design of the dual closed-loop flow rate controller based on h∞ optimization theory for computer...

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Systems & Control Letters 52 (2004) 85–98www.elsevier.com/locate/sysconle

Design of the dual closed-loop &ow rate controller based on H∞optimization theory for computer networks�

Wei Shen∗, Hui-he ShaoInstitute of Automation, Shanghai Jiaotong University, No. 1954, Hua Shan Road, Shanghai 200030, PR China

Received 29 March 2003; received in revised form 21 October 2003; accepted 14 November 2003

Abstract

Aiming at the de5ciency against disturbance of &ow rate controllers proposed formerly, the dual closed-loop &ow ratecontroller based on H∞ optimization theory is proposed for the congestion control in computer networks. The designagainst disturbance is separated from the design of setpoint response to enhance the abilities against disturbance andmis-modeling. The designed optimization controller can implement Dahlin algorithm approximately, which can overcomethe adverse e8ect caused by time delay e8ectively. Moreover, the de5nite formula is given between the stable margin andparameters of the controller. The proposed scheme can make the bu8er queue level in bottleneck node stabilized on asetpoint quickly and the link bandwidth shared weighted-fairly. A number of simulations are done to verify the theoreticalanalysis and demonstrate the performance of the designed controller. Besides, the relationship between the robustness andthe transient response of the &ow rate control system is analyzed qualitatively.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Computer networks; Flow rate control; H∞ optimization; Dual closed-loop

1. Introduction

Along with the high-speed development of computer network technology, the requirements for real-timemulti-media communication and the network load are all on the increase, and then the congestion control innetworks will become more and more important. However, the increasing de5ciencies of the current controlschemes have already been shown. It is necessary that the current schemes are replaced by the more e8ectiveones.

The control theory is rather profound and developed. The network performances can be improved greatlyby the design of the network controller according to this theory, which has been shown su?ciently by manyresearches in this 5eld at present. For instance, in papers [5,6] proportional derivative (PD) controller is usedfor controlling source rates so that the response to the changes of the network state is quickened. In [4,10,12]proportional integral (PI) controller is adopted so that the source response is quickened as well as the packet

� The research work in this paper was supported by national project 973 that is a key fundamental research and development projectunder grant G1998030415.

∗ Corresponding author. Tel.: +86-21-62933427 Ext. 13.E-mail addresses: [email protected], [email protected] (W. Shen).

0167-6911/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2003.11.003

mailto:[email protected]

mailto:[email protected]

86 W. Shen, H.-h. Shao / Systems & Control Letters 52 (2004) 85–98

queue level in bu8er is stabilized on a set-point in steady state. In a similar way, PID controller is adopted in[9], as a result, the &uctuation of bu8er queue level is decreased greatly, the link utilization is increased, andthe delay jitter of the packet queuing becomes smaller. In [1], the robust PID controller is designed for thenetwork &ow rate control by applying the classical Nyquist method, which can meet the actual requirementsof the network control better. However, these PID controllers mentioned above are all single closed-loop,which cannot make both the set-point response and the disturbance response well at the same time. In [7], theSmith compensation principle is applied for the &ow rate control, which can overcome the adverse e8ect ofthe propagation delay better, but the scheme is also single closed-loop. It is mainly for the set-point response,namely the e8ect against disturbance is not good, and sensitive to the estimated error of time delay [11].

It is di?cult to predict the propagation delay exactly, so if the controller is sensitive to mis-modeling,namely the controller robustness is not good, the control performance will be reduced greatly in the actualapplication. Moreover, because it is di?cult to measure the actual bandwidth exactly, the bandwidth is as adisturbance in the designed &ow rate control system [7]. As long as the controller has the ability to respondto the disturbance quickly, the source rates can respond to the changes of the bandwidth in time. Thus, notonly can the congestion be avoided, but also the bandwidth can be utilized su?ciently. Therefore, it is veryimportant for the designed controller to have a good ability against disturbance and a good robustness.

