performance of robust controller for dfim when the rotor angular speed is treated as a time-varying parameter

Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011
Performance of robust controller for DFIM when the rotor angular speed is
treated as a time-varying parameter
Nguyen Tien Hung
1
, Ngo Duc Minh
2
1
Thainguyen University of Technology, email:
2
Thainguyen University of Technology, email:


Abstract: This paper describes the design of a
robust current controller for doubly-fed induction
machines (DFIM), in which the rotor angular speed
is considered as an uncertain parameter. The
robust controller is then synthesized to guarantee that
the -norm of the closed-loop system is smaller
than some given number for different frozen values of
. Next, the robust performance of the robust
controller with respect to other rotor angular speeds is
investigated for both constant and fast parameter
variations. Some simulation results are given to
demonstrate the performance and robustness of the
control algorithm.

1. Introduction
In the literature, the control structure of DFIM

including PI current controllers is described in [1],
[2], [3], [4]. In some cases, the cross coupling term in
the rotor equations that includes the mechanical
angular speed is eliminated by adding a feed-forward
term to the output of the q-axis controller [2], [5]. The
rotor mechanical angular speed is treated as an
scheduling parameter that is used for these
compensators. In these situations the difficulties of
the nonlinear dynamics of the doubly-fed induction
machine are not taken into account, i.e., the model of
the machine is linearized and it is assumed that the
machine parameters required by the control algorithm
are precisely known. Clearly, such controller designs
might result in a closed-loop behavior that is highly
sensitive to a change in operating conditions and/or
parameters.
In this work, a mixed loop shaping -design for the
rotor current control loop at fixed frozen values of the
rotor angular speed is presented first. Then the
performance of the closed-loop system with
controller designed for different frozen values of
for other rotor angular speeds is investigated. The
performance analysis is also extended for the case
with the face of the stator voltage action. As a further
investigation, the designed controller for a frozen
values of is tested for a fast variation of the rotor
speed along the whole parameter interval.

2. Preliminaries
2.1 Notations

Let denote the space of square-integrable signals
defined on the interval . A matrix is called
symmetric if it is real and satisfies . The set
of all symmetric matrices will be denoted by
. A transfer function with a state-space
realization will be denoted by



2.2 Linear matrix inequalities
A linear matrix inequality (LMI) has the form

(1)

where denotes the vector of decision
variables and .

2.3 The -norm
Consider a linear input-output system that is
described by

(2)

and whose transfer matrix is given by



If is stable and if we choose the initial condition
to be zero, defines a linear map on
with a finite energy gain defined as

It is well-known that the energy-gain of coincides
with the -norm of the corresponding transfer
matrix given by



where stands for the largest singular value of
the complex matrix matrix .

2.4 The bounded real lemma
It is not possible to explicitly compute in terms
of the realization matrices. Instead, one can
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characterize stability of and the validity of the
inequality

(3)

as an LMI in some auxiliary matrix and , which is
one version of the celebrated bounded real lemma.
Indeed, it can be shown [6] that is stable and that
(3) holds if and only if

and there exits some such that the
Riccati inequality

(4)

is satisfied. By the Schur lemma (4), these conditions
are equivalent to following system of LMIs [7]

(5)

This result is referred to as the bounded real lemma.
Yet another application of the Schur lemma allows to
rewrite these inequalities with into the
following form [8], [9]:

(6)

Note that (6) are LMI constraints on and . This
allows to determine the infimal for which (3) is
true, and hence in turn the value , by
minimizing over the constraint (6) which is a
standard LMI problem. Let us now show how this
procedure of analysis can be successfully generalized
to synthesizing controllers.

2.5 performance
A standard setup for control is presented in
Figure 1, where represents the generalized
disturbances, the controlled variable, the control

input and the measurement output, while is a
linear time-invariant system described as



w
z
u
y
P
K

Figure 1. The interconnection of the system

The goal in control is to find a stabilizing linear
time-invariant (LTI) controller that minimizes the
norm of the closed-loop system

(7)

where is lower linear fractional
transformation of and , which is nothing but the
closed-loop transfer function in Figure 1.

2.6 Sub-optimal control
Let us now consider a generalized plant where
weights are incorporated already as follows

(8)

If the linear time-invariant controller is expressed
as
(9)

the closed-loop system admits the following
state-space description:

(10)
where

(11)

In practice, the control problem is rather
concerned with finding an LTI controller which
renders stable and such that

(12)

holds true [11], where is a given number that
specifies the performance level. This is the so-called
sub-optimal problem.

