In this paper, we propose a singular perturbationlike approach to EDFA gain controller design and analysis. Considering a threelevel model of EDFA, a gain controller containing a state observer and a channel add/drop estimator is designed based on a singular perturbation  like concept. The proposed design methodology is shown to be effective and advantageous not only in theoretically verifying the asymptotic stability of systems with multitime scales such as EDFA but also in designing an asymptotic estimator for channel add/drops which does not satisfy the matching condition.
1. Introduction
EDFA(Erbium Doped Fiber Amplifier) is widely used for the amplification of channel signals in a WDM optical network. In an EDFA, it is important to maintain the gain of each channel when channel add/drops or active rearrangements of the network occur. The change of the number of channel signals called a channel add/drop causes a change of the amplifier gain of each channel signal due to the cross gain saturation effect
[1]
.
There have been suggested several methods to handle this issue. One of them uses an EDFA output as feedback signal in an optical feedback control loop
[2]
. It, however, has the drawback that the frequency of channel add/drops should be less than that of the relaxation oscillation frequency of the EDFA, which is several hundred hertz. On the other hand, the gain fluctuation due to channel add / drops can be effectively compensated for by controlling the pump laser output electrically according to the EDFA output signal level
[3]
. In the previous papers
[4

7]
, we proposed a novel technique which minimizes the gaintransient time effectively under the assumption that the rate of Erbium ions at level 3 converges relatively fast to the desired equilibrium compared with the one at level 2. A simplified twolevel EDFA model was considered to design a gain controller and a disturbance observer (DOB) technique
[8
,
9
,
10]
, and a proportional / integral (PI) controller was applied to the control of the EDFA gain in WDM add/drop networks. However, in order to compensate for the gain fluctuation due to channel add/drops as fast as in the order of microseconds, a full threelevel model should be considered and a nominal gain controller should be designed considering the state of the population of Erbium ions at level 3. In a simplified twolevel model, the matching condition is satisfied and channel add / drops can be easily controlled by a disturbance observer. However, the matching condition is not satisfied by the threelevel model, so a new EDFA gain controller design methodology based on the three level model is necessary. In
[5]
, a PID gain control algorithm considering the threelevel EDFA model was applied to a nominal control. Since a channel add / drop compensator was still designed using a DOB based on a simplified twolevel model, theoretical analysis of asymptotic stability could not be provided rigorously.
In this paper, a theoretical design and analysis of EDFA gain control system is carried out based on a mathematical three level EDFA model
[11]
using a singular perturbation technique
[12]
. In order to compensate for channel add / drop effects, a channel add/drop estimator is designed based on an internal model of EDFA, and an EDFA gain controller is proposed combining a state observer with the channel add/drop estimator. With successive applications of time scale separation to the designed EDFA control system, a singular perturbation technique gives a theoretical performance analysis of the proposed EDFA gain control algorithm even in the case that the matching condition is not satisfied. Through simulations, the practicality of the proposed control algorithm is also confirmed.
2. Design of EDFA Gain Control System
 2.1 Threelevel EDFA Model
In order to design an EDFA gain controller, the following threelevel model is considered
[11]
. The energy level of EDFA is shown in
Fig. 1
and the equations for the threelevel process are given as
Models of EDFA
Where Γ
_{21}
, Γ
_{32}
are positive constants;
ϕ_{s}, ϕ_{p}
are photon flux densities per second of the signal and the pump;
σ^{e}_{s}
,
σ^{a}_{s}
,
σ^{e}_{p}
,
σ^{a}_{p}
are absorption and emission cross section of the signal and the pump (
σ
^{T}
=
σ
^{e}
+
σ
^{a}
); and
N_{1}
,
N_{2}
, and
N_{3}
are the number of erbiumions at each energy level (
N
=
N_{1}
+
N_{2}
+
N_{3}
= 1). The power
P_{s}
of the signal and the power
P_{p}
of the pump obey the following equations:
where
ρ
is the Erbium density, and Γ
_{s}
and Γ
_{p}
are respectively the geometric correction factor for the overlap between the power and the erbiumions.
