L2 Nonlinear Control of EDFA with ASE

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L2 Nonlinear Control of EDFA with ASE

• By Nem Stefanovic
• Supervisor Prof. Lacra Pavel

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Outline

• I. Introduction
• II. Model and Controller Derivation
• III.Results
• IV.Future Work
• V. References

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I. INTRODUCTION

• EDFA, ASE, and L2 Definition

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EDFA device23
Signal in
Signal out
pump
Erbium Doped Fiber

• Silica Fiber doped with Er3
• Optical signal enters and is amplified at the
  output
• A pump laser used for amplification

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EDFA Pictures and Components
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EDFA physics23
E3
E2
E1

• Laser excites Erbium ions into higher energy
  levels
• Stimulated emission from E2 to E1 amplifies
  signal

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Absorption/Emission Spectrum
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Amplified Spontaneous Emission

• Spontaneous emission from E2 to E1 is incoherent
  and random in direction, polarization, phase2
• Amplified by the EDFA just like the input signal
• Appears as noise in the output

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Manifestation of ASE

• Small input powers with high inversion
• Large channel drops
• Any large disturbances - ASE is crucial to
  disturbance and robustness analysis

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Optical Network Model12
OA
OA
OA
MUX
DEMUX

• Channels multiplexed by WDM and transmitted long
  distances using OAs
• Currently static connections, need dynamic
  reconfiguration capabilities1
• Sensitive to uncertainties in the system and
  model, not robust

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Control Motivation145

• Gain through channels depends on average
  inversion
• Changes in input power alters average inversion,
  which alters channel gains
• We want to maintain constant gain across all
  channels

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L norm

• Define an Lp norm, .Lp, as 17
• uLp (?0Tu(t)pdt)1/p lt ?
• where the space of u is said to belong to Lp (ie.
  u ? Lp)
• The special case of L? is expressed as 17
• uL? supu(t) lt ?
• for 0?t ?T

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L2 Gain

• Take the nonlinear system
• dx/dt f(x) g(x)u
• y h(x) d(x)u
• 8 L2 gain ? ? if
• ?0Ty(t)2dt ? ?2?0Tu(t)2dt
• for initial state x(0) 0 and u ? L2 0,T.

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General L2 Control Problem

• The general L2 control problem can be stated as
  8.
• Given input affine plant description,
• dx/dt f(x) g1(x)w g2(x)u
• z h1(x) d11(x)w d12(x)u
• y h2(x) d21(x)w d22(x)u
• Then, find a controller K, such that
• i)Fl(G,K) is asymptotically stable for w 0.
• ii)Fl(G,K) has L2 gain from w to z less than or
  equal to number ?

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Control Diagram
w
z
G
y
u
K

• Fl(G,K) represents the system from w to z
• with controller applied from y to u

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Full Information Problem14

• States and disturbances available in output

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Theorem 17
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Hamilton-Jacobi Inequality8
We
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Linearized System9

• Define a linearized system,
• dx/dt Fx G1w G2u
• y Hx
• u Lx
• If we can solve the problem for the linearized
  system, theory guarantees a neighborhood around
  x0 where L2 problem is satisfied in nonlinear sys.

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Theorem 2 9

• For (F,G2) stabilizable and (H,F) detectable, and
  let ?gt0. Then, infL TLlt?, where FGL is
  asymptotically stable and TL2 supd(y2
  u2 )/ d2
• iff there exists symmetric solution P ? 0 of
• FTP PF - P(G2G2T- ?-2G1G1T)P HTH
  0
• satisfying ?(F- G2G2TP ?-2G1G1TP) ? C-
• Also, one possibility is L -G2TP.

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II. Model and Controller Derivation

• Common Model, ASE model, FI problem and Theory,
  Control Design

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Common EDFA Model

• EDFA equations14
• dx/dt -x/? - 1/(?sL)?ke( (akgk)x-ak)L
  1Qin,k(t)
• Qout,1(t) e((a1g1)x-a1)L Qin,1(t)
• Qout,N(t) e((aNgN)x-aN)L Qin,N(t)

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Inclusions of Common Model

• State equation describes3
• i)absorption
• ii)emission
• iii)spontaneous emission
• on Erbium populations in ground and excited
  states
• Output equations describe channel gains

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Common Control Technique

• Linearization about an equilibrium point with PID
  control
• Scheduling between Linear controllers
• Heuristics

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Linear Control Switching Criticism

• Real EDFA behaviour is HIGHLY nonlinear
• Linear system approximation is only valid in a
  small neighborhood
• Must design multiple controllers
• No systematic approach for switching (based on
  measurement)
• Works! But could be better...

