A Study of Quantitative Design Method of Adaptive
Current Control System with Armature Resistance
Identification Function
Jun-ichi Itoh

Yuki Nakajima

Dept. of Electrical, Electronics and Information Engineering
Nagaoka University of Technology
Nagaoka Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Dept. of Electrical, Electronics and Information Engineering
Nagaoka University of Technology
Nagaoka Niigata, Japannyuuki@stn.nagoakaut.ac.jp

Abstract— This paper clarified quantitative design method ofadaptive current control system with armature resistanceidentification function. Additionally, it was proved in this design
method that response error occurs between design and actual
responses. However, it was confirmed from analysis that theerror does not depend on motor parameters. As a result, theresponse error can be compensated by adding margin into thedesign value.

Furthermore, in case that the position estimation method basedon a minimum order of observer is used, a method to identify
the armature resistance is required separately because theobserver cannot be added with additional armature resistanceidentifier into the system [9].

Keywords- IPMSM; armature resistanceadaptive current control; design methodI.

identification;

INTRODUCTION

Recently, the sensorless vector control has been attracted
attention intensely. The sensorless vector control is a low cost
structure but achieves high reliability in the motor drive
system. Therefore, many sensorless vector control methods
have been proposed and studied [1-6].
In the sensorless vector control, accurate motor parameters
are required to achieve high performance because theseparameters are needed to estimate magnetic pole position.
However, these motor parameters are different from one toanother even it is a same type motor. Therefore, a mismatchoccurs between the actual armature resistance value and theresistance value which is applied in the sensorless vector
control. Additionally, the motor parameters are varied
according to the operating condition. For example, thearmature resistance value is varied to the temperature.
Therefore, if the motor temperature rises, mismatch error will
occur. These mismatches will cause a poor control
performance especially at low speed. Furthermore, highresponse resistance identification is important in order tostabilize the motor at low speed drive because the error of
output voltage caused by the dead-time is able to takeequivalent resistance.
Ref. 7 has proposed a speed and armature resistanceidentification method based on an adaptive full-order observer
to solve the problem. However, this method has a problem thatthe armature resistance identification system is unstable [8].

In order to solve these problems, an adaptive current
control system with armature resistance identification function
was suggested [10]. This method is a simple configuration
similar to PI current controller. Stability of this method has
been ensured by Lyapunov stability theory. Additionally, thissystem is decoupled with the position estimation system; eachidentification system is independent to another. Therefore, thismethod is possible to combine with a position estimation
method.
However, the design procedure for this method has notbeen discussed. In this method, clarification of the design
method is important because this method includes the current
control system. Additionally, in case where the application has
an unstable load such as air conditioner or waving machine, theidentification system needs to regulate the responses quick andaccurate because the identification operation uses current
control error. In another word, when current value converges tocommand value, identification value converges the actual value.
Furthermore, high response identification is needed in order toobtain high accuracy identification function because resistancevalues are varied by equivalent resistance due to the dead-time.
In this paper, a quantitative design method of the adaptive
current control system features to identify armature resistanceis discussed. The parameter design of this system needs toconsider the response of the current control system because theidentification system and the current controller are coupled.
This paper is organized as follows: firstly, the adaptive current
control system is simplified by linearization to derive thetransfer function of a standard form to a second-order system.
As a result, design equations of the current control system arederived. Secondly, the parameters that cause the error between
design value and actual response are studied. From the result, it
is clarified that the error is independent from the motorparameters. Finally, the accuracy of this design method is
confirmed by simulation and experimental results. In thesimulation, step response and operation with load oscillation
are confirmed. In the experimentation, the step response isconfirmed by using a 1.5kW IPMSM and a 2-level inverter.
A QUANTITATIVE DESIGN METHOD OF AN ADAPTIVECURRENT CONTROL SYSTEM

II.

g R̂
s

A. Adaptive Current Control System Features Identify
Functions With Armature Resistance
Figure 1 shows a block diagram of the adaptive current
control system, where Kd and Kq are the proportional gain, g is
the adaptive gain, id* and iq* are the dq-axis current command
value, id and iq are the dq-axis current detection value, eid andeiq are the current control error, vd* and vq* are the dq-axis
voltage command value, vRq and vRd are the compensatedvoltage value of voltage drop in armature resistance, ωrm is theangular velocity of the motor, and KE is the back electromotive
force of the motor. This control method identifies the armature
resistance value by the use of the current control error.

