to solve this instability, a simply modified current controller is proposed in this paper. To
guarantee both robust stability and current control performance simultaneously, this paper
employees two degree of freedom (2DOF) structure fot the current controller, which can
enlarge stable region and maintain its performance (Hasegawa et al. (2007)). Finally, some
experiments with a disturbance observer for sensor-less control show that the proposed
current controller is effective to enlarge high-speed drives for IPMSM sensor-less system.
2. IPMSM model and conventional controller design
IPMSM on the rotational reference coordinate synchronized with the rotor magnet (d −q axis)
can be expressed by

v
d
v
q

=

R
+ pL
d
−Pω
rm
L
q
Pω
rm
L
d
R + pL
q

d
− Pω
rm
L
q
i
q
,(2)
v
∗
q
= v

q
+ Pω
rm
(L
d
i
d
+ K
E
) ,(3)
where v

d
and v

q
are obtained by amplifying current control errors with proportional - integral

) ,(5)
in which x
∗
means reference of x. From (1) to (5), feed-back loop for i
d
and i
q
is constructed,
and current controller gains are often selected as follows:
K
pd
= ω
c
L
d
,(6)
K
id
= ω
c
R ,(7)
K
pq
= ω
c
L
q
,(8)
K
iq

− L
q
are
employed.
508
Robust Control, Theory and Applications
S
N
d
q
re
re
Fig. 1. Coordinates for IPMSMs
IPMSM
on d-q axis
re
Current Regulator
on axis
Current Regulator
on axis
)(⋅R
)(⋅R
1
+
ii
**
ii
qd
ii
qd

+ L
γ
p −Pω
rm
L
δ
+ L
γδ
p
Pω
rm
L
γ
+ L
γδ
pR+ Pω
rm
L
γδ
+ L
δ
p

i
γ
i
δ

+ Pω
rm

q
) sin
2
Δθ
re
,
L
γδ
=
L
d
− L
q
2
sin 2Δθ
re
.
It should be noted that the equivalent resistances on d axis and q axis are varied as ω
rm
increases when L
γδ
exists, which is caused by Δθ
re
.Asaresult,Δθ
re
forces us to modify the
509
Robust Current Controller Considering Position Estimation Error for Position
Sensor-less Control of Interior Permanent Magnet Synchronous Motors under High-speed Drives
current controllers (2) – (5) as follows:

γ
=
K
pd
s + K
id
s
(i
∗
γ
−i
γ
) , (13)
v

δ
=
K
pq
s + K
iq
s
(i
∗
δ
−i
δ
) . (14)
3.2 Closed loop system of current control and s tability analysis
This subsection analyses robust stability of the closed loop system of current control. Consider

+ L
γ
p ΔZ
γδ
(p, ω
rm
)
ΔZ
δγ
(p, ω
rm
) R + Pω
rm
L
γδ
+ L
δ
p

i
γ
i
δ

+Pω
rm
K
E

−sin Δθ

+ L
γδ
p ,
ΔZ
δγ
(p, ω
rm
)=Pω
rm
(L
d
− L
q
) sin
2
Δθ
re
+ L
γδ
p .
It sh ould be noted that the decoupling controller fails to perfectly reject coupled terms because
of Δθ
re
. In addition, with current controllers (13) and (14), the closed loop system can be
expressed as shown in Fig.3, the transfer function (16) is obtained with the assumption
pΔθ
re
= 0, pω
rm
= 0 as follows:

F
γδ
(s)=
ΔZ
γδ
(s, ω
rm
) ·s
L
γ
s
2
+(K
pd
+ R − Pω
rm
L
γδ
)s + K
id
,
F
δγ
(s)=
ΔZ
δγ
(s, ω
rm
) ·s
L

id
,
G
δ
(s)=
K
pq
·s + K
iq
L
δ
s
2
+(K
pq
+ R + Pω
rm
L
γδ
)s + K
iq
.
Figs.4 and 5 show step responses based on Fig.3 with conventional controller (designed with
ω
c
= 2π ×30 rad/s) at ω
rm
=500 min
−1
and 5000 min

0.5
1
Time sec]
Current i
γ
[A]
(a) γ axis current response (b) δ axis current response
Fig. 4. Response with the conventional controller (ω
rm
= 500 min
−1
)
was intentionally given by Δθ
re
= −20
◦
. i
∗
δ
was stepwise set to 5 A and i
∗
γ
was stepwise kept
to the value according to maximum torque per current (MTPA) strategy:
i
∗
γ
=
K
E

