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 gain-transient 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 two-level 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 micro-seconds, a full three-level 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 two-level 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 three-level 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 three-level EDFA model was applied to a nominal control. Since a channel add / drop compensator was still designed using a DOB based on a simplified two-level 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 Three-level EDFA Model
In order to design an EDFA gain controller, the following three-level model is considered [11]. The energy level of EDFA is shown in Fig. 1 and the equations for the three-level process are given as
Fig. 1.Models of EDFA
Where Γ21, Γ32 are positive constants; ϕs, ϕp are photon flux densities per second of the signal and the pump; σes, σas, σep, σap are absorption and emission cross section of the signal and the pump (σT = σe + σa); and N1, N2, and N3 are the number of erbium-ions at each energy level (N = N1 + N2 + N3 = 1). The power Ps of the signal and the power Pp of the pump obey the following equations:
where ρ is the Erbium density, and Γs and Γp are respectively the geometric correction factor for the overlap between the power and the erbium-ions.
Define a reservoir ri (t), i = 2,3 that represents the number of excited Erbium-ions at each level and the EDFA gain of the k-th channel as follows:
where L is the length of the Erbium-doped fiber, A is the cross-section area of erbium-doped fiber core, and and are respectively the k-th 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 three-level EDFA model equations from definitions of reservoir ri(t), i = 2,3 and
where N is the number of channels, Gp(t) is the gain of input pump channel and
Suppose that the k-th channel gain Gk(t) should be maintained to be a desired constant channel gain Then, the state variable r2 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 so-called 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 Gp(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 r3(t) and the disturbance and ki , 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 r3(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 r3(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 Gk 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 AD is to be chosen later. Here we use a first-order 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 r3(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 L1 and L2 are observer gains. The observer gains L1 and L2 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 multi-time scaled systems.
3.1 Reduced dynamics of time-scaled closed-loop system
Since the estimator should have a faster performance than the controller, the control system designed in the previous section is considered as a multi-time 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 AD 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 k2 is chosen so that (Γ32 + k2 ) becomes much larger than Γ21, a singular perturbation procedure is applied to (30)-(35) by letting . Let k3 be chosen as
Then we obtain the following reduced dynamics.
where
and
and and satisfy the following:
The design parameter AD 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 L1 and L2 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 Σ3 :
From (47), it is obvious that
where
The performance of the control system is determined by a desired bandwidth λ, and the controller gain k1 is chosen as
for given λ and k2. Choice of the controller gain k2 is discussed in the next section.
3.2 Stability analysis
Stability analysis is carried out by showing the asymptotic stability of each system Σi, i = 1,2,3 successively using asymptotic stability of systems Σm, 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 tn, n = 1,⋯, M. So, if we define tM+1 = ∞, each channel input is zero or a positive constant for all t ∈ [ti, ti+1), i = 1,⋯, M. So, the time derivative of each channel input is zero for all t ∈ (ti, ti+1), i = 1,⋯, M and the following equation holds for any ε ＞ 0 :
Step 1. Stability analysis of Σ3
In order to show that the system Σ3 is asymptotically stable, we define the error variables between the system Σ3 and the system Σ4 as follows:
Then,
It is obvious that the system (53)-(55) is asymptotically stable. So, there exist positive numbers α3 and β3 such that
Step 2. Stability analysis of Σ2 using the asymptotic stability of Σ3 and Σ4
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 r2, Gk(t) and defined in (41) are bounded. So, is also bounded from (40) such that there exists a positive constant BC satisfying
Therefore, from (57), (58), (59), and (61), there exists a positive constant BE such that the following inequality holds.