Fig.1

1
Ra + sLqgs(a) id=0 model.

A voltage control equation of the adaptive current control
system is given by (1).
⎡v ⎤ ⎡ Rˆ
⎢ *⎥ = ⎢⎢⎣vq ⎥⎦ ⎣ωre Ld*d− ωre Lq ⎤ ⎡id ⎤ ⎡ K d⎥⎢ ⎥ + ⎢Rˆ ⎦ ⎣iq ⎦ ⎣ 0()

Rˆ = g ∫ i d eid + iq eiq dt(2)

Equation (2) is derived by Lyapunov stability theory. This
control system ensures the system stability as long as thecondition of (2) is satisfied.
B. Quantitative Design Method of Adaptive Current Control
SystemFigure 2 shows the control block diagram that linearized
the nonlinear term of figure 1, where, iqs is the q-axis current
steady state value, F(s) is the command filter which transformthe transfer function into a second order system. First of all,
id=0 control is supposed to ignore in the coupling term between
d and q axis as shown in Figure 1. Second, the nonlinear term
is linearized. vRq as shown in Fig.2(a) is given by (3).

(gi q eiqs1
Ra + sLq0 ⎤ ⎡eid ⎤ ⎡ 0 ⎤ (1)+K q ⎥⎦ ⎢⎣eiq ⎥⎦ ⎢⎣ωrm K E ⎥⎦where Ra is the armature resistance value, R̂ is armature
resistance identification value, and Ld and Lq are the dq-axis
synchronous inductance. This control system consists of threesystems. First is the compensation system for armature
resistance and coupling term as shown in the first term of (1).
Second is the proportional control system as shown in thesecond term of (1). Third is the compensation system of backelectromotive force as shown in the third term of (1). In
addition, the equation to identify the armature resistance isgiven by (2).

v Rq = i q

Adaptive current control system.

)

(3)

g
s

(b) Linearization model.
Fig.2

Linier approximated control block diagram

Equation (3) is a nonlinear equation because there is amultiplication between some variables such as iq and eiq in thisequation. Here, equation (3) is linearized on the supposition
that d axis current is nearly a steady state. When (3) isseparated into the steady term and ripple term, vRq is obtainedby (4).

(

v Rq = i qs + Δi q

) gs (i e + Δi e + i Δe + Δi Δe )qs iqs

q

iqs

qs

iq

q

iq

(4)

where, iqs and eiqs are the steady term, Δiq and Δeiq are theripple term. The product of the ripple components Δiq and Δeiq
can be neglected because the value is small. In addition, Δiq
equals to zero when the q-axis current is controlled to constant.
Then, linearized vRq (=vRq’) is obtained by (5).
v Rq ' = i qs(

g
i qs e iq
s

)

(5)

Equation (5) is a linear equation because the iqs is a constantvalue.
Finally, the command value filter given by (6) is added sothat it transforms into a second order system (Fig.2(b)).

F (s ) =1
Kq
2

i qs g(6)

From figure 2(b), transfer function G(s) is obtained by (7).
G (s ) =2(

1
Ra + sLq

s +1

i qs g

)

g
s

g
s

(7)

2

L q s 2 + s R a + K q + i qs g

Here, the variables of (7) is compared with the variables inthe transfer function of second order system G’(s) given by (8).
G ' (s ) =Fig.3 Linear approximated control block diagram with linearωn2
s 2 + 2ζω n s + ω n(8)2

As the result, the proportional gain Kq and adaptive gain g
can be obtained by (9) and (10) respectively.
K q = 2 ζω n L q − R ag=(a) 800W SPMSM.

(10)

2

With any ωn and ζ value, Kq and g are calculated by (9) and(10). Therefore, it is possible to design any response waveformon this control system.
III.

Table.1 Motor Parameters used in simulations.

(9)ω n 2 Lqi qs

approximation error model.

(b) 750W IPMSM.

MATHEMATICAL ANALYSIS OF THE ERROR OCCURRING
BY LINEARIZATION

A. Response Error Occurring By Linearization
This design method uses linearization to derive the transfer
function. Therefore, the block diagram used to derive design
the equation is technically different from the actual current
control system. The difference becomes larger when the stateof current control system is differed from steady state, which isprecondition of linearization. This factor refers that current
response waveform becomes different from design value whenlarge variation input occurs on the current command.
Therefore, a mathematical method is used to analyze thedifference between the actual current and block diagram value,
to clarify the parameters depend on the response error.

(c) 3.7kW IPMSM.

B. Mathematical Analysis of the Errors
Figure 3 shows the linearization model added with error
terms. The error equation is derived as following.
From the comparison between (3) and (5), the relationshipbetween vRq and vRq’ is obtained as (11).
⎛ Δi q ⎞⎛⎟⎜ v Rq '+i qs g Δi q Δeiq ⎞⎟v Rq = ⎜1 +
⎜
⎟
i
s
⎠
qs ⎠⎝
⎝

(

)

(11)

From (11), it can be confirmed that the added error andproportional error are including the compensated voltage ofvoltage drop which is caused by the armature resistance.

Fig.4 Step response waveform.