2.7 controller synthesis
Using the bounded real lemma for (12), the matrix
is stable and (12) is satisfied if and only if the LMI

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(13)

holds for some . Unfortunately this inequality
is not affine in and in the controller parameters
which are appearing in the description of , , , .
However, a by now standard procedure [8], [9], [12],
[13] allows to eliminate the controller parameters
from these conditions, which in turn leads to convex
constraints in the matrices and that appear in the
partitioning of

(14)

According to that of in (11) one then arrives at the
following synthesis LMIs for -design [14]:

(15)
(16)
(17)

where and are basis matrices for the
subspaces

(18)

respectively.
Note that these inequalities are defined by open-loop
system parameters only, and that they depend affinely
on the design variables and . Hence we can
directly minimize over these LMIs in order to

compute the best possible level with (12) that can
be achieved by a stabilizing controller.
After having obtained and that satisfy (15)-(17)
for some level , the controller parameters can be
reconstructed by using the projection lemma [8]. This
procedure for -synthesis is implemented in the
robust control toolbox [15].

2.8 Mixed sensitivity approach
Figure 2a shows a simple feedback control system.
This interconnection can be recast into a standard
setup for control as depicted in Figure 2b. For
this control configuration, engineers are usually
interested in some specific transfer functions. In
particular, is the sensitivity function
which describes the influence of the external
disturbance to the tracking error .
is the complementary
sensitivity function which describes the influence of
the reference signal to the system output . Finally,
is the transfer function from to the control
input that indicates control activity [16].

++
¡¡
w
y
u
z
K

G

(a)

++
¡¡
w
z
G
u
P
y
K

(b)

Figure 2. General feedback control configuration

In general, performance of the closed-loop system
that is specified by norm of the channel in
(7) can be formulated as a multi-objective problem
(see Figure 3). This leads to the minimization of

(19)

The multi-variable loop shaping with various
specifications (19) is the so-called the mixed
sensitivity design approach.
++
¡¡

w
y
K
u
G
1
z
2
z
3
z
z


Figure 3. Mixed sensitivity control

It is well-known in the literature that the transfer
functions , , and need to be small in
magnitude in order to achieve good command
tracking and robust stability. However, the well-
known constraint reveals that these
requirements can not be achieved simultaneously over
the whole frequency range. However, the use of
frequency filters or weighting functions opens up the
possibility to minimize the magnitudes of , , and
over different frequency ranges [17]. Hence, in
practice, instead of minimizing (19) one rather
determines a stabilizing LTI controller that
minimizes the cost function

(20)
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where , , are suitably chosen weighting
functions (Figure 4).

++
¡¡
w
y
u
K
G
S
W
P
W
T
W
z


Figure 4. Weighting functions.

3. The system representation
In this paper, the mechanical angular speed is
considered as a time-varying parameter. This

particular choice is motivated by the fact that ,
which causes the system to be nonlinear, can be
measured online. The value of the rotor angular speed
varies by around the synchronous speed
, i.e.

(21)

where is the nominal speed, is the
variation of the rotor angular speed around its
nominal value, and is the normalizing factor that
maps the uncertain element into a normalized
uncertain element such that .
In the normal operation of the DFIM, the nominal
speed is close to the synchronous speed .
Hence, if we denote the ratio of the nominal speed
and the synchronous speed by , i.e. we
can write

and
(22)
Here, is a scaling factor that allows to present the
variation of the rotor speed around the
synchronous speed . From that point, the deviation
of the rotor speed by from the nominal speed
can be expressed as

(23)

The representation of in (23) provides a flexible

choice of the rotor speed range in the controller
design for the DFIM. As a result, the system matrices
presented in [18] can now be rewritten as
follows:

where
and
(24)
in which
(25)
(26)

The DFIM model [18] reads as

(27)

where

(28)
(29)
In (28) and (29), and represent the input and
output signals of the disturbance channel
corresponding to the time-varying parameter .

Equations (27), (28), and (29) in combination with the
output equation in [18] can now be expressed as

(30)

(31)

where is an unity matrix, is an zero
matrix, . is also called the
perturbation block.
Since the two last rows of the matrices and
are zero, let , , and
we can write
(32)

where , and are two vectors, is a
matrix.
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Equations (30) and (31) can be easily simplified as

(33)

where





Let be the transfer function with the state-space
realization (33), i.e.