Define a reservoir
r_{i}
(
t
),
i
= 2,3 that represents the number of excited Erbiumions at each level and the EDFA gain of the kth channel as follows:
where
L
is the length of the Erbiumdoped fiber,
A
is the crosssection area of erbiumdoped fiber core, and
and
are respectively the kth channel input power and output power. Without loss of generality, each channel input power is assumed to be an average power that is a positive constant until channel signals drop. Then, by integrating (1)(3) along the whole length of EDF, we can obtain the following threelevel EDFA model equations from definitions of reservoir
r_{i}
(
t
),
i
= 2,3 and
where
N
is the number of channels,
G
_{p}
(
t
) is the gain of input pump channel and
Suppose that the kth channel gain
G
_{k}
(
t
) should be maintained to be a desired constant channel gain
Then, the state variable
r
_{2}
in the EDFA model Eqs. (8) and (9) must satisfy
at the steady state or equilibrium. Define an error variable as
Then, the error dynamics are written as
Our goal is to design a stabilizing controller for the system described by (14) and (15). The term
consists of channel signals and varies according to channel add/drops which is not predictable in advance and is considered as a disturbance. So a disturbance observer technique can be adopted to reject the influence of channel add/drops on the channel gain variation. However, the control input
does not appear in the same equation with this term and thus it is nontrivial to compensate for this channel add/drops. The system (14) and (15) does not satisfy the socalled matching condition. In order to overcome this difficulty, we employ a singular perturbation method. If the dynamics (14) can be made much faster than the dynamics (15) by a control, a singular perturbation can be applied and we can reduce the dynamics such that the reduced dynamics satisfy the matching condition. We can then design a stabilizing controller for this reduced dynamic system using error state feedback and a disturbance estimator.
 2.2 EDFA gain controller
In order to design a stabilizing controller for the system in (14) and (15), let us make the following assumption:
(A1)
The gain
G
_{p}
(
t
) of the input pump channel is measurable.
Then, a stabilizing controller for the error dynamics (14) and (15) is designed as follows.
where
and
are respectively estimation variables of the state
r_{3}
(
t
) and the disturbance
and
k_{i}
,
i
= 1, 2, 3 are positive constants. In (16), the term
is to make the dynamics in (14) much faster than the one in (15). Since
r_{3}
(
t
) is not measurable, we use its estimated value
instead. The term
is to reject the term
c
(
t
). So we need to design a state estimator for
r_{3}
(
t
) and a disturbance observer for
c
(
t
).
 2.3 Design of a channel add/drop estimator
As mentioned in the previous section, we need to estimate the term
in (15) including channel add/drops. It usually costs a lot to measure all the channel powers
and all the channel gains
G_{k}
of the EDFA in optical networks. So it is inevitable to estimate the term
. In order to estimate this, we consider the following internal nominal model of the EDFA:
Define
Then, we obtain the following equations:
Notice that the system (21)(23) has stable zero dynamics. So we have the following transfer function between the channel add/drop input
and the output
where
L
[·] denotes the Laplace transform. Define a filter
Q
(
s
) by
where
is the estimated output of
c
(
t
). We have the following relation between the channel add/drop signal
c
(
t
) and its estimate
:
where the positive constant
A_{D}
is to be chosen later. Here we use a firstorder linear model for the resultant channel add/drop estimator for convenience, but any higher order model can be equally used.
 2.4 Design of a state observer
In order to stabilize the error system given by (14) and (15) with error state feedback, we need to estimate the state variable
r
_{3}
(
t
). Usually a state estimator can be easily designed if the system is observable, but its design becomes nontrivial when the term
in the EDFA model given by (8)(10) cannot be measured. However, it is possible to design a state estimator that guarantees asymptotic estimation performance even when an unknown term
is present, if we use the channel add/drop estimator proposed in the previous section. Now we propose the following state estimator for the system (8)(10):
where
L
_{1}
and
L
_{2}
are observer gains. The observer gains
L
_{1}
and
L
_{2}
are chosen such that
is stable,
3. Theoretical Analysis : A Singular Perturbation Approach
In this section, we introduce a singular perturbation approach to stability analysis, which provides a systematic procedure for analysis of multitime scaled systems.
 3.1 Reduced dynamics of timescaled closedloop system
Since the estimator should have a faster performance than the controller, the control system designed in the previous section is considered as a multitime scaled system. So a singular perturbation method can be applied to the analysis of the EDFA gain control system designed in Section II.