Here, equation (11) does not include the motor parameters.
Therefore, the occurred error is not depending on the motorparameters. In addition, the terms for the error units are
infinite. As a result, the error is not depending on the rated
value of the either.
Based on the above equations, it can be assumed that the
response error occurring at the linearization is independent to
the motor.
IV.

SIMULATION RESULTS

In this section, simulation results are provided to confirm
the step response waveform of the q-axis current. Table 1
shows the simulation conditions. These three motors aredifferent in types of motor and rated power.
A. Accuracy Design in the Proposed Design Method
Figure 4 shows the step response of the q-axis current
waveform when the q-axis command value is controlled from0.95 p.u. to 1.00 p.u.. The motor shown in Table 1(a) is used inthis simulation result. In addition, iqs is 1.00p.u. and the load isa constant speed at 0 p.u.. Moreover, the control system isdesigned as ζ is 0.7 and ωn is 4000rad/s by using (9) and (10).
From figure 4, it is confirmed that the step responses of the qaxis current waveform follows accurately to the step response
waveform of the second order system. Furthermore, based on
the overshoot waveform and the time to reach peak value (peakof overshoot), these can calculate that ζ is 0.68 and ωn is 3969
rad/s. Comparing to the design value, the error ratio is
approximately -0.8%.
Figure 5 shows a comparison on the step response of q-axis
current command with change in the step width operating
under a different angular frequency. Here, the ωn_cal is thecalculation value of the ωn from the response waveform, andthe ωn_des is the design value of the ωn. From figure 5, it isconfirmed that the error between the ωn_cal and the ωn_desbecomes larger when the step width of the q-axis current
command value becomes larger. This phenomenon agrees withthe supposition in the chapter III.B. Thus, the response error
becomes larger when the state of the current control system isdiffered from the steady state because the linearization method
assumes the current is nearly to steady state, iq = iqs.
Furthermore, the result shows the trend of error is constant with
the natural angular frequency

Fig.5 Natural angular frequency
between design and simulated value.

1.00
iq*
0.50

0

iq
iq_lin+err

iq_lin

0

0.5

1.0

1.5

2.0

[ms]

2.0

[ms]

0.001

0
iq-iq_lin+err-0.001
0

0.5

1.0

1.5

Fig.6 Step response waveformwith considering error term model.

B. Analysis of the response error occurring by thelinearization
Figure 6 shows the comparison on the step response of qaxis current command between the block diagram figure 1 andfigure 3. Here, iq* is varied from 0.1 p.u. to 1.0 p.u.. In addition,
iqs is 1.0p.u. and control system is designed as ζ=0.7 ，
ωn=8000rad/s. Furthermore, in figure 6, iq_lin is the step
response waveform of the linearization model (Fig.2(b)), and
iq_lin+err is the step response waveform of the block diagram that
added with the error term (Fig.4).
When id=0 control is used, the actual current response
waveform obtained from the block diagram (Fig.1) is differedfrom the current response waveform obtained from linearizedblock diagram (Fig.2(b)). Here, these are confirmed that
ζ=0.70 and ωn=8000 rad/s from the overshoot and the time toreach peak of the response waveform. From the above, whenFig.7 Natural angular frequency error
between design and simulated value as three motors.

the step width of the q-axis current command value is large, it
can be confirmed that the actual response waveform becomesdifferent from the designed response.

In addition, the step response waveform as the block
diagram of figure 4 is confirmed to the actual current response
waveform. Therefore, the cause of the current response error isthe added and proportional error term as shown in equation
(11).

V.

q-axis current
iq [p.u.]

C. Variation of armature resistance identification value withdesign of current control system
Figure 8 shows the results of armature resistanceidentification value R̂ with and without the proposed design
control. Figure 8(a) shows a result that the proposed control is
not included, Kq is 3.0 and g is 15.0. On the other hands,
Figure 8(b) demonstrates a result that included the proposed
control system, the design values are follows ζ is 0.7 and ωn is
4000rad/s by using (9) and (10), respectively. When the load
is fluctuating, q-axis current is changing and R̂ becomesdifferent. If the control system is not designed correctly, then
the R̂ will not converge. In another words, armature resistancevalue cannot be identified. On the other hands, if the control
system is designed correctly, the R̂ can be converged and thearmature resistance value can be identified. Therefore, it canclarify that the identification system is necessary in order todesign the system correctly.

(a) Kq=3.0 g=15.0 (not using command filter)

Ra identification value[p.u.]

Figure 7 shows a comparison result with three types ofmotors. The step width of q-axis current command is varied
according to the natural angular frequency. From figure 6, it isconfirmed that the error in the natural angular frequency error
becomes larger when the step width of the q-axis current
command value becomes larger. In addition, the trend of theerror remains constant even the motor has changed. Again, it isconfirmed that the error in natural angular frequency error rate
is not depending on the motor parameters. This result agrees
with chapter III.B. Therefore, we can achieve the response by
design using the actual design value increased to 1.13 times ofrequired natural angular frequency.

