3/8/ IMA Workshop Passivity Approach to Dynamic Distributed Optimization for Network Traffic Management John T. Wen, Murat Arcak, Xingzhe Fan Department of Electrical, Computer, & Systems Eng. Rensselaer Polytechnic Institute Troy, NY 12180

3/8/ IMA Workshop Network Flow Control Problem Design source and link control laws to achieve: stability, utilization, fairness, robustness. RfRf RbRb q Nq N x   N y Ly L p Lp L Forward routing matrix (including delays) Return routing matrix (including delays) N sourcesL links Source control. link control diagonal Adjust sending rate based on congestion indication (AIMD, TCP Reno, Vegas) AQM: Provide congestion information (RED, REM, AVQ) Optimization approach: Kelly, Low, Srikant, …

3/8/ IMA Workshop Passivity A System H is passive if there exists a storage function V(x)  0 such that for some function W(x)  0 H uy x If V(x) corresponds to physical energy, then H conserves or dissipates energy. Example: Passive (RLC) circuits, passive structure, etc.

3/8/ IMA Workshop Passivity Approach: Primal RTRT R h + - -p p -qxy y Kelly’s Primal Controller + - -(q-q * ) -(U’(x)-U’(x * )) s -1 I N K g1g1 RTRT R + - -(p-p * ) p-p * -(q-q * )x-x * y-y * s I L s -1 I L h1h (q-q * ) -(U’(x)-U’(x * )) (D+C(sI-A) -1 B) s -1 I N g1g1

3/8/ IMA Workshop Extension First order dual: (y-y *) to (p-p * ) is also passive Implementable using delay and loss Passive decomposition is not unique: For first order source controller, the system between –(p-p * ) and (y-y * ) is also passive. RTRT R + - -(p-p * ) p-p * -(q-q * )x-x * y-y * x=K(U’(x)-q) + x. h1h1 If U’’<0 uniformly (strictly concave), contains a negative definite term in x-x * --- important for robustness!

3/8/ IMA Workshop Passivity Approach: Dual Low’s Dual Controller RTRT R -U’ q -q p x y x = (y-c) + b b. =  (y-c+  b) + p p. RTRT R g q-q * -(q-q * ) p-p * x- x * y-y * x- x * =  (y-c+  b) + p p. = (y-c) + b b. q. q. - s -1 I L sI L p. y - c D+C(sI-A) -1 B y - c p.

3/8/ IMA Workshop Passivity Approach: Primal/Dual Controller RTRT R + - -p p -qxy y x= K (U’(x)-q)) + x. p =  (y - c) + p. Consequence of passivity of first order source controller and first order link controller: combined dynamic controller is also stable. Generalizes Hollot/Chait controller and easily extended to Kunniyur/Srikant controller.

3/8/ IMA Workshop Simulation: Primal Controller.25 sec delay (A1:Kelly) (B1:Passive)

3/8/ IMA Workshop Simulation: Dual Controller 1 sec delay (A2:Low/Paganini) (B2:Passive)

3/8/ IMA Workshop Robustness in Time Delay Passivity approach provides Lyapunov function candidates to compute quantitative trade-offs between disturbance and performance, and stability bounds on delays.

3/8/ IMA Workshop Extension to CDMA Power Control Passivity approach is applicable to other distributed optimization problems: minimize power subject to the signal-to-interference constraint.

3/8/ IMA Workshop Extension to Multipath Flow Control Traffic demand in multipath flow control can be incorporated as additional inequality constraints. Same passivity analysis applicable with demand pricing feedback based on r-Hx. x1x1 x2x2 x1x1 x2x2 x3x3 x4x4 x5x5 x3x3 x4x4 x5x5 z 1 =x 1 +x 2 ≥ r 1 z 2 =x 3 +x 4 +x 5 ≥ r 2