Sensor-less Control of Interior Permanent Magnet Synchronous Motors under High-speed Drives
0 0.05 0.1 0.15 0.2
0
5
10
Time [sec]
Current i [A]
0 0.05 0.1 0.15 0.2
0
2
4
6
8
10
Time [sec]
Current i [A]
(a) γ axis current response (b) δ axis current response
Fig. 5. Response with conventional controller (ω
rm
= 5000 min
−1
)
fail to be successfully regulated to each reference in high-speed region because of Δθ
re
,which
is often visible in position sensor-less control systems.
Figs.6 and 7 show poles and zero assignment of G
γ
(s) and G
δ

+ R + Pω
rm
L
γδ
> 0 . (19)
Fig.8 shows stable region by conventional current controller, which is plotted according to (18)
and (19). The figure shows that stable speed region tends to shrink as motor speed increases,
even if position error Δθ
re
is extremely small. It can also be seen that the stability condition on
γ axis (18) is more strict than that on δ axis (19) because of K
pd
< K
pq
, in which these gains
are given by (6) and (8), and L
d
< L
q
in general. To solve this instability problem, all poles of
G
γ
(s) and G
δ
(s) must be reassigned to stable region (left half plane) even if there exists Δθ
re
.
This implies that equivalent resistances in γ
−δ axis need to be increased.
4. Proposed current controller with 2DOF structure

(s) and G
δ
(s) at ω
rm
= 500 min
−1
Imaginary Axis
Real Axis
Poles
Zero
Imaginary Axis
Real Axis
Poles
Zero
(a) G
γ
(s) (b) G
δ
(s)
Fig. 7. Poles and zero assignment of G
γ
(s) and G
δ
(s) at ω
rm
= 5000 min
−1
Position Error [deg.]
Speed [min ]
-15000

γ
=
K
pd
s + K
id
s
(i
∗
γ
−i
γ
) −K
rd
i
γ
, (20)
v

δ
=
K
pq
s + K
iq
s
(i
∗
δ
−i


γδ
(s)
F

δγ
(s) 1

−1

G

γ
(s) ·i
∗
γ
G

δ
(s) ·i
∗
δ

514
Robust Control, Theory and Applications
Fig. 10. Current control system with K
rd
and K
rq
where

(s, ω
rm
) ·s
L
δ
s
2
+(K
pq
+ L
rq
+ R + Pω
rm
L
γδ
)s + K
iq
,
G

γ
(s)=
K
pd
·s + K
id
L
γ
s
2

γδ
)s + K
iq
.
From these equations, stability criteria are given by
K
pd
+ K
rd
+ R −Pω
rm
L
γδ
> 0 , (22)
K
pq
+ K
rq
+ R + Pω
rm
L
γδ
> 0 . (23)
The effect of K
rd
and K
rq
is described here. It should be noted from stability criteria (22) and
(23) that these gains are injected in the same manner as resistance R, so that the current control
loop system with K

the proposed current controller, PI gains are selected in the same manner as occur in the
conventional design:
K
pd
= ω
c
L
d
, (24)
K
id
= ω
c
(R + K
rd
) , (25)
K
pq
= ω
c
L
q
, (26)
K
iq
= ω
c
(R + K
rq
) . (27)

c
.
Therefore, the proposed design can improve robust stability by only proportional gains K
rd
and K
rq
while maintaining closed loop dynamics of the current control. This is why the
authors have chosen to adopt 2DOF control.
4.4 Design of K
rd
and K
rq
, and pole re-assignment results
As previously described, re-assigned poles by proposed controller (= −
R+K
rd
L
d
, −
R+K
rq
L
q
)can
further be moved to the left in the s
−plane as larger K
rd
and K
rq
are designed. However,

= −R + ω
c
L
d
, (30)
K
rq
= −R + ω
c
L
q
. (31)
Based on this design, characteristics equation of the proposed current closed loop (the
denominator of G