In order to meet the requirements of the network control better, this paper proposes a &ow rate controller thathas dual closed-loop to enhance the abilities against disturbance and model-mismatch, which is on the basisof internal-model control principle. Moreover, the H∞ optimization control theory is used for the controllerdesign. In this paper, the rate-based feedback &ow control for the scenario of multiple sources and singlebu8er is researched, and the control for the available bit rate (ABR) communication in asynchronous transfermode (ATM) networks is taken as an example to do the simulation, which is compared with a PID &ow ratecontroller that is single closed-loop. The simpleness, e8ectiveness and robustness are as a basic rule for theproposed scheme in order to make it have practical use value.

2. Network �ow rate model

The rate-based feedback packet switching network about the scenario of multiple sources and single bu8eris shown in Fig. 1. Presumption: (1) Each switching node sets a 5rst in-5rst out bu8er for each outgoing link,which is shared by all the packets passing the outgoing link. (2) The processing capacity of each switchingnode is much larger than the transmission capacity of its links, namely the network congestion is only causedby the transmission capacity of links. The &ow rate controller is set in switching node. The controller’s inputis the di8erence between the bu8er queue level and its set-point, and the output is just the required sourcerates. It can be seen that there are a forward time delay and a backward time delay in the &ow rate control.

qr

qr

q(t)

q(t)Source1

Sourcen

Flow RateController

Link Serve

Bottleneck Switching Node

backwarddelay1

backwarddelayn

forwarddelay1

forwarddelayn

Buffer

+-

Fig. 1. Schematic diagram of the rate-based feedback network &ow control—multiple sources and single bu8er scenario.

W. Shen, H.-h. Shao / Systems & Control Letters 52 (2004) 85–98 87

The &ow rate model in computer networks can be determined by the following di8erential equation [7]:

dq(t)dt

=n∑i=1

ui(t − �i) − d(t); (1)

d(t) =

bw(t); q(t)¿ 0;

min

[n∑i=1

ui(t − �i); bw(t)

]; q(t) = 0;

where q(t) is the packet queue level in bu8er; ui(t) is the ith required source rate determined by the &owrate controller; d(t) is the bu8er output rate, namely the service bandwidth of the outgoing link; bw is thetransmission bandwidth of the outgoing link; � is a sum of the forward and backward time delays and n isthe number of the sources whose packets enter into the same bu8er.

In computer networks, the queue dynamics is a discrete process. In order to simplify the analysis and designin this paper, modeling the queue dynamics is as a continuous process. However, this is an approximation.

3. Design of the �ow rate controller

The basic idea about rate-based feedback &ow control is as follows: In each switching node, setting a &owrate controller for each outgoing link. According to the current packet queue level in outgoing link bu8er, thesource rates of packets passing the outgoing link are controlled in time so that the queue level will be kept ina de5nite range. Thus, the packet loss caused by the bu8er over&ow will be avoided and the link bandwidthwill be utilized su?ciently.

3.1. Network 3ow rate feedback control system

According to the feedback mechanism of the rate-based &ow control and the network &ow rate model, theblock diagram of the designed &ow rate control system is shown in Fig. 2.

The multiplex of several kinds of services exists in links. The bandwidth used for the low priority level ischanged frequently, which is di?cult to be measured, such as the ABR communication in ATM networks, sothe service bandwidth d(t) is as an immeasurable disturbance in the &ow rate control system. The outputs ofthe controller ui(t) are control variables, i.e. required source rates. The packet queue level q(t) is a controlledvariable, i.e. output variable. The set-point of the queue level qr(t) is an input variable.

Flow Rate Controller

∑ 1αie s

1 s

+

+

+

-

+

-

-+

-

Gc2(s)

Gc1(s)α1

αn

e

e

-τ1s

-τns

d(t)

qr(t)

u1(t)

un(t)

q(t)u(t)

i=1

n -τis~

Fig. 2. Block diagram of the network &ow rate feedback control system—multiple sources and single bu8er scenario.