(34)

The system can then be generally described by

(35)

where is the transfer function mapping ,
and is the transfer function
from to .
The linear fractional transformation (LFT)
representation of the system is depicted as shown in
Figure 5.

w

z



s
v
r
v
r
y
rc
G


Figure 5. LFT representation of the system

4. control of the system

In this section we start with -synthesis for
mentioned frozen values of the rotor speed . Then
the performance of the LTI controller designed for a
fixed value of is evaluated with other constant
values of as well as with a fast variation of the
rotor speed along the parameter range.

4.1 The control configuration
With the LFT representation of the plant as shown in
Figure 5 we can now derive a standard control
structure for the synthesis of an -controller as
depicted in Figure 6. Here, is the LTI part of the
plant as given in (34), is the uncertainty block as
given in (32), is the controller that is to be
designed.
+
¡
w

z



s
v
r
v
r
e
ref

r
i
rc
K
rc
G
r
y


Figure 6. Structure of the closed-loop system in
design

In this configuration, is the
reference input, is the controller
output, is the controlled output, and
is
the controller input which is equal to the tracking
error. In this case, the transfer function from the
reference input to the tracking error will be
. The transfer function from
reference inputs to controlled outputs is denoted by
, i.e.

(36)

4.2 loop shaping design
The interconnection of the system used for the
controller synthesis is shown in Figure 7. The external
control input consists of stator voltages and

reference rotor currents
. The controller
output is . The controller input or
tracking error is
. The
controlled variable is . Note
that the components of the external
control inputs are considered as disturbances and their
influences on the controlled outputs must be reduced
as much as possible.
The weighting function is
used to shape the function which is corresponding
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to the transfer function from the reference input
to the tracking error . is kept large over the low
frequency range for tracking. The weighting function
is used to shape the transfer
function from the external control input to the
controlled output . The selection of the weighting
function is not only intended to keep the closed
loop bandwidth at a desired value, but also to reject
the effects of the components and on the
controlled outputs as discussed above. Note that a
large bandwidth corresponds to a faster rise time but
the system is more sensitive to noise and to parameter
variations [16].
+

¡¡
+
¡¡
ref
rd
i
ref
rq
i
rc
w
sd
v
sq
v
rn
G
rd
v
rq
v
rc
K
rd
i
rq
i
rcd
e
rcq

e
rtd
W
rtd
z
rtq
W
rtq
z
rsd
W
rsd
z
rsq
W
rsq
z
rc
z


Figure 7. The interconnection of the system

The standard control problem is to find a
stabilizing LTI controller at fixed frozen values
of such that the -norm of the channel
is smaller than a given number .

at fixed frozen values of .
The set of 620kW DFIM parameters is applied for the
controller synthesis. During the controller design
stage, a trial-and-error-repetition technique is used in
order to achieve the desired performance
specifications by adjusting the weighting functions.
The design steps are repeated until we are able to
meet the required performance specifications. Finally,
the following weighting functions were obtained:

(37)
(38)
For the chosen frozen value of (at under-
synchronous speed), the controlled system with
current controller with the above given weighting
functions achieves a norm of 0.36.

4.3 Simulation results with the current
controller
Figure 8 shows the frequency responses of the
controlled system with the current controller and
the inverse of the weighting functions , and
. Figure 8a,b show the relevant magnitude plots
of the complementary sensitivity and sensitivity
functions of the closed-loop system with the
performance requirements specified by and .
The blue-thick curve shows the response of the output
with respect to the reference inputs . This
curve corresponds to the transfer function (see
equation (36)). Similarly, the red-thick curve shows

the response of the output with respect to the
reference inputs and it corresponds to the transfer
function . Meanwhile the black-solid curve
shows the influence of the reference input on the
output corresponding to the transfer functions
, and the green-solid curve shows the influence
of the reference input on the output
corresponding to the transfer functions . The
inverse of the weighting functions , and
(see Figure 7) are depicted by dotted lines A, and B in
Figure 8a, while the inverse of the weighting
functions , and are depicted by dotted lines
C, and D in Figure 8b, respectively. The influences of
the stator voltage on the controlled outputs and
controller inputs are show in Figure 8c,d with the
same color and line styles.