γ
(s) and G

δ
(s) ) is expressed under Δθ
re
= 0by
Ls
2
+ 2ω
c
Ls + ω
2
c
L = 0,

. It can be seen from this stability map that
operation point P can be stabilized by the proposed current controller with 2DOF structure,
despite the fact that the conventional current regulator fails to realize stable control and
current diverges, as shown in the previous step response.
Fig.12 demonstrates that stable step response can be realized under ω
rm
=5000 min
−1
and
Δθ
re
= −20
◦
. These results demonstrate that robust current control can experimentally be
realized even if position estimation error Δθ
re
occurs in position sensor-less control.
516
Robust Control, Theory and Applications
-15000
-10000
-5000
0
5000
10000
15000
-40 -30 -20 -10 0 10 20 30 40
Position Error [deg.]
Stable region
by proposed controller

14bit
A/D
14bit
A/D
COUNTER
PE
IPMSM
LATCH
INVERTER
DRIVER
PWM
Pattern
Dead Time
FPGA
3φ
AC200[V]
θ
re
i
v
TMS320C6701
DSP
LEAD LAG
TMS320C6701 SYSTEM BUS
FPGA
v
*
Fig. 13. Configuration of system setup
5.2 Robust stability of current control to rotor position error
The first experiment demonstrates robust stability of the proposed 2DOF controller. In

−1
are illustrated in Figs.16 and 17. In the case of conventional
controller, current control system became unstable at Δθ
re
= −10
◦
as shown in Fig.16. Fig.17
shows results of the proposed 2DOF controller, in which currents were also tripped at Δθ
re
=
−
21
◦
.AllΔθ
re
to show unstable phenomenon is met to (18) and (19), which describes that
the robust stability analysis discussed in the previous section is theoretically feasible. This
robust stability cannot be improved upon as far as the proposed strategy is applied. In other
words, furthermore robust stability improvement necessitates higher cut-off frequency ω
c
,
which forces us to employ high-performance processor.
5.3 Position sensor-less control
This subsection demonstrates robust stability of current control system when position
sensor-less control is applied. As the method for p osition estimation, the disturbance observer
based on the extended electromotive force model ( Z.Chen et al. (2003) ) was utilized for
all experiments. Rotor speed estimation was substituted by differential value of estimated
518
Robust Control, Theory and Applications
sec1

i
γ
i
o
40
1
min4000
−
o
0
1
min0
−
A0
A0
re
θΔ
A0
A8
rm
ω
Fig. 14. Current control characteristics by conventional controller at 4500min
−1
δ
δ
γ
Fig. 15. Current control characteristics by proposed controller at 4500min
−1
δ
δ

A0
re
θΔ
A0
A8
rm
ω
o
40
sec0.2
*
i
i
i
1
min4000
−
o
0
1
min0
−
A0
A0
re
θΔ
A0
A8
rm
ω

.
Also, these figures show that Δθ
re
of about − 40
◦
is steadily caused because of estimation
delay in disturbance observer. Needless to say, this error can be compensated since DC
component of Δθ
re
can be obtained in advance according to motor speed and LPF time
constant in disturbance observer. Δθ
re
cannot be compensated, however, at the transient time.
520
Robust Control, Theory and Applications
sec0.5
*
i
i
i
1
min4000
−
o
0
1
min0
−
A0
A0

to 6500min
−1
by the
conventional PI current controller and proposed controller(designed with ω
c
= 200rad/s),
respectively. 20% motoring load was given by the induction motor in these experiments.
It turns out from Fig. 20 that current control begins to oscillate at 0.7sec due to Δθ
re
,and
then the amplitude of current oscillation increases as speed goes up. On the other hand, the
proposed current controller (Fig. 21) makes it possible to realize stable step response with the
assistance of the robust current controller to Δθ
re
.
It should be noted that these experimental results were obtained by the same sensor-less
control system except with a dditional gain and its design of the proposed current controller.
Therefore, these sensor-less control results show that robust current controller enables us to
improve performances of t otal control system, and it is important to design robust current
controller to Δθ
re
as well as to re alize precise position estimation, which has been surveyed by
many researchers over several decades.
521
Robust Current Controller Considering Position Estimation Error for Position
Sensor-less Control of Interior Permanent Magnet Synchronous Motors under High-speed Drives
6. Conclusions
This paper is summarized as follows:
1. Stability analysis has been carried out while considering its application to position
sensor-less system, and operation within stable region by conventional current controller