)(1 sGc s1 )(tq

)(td

)(tqr

s1 )(2 sGc

=

n

i

-τis

1

=

n

i

αi e1

)(1 tx

)(tdc

)(2 tx∑

∑ αi e

-τis +-

+-

+

-

Fig. 3. Equivalent network &ow rate feedback control system when model-match—multiple sources and single bu8er scenario.

In Fig. 2, Gc1(s) and Gc2(s) are two sub-controller transfer functions in the &ow rate controller. 1=s is thetransfer function of an integrator. e−�is is the transfer function of time delay. �i is the actual delay. �̃i is theestimated delay. �i is the weight coe?cients for the bandwidth allotment. i = 1; 2; : : : ; n, which shows thatthere are n communications passing the same bu8er.

In this paper, Laplace transform is used as a mathematic tool for the analysis and design of the &ow ratecontrol system.

The &ow rate controllers proposed formerly are mainly for the set-point response, whereas their abilityagainst disturbance is not good. In order to overcome the de5ciency, this paper introduces an internal-modeland adds a sub-controller Gc2(s) to give prominence to the disturbance. It can be seen in Fig. 2 that the inputof Gc2(s) re&ects the e8ect of the disturbance and model-mismatch. Then the abilities against disturbance andmodel-mismatch can be improved by the design of Gc2(s).

The transfer function from the set-point to the output is

Gr(s) =(1=s)Gc1(s)

(1 + (1=s)Gc2(s)

∑ni=1 �ie

−�̃is)∑ni=1 �ie

−�is

1 + (1=s)(Gc1(s) + Gc2(s) + (1=s)Gc1(s)Gc2(s)

∑ni=1 �ie

−�̃is)∑n

i=1 �ie−�is : (2)

The transfer function from the disturbance to the output is

Gd(s) = − 1=s1 + (1=s)

(Gc1(s) + Gc2(s) + (1=s)Gc1(s)Gc2(s)

∑ni=1 �ie

−�̃is)∑n

i=1 �ie−�is ; (3)

where∑n

i=1 �i = 1; 06 �6 1.The total output is a sum of the set-point response and the disturbance response, namely

q(s) = qr(s)Gr(s) + d(s)Gd(s); (4)

where q(s); qr(s) and d(s) are the Laplace transforms of q(t); qr(t) and d(t), respectively.When the model matches with the actual status, namely �̃i = �i; i = 1; 2; : : : ; n,

Gr(s) =(1=s)Gc1(s)

∑ni=1 �ie

−�is

1 + (1=s)Gc1(s)∑n

i=1 �ie−�is ; (5)

Gd(s) = − (1=s)(1 + (1=s)Gc2(s)∑n

i=1 �ie−�is)−1

1 + (1=s)Gc1(s)∑n

i=1 �ie−�is :

Here, the equivalent control system is shown in Fig. 3.Obviously, the design against disturbance is separated from the design of the set-point response, and then

the design and set of Gc2(s) will not a8ect the set-point response.


3.2. H∞ optimization design

Gc1(s) is a sub-controller in main loop. Its performance will a8ect the whole performance of the controlsystem directly, so H∞ optimal control theory is used for the Gc1(s) design in this paper. In the feedbackcontrol system shown in Fig. 3, the sub-controller Gc1(s) can be expressed as follows [8,13]:

Gc1(s) =L(s)

1 − P(s)L(s); (6)

where P(s)= (1=s)∑n

i=1 �ie−�is; L(s)∈RH∞, namely it is a stable and proper real rational function. Because