10
0
10
1
10
2
10
3
10
4
10
5
10

6
-120
-100
-80
-60
-40
-20
0
20
40
Magnitude (dB)
Closed-loop performance of reference inputs to outputs
Frequency (rad/sec)
A
B

(a)
10
0
10
1
10
2
10
3
10
4
10
5
10

6
-150
-100
-50
0
50
100
Magnitude (dB)
Closed-loop performance of reference inputs to control errors
Frequency (rad/sec)
C
D

(b)
10
0
10
1
10
2
10
3
10
4
10
5
10
6
-140
-120

-100
-80
-60
-40
-20
0
Magnitude (dB)
The eff ects of stator voltages to outputs
Frequency (rad/sec)

(c)
10
0
10
1
10
2
10
3
10
4
10
5
10
6
-140
-120
-100
-80
-60

-40
-20
0
Magnitude (dB)
The eff ects of stator voltages to control errors
Frequency (rad/sec)

(d)

Figure 8. Performance of the controlled system with
current controller in the frequency domain for
.

It is clear in Figure 8 that the sensitivity and
complementary sensitivity functions are below the
inverse of the performance weighting functions. The
bandwidths corresponding to the channels
and are about rad/s. The gains of
the frequency responses of the stator voltages to
controlled outputs and controller inputs are all smaller
than -10db. This indicates that the controlled system
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has good disturbance rejection with respect to the
stator voltage . The overshoots of the channels
and are about 15%. Moreover,
the gains corresponding to the frequency responses of
the channels and are smaller

than -22db. This means that the cross-coupling
interaction between and remains quite small, or
in other words, the rotor current components can be
considered to be no influence on one another. As a
result, the characteristics of electrical torque and
power factor responses are not deteriorated.

0 1 2 3 4 5
x 10
-3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Closed-loop performance of reference inputs to outputs
time (s)


i
rd
ref
 i
rd
i
rd
ref

 i
rq
i
rq
ref
 i
rq
i
rq
ref
 i
rd

(a)
0 1 2 3 4 5
x 10
-3
-0.2
0
0.2
0.4
0.6
0.8
1
Closed-loop performance of reference inputs to control errors
time (s)


i
rd

ref
 e
rcd
i
rd
ref
 e
rcq
i
rq
ref
 e
rcq
i
rq
ref
 e
rcd

(b)
0 0.002 0.004 0.006 0.008 0.01
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
The effects of stator voltages to outputs
time (s)

v
sd
ref
 i
rd
v
sd
ref
 i
rq
v
sq
ref
 i
rq
v
sq
ref
 i
rd

(c)
0 0.002 0.004 0.006 0.008 0.01
-0.05
0
0.05
0.1
0.15

0.2
0.25
The effects of stator voltages to control errors
time (s)



(d)

Figure 9. Performance of the controlled system with
current controller in the time domain for
.

Figure 9 shows the time responses of the controlled
system for a step input. The solid line in Figure 9a
shows the response of the output with respect to
the reference input and it corresponds to the
transfer function in equation (36). The dashed
line shows the response of the output with respect
to the reference input and it corresponds to the
transfer function in equation (36). The dotted
curve shows the influence of the reference input
on the output corresponding to the transfer
function , and the dash-dotted curve shows the
influence of the reference input on the output
corresponding to the transfer function . The solid
line in Figure 9b shows the response of the control
error with respect to the reference input . The
dashed line shows the response of the control error
with respect to the reference input . The

dotted curve shows the influence of the reference
input on the control error , and the dash-
dotted curve shows the influence of the reference
input on the control error . The influences of
the stator voltages on the controlled outputs and
control errors are also show in Figure 9c,d with the
same line styles.

0 1 2 3 4 5
x 10
-3
-0.2
0
0.2
0.4
0.6
0.8
1
time (s)
Closed-loop performance of reference inputs to outputs


i
rd
ref
 i
rd
(
m
= 0.63 

s
)
i
rd
ref
 i
rq
(
m
= 0.63 
s
)
i
rq
ref
 i
rq
(
m
= 0.63 
s
)
i
rq
ref
 i
rd
(
m
= 0.63 

s
)
i
rd
ref
 i
rd
(
m
= 0.9 
s
)
i
rd
ref
 i
rq
(
m
= 0.9 
s
)
i
rq
ref
 i
rq
(
m
= 0.9 

s
)
i
rq
ref
 i
rd
(
m
= 0.9 
s
)

