determining the nominal plant Pn(s)
close to the plant P(s). On the other hand, since Q(s)
is the sensitivity to the sensor noise, Q(s) must be determined,
so that influence of the sensor noise does not become excessively large.
Therefore, the bandwidth of the low-pass filter Q(s) is limited
by the bandwidth of the sensor noise γ.


## 3. Experimental Results

To confirm the validity of the proposed control
algorithm, simulations and experiments are carried out. The low-speed operation
is especially focused on the simulation and experiment. The hardware consists
of a 3-phase, 380 V, 50 Hz, 4-pole, 3 kW-induction motor, and power circuit
with a matrix converter. A dual controller system consisting of a 32-bit DSP
(ADSP 21062) and a 16-bit microcontroller (80C167), in conjunction with a
12-bit A/D converter board is used to control the matrix converter-based
induction motor drive [5]. Figure 3 shows the experimental results of the current
control with and without the nonlinearity compensation in the low-speed region.
The motor is operated at 100 rpm. It can be seen from Figure 4 that the 5th and 7th harmonic
currents are reduced by 61% and 75% with the proposed
compensation method using the disturbance observer.Experimental result of current control at 100 rpm (speed and phase current).
(a)
With the proposed compensation(b)
Without compensationFigure 4
Comparison of output current harmonics at 100 rpm.


## 4. Conclusion

In order to realize high-performance control of
induction motor drives fed by matrix converter, a new online compensation
method for a nonlinearity of matrix converter drives has been proposed in this
paper. The proposed method using disturbance observer compensates the
nonlinearity of a matrix converter drive such as commutation delay, turn-on and
turn-off time of switching device, and an on-state switching device voltage
drop in online manner. Experimental results show that the
proposed compensation method can provide good compensation performance.



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*Source: 215782-2008-11-24.xml*