H∞-performance analysis of robust controllers designed for AQM

(1)

H"-Performance Analysis of Robust Controllers

Designed for AQM'

Peng Yan

Dept. of Electrical Engineering

The Ohio State University 2015 Neil Ave. Columbus, OH 43210

yanp0ee.eng.ohio-state.edu

Abstract

It has been shown that the TCP connections through the congested routers with the Active Queue Management

(AQM) can be modeled as a nonlinear feedback system. In

this paper, we design

H"

robust controllers for AQM based

on the linearized TCP model with time delays. For the linear system model exhibiting LPV nature, we investigate the

H--performance with respect to the uncertainty bound of

RTT(round trip time). The robust controllers and the corresponding analysis of H"-performance are validated by simulations in different scenarios.

1 Introduction

Active Queue Management has recently been proposed in

[I] to support the end-to-end congestion control for TCP

traffic regulation on the Internet. For the purpose of alle-

viating congestion for IP networks and providing some no-

tion of quality of service (QoS), the AQM schemes are de-

signed to improve the Internet applications. Earliest efforts on AQM (e.g. RED in [2]) are essentially heuristic without

systematic analysis. The dynamic models of TCP ( [ 9 , 121)

make it possible to design AQM in the literature of feed-

back control theory. We refer to [ 1 I] for a general review of

Internet congestion control.

In [ 121, an TCP/AQM model was derived using delay dif-

ferential equations. They further provided a control theoretic analysis for RED where the parameters of RED can

be tuned as an AQM controller [4]. In [ 5 ] , a Proportional-

Integral controller was developed based on the linearized model of [12]. Their controller could ensure robust stability

of the closed loop system in the sense of gain-phase margin

of the PI AQM [5, 61. A challenging nature in the design

of AQM is the presence of a time delay, which is called

R T T (round trip time). To further complicate the situation, the linearized TCWAQM model is linear parameter varying

(LPV), with RTT being the scheduling parameter. In the

'This work is ruppomd by the Nalional Science Fundation undergranl ZHitay Ozbay is on leave from The Ohio Stale University.

number ANI-0073725

0-7803-7896-2/03/$17.00 02003 IEEE

Hitay Ozbay*

Dept. of Electrical & Electronics Engineering

Bilkent University Bilkent, Ankara, Turkey TR-06533

ozbay@ee.eng.ohio-state.edu

present paper, robust AQM controllers are developed based

on the

H"

control techniques for SISO infinite dimensional

systems [3, 151. We also analyze the

b'-

performance for

the robust controllers with respect to the uncertainty bound of the scheduling parameter RTT. Our results show that a

smaller operating range of RTTresults in better

H -

perfor-

mance of the AQM controller, which indicates that switching control among a set of robust controllers designed at se- lected smaller operating ranges can have better performance

than a single

H"

controller for the whole range.

The paper is organized as follows. The mathematical model

of TCPIAQM is stated in Section 2, where the linearized

LPV system with time delays is described. In Section 3, An

H-

optimization problem is formulated, where the para-

metric uncertainties are modeled and the robust controllers

are obtained. We investigate in Section 4 the

H"

perfor-

mance of the robust AQM controllers. MATLAB simula-

tions are given in Section 5 to validate our design and anal-

ysis, followed by concluding remarks in Section 6.

'

2 Mathematical Model of TCPlAQM

In [IZ], a nonlinear dynamic model for TCP congestion control was derived, where the network topology was as-

sumed to be a single bottleneck with N homogeneous

TCP flows sharing the link. The congestion avoidance

phase of TCP can he modeled as AlMD (additive-increase

and multiplicative-decrease), where each positive ACK in- creases the TCP window size W ( f ) by one per R7T and a

congestion indication reduces W ( f ) by half. Aggregating N

TCP flows through one congested router results in the fol-

lowing TCP dynamics [ 12.61:

where R(f) is the RTT, 0

5

p ( f )

5

1 is the marking proba- bility, q ( f ) is the queue length at the router, and C is the link

(2)

capacity. Note

df)

R ( t ) = Tp

+

c

where T, is the propagation delay and q ( f ) / C is the queuing

delay.

Assume N ( t ) = N and C ( t ) = C , the operating point of (1)

is defined by

W

= 0

Roc

WO = - N (3) L (4)

-

W,2 ' PO =

It can be shown that an uncertainty bound WZ(e,'*') satisfy- ing

IAPe(s)Js=jo

5

JWz(Bopel(s)J,T=jw V o E R' (12)

WJeo'Ae)(s)

= a + b s + c s 2 (13)

where a , b and c are defined in (35) (see the appendix for the

details of derivation). Note that once Bo and AB are fixed,

these coefficients are fixed.