(b) Kq=20.7 g=899.5(ζ =0.7 ωn=4000rad/s).
Fig.8 Response of load oscillation

EXPERIMENTAL RESULTS

Figure 9 shows the experimental schematic. In theexperimental setup, the AC source is connected to the dioderectifier stage as an input power, and an IPM motor isconnected to the inverter stage. Furthermore, an induction
motor is connected to the IPM motor stage as a load. Table 2
shows the motor parameter using in the experiment. The motoris 1.5 kW and 2000rpm.
A. Response Error Occurring By Linearization
Figure 10 shows the step response of q-axis current
waveform when the iq* is varied from 0.5p.u. to 0.7p.u.. In thiscase, ζ=0.7, and ωn is varied from =1000 rad/s, 2000 rad/s, and4000 rad/s. Here, id=0 control is used, iqs is 1.0p.u.. In addition,
the load was a constant speed at 0 p.u.. From figure 10, it isconfirmed that the id is controlled to nearly 0 p.u.. Additionally,
it is confirmed that the step response waveform of iq
corresponds to the step response waveform of the second ordersystem even the ωn is changed. Furthermore, the setting time ofiq becomes half when the design value of natural angularfrequency is double. From the above, the validity of theclarified design method in the chapter II can be confirmed.

Fig.9 Experimental system.
Table.2 Motor Parameters used in experimentations.

B. Analysis of the response error occurring by thelinearization
Table 3 shows the natural angular frequency calculated
from the overshoot and time to reach peak of overshoot basedon figure 10. From table 3, the error rate of natural angularfrequency is lesser than 10%, which demonstrate a highaccuracy in the designed system. However, the error rates ofnatural angular frequency are varied with the design value ofnatural angular frequency. In considered to this subject, theinfluence of discretization in the controller is investigated.
Figure 11 shows the comparison results of the naturalangular frequency between simulation and experimental results.
Note that the simulation result calculated from simulator,
which the simulation condition is confirmed similar to theexperimental condition. The simulation results are shown incontinuous and discrete system for each design. From figure 11,
the trend of natural angular frequency obtained by experimentand simulation are nearly matched. In addition, the naturalangular frequency errors occur between the design value andthe simulation result in the continuous system is approximately
at 25% difference. This trend is similar to the error of naturalangular frequency as shown in figure 5. Furthermore, when thecontroller is discretized, variable of the actual natural angularfrequency is confirmed. Accordingly, it is considered that thedivergence between experimental result and analysis value iscaused by the discretization.
Table 4 shows finite values of the resistance identification.
For the experimental result, it is confirmed that resistanceidentification value is converged at approximately 1.3p.u..
This result shows that the true resistance value duringexperimentation is around 1.3 times higher than the nominal
resistance value. Here, the resistance value measured
immediately after the experimentation is around 1.2 times
higher than the nominal resistance value. The cause ofdifference between the measured value and the nominal valueof armature resistance is temperature which is the heat causedby the current.

(a) ωn=4000rad/s ζ=0.7(b) ωn=2000rad/s ζ=0.7

Figure 12 shows the comparison result on a 10% setting
time of resistance identification value between the simulationand experimental results. Here, the simulation result is obtainedby the discretization model. From figure 11, it is confirmed thatthe setting time of identification value is well matched with thesimulation result. This result shows that the time to converge
the resistance identification value can set to be an arbitrarily.
VI.

CONCLUSION

In this paper, the quantitative design method of the adaptive
current control system functions to identify armature resistancehas discussed. Firstly, the quantitative design method of theadaptive current control system was clarified. Secondly, thecause of the error in the design was analyzed. Finally, thefollowing shows the conclusion based on the simulation andexperimental results.
1) When the current is nearly to steady state, it was
confirmed that the error ratio of natural angular frequency isapproximately -0.8% comparing to the design value.
2) The error of natural angular frequency occurs
depending on the step width in the linearization. However, thiserror is possible to compensate because this error is notdepending on the motor parameters.
3) The validity of the design method has confirmed insimulation. Additionally, it was clarified that the naturalangular frequency is not depending on the motor parameter.

(c) ωn=1000rad/s ζ=0.7
Fig.10 Step response waveform as the experimentation.
Table.3 natural angular frequency calculated from experimental
waveform.

4) From the experimental results, it was confirmed that thestep response waveform of the iq corresponds to the step
response waveform of the standard form of the second ordersystem. In addition, the error of natural angular frequency islesser than 10%.

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Fig. 11 Comparison experimental resultswith simulation results.
Table.4 Armature resistance identification value
obtained by experiment.

Fig.12 Setting time of armature resistanceidentification value.

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