L 2 Nonlinear Control of EDFA with ASE

1. L2 Nonlinear Control of EDFA with ASE By Nem Stefanovic and Lacra Pavel University of Toronto Nem Stefanovic and Lacra Pavel

2. Outline I. Introduction II. EDFA Model III. L2 Control IV. Simulations V. Significant Results Nem Stefanovic and Lacra Pavel

3. EDFA device Signal in Signal out pump Erbium Doped Fiber • Silica Fiber doped with Er3+ • Optical signal amplified at output • A pump laser used for amplification Nem Stefanovic and Lacra Pavel

4. EDFA Pictures and Components Nem Stefanovic and Lacra Pavel

5. EDFA physics E3 E2 E1 • Laser excites Erbium ions into higher energy levels • Stimulated emission from E2 to E1 amplifies signal Nem Stefanovic and Lacra Pavel

6. Amplified Spontaneous Emission • Spontaneous emission is incoherent and random in direction, polarization, phase • Amplified by the EDFA just like the input signal • Appears as noise in the output Nem Stefanovic and Lacra Pavel

7. Optical Network Model • Channels multiplexed by WDM • Static connections, need dynamic reconfiguration • Sensitive to uncertainties in signals and model, not robust OA OA OA MUX DEMUX Nem Stefanovic and Lacra Pavel

8. Control Motivation • Channel gains change with input power variation • Bit error occurs with power transients • Power transient speeds increase with daisy-chained EDFAs Nem Stefanovic and Lacra Pavel

9. Linear Control Switching Criticism • EDFA behaviour is HIGHLY nonlinear • Linear approximation only valid in neighborhood • Must design multiple controllers • No systematic switching (based on heuristics) • Works! But could be better... Nem Stefanovic and Lacra Pavel

10. Common EDFA Model • EDFA equations: Where, and Nem Stefanovic and Lacra Pavel

11. Improved EDFA model Nem Stefanovic and Lacra Pavel

12. 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 Nem Stefanovic and Lacra Pavel

13. Static Term Plots Nem Stefanovic and Lacra Pavel

14. Dynamic Channel Drop50% Nem Stefanovic and Lacra Pavel

15. Dynamic Channel Drop100% Nem Stefanovic and Lacra Pavel

16. L2 Control Motivation • Restrict state, x, directly • One nonlinear controller • Readily extends into robust analysis Nem Stefanovic and Lacra Pavel

17. L2 Gain • Take the nonlinear system: dx/dt = f(x) + g(x)u y = h(x) + d(x)u • L2 gain  if 0T||y(t)||2dt 20T||u(t)||2dt for initial state x(0) = 0 and u  L2 [0,T]. Nem Stefanovic and Lacra Pavel

18. L2 Control Problem w z G • Find a controller K, such that: i)Fl(G,K), the system G from w to z with K applied, is asymptotically stable for w = 0. ii)Fl(G,K) has L2 gain   from w to z y u K Nem Stefanovic and Lacra Pavel

19. Full Information Problem • Two conditions must be satisfied to solve the FI problem: Nem Stefanovic and Lacra Pavel

20. Solution to First Condition • The HJI is too complex to solve by hand • No commercial software to numerically solve this equation!  Write MATLAB library to do it! • Use Taylor Series Approximation from Lukes Nem Stefanovic and Lacra Pavel

21. Nonlinear Design • Explicit solution for Linear problem • V(x) is NOT known in advance • We CAN infer validity in nbhd, where V(x)>0 • We increase  until valid solution Nem Stefanovic and Lacra Pavel

22. Closed Loop Simulation Nem Stefanovic and Lacra Pavel

23. Significant Results • Developed new EDFA model with ASE • Heuristic switching was replaced by one L2 nonlinear controller • Achieved smoother, faster response Nem Stefanovic and Lacra Pavel

24. Thank You Nem Stefanovic and Lacra Pavel