Combining the robust stability and the nominal tracking performance condition, we come up with a two block infinite

dimensional

H -

optimization problem as follows:

is

"

Minimize y, such that robust controller CO($) is stabilizing

Po(s) and

Let Sq := q

-

qo and 6 p := p - po. the linearization of ( I ) results in the following LPV time delay system, [ 6 ] ,

K(B)e-"(@)s (14) (5) ~ 6q(s) ,- .- P&) =

S P ( 4

( r , ( e ) s + I)(TZ(B)S+ 1) where ~ 3 0 3 K(B) =

-

4N2

and 0 = R ( r ) E IT,,

T,

+qmyr/C] is the scheduling param-

eter of ( 5 ) where q m m is the buffer size. Note that we em-

ploy

L{f(t,B)le=s}

= &(S) to describe theLPV dynamic

equations in Laplace domain at fixed parameter values.

3

H"

Controller Design for AQM

Consider the nominal svstem

where

So(.) = (1 +Po(s)co(s))-'

To($)

= 1 -So(s) = Pfl(s)Co(s)(l +Pfl(s)co(s))-',

and

Wl

(s) = I/s is for good tracking of step-like reference

inputs.

By applying the formulae given in [I51 and [3], the optimal

solution to (14) can be determined as follows:

where

P5?

A ( s ) =

-

S

and F ( s ) is a finite impulse response (FIR) filter with time domain response

where 80 = Rg is the nominal

RTT.

We would like to design

a robust AQM controller CO(S)

for

the nominal plant (10)

so

where

that

P = h

(i) Co(s) robustly stabilizes P&) for VB E

0

:= [BO

-

AB, Bo

+

AB];

(bz - 2QC)TZ - C 2

(ii) The closed loop nominal system has good tracking of the desired queue length q o which is a step-like signal.

-

with x the unique positive root of

Notice that the plant ( 5 ) can be written as _{b Z - 2 a c - a 2 $} _(2ac_-_b2)$

₊

_c2

X

c4.p

x3

+

c*y* x -

(?

-aZ)

pe(s) =po(s)(l ape(^)) (11)

-

(U"

-a2

= 0 (19)

where APe(s) is the multiplicative plant uncertainty.

Proceedings of the American Control Conference

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The optimal

H"

performance cost y is determined as the largest root of

Note that an internally robust digital implementation of the

Hm

AQM controller (15) includes a second-order term which is cascaded with a feedback block containing an FIR filter F ( s ) . The length of the FIR filter is h(Oo)/T,, where

Ts is the sampling period.

4 H"-Performance Analysis

As shown in Section 3, the

H -

AQM controller (15) is de-

signed for Pe(s)Je=e, and allows for 8 € 0 =

[e

-

AO, 8

+

Ae]. In this section, we would like to investigate the

H"-performance for the corresponding closed loop system, which indicates the system robustness and system response.

0, - A8 00 Oo

+

as

... 0 4

C O ( S )

*... ...

Figure 1: Partition of 0 by 01 and 02

Define the H"-performance of controller Co(s) with respect

to P&) as follows:

for any 8 E 0 = [eo

-

A8, 80

+

Ae], where

S ( 4 = (1 +Pe(s)Co(s))-l, (22)

here the term IWi"Ae)(jo)Po(jw)/ can be seen as a bound

on the additive plant uncertainty. Furthermore, we define

which corresponds to the worst system response of con-

Notice that a smaller

6;

means better performance of the

robust controller within the operating range 0.

Particularly, we are interested in the scenario depicted

in Fig.], where 0 is equally partitioned by 0 1 = [el -

troller CO(.) for plant P ~ ( s ) with V8 E [eo - A9, Bo

+

AQ].

4191

A%, 8 1 + A 8 1 ] a n d 0 2 = [ e 2 - ~ e 2 , &+A82].withA81 =

A92 =

y .

For 8 E

e:,

i = 1,2, we design

H -

con-

troller Ci(s) obeying (15) with the nominal plant Pi($) :=

Pe(s)le=ei. Similar to (21) and(23), we have

for any

e

E Oi i = 1,2, and

where Sj(s) = (1

+

P&)Cj(s))-' is defined similarly to

(22).

In what follows, we provide numerical analysis of the

H -

performance with respect to the operating ranges and cor-

responding controllers shown in Fig.1. Assume N = 150,

C = 500, A8 = 0.2, and 00 = 0.5, the

H-

performance

yco(e) andyc,(O), i = 1,2canbenumericallyobtainedfrom

(21) and (24). As depicted in Fig.2, it is straightforward to

have

m a x ( c ' , e ) = 24.4

<

6;

= 104.4

which means that the partition of Fig.] can improve system

performance in the sense of smaller

H-

performance cost.

In fact, it is a general trend that

ma@>*)

<

Yg,

(26)

which can be further verified by Fig.3, Fig.4, and Fig.5,

where N is chosen from 100 to 200, C from 400 to 600.

. . .

