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
2/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
3/16
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]
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
4/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
5/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
6/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
7/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
8/16
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)
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
9/16
Complete Stability Region SR
SR = (S0 \ SN) \ SL
S0 \ SN
SR
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
10/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
11/16
Example 1
N = 75; d0 = 0.15 sec; C = 1250 pkt/sec
*[Hollot et al 2001]
*
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
12/16
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
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
13/16
PIP Controller [Heying et al 2002]
q(s)G(s) P(s)
q0
+ _
e u
Kh
+ _
PI
PIP Kh
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
14/16
Example 2 [Heying et al 2002]
N = 60; d0 = 0.22 sec; C = 1250 pkt/sec
*[Heying et al 2002]
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
15/16
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)
Ahmad Al-Hammouri Parameterizing PI Congestion Controllers FeBID’06
16/16
Thank You
Questions Comments