Salient-pole PMSM Based on Extended EMF in Rotating Reference Frame, IEEE
Trans. on on Industry Applications 38(4): 1054 – 1061.
Z.Chen, M.Tomita, S.Doki & S.Okuma (2003). An Extended Electromotive Force Model for
Sensorless Control of Interior Permanent-Magnet Synchronous Motors, IEEE Trans.
on Industrial Electronics 50(2): 288 – 295.
522
Robust Control, Theory and Applications
João Marcos Kanieski
1,2,3
, Hilton Abílio Gründling
2
and Rafael Cardoso
3
1
Embrasul Electronic Industry
2
Federal University of Santa Maria - UFSM
3
Federal University of Technology - Paraná - UTFPR
Brazil
1. Introduction
The most common approach to design active power filters and its controllers is to consider
the plant to be controlled as the coupling filter of the active power filter. The load dynamics
and the line impedances are usually neglected and considered as perturbations in the
mathematical model of the plant. Thus, the controller must be able to reject these perturbations
and provide an adequate dynamic behavior for the active power filter. However, depending
on these perturbations the overall system can present oscillations and even instability. These
effects have been reported in literature (Akagi, 1997), (Sangwongwanich & Khositkasame,
1997), (Malesani et al., 1998). The side effects of the oscillations and instability are evident in
damages to the bank of capacitors, frequent firing of protections and damage to line isolation,

The schematic diagram of the power quality conditioning device, consisting of a DC source
of energy and a three-phase/three-legs voltage source PWM inverter, connected in parallel to
the utility, is presented in Fig 1.
Fig. 1. Schematic diagram of the power quality conditioning device.
The Kirchoff’s laws for voltage and current, applied at the PCC, allow us to write the 3
following differential equations in the ”123” frame,
v
1N
= L
f
di
F1
dt
+ R
f
i
F1
+ v
1M
+ v
MN
, (1)
v
2N
= L
f
di
F2
dt
+ R

dqO
=
2
3
⎡
⎢
⎢
⎣
sin
(
ωt
)
sin

ωt −
2π
3

sin

ωt
−
4π
3

cos
(
ωt
)
cos

v
q
v
O

T
= C
123
dqO

v
1
v
2
v
3

T
, (5)

i
d
i
q
i
O

T
= C
123


T
. (7)
The inverse process is given in equations (8)-(10),

v
1
v
2
v
3

T
= C
dqO
123

v
d
v
q
v
O

T
, (8)

i
1
i


d
d
d
q
d
O

T
, (10)
where,
C
dqO
123
= C
123
dqO
−1
=
3
2
C
123
dqO
T
, (11)
and d is the switching function (Kedjar & Al-Haddad, 2009). As it is a three-phase/three-wire
system, the zero component of the rotating frame is always zero, thus the minimum plant
model is then given by Eq. (12)
d

−ω
ω
R
f
L
f
⎤
⎦
, B
= −

v
dc
L
f
0
0
v
dc
L
f

,
E
=

1
L
f
0

Fig. 3. Open loop system with variable line inductance
distorted waveform. It can be visualized in Fig. 4, that shows the voltage signals v
123
at the
PCC, for a line inductance of L
S
= 2mH.
Fig. 4. Open loop voltages at the PCC with line inductance of L
S
= 2mH.
526
Robust Control, Theory and Applications
Fig. 5 shows now an extreme case, with line inductance of L
S
= 5mH, it is also visually
perceptible the significant growth on the voltage harmonic content.
Fig. 5. Open loop voltages at the PCC with line inductance of L
S
= 5mH.
Concluding, the voltages at the PCC have its dynamic substantially dependent on the line
impedance. In other hand, the system dynamic is directly associated with the PCC voltages.
Therefore, the control of this kind of system strongly depends on the behavior of the voltages
at the PCC. As the output filter of the Voltage Source Inverter (VSI) has generally well-known
parameters (they are defined by the designer), which are at most fixed for the linear system
operation, one of the greatest control challenges of these plants is associated with the PCC
voltages. The text that follows is centered on that point and proposes an adaptive and a fixed
robust algorithm in order to control the chosen power conditioner device, even under load
unbalance and line with variable or unknown impedance.
3. Robust Model Reference Adaptive Control (RMRAC)
The RMRAC controller has the characteristic of being designed under an incomplete