From (14)  (16) and (27)  (29), we obtain the error equations of the closed loop system as follows:
where
From (26), we obtain the following equation for the channel add/drop estimator:
If the design parameter
A_{D}
in (35) is chosen so that the dynamics given in (35) is faster than any other subsystems (30)(33), then the system (35) is stabilized very fast and
immediately converges to
c
(
t
).
Since
k
_{2}
is chosen so that (Γ
_{32}
+
k
_{2}
) becomes much larger than Γ
_{21}
, a singular perturbation procedure is applied to (30)(35) by letting
. Let
k
_{3}
be chosen as
Then we obtain the following reduced dynamics.
where
and
and
and
satisfy the following:
The design parameter
A_{D}
in (35) or (40) is chosen so that the dynamics given by (35) or (40) is faster than the other subsystem dynamics (37)(39). Then the system (40) is stabilized very fast and
immediately converges to
. So, if we apply a singular perturbation method again to (37)(40) as
, we have the following reduced dynamics:
Since the design parameters
L
_{1}
and
L
_{2}
of the state estimator are designed so that its performance is much faster than error state feedback control and stable, the estimation error states
and
of the reduced dynamics in (45) and (46) decay to zero much faster than
. So, as
we have the following reduced dynamics for the system Σ
_{3}
:
From (47), it is obvious that
where
The performance of the control system is determined by a desired bandwidth
λ
, and the controller gain
k
_{1}
is chosen as
for given
λ
and
k
_{2}
. Choice of the controller gain
k
_{2}
is discussed in the next section.
 3.2 Stability analysis
Stability analysis is carried out by showing the asymptotic stability of each system Σ
_{i}
,
i
= 1,2,3 successively using asymptotic stability of systems Σ
_{m}
,
m
=
i
+1,⋯,4. In order to show the asymptotic stability, we assume that channel add/drops are not persistent. That is, channel add/drops are assumed to occur finitely many times. We make the following assumption.
(A2)
The number of Channel add/drops, M, is finite.
Define the instants at which channel add/drops occur by a time sequence
t_{n}
,
n
= 1,⋯,
M
. So, if we define
t
_{M+1}
= ∞, each channel input
is zero or a positive constant for all
t
∈ [
t_{i}
,
t
_{i+1}
),
i
= 1,⋯,
M
. So, the time derivative of each channel input is zero for all
t
∈ (
t_{i}
,
t
_{i+1}
),
i
= 1,⋯,
M
and the following equation holds for any
ε
＞ 0 :
Step 1.
Stability analysis of Σ
_{3}
In order to show that the system Σ
_{3}
is asymptotically stable, we define the error variables between the system Σ
_{3}
and the system Σ
_{4}
as follows:
Then,
It is obvious that the system (53)(55) is asymptotically stable. So, there exist positive numbers
α
_{3}
and
β
_{3}
such that
Step 2.
Stability analysis of Σ
_{2}
using the asymptotic stability of Σ
_{3}
and Σ
_{4}
Define an error vector
by
From (37)  (40) and (44)  (46), the error system is described as follows:
Since the length of EDF,
l
is finite, the reservoir
r
_{2}
,
G_{k}
(
t
) and
defined in (41) are bounded. So,
is also bounded from (40) such that there exists a positive constant
B_{C}
satisfying
Therefore, from (57), (58), (59), and (61), there exists a positive constant
B_{E}
such that the following inequality holds.
From (41) , (43), (51) and (62),
and
Thus, for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
,
where
Meanwhile, the reservoir
r
_{2}
is bounded and
G_{k}
(
t
) is also bounded since the length of EDF,
l
is finite. So, there exist positive constants Π
_{m}
and Π
_{M}
such that
Define a matrix
A
_{3}
by
Since
A
_{3}
is stable for any positive numbers
k
_{1}
and
k
_{2}
, there exist positive definite symmetric matrices
P
and
Q
satisfying the following Lyapunov equation
Define a Lyapunovlike function
V
_{3}
by
where
Then, from (58) and (65) , for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
where
B_{c}
= [1 1 0]
^{T}
. Since
is positive and bounded as in (67), we have the following inequality for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
).
where
For a positive number
, choose a design parameter
A_{D}
as follows
where
From (67), it follows that
where
Then,
is positive definite and satisfies
So, the Lyapunov function
V
_{3}
satisfies the following inequality for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
.
where
λ_{m}
(⋅) and
λ_{M}
(⋅) are respectively the smallest and the largest eigenvalue of the associated matrix. Dividing (80) by
leads to the following inequality:
for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
. It follows from (48), (56), and (81) that
for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
). So, there exist positive numbers
and
for each
i
∈ [1,
M
] such that
From (70) and (71), there exists a positive constant
B_{X}
_{3}
such that the following inequality holds.
