Parameterizing PI Congestion Controllers

Ahmad T. Al-Hammouri, Vincenzo Liberatore, Michael S. Branicky

Case Western Reserve University

Stephen M. PhillipsArizona State University

April 3, 2006

Support by: NSF CCR-0329910, Department of Commerce TOP 39-60-04003, NASA NNC04AA12A, and an OhioICE Training grant

Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06

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Contributions of the Paper

Complete stability region for PI Presents examples that show

Different stable PI parameters exhibit widely different control performance

Neglecting delays in control design leads to unstable systems

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TCP-AQM Control Loop

x(t)=cwnd/RTT

tf

q(t)f(q(t))q`=Σx(t) - C

p(t)

tb

Controller G(s)

On Ackcwnd += 1/cwndOn losscwnd /= 2

( )( )sdB

es s

Plant

P(s)=[Hollot et al 2001]

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PI AQM [Hollot et al 2001]

PI vs RED Control signal:

.

Frequency transfer fn:

.

0( ) ( ) ( )

tp iu t k e t k e d

( )i

pk

G s ks

q(s)G(s) P(s)

q0

+ _e u

P

PI

Integral term eliminatesthe steady-state error

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Parameterizing PI AQM

Problem: Determine the entire region of stabilizing

kp and ki values Objectives:

Stable closed-loop system Enhanced closed-loop performance

Steady-state error, convergence time, overshoot

Challenges: Delays

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Contributions of the Paper

Complete stability region for PI Presents examples that show

Different stable PI parameters exhibit widely different control performance

Neglecting delays in control design leads to unstable systems

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p i{(k , k ) :

( )lim 1} =

( )( )

p i

s

k s k B

s s s

Complete Stability Region SR [Silva et al 2005]

SR = (S0 \ SN) \ SL

S0 : Stability region for

the delay-free system P0(s) =

SN : .

p i

0

{(k , k ) :

lim|G(s)P (s)| 1}s

( )( )

B

s s S0 \ SNS0

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Determination of SL

Set of kp and ki’s that destabilizes the closed-loop for delays less than d0

For given kp and ki

Find d that gives the blue curve If (d ≤ d0), (kp,ki ) SL

Else, (kp,ki ) SL

Sweep (kp,ki ) S0

-1

Im{s}

x

d++

Re{s}

Nyquist Plot ofG(s)P(s)

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Complete Stability Region SR

SR = (S0 \ SN) \ SL

S0 \ SN

SR

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Contributions of the Paper

Complete stability region for PI Presents examples that show

Different stable PI parameters exhibit widely different control performance

Neglecting delays in control design leads to unstable systems

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Example 1

N = 75; d0 = 0.15 sec; C = 1250 pkt/sec

*[Hollot et al 2001]

*

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Complete stability region for PI Presents examples that show

Different stable PI parameters exhibit widely different control performance

Neglecting delays in control design leads to unstable systems

Contributions of the Paper

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PIP Controller [Heying et al 2002]

q(s)G(s) P(s)

q0

+ _

e u

Kh

+ _

PI

PIP Kh

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Example 2 [Heying et al 2002]

N = 60; d0 = 0.22 sec; C = 1250 pkt/sec

*[Heying et al 2002]

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Future Work

Conduct packet-level simulations (ns-2) Define a “Networks Performance”

objective function Optimize the objective function over the

stability region Analyze the queue nonlinearity (i.e.

truncation)

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Thank You

Questions Comments