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L 2 Nonlinear Control of EDFA with ASE. By Nem Stefanovic and Lacra Pavel University of Toronto. Outline. I. Introduction II. EDFA Model III. L 2 Control IV. Simulations V. Significant Results. EDFA device. Signal in. Signal out. pump. Erbium Doped Fiber.
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L2 Nonlinear Control of EDFA with ASE By Nem Stefanovic and Lacra Pavel University of Toronto Nem Stefanovic and Lacra Pavel
Outline I. Introduction II. EDFA Model III. L2 Control IV. Simulations V. Significant Results Nem Stefanovic and Lacra Pavel
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
EDFA Pictures and Components Nem Stefanovic and Lacra Pavel
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
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
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
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
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
Common EDFA Model • EDFA equations: Where, and Nem Stefanovic and Lacra Pavel
Improved EDFA model Nem Stefanovic and Lacra Pavel
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
Static Term Plots Nem Stefanovic and Lacra Pavel
Dynamic Channel Drop50% Nem Stefanovic and Lacra Pavel
Dynamic Channel Drop100% Nem Stefanovic and Lacra Pavel
L2 Control Motivation • Restrict state, x, directly • One nonlinear controller • Readily extends into robust analysis Nem Stefanovic and Lacra Pavel
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 20T||u(t)||2dt for initial state x(0) = 0 and u L2 [0,T]. Nem Stefanovic and Lacra Pavel
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
Full Information Problem • Two conditions must be satisfied to solve the FI problem: Nem Stefanovic and Lacra Pavel
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
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
Closed Loop Simulation Nem Stefanovic and Lacra Pavel
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
Thank You Nem Stefanovic and Lacra Pavel