Therefore, the asymptotic stability is satisfied since (83) holds for
t
∈ [
t
_{M}
+
ε
, ∞).
Step 3.
Stability Analysis of Σ
_{1}
using the stability of Σ
_{2}
, Σ
_{3}
, and Σ
_{4}
.
Define an error vector
and an error variable
by
where
Then, we have the following error equations:
Rewriting (31) and (32) , we obtain
It follows from (91) and (92) that (87) is described by
From the analysis in step 1 and step 2,
E
_{2}
,
,
,
,
and
are bounded. Using the same arguments in step 2, we can also show that
and
are bounded because
c
(
t
) in (30) is bounded. Since Γ
_{21}
,Γ
_{32}
,
k
_{1}
,
k
_{2}
and
k
_{3}
are positive constants and the observer gains
L
_{1}
and
L
_{2}
are chosen such that
is stable, the error equations given by (88)(90) and (93) satisfy the BIBO stability. Therefore, the error state vector
and
defined by (85) are bounded and there exist positive constants
and
such that
As in step 2, we now consider the performance for
t
∈ (
t_{i}
,
t
_{i+1}
),
i
= 1, ⋯,
M
. Since each channel input
is zero or positive constant for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
, the time derivative of
c
(
t
) is given by
where
As in step 2, the following error equation holds for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
.
Define a Lyapunovlike function
V
_{2}
by
where
Then, it follows from (36), (49), (69), (71), (73), and (74) that for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
,
where
By (79), for all
t
∈ [
t_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
,
where
Let
In order for (
) to be positive definite for all
t
≥ 0, the following inequality must be satisfied:
Define
σ_{M}
by
Since Π
_{1}
(
t
) is positive and bounded as in (67),
σ_{M}
is obtained for each
k
_{2}
when Π
_{1}
(
t
) = Π
_{m}
or Π
_{1}
(
t
) = Π
_{M}
. Let us define
σ
_{1}
and
σ
_{2}
as
If
k
_{2}
is chosen such that
and
then (
) is positive definite. Since the inequalities (107) and (108) are of third order in
k
_{2}
if ( Γ
_{32}
+
k
_{2}
)
^{2}
is multiplied to both sides of the inequalities, there always exists a
k
_{2}
satisfying (107) and (108). Then, for all
t
∈ [
t
_{i}
+
ε
,
t
_{i+1}
),
i
= 1, ⋯,
M
,
Using the same arguments as in step 2, it can be shown from (51), (55), (56), (71), (82), and (109) that there exist positive numbers
and
satisfying
From (94), there exists a positive constant
B_{X}
_{2}
such that
Thus, asymptotic stability is achieved since (110) holds for
t
∈ [
t_{M}
+
ε
, ∞). Hence,
and from (55), (83), and (110)
as
t
→ ∞. By (10) and (12),
as
t
→∞. This completes the stability analysis of Σ
_{1}
.
4. Simulations
In order to analyze the performance of the proposed method, computer simulations are carried out. In the simulations, the wavelength of the pump laser is 980nm. As for signals, two channel signals with 1552.4 nm and 1557.9 nm wavelengths are applied to the system. The signal power of each channel is 0.316mW. In the simulations, the desired channel 1 signal gain is set to 6.6897. The other EDFA system parameters in (8), (9), and (10) are given as follows.
Since the gain control is desired to be achieved within a microsecond, the controller gains
k
_{1}
and
k
_{2}
are chosen as follows
so that the natural frequency of the resultant secondorder closed loop system may become
ω_{n}
= 10
^{7}
(
rad
/ sec). Thus,
k
_{3}
is set to be 20.9999 from (36). Next, observer gains
L
_{1}
and
L
_{2}
are designed such that the bandwidth of the observer system is almost three times larger than that of state feedback control system. Observer gains
L
_{1}
and
L
_{2}
are given by
Finally, the channel add/drop estimator gain
A_{D}
in (25) or (26) is selected to be
A_{D}
= 1.5×10
^{8}
in order for the channel add/drop estimation to be fast enough compared with other controller and observer. In this case,
A_{D}
= 1.5×10
^{8}
is chosen such that the bandwidth is 5 times larger than that in state observer.