0
(s)
(13)
where u represents the control input of the system and i
F
is the output variable of interest as
shown in Fig. 1.
527
Robust Algorithms Applied for Shunt Power Quality Conditioning Devices
⇒ Assumptions for the Plant
H1.Z
0
is a monic stable polynomial of degree m(m ≤ n −1),
H2.R
0
is a monic polynomial of degree n;
H3.The sign of k
p
> 0 and the values of m, n are known.
For the unmodeled part of the plant it is assumed that:
H4.Δ
m
is a stable transfer function;
H5.Δ
a
is a stable and strictly proper transfer function;
H6.A lower bound p
0
> 0 on the stability margin p > 0 for which the poles of Δ
m

m
and Δ
a
. In (14), r is a uniformly
limited signal.
⇒ Assumptions for the model reference:
M1.Z
m
a monic stable polynomial of degree m(m ≤ n −1);
M2.R
m
is a monic polynomial of degree n.
The plant input is given by
u
=
θ
T
ω + c
0
r
θ
4
(15)
where θ
T
=

θ
T
1

=
α( s)
Λ(s)
u and ω
2
=
α( s)
Λ(s)
y. (16)
⇒ Assumptions for the signals ω
1
and ω
2
:
R1.The polynomial Λ in (16) is a monic Hurwitz of degree n
−1, containing stable eigenvalues.
R2.For n
≥ 2, α 

s
n− 2
, , s,1

T
and for n = 1, α  0.
For the adaptation of the control action parameters, the following modified gradient algorithm
was considered
˙
θ
= −σPθ−

0
−1

if M
0
≤

θ

<
2M
0
σ
0
if

θ

>
2M
0
(18)
where σ
0
> 0 is a parameter of design. P = P
T
> 0, ε = y − y
m
+ θ
T

with m
0
> δ
1
/δ
0
, δ
1
≥ 1 and δ
0
> 0.
The normalization signal m is the parameter which ensures the robustness of the system.
Looking to Eq. (15)-(19), it can be seen that when the control action u, the plant output y or
both variables are large enough, the θ parameters decreases and therefore the control action,
which depends on the θ parameters, also has its values reduced, limiting the control action as
well as the system output in order to stabilize the system.
3.3 RMRAC applied for the power conditioning device
In the considered power conditioning system, as shows Eq. (12), there is a coupling between
the "dq" variables. To facilitate the control strategy, which should consider a multiple input
multiple output system (MIMO), it is possible to rewrite Eq. (12) as
L
f
di
d
dt
+ R
f
i
d
= L

u
d
= L
f
ωi
q
−v
dc
d
nd
+ v
d
(21)
and
u
q
= −L
f
ωi
d
−v
dc
d
nq
+ v
q
, (22)
the MIMO tracking problem, with coupled dynamics, is transformed in two single input single
output (SISO) problems, with decoupled dynamics. Thus, currents i
d

q
ω
q
+ c
0
r
q
θ
4q
. (24)
The PWM actions (d
nd
and d
nq
), are obtained through Eq. (21) and (22) after computation of
(23) and (24).
529
Robust Algorithms Applied for Shunt Power Quality Conditioning Devices
3.3.1 Design procedure
Before starting the procedure, lets examine the hypothesis H1, H2, M1, M2, R1 and R2. Firstly,
as the nominal system, accordingly to Eq. (12), is a first order plant. The degrees n and m are
then defined by n
= 1 and m = 0. Therefore, the structure of the model reference and the
dynamic of signals ω
1
and ω
2
can be determined. By M1 and M2, the model reference is also
a first order transfer function W
m

i
q
+ r
q
θ
4q
. (27)
From the information of the maximum order harmonic, which has to be compensated by
the power conditioning device, it is possible to design the model reference, given in Eq. 25.
Choosing, for example, the 35
th
harmonic, as the last harmonic to be compensated, and W
m
(s)
with unitary gain, the model reference parameters become ω
m
= 35 ·2 ·π ·60 ≈ 13195
rad
s
and
k
m
= 1. Fig. 6 shows the frequency responses of the nominal plant of a power conditioning
device, with parameters L
f
= 1mH and R
f
= 0.01Ω and of a model reference with the
parameters aforementioned.
Fig. 6. Bode diagram of G

dc
550VL
S
5uH /2mH /5mH
θ
d
(0)[−1.02, 0.53]
T
P diag{0.99, 0.99}
θ
q
(0)[−1.02, 0.53]
T
k
m
1
c
0
1 ω
m
13195
rad
s
L
L
2mH L
L1
2mH
R
L