From (41) , (43), (51) and (62),
and
Thus, for all t ∈ [ti + ε, ti+1), i = 1, ⋯, M,
where
Meanwhile, the reservoir r2 is bounded and Gk(t) is also bounded since the length of EDF, l is finite. So, there exist positive constants Πm and ΠM such that
Define a matrix A3 by
Since A3 is stable for any positive numbers k1 and k2, there exist positive definite symmetric matrices P and Q satisfying the following Lyapunov equation
Define a Lyapunov-like function V3 by
where
Then, from (58) and (65) , for all t ∈ [ti + ε, ti+1),
where Bc = [1 1 0]T. Since is positive and bounded as in (67), we have the following inequality for all t ∈ [ti + ε, ti+1).
where
For a positive number , choose a design parameter AD as follows
where
From (67), it follows that
where
Then, is positive definite and satisfies
So, the Lyapunov function V3 satisfies the following inequality for all t ∈ [ti + ε, ti+1), i = 1, ⋯, M.
where λm(⋅) and λM(⋅) are respectively the smallest and the largest eigenvalue of the associated matrix. Dividing (80) by leads to the following inequality:
for all t ∈ [ti + ε, ti+1), i = 1, ⋯, M. It follows from (48), (56), and (81) that
for all t ∈ [ti + ε, ti+1). So, there exist positive numbers and for each i ∈ [1, M] such that
From (70) and (71), there exists a positive constant BX3 such that the following inequality holds.
Therefore, the asymptotic stability is satisfied since (83) holds for t ∈ [tM + ε, ∞).
Step 3. Stability Analysis of Σ1 using the stability of Σ2, Σ3, and Σ4.
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, E2, , , , 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 Γ21,Γ32 , k1, k2 and k3 are positive constants and the observer gains L1 and L2 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 ∈ (ti, ti+1), i = 1, ⋯, M. Since each channel input is zero or positive constant for all t ∈ [ti + ε, ti+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 ∈ [ti + ε, ti+1), i = 1, ⋯, M.
Define a Lyapunov-like function V2 by
where
Then, it follows from (36), (49), (69), (71), (73), and (74) that for all t ∈ [ti + ε, ti+1), i = 1, ⋯, M,
where
By (79), for all t ∈ [ti + ε, ti+1), i = 1, ⋯, M,
where
Let In order for () to be positive definite for all t ≥ 0, the following inequality must be satisfied:
Define σM by
Since Π1(t) is positive and bounded as in (67), σM is obtained for each k2 when Π1(t) = Πm or Π1(t) = ΠM. Let us define σ1 and σ2 as
If k2 is chosen such that
and
then () is positive definite. Since the inequalities (107) and (108) are of third order in k2 if ( Γ32 + k2)2 is multiplied to both sides of the inequalities, there always exists a k2 satisfying (107) and (108). Then, for all t ∈ [ti + ε, ti+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 BX2 such that
Thus, asymptotic stability is achieved since (110) holds for t ∈ [tM + ε, ∞). Hence,
and from (55), (83), and (110)
as t → ∞. By (10) and (12),
as t →∞. This completes the stability analysis of Σ1.
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 k1 and k2 are chosen as follows
so that the natural frequency of the resultant second-order closed loop system may become ωn = 107 (rad / sec). Thus, k3 is set to be 20.9999 from (36). Next, observer gains L1 and L2 are designed such that the bandwidth of the observer system is almost three times larger than that of state feedback control system. Observer gains L1 and L2 are given by
Finally, the channel add/drop estimator gain AD in (25) or (26) is selected to be AD = 1.5×108 in order for the channel add/drop estimation to be fast enough compared with other controller and observer. In this case, AD = 1.5×108 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 three-level model is superior to that of the controller designed based on a simplified two-level 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 kC 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 three-level model shows faster settling performance. The control based on the two-level model shows oscillation and longer settling performance because it considers the model simplified by ignoring the level three state.
Fig. 2.Comparison of gain control performance
Fig. 4.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.
Fig. 3.Gain control performance over channel add/drops
Fig. 5.Channel drop case
Fig. 6.Channel drop case.
Fig. 7.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 three-level 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 well-known disturbance observer technique plays an effective role in guaranteeing the desired performance when channel add/drops occur.
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