:

. . . . : ... .... . . . . 0 . . . . .... . . . I , 8.3 0 % 0.4 0.45 0.5 055 0 6 0.65 0.7

Figure 2:

H-

performance with respect to f~

Remark: Based on the observation of better performance

obtained by the partition shown in Fig.1, it is natural to

consider switching robust control among a set of

H"

con-

trollers, each of which is designed for a smaller operating range. This will be an interesting extension of the present work.

Proceedings of the American Control Conference Denver. Colorado June 46. zoo3

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Figure 3: Performance cost

$:

w.r.t. N and C . . . .... . . . . .. . . : ...., . . ,.. .. . Zi

<"-

2d I 24 mo

Figure 4: Performance cost

y,!:'

w.r.1. N and C

5 Simulations

The closed loop system with the determined controllers

is implemented in MATLABlsimulink to validate the con-

troller design as well as the

H -

performance analyzed in

Section 4. We assume the TCP Row number N = 150. the

link capacity C = 500 packetslsec. The propagation delay

Tp is set to be 0.3 sec and the desired queue size is qo = 100

packets. Therefore, the nominal R 7 T is 0.5 sec (Bo = O S ) ,

which is straightforward from (2). We use AB = 0.2 in the

design of

CO(S)

and A01 = A02 = 0.1 in

Cl(s)

and CZ(S).

The following three scenarios are considered:

Assuming the plant is the nominal one, i.e. Pe(s) =

Po(.). we implement controller Co(s) as well as CI (s)

and Cz(s). It is shown in Fig.6 that the three con-

trollers can stabilize the queue length because the

nominal value BO is within the operating range of 0,

01, and 0 2 . Note that the system response ofCO(.) is

better than the other two due to the fact that it achieves

the optimal

H -

perfomance at Bo.

Assuming 6 = 00 - AB = 0.3, we implement con-

troller CO and Cl (CZ is not eligible in this scenario).

. . . . . . .. . . . . . . . .. . .

Figure 5: Performance cost

y,!?

w.r.1. N and C

llme I yecon@

Figure 6: System responses of CO, C, and C, at 0 = 0, = 0.5

As depicted in Fig.7, CO and C1 can robustly stabilize the queue length. Observe that the system response

of CI is better because it has much smaller

H -

per-

formance cost, which has been shown in Section 4.

0 Similarly, w e choose 9 = Bo

+

AB = 0.7 and repeat

the simulation for controller CO and Cz (Cl is not el-

igible). As depicted in Fig.8. the two controllers can robustly stabilize the queue length and their system

responses coincide with the

H"

performance

analy-

sis given previously.

The above simulations show that our robust AQM con-

trollers have good performance and robustness in the presence of parameter uncertainties. Meanwhile, the system re-

sponses also affirm a good coincidence with the

ff-

perfor-

mance analysis in Section 4.

Proceedings of the American Control Conference

Denver, Colorado June 4-6, 2003

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€9 . . . . ._. _. ... _. _. 1 ..< ' . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . m ; ~ 30. !

-

::.. . ._. ._. . . . . . . . . .

E ' ; ; . :

a ;

s : . . : .

_{. . .} : . . . . . . . . . . . .. .. . . . . . . . . I la=jo where I Ir=jo and

Figure 8: System responses of CO and C2 at 0 = 00 +A0 = 0.7 _{( T , P ) s +}l)(TZ(e)s+ 1)

>

max(T;,T;)

S 3 = j o

where

6 Concluding Remarks

which are straightforward from (7) and (8). Thus

We provided in this paper the guidelines of designing ro-

bust controllers for AQM, where the

H -

techniques for in-

finite dimensional systems were implemented. The

H - -

performance was numerically analyzed with respect to

the hound of the scheduling parameter 8. It was shown

that smaller uncertainty bound could result in better

H"-

performance of the corresponding closed loop systems.

Simulations were conducted to validate the design and anal-

ysis. A challenging extension of the present work is to con-

('2.8) IAhI max(T;,T;) ' q e .T+I (r2(e)s+il

I(

;

s=jo Recall

mi

:= TI(e)Tz(e)

- ~ ~ ( e ~ ) ~ ~ ( e ~ )

-

sider switching

H -

control, where the system performance - ( T I ( ~ o ) + A T I ) ( T z ( ~ o ) + A T z ) -TI(%)Tz(~o)

can b e improved in a larger operating range. = ATIATz

+

TI (0o)ATz

+

Tz(0o)ATt. (29)

Proceedings of the American Control Conference Denver. Colorado June 4-6.2003

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Invoking (27) and (28), we have IPe(s) -po(~)l.s=jo

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(32)

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(34)

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where with 9 17-928, June 2002. W,(Bo’Ae) ( s ) = a

+

6s

+

cs2 A(%,Ae) A ( e , , ~ e ) ( f i

(eo) +

weo))

a = - - K(00) b = K(%) (35) A ( @ 0 , 4 e ) ~ l ( e 0 ) T 2 ( 0 0 ) c = K ( W

Proceedings of the American Control Conference