passivity approach to dynamic distributed optimization for network traffic management
DESCRIPTION
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. Network Flow Control Problem. - PowerPoint PPT PresentationTRANSCRIPT
3/8/2004 -- 1 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
3/8/2004 -- 2 IMA Workshop
Network Flow Control ProblemDesign source and link control laws to achieve:
stability, utilization, fairness, robustness
. .
Rf
Rbq N
x N y L
p L
: capacityc: queueb
Forward routing matrix (including
delays)
Return routing matrix (including delays)
N sources L links
Source control
. .
link control
diagonal diagonal
Adjust sending rate based on congestion indication
(AIMD, TCP Reno, Vegas)
AQM: Provide congestion information
(RED, REM, AVQ)
0max ( ) , i i
xi y
U x Rx c
0 0
minmax ( ) ( )i ip xi
U x p y c
Optimization approach: Kelly, Low, Srikant, …
3/8/2004 -- 3 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
Hu y
x
( ) TV W x u y
If V(x) corresponds to physical energy, then H conserves or dissipates energy. Example: Passive (RLC) circuits, passive structure, etc.
3/8/2004 -- 4 IMA Workshop
Passivity Approach: Primal
RT R
h
( ( ) )xx K U x q +
-
-p
p
-q x y
y
Kelly’s Primal Controller
+
-
-(q-q*)
-(U’(x)-U’(x*))s-1 IN
K
g1
RT R+
-
-(p-p*)
p-p*
-(q-q*) x-x*
y-y*
s IL
s-1 ILh1
+
-
-(q-q*)
-(U’(x)-U’(x*))
(D+C(sI-A)-1B)
s-1 INg1
x
x
( '( ) )xx K U x q
x
y
y
*
* * * *
Lyapunov Function (Kelly, Mauloo, and Tan '98):
( ( ) ( )) ( ) ( ( ) ( ))yT
i i i i yi
V U x U x q x x h h y d
*
* * * *
Lyapunov Function:
1( ( ( ) ( )) ( )) ( ( ) ( ))2
yTi i i i i i i i i i y
i
V P U x U x q x x h h y d
3/8/2004 -- 5 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.
RT R+
-
-(p-p*)
p-p*
-(q-q*) x-x* y-y*
y-y*
x=K(U’(x)-q)+x
.
h1
* 1 *
Lyapunov Function:
1( ) ( )
2TV x x K x x
If U’’<0 uniformly (strictly concave), contains a negative definite term in x-x* ---important for robustness!
V
* 1 *
Lyapunov Function:
1( ) ( )
2TV p p p p
p
b
c max max
max
or ( and )( )
and 0 b
b b b b y cy cb
b b y cy c
3/8/2004 -- 6 IMA Workshop
Passivity Approach: DualLow’s Dual Controller
RT R
-U’-1+
-
q
-q
px
y
x
= (y-c)+bb
.
= (y-c+ b)+pp
.
RT R
g1-1+
-
q-q*
-(q-q*)
p-p*
x-x*
y-y*
x-x*
= (y-c+ b)+pp
. = (y-c)+bb
.q.
q.
- s-1IL
sIL
p .
y - c ( )s
s
D+C(sI-A)-1By - c
p .
*
2 * ' 1
Lyapunov Function:
( *) ( ( ))2
i
i
qTmm i iq
im
cV c y p x U d
b
*
2* ' 1
Lyapunov Function (Paganini 02):
( *) ( ( ))2
i
i
qTi iq
i
bV c y p x U d
*
* ' 1
First Order Law
( )
Lyapunov Function:
( *) ( ( ))i
i
p
qTi iq
i
p y c
V c y p x U d
3/8/2004 -- 7 IMA Workshop
Passivity Approach: Primal/Dual Controller
RT R+
-
-p
p
-q x y
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.
3/8/2004 -- 8 IMA Workshop
Simulation: Primal Controller1
1
loop gain: '( )( ( ) ( )) ( )
0.1( 1)( ) (0.1) or ( )
20
T
i i i i i
Rh Rx sI W s U x W s R
sW s k D C sI A B
s
.25 sec delay
(A1:Kelly)
(B1:Passive)
A1: 21.9rad/s, PM 108.2
B1: 8.4rad/s, PM 155.5
ogc
ogc
max
max
max
LTI PM/
A1: 0.086sec
B1: 0.322sec
gcT
T
T
3/8/2004 -- 9 IMA Workshop
Simulation: Dual Controller
1 sec delay
(A2:Low/Paganini)
(B2:Passive)
1 1 1
1
loop gain: ( ) ( ( ) )
2 2( ) 2 or 2
1
TRU x R s D C sI A B
D C sI A Bs s
A2 : 0.79rad/s, PM 4.5
B2: 0.59rad/s, PM 62.4
ogc
ogc
max
max
max
LTI PM/
A2 : 0.1sec
B2: 1.85sec
gcT
T
T
3/8/2004 -- 10 IMA Workshop
Robustness in Time Delay
typ p h y
x
x K U x q
R lisR e
• Passivity approach provides Lyapunov function candidates to compute quantitative trade-offs between disturbance and performance, and stability bounds on delays.
Rq x
x
x K U x q
p h y
TR_
yp2d
1d
* 1 *
Lyapunov Function:
1( ) ( )
2TV x x K x x
3/8/2004 -- 11 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.
w p
_
2
1y
y
Th
yq
1
M
2
h
Base Station
Mobiles
CDMA Power Control:
2
i
+
i i i ii i i i
i k kpk
dF pp = -λ + q q
dp L p
2
max ( ( )) ( )
SIR: ( )
( ) ln( )
i i i ip
i ii
k kk i
i i i i i
U p F p
L pp
L p
U L
3/8/2004 -- 12 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.
0max ( ) subject to ,i i
xi zy
U x Rx c Hx r
x1
x2
x1
x2
x3 x4 x5
x3 x4x5
z1=x1+x2 ≥ r1
z2=x3+x4+x5 ≥ r2