L(s) is a freedom parameter, a freedom degree is provided for the optimal design of the controller.It is di?cult to deal with the pure time delay strictly in the controller design. A practical way which is

widely used is to approximate the time delay by rational-function [3]. In this paper, the time delay term e−�s

is expanded in Maclaurin series, and 5rst-order approximation is adopted, namely e−�s ≈ 1 − �s.In Fig. 3, the transfer function from the set-point qr(t) to the tracking error x1(t), namely sensitivity function

is

S(s) = (1 + P(s)Gc1(s))−1: (7)

Minimizing its ∞-norm is as a performance objective, i.e. min‖W (s)S(s)‖∞, where W (s) is a weightfunction. The set-point is a step signal, so W (s) is selected as 1=s. By Eqs. (6) and (7), we can obtain

min‖W (s)S(s)‖∞ = min‖W (s)(1 − P(s)L(s))‖∞:Obviously, the ∞-norm can be minimized by selecting the freedom parameter L(s).

According to the maximum modulus theorem, we can get [2,13]

‖W (s)(1 − P(s)L(s))‖∞¿ |W (s)|s=(∑ni=1 �i�i)

−1 ;

and then

min‖W (s)(1 − P(s)L(s))‖∞ =n∑i=1

�i�i:

Therefore the optimal L(s) is got as follows:

Lop(s) = s

In order to make Lop(s) stable and proper, a 5rst-order low pass 5lter is introduced. Then we get the subop-timum

L(s) = Lop(s)1

�s+ 1=

s�s+ 1

; (8)

where � is a constant that is more than zero. It is obvious that the optimal L(s) can be obtained when � tendtowards zero.

After Eq. (8) is substituted into Eq. (6), we obtain the optimal controller:

Gc1(s) =1

�+∑n

i=1 �i�i: (9)

It can be seen that the optimal controller Gc1(s) is a proportional controller, which is very simple and onlyhas a parameter.

By Eqs. (9) and (5), we can obtain the closed-loop transfer function from the set-point to the output shownin Fig. 3 is

Gr(s) ≈ 1�s+ 1

n∑i=1

�ie−�is: (10)


It is obvious that the obtained optimal controller implements Dahlin algorithm approximately, which canovercome the adverse e8ect of the time delay e8ectively and has a good robustness [11]. Here, the characteristicequation of the closed-loop control system is �s + 1 = 0, there is no pure time delay term in the equation,so the adverse e8ect caused by the delay can be eliminated. From the point of view of the control theory,the critical gain will be reduced if there is a pure time delay term in the characteristic equation. This meansthat the controller gain must be reduced to guarantee the control system stable, and then the response willbecome slower, that is to say, the source rate cannot respond to the changes of network state quickly, whichis insu8erable to high-speed communication networks. However, if the gain is not reduced, the control systemwill become unstable. This will lose the control for source rates, which will cause either link bandwidth idleor bu8er over&owed and a lot of packets lost. In such a case, the packet loss-resending mechanism will makethe network enter a vicious circle. Finally the network will be congested.

To simplify the &ow rate controller as much as possible, the proportional control is adopted for thesub-controller in the disturbance loop, i.e. Gc2(s) = kP2, where kP2 is a constant and more than zero.

The Gc1(s) determined by Eq. (9) and Gc2(s) = kP2 are substituted into Eqs. (2) and (3), we obtain

lims→0

Gr(s) = 1 and lims→0

Gd(s) = 0

and then lims→0 q(s) = lims→0 qr(s) can be obtained by Eq. (4).According to the 5nal-value theorem, we can obtain

limt→∞ q(t) = lim

s→0sq(s) = lim

s→0sqr(s) = lim

t→∞ qr(t):

The above equation shows that as long as the designed control system is stable, the e8ect caused by thedisturbance can be eliminated completely in the steady state, namely the set-point can be followed with nosteady-state error, whether the estimated delay is exact or not.