(a)
0 1 2 3 4 5
x 10
-3
-0.2
0
0.2
0.4
0.6
0.8
time (s)
Closed-loop performance of reference inputs to control errors



(b)
0 0.002 0.004 0.006 0.008 0.01

-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
time (s)
The effects of stator voltages to outputs



(c)
0 0.002 0.004 0.006 0.008 0.01
-0.05
0
0.05
0.1
0.15
0.2
0.25
time (s)
The effects of stator voltages to control errors


v
sd
ref
 e
rcd

(
m
= 0.63 
s
)
v
sd
ref
 e
rcq
(
m
= 0.63 
s
)
v
sq
ref
 e
rcq
(
m
= 0.63 
s
)
v
sq
ref
 e
rcd

(
m
= 0.63 
s
)
v
sd
ref
 e
rcd
(
m
= 0.9 
s
)
v
sd
ref
 e
rcq
(
m
= 0.9 
s
)
v
sq
ref
 e
rcq

(
m
= 0.9 
s
)
v
sq
ref
 e
rcd
(
m
= 0.9 
s
)

(d)

Figure 10. Performance of the controlled system with
current controller for frozen value
for .

Note that the current controller is designed with a
fixed frozen value of the rotor angular speed .
Hence, the obtained performance is not guaranteed for
the whole region of variation of . However, we can
further investigate the performance of the closed-loop
system with the controller designed for the frozen
value for other angular rotor speeds. In
the following investigation, we consider the

performance of the controlled system with the rotor
speed variations by from the rotor nominal
speed ( ), i.e.
. In order to do so we
synthesized two controllers for the frozen
parameter values
using the same weighting functions as in (37) and
(38). Then we plot the time responses of the closed-
loop system with the controller designed for the
frozen value applying for the case where
and , respectively. The
time responses of the closed-loop system with the
local controllers designed for the frozen values
and are also plotted on
each figure for the purpose of comparison of the
achieved performance among these controllers.
Figure 10 shows the performance of the closed-loop
system at with two controllers
designed for the frozen value and
, respectively. The thick-solid lines are
related to the designed controller for the frozen
value . The thin-lines are related to the
local controller designed for . As
can be seen from Figure 10a, the time responses of the
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outputs and with respect to the step change of
the reference inputs and , respectively, of the

closed-loop system with the controller designed
for the frozen value are maintained for
the frozen values if compared with that
in Figure 9. In addition, these curves are almost the
same with that of the local controller designed for
. The same conclusion can also be
drawn for the curves related to the time responses of
the control errors , with respect to the step
change of the reference inputs , (Figure 10b),
and the influences of the stator voltages , on
the controlled outputs , (Figure 10c) and control
errors , (Figure 10d), respectively. However,
the remarkable difference in the performance among
these controllers is indicated by their cross-
coupling interactions. The effects of the stator
voltages and to the outputs , (Figure
10c) and to the control errors , (Figure 10d),
respectively, in the case of the controller
designed for the frozen value are larger
than that of the local controller designed for
.

0 1 2 3 4 5
x 10
-3
-0.2
0
0.2
0.4
0.6

0.8
1
time (s)
Closed-loop performance of reference inputs to outputs


i
rd
ref
 i
rd
(
m
= 1.17 
s
)
i
rd
ref
 i
rq
(
m
= 1.17 
s
)
i
rq
ref
 i

rq
(
m
= 1.17 
s
)
i
rq
ref
 i
rd
(
m
= 1.17 
s
)
i
rd
ref
 i
rd
(
m
= 0.9 
s
)
i
rd
ref
 i

rq
(
m
= 0.9 
s
)
i
rq
ref
 i
rq
(
m
= 0.9 
s
)
i
rq
ref
 i
rd
(
m
= 0.9 
s
)

(a)
0 1 2 3 4 5
x 10

-3
-0.2
0
0.2
0.4
0.6
0.8
time (s)
Closed-loop performance of reference inputs to control errors


i
rd
ref
 e
rcd
(
m
= 1.17 
s
)
i
rd
ref
 e
rcq
(
m
= 1.17 
s

)
i
rq
ref
 e
rcq
(
m
= 1.17 
s
)
i
rq
ref
 e
rcd
(
m
= 1.17 
s
)
i
rd
ref
 e
rcd
(
m
= 0.9 
s

)
i
rd
ref
 e
rcq
(
m
= 0.9 
s
)
i
rq
ref
 e
rcq
(
m
= 0.9 
s
)
i
rq
ref
 e
rcd
(
m
= 0.9 
s

)
(b)
0 0.002 0.004 0.006 0.008 0.01
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
time (s)
The effects of stator voltages to outputs


v
sd
ref
 i
rd
(
m
= 1.17 
s
)
v
sd
ref
 i
rq
(

m
= 1.17 
s
)
v
sq
ref
 i
rq
(
m
= 1.17 
s
)
v
sq
ref
 i
rd
(
m
= 1.17 
s
)
v
sd
ref
 i
rd
(

m
= 0.9 
s
)
v
sd
ref
 i
rq
(
m
= 0.9 
s
)
v
sq
ref
 i
rq
(
m
= 0.9 
s
)
v
sq
ref
 i
rd
(

m
= 0.9 
s
)

(c)
0 0.002 0.004 0.006 0.008 0.01
-0.05
0
0.05
0.1
0.15
0.2
0.25
time (s)
The effects of stator voltages to control errors


v
sd
ref
 e
rcd
(
m
= 1.17 
s
)
v
sd

ref
 e
rcq
(
m
= 1.17 
s
)
v
sq
ref
 e
rcq
(
m
= 1.17 
s
)
v
sq
ref
 e
rcd
(
m
= 1.17 
s
)
v
sd

ref
 e
rcd
(
m
= 0.9 
s
)
v
sd
ref
 e
rcq
(
m
= 0.9 
s
)
v
sq
ref
 e
rcq
(
m
= 0.9 
s
)
v
sq

ref
 e
rcd
(
m
= 0.9 
s
)

(d)

Hình 11. Performance of the controlled system with
current controller for frozen value
for

The performance of the closed-loop system at
with two controllers designed for
the frozen value and ,
respectively, is shown in Figure 11. The thick-solid
lines are related to the designed controller for the
frozen value . The thin-lines are related to
the local controller designed for .
Similarly to the previous simulation, the time
responses of the closed-loop system with the
controller designed for the frozen value
and the local controller designed for
corresponding to the step change of the
reference inputs , , and to the effect of stator
voltages , are almost the same, except their
cross-coupling interactions. In the case of the

controller designed for the frozen value ,
the influences of the stator voltages and to the
outputs , (Figure 11c) and to the control errors
, (Figure 11d) are larger than that of the local
controller designed for .
Obviously, the performance of the the controller
designed for the frozen value for other
rotor angular speeds are not maintained because of the
cross-coupling interactions between the stator
voltages , and the outputs , as well as the
stator voltages , and the control errors ,
. This may cause a large tracking error for the
controlled system since the stator voltages and
are the input disturbances.

0 1 2 3 4 5
x 10
-3
690
700
710
720
730
740
750
760
d component of the rotor currents
time (s)
Ampere


m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(a)
0 1 2 3 4 5
x 10
-3
0
50
100
150
200
250
300
350
400
q component of the rotor currents
time (s)
Ampere


m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(b)
0 1 2 3 4 5
x 10
-3
690
700
710
720
730
740
750
760
d component of the rotor currents
time (s)
Ampere


m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(c)
0 1 2 3 4 5
x 10
-3
0
50
100
150
200
250
300
350
q component of the rotor currents
time (s)
Ampere


m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(d)
0 1 2 3 4 5
x 10
-3
690
700
710
720
730
740
750
760
770
d component of the rotor currents
time (s)

Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(e)
0 1 2 3 4 5
x 10
-3
0
50
100
150
200
250
300
350
q component of the rotor currents
time (s)

Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(f)

Hình 12. Performance of the controlled system with
current controllers designed for (a,
b), for (c, d), and for (e, f)
at different constant values of .

In order to evaluate the performance of the closed-
loop system with controller designed for different
frozen values of for other rotor angular speeds in
the face of the stator voltage action, we performed the
simulations with the set value of and the
set value of as shown in Figure 12. The
time responses of the (Figure 12a) and (Figure

12b) components of the rotor currents achieved by the
384
Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

VCCA-2011
designed controller for the frozen value
are plotted by the solid curves. While
the dashed and and the dash-dotted curves show the
performance of this controller for the value
, and , respectively. These
figures reveal that the tracking errors of the and
components of the rotor currents achieved by this
controller are increased for and
. This is because of the cross-coupling
interactions between the stator voltages , and
the outputs , as well as the stator voltages ,
and the control errors , become larger for
bigger rotor angular speeds as presented in the
previous simulation. Figures. 12c and 12d show the
time responses of the and components of the rotor
currents achieved by the designed controller for
the frozen value (solid curves) for the
value (dashed curves), and
(dash-dotted curves), respectively.
Figures. 12e and 12f show the time responses of the
and components of the rotor currents achieved by
the designed controller for the frozen value
(solid curves) for the value
(dashed curves), and
(dash-dotted curves), respectively. These figures

reveal that performance of the controller
designed for a frozen value of is not guaranteed
for these other values of .