Firstly, we show that the performance of the proposed controller based on the threelevel model is superior to that of the controller designed based on a simplified twolevel model. In order to show this, we consider the following simple error state feedback controller including the same channel add/drop estimator.
As in the selection of the gains of the proposed controller, the gain
k_{C}
in (114) is also chosen as large as possible so that the gain control of the resultant first order control system can be achieved within a microsecond. For example,
Fig. 2
shows the graphs of the controlled gain of channel 1 signal when channel add/drop occurs at every microsecond as in
Fig. 4
. As expected, the proposed observer based controller designed based on the threelevel model shows faster settling performance. The control based on the twolevel model shows oscillation and longer settling performance because it considers the model simplified by ignoring the level three state.
Comparison of gain control performance
Channel add/drop estimation
The proposed controller guarantees the desired performance with 0.8 usec settling time, but the simplified control cannot guarantee the desired settling performance with a settling time longer than 1 usec.
Fig. 3
is an enlarged version of
Fig. 2
to show the gain control results and the influence on the gain due to channel add/drops was effectively compensated for within 1 usec as expected. The channel add/drop estimation performance is shown in
Fig. 4
. The channel add/drop estimation should be fastest compared with gain control and state observation and
Fig. 4
shows that the estimation is done abruptly. In more details,
Fig. 5
and
Fig. 6
show the channel add/drop estimation results in case of channel drop and channel add. In both cases, the channel add/drop estimation is achieved in 0.03 microsecond and the channel gain is stabilized within 1 microsecond.
Fig. 7
shows the results of state estimations. As we intended, the state estimation is done in 0.15 microsecond which is 5 times larger than channel add/drop estimation.
Gain control performance over channel add/drops
Channel drop case
Channel drop case.
State estimation
5. Conclusion
In this paper, a systematic design methodology of an EDFA gain controller has been proposed based on singular perturbation and observer technique. The threelevel EDFA model has been fully considered without any simplification, and time scaling design approach based on singular perturbation technique has been applied.
Theoretical stability analysis has been carried out thoroughly. Through computer simulation, it is shown that the performance of the proposed EDFA controller is superior to that of the controller designed based on the simplified two level model. The computer simulation also shows that the wellknown disturbance observer technique plays an effective role in guaranteeing the desired performance when channel add/drops occur.
Acknowledgements
This research was supported by Hallym University Research Fund 2014 (HRF201404011) and by Midcareer Researcher Program (No.20110013091) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology.
BIO
SeongHo Song He received the B.S, M.S, and Ph. D degree in measurement and control engineering from Seoul National University. His research interests are nonlinear control, aerospace engineering, mechatronics and vision systems.
Dong Eui Chang He received the B.S degree in control and Instrumentation engineering and the M.S. degree from electrical engineering, both, from Seoul National University and the Ph.D. in control & dynamical systems from the California Institute of Technology. He is currently associate professor in applied mathematics at the University of Waterloo, Canada. His research interests lie in control, mechanics and various engineering applications.
Kwang Y. Lee He received his B.S.degree in Electrical Engineering from Seoul National University, Korea, in 1964, M.S. degree in Electrical Engineering from North Dakota State University, Fargo, in 1968, and Ph.D. degree in System Science from Michigan State University, East Lansing, in 1971. He has been with Michigan State, Oregon State, Univ. of Houston, the Pennsylvania State University, and Baylor University where he is currently a Professor and Chair of Electrical and Computer Engineering. His interests include power system control, operation, planning, and intelligent system applications to power systems. Dr. Lee is a Fellow of IEEE, Editor of IEEE Transactions on Energy Conversion, and Former Associate Editor of IEEE Transactions on Neural Networks. He is also a registered Professional Engineer.
HoChan Kim He received the B.S., M.S., and Ph.D. degrees in Control & Instrumentation Engineering from Seoul National University in 1987, 1989, and 1994, respectively. Since 1995, he has been with the Department of Electrical Engineering at Jeju National University, where he is currently a professor. He was a Visiting Scholar at the Pennsylvania State University in 1999 and 2008. His research interests include wind power control, electricity market analysis, and grounding systems.
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