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EDFA Physical Equations3
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Simplifying Assumptions3

• Erbium populations distributed uniformly in a
  disk about the fibre cross-section
• Erbium populations distributed uniformly across
  length of fibre
• ? We get an average inversion level across EDFA
  length

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ASE Model Derivation
Where,
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ASE Model Derivation(Contd...)

• Let,
• from the propagation equation, we get

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Final State Space ASE model
Where,
and
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ASE Model Discussion

• Extra linear and Nonlinear ASE terms appear in
  state equation
• Extra nonlinear ASE term appears in output
  equation
• Stiff differential equation
• Wide difference in magnitude between terms

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FI Problem Formulation
Design performance output, z, to satisfy
conditions in Theorem 1 and to attenuate state,
x, and input, u
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Disturbance Formulation

• Let x0 represent equilibrium average inversion,
  Let u0 represent pump power equilibrium
  Let w0 represent channel power equilibrium
• Define w as disturbance, where w is a column
  matrix of channel powers

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EDFA HJI
Where,
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Solution to EDFA HJI

• The HJI is too complex to solve by hand
• No commercial software to numerically solve this
  equation!
• ? Write MATLAB library to do it!
• Methodology uses Taylor Series Approximation as
  outlined by Lukes 18

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Linear Design

• The HJE becomes a Riccati Equation for the Linear
  
• system9

• For QFI ? 0,

Note A,QFI,R,G1,G2 are constant values
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Scaling Design

• L2 gain design would not be useful if we could
  not choose an arbitrary ?
• Let
• which gives,

and
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Scaling Discussion

• We have full control over G1 and G2, and thus, ?
• Notice x is NOT affected by scaling
• A, R are NOT affected by scaling

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Final Linear Design

• We obtain full expression for L2 gain,
• Set the scaling to represent the worst case
  scenario of all channels being dropped,
• We want ? 10-8, which gives

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Nonlinear Design

• V(x) is NOT known in advance because it is
  computed numerically
• We CAN infer validity in nbhd, where it does
  satisfy V(x)gt0, and it has parabolic looking
  shape
• If QFI(x) lt 0, the geometry of V(x) can give
  valid solution for
• We increase ? until valid solution

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Final Nonlinear Design

• Starting from ? 10-8, we increase ? by orders
  of 10
• The conditions of Theorem 1 are satisfied when
• ? 10-4

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Function Plots
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More Function Plots
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III. Results

• Steady State, Dynamics, Channel Drop Cases

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Simulation Parameters

• 32 input channels with 1 pump channel
• C-band input signals and attributes
• length, L 13m.
• Pk,in 42.6 ?W
• Ppump 150mW
• ? x0 0.60725, average inversion level

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Steady State Terms in ASE Model

• Decompose the state equation into 4 major terms
• Term1 (Linear) Spontaneous Emission
• Term2 (N.L.) Power Input
• Term3 (Linear) ASE
• Term4 (N.L.) ASE

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Steady State Simulation
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Dynamic Channel Drop50
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Dynamic Channel Drop100
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Closed Loop100 Channel Drop
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Controller Simulation
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IV. Future Work

• Output feedback, Robust control, Elimination of
  Taylor Approximation

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Output Feedback Extension

• Solution works only if the state is available as
  output
• Full output feedback theory exists
• Another option is to use an observer

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Robust Control for EDFA

• L2 control by itself only guarantees disturbance
  attenuation
• Robust control simplifies down to L2 control
  problem
• Ideally, EDFA controller must be robust to system
  uncertainty as well

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Design for Valid V(x) in Advance

• Could we find a direct approach to solving the
  nonlinear L2 control problem like in the linear
  control problem?
• Can we exploit polynomial form in Taylor
  approximation to guarantee valid V(x)?
• Since we only have 1 state, maybe we can use a
  graphical approach to get a nonlinear controller

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Remove Taylor Approximation

• Taylor approximation causes loss of information
  about the system
• May limit controller operation to within a
  neighborhood

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Taylor Approximation Comparison
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V. References
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References (Contd)
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References (Contd 2)