0 0.002 0.004 0.006 0.008 0.01
0
100
200
300
400
500
600
700
800
d component of the rotor currents
time (s)
Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17

s

(a)
0 0.002 0.004 0.006 0.008 0.01
0
100
200
300
400
500
600
700
800
d component of the rotor currents
time (s)
Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(b)
0 0.002 0.004 0.006 0.008 0.01
0
50
100
150
200
250
300
350
400
d component of the rotor currents
time (s)
Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(c)
0 0.002 0.004 0.006 0.008 0.01
0
50
100
150
200
250
300
350
400
d component of the rotor currents
time (s)
Ampere



m
= 0.63
s

m
= 0.9
s

m
= 1.17
s

(d)

0 0.002 0.004 0.006 0.008 0.01
200
250
300
350

m
time (s)
rad/s

(e)
0 0.002 0.004 0.006 0.008 0.01
200
250
300
350

m
time (s)
rad/s

(f)

Hình 13. Performance of the controlled system with
three current controllers designed for
, , and ,
respectively, with fast variations of the rotor speed.

For further investigation, a simulation with the
controller designed for a frozen values of for a

fast variation of the rotor speed along the whole
parameter interval is carried out. We consider three
local controllers designed for the frozen values
, , and as
above. The parameter trajectory is given by the step
response of the rotor speed. Figures. 13a,c,e show the
behaviors of the and components of the rotor
currents when the rotor angular speed increases from
70% to 130% of the nominal speed of the rotor
, where (rad/s),
. Conversely, the behaviors of the and
components of the rotor currents when the rotor
angular speed decreases from 130% down to 70% of
the nominal speed of the rotor are shown in Figures.
13b,d,f. These figures reveal that the controllers
do not guarantee tracking during the fast parameter
transition. The control error increases along the
parameter trajectory and reaches the largest value at
the end of it.

5. Conclusions
This paper briefly recapitulated the theory of the
mixed loop shaping -design for the rotor current
controller for DFIMs at some fixed frozen values of
the rotor angular speed. The performance of these
current controllers has been investigated for different
values of the mechanical angular speed varied by
% from the rotor nominal speed. The simulation
results showed that the performance of the
controller designed for a frozen value of was not

completely guaranteed for other rotor angular speeds.
An important point that is needed to be emphasized in
this particular case is that the performance of the
controller is considerably changed for fast parameter
variations. In order to get better performance level for
the controlled system, the designed controller has to
adapt to changing of the rotor angular speed. In that
sense, the rotor angular speed can be adopted as a
gain-scheduling parameter.

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Hội nghị toàn quốc về Điều khiển và Tự động hoá - VCCA-2011

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Dr. Ngo Duc Minh was

born in Lang son, Vietnam,
in 1960. He received the
B.S. degree from
Thainguyen University of
Technology in 1982 in
Electrical Engineering,
M.S. degree from Hanoi
University of Technology
in 1997 in Electrical
Engineering and in
Industrial Information Technology, and Ph.D degree
from Hanoi University of Technology in 2010 in
Atutomation Technology. He is currently a vice-chair
of the Education department of Thainguyen
University of Technology. Dr. Minh’s interests are in
the areas of high voltage technology, hydrolic power
plant, power supply, control of electric power
systems, FACTS, BESS, AF, PSS equipments, new
and renewable energy technologies, distribution
power systems.

Nguyen Tien Hung was
born in Thainguyen,
Vietnam. He received the
B.S. degree from
Thainguyen University of
Technology in 1991 and
M.S. degree from Hanoi
University of Technology in
1997, both in Electrical

Engineering. He is currently
a Ph.D candidate at Delft Center for Systems and
Control (DCSC), Delft University of Technology, the
Netherlands. His main research interests include
topics in robust control, linear parameter varying
control of nonlinear systems, gain-scheduling design,
and their applications in electrical systems.
386