3.3. Stability and stable margin

By Eq. (10), we know that the main loop is input–output stable for any �¿ 0, and by the control systemshown in Fig. 3, we get[

x1

x2

]=

1s+ Gc1(s)

∑ni=1 �ie

−�is

s 1

sGc1(s)n∑i=1

�ie−�is −s

[qr

dc

]

≈ 1(1 − kP1

∑ni=1 �i�i

)s+ kP1

s 1

kP1sn∑i=1

�ie−�is −s

[qr

dc

];

where kP1 is the proportional gain of Gc1(s), i.e. kP1 = (�+∑n

i=1 �i�i)−1.

Obviously, when kP1¡ (∑n

i=1 �i�i)−1, i.e. �¿ 0, the four transfer functions between x1(t); x2(t) and

qr(t); dc(t) are all stable, all the internal signals are bounded if all the external input signals are bounded,that is to say, the main loop is not only input–output stable but also internal stable for any �¿ 0.

In Fig. 3, the closed-loop transfer function from d(t) to dc(t) is

Gd−dc (s) =

(1 +

kP2

s

n∑i=1

�ie−�is)−1

:

Then the characteristic root is

s ≈ kP2

kP2∑n

i=1 �i�i − 1:


Therefore, when proportional gain kP2¡ (∑n

i=1 �i�i)−1, the characteristic root is on the left-hand s-plane, and

then the transfer function Gd−dc (s) is stable. In other words, dc(t) is bounded certainly as long as d(t) isbounded.

In a practical control, the designed control system must not only be stable but also have a certain stablemargin to guarantee the robustness. In Fig. 3, the open-loop transfer function of the main loop is

GO(s) =kP1

s

n∑i=1

�ie−�is ≈ kP1

s

(1 − s

n∑i=1

�i�i

):

The amplitude margin of the loop is

A(!g) = 20 lg|GO(j!g)|−1 ≈ 20 lg

∣∣∣∣∣−kP1

n∑i=1

�i�i − jkP1

!g

∣∣∣∣∣−1

(dB);

where !g is the phase across frequency, i.e. “GO(j!g) = −180◦.Then it can be obtained that the main loop has the amplitude margin above A dB when

kP16

(10A=20

n∑i=1

�i�i

)−1

; i:e: �¿ (10A=20 − 1)n∑i=1

�i�i:

Usually the amplitude margin above 6 dB is adopted in practice.In a similar way, it can be got that the disturbance loop in Fig. 3 has the amplitude margin above A dB

when

kP26

(10A=20

n∑i=1

�i�i

)−1

:

It can be seen that there are only two parameters, namely � and kP2, to be set in the &ow rate controller,which can be regulated according to the required stability, robustness and rapidity of the control system.Because the stability and robustness are in contradiction with the rapidity, it is necessary to consider themelectrically when the parameters are selected. By Eq. (10), we can know intuitively that � is a time constantof the low pass 5lter, which turnover frequency is 1=�. The less the �, the wider the passband of the controlsystem, and then the faster the transient response. However, the system also becomes more sensitive to noisesand model-mismatch, namely the stability and robustness are worse. Whereas, the more the �, the strongerthe high-frequency attenuation, and then the slower the response. Here the system is also more slowwitted tonoises and model-mismatch, namely the stability and robustness are better.

3.4. Bandwidth allotment

Because the set-point of bu8er queue level is a constant in &ow rate control, in the steady state accordingto Eq. (1) we can obtain

limt→∞

(n∑i=1

ui(t − �i) − d(t))

= 0:

Because of ui(t) = �iu(t); i= 1; 2; : : : ; n,∑n

i=1 �iu(∞) = d(∞), and then u(∞) = d(∞). Therefore we canobtain ui(∞) = �id(∞); i = 1; 2; : : : ; n.

It is obvious that the &exible allotment of the bandwidth for each source can be got by the set of weightcoe?cients �i.

When �i = 1=n, ui(∞) = (1=n)d(∞); i = 1; 2; : : : ; n, namely in the steady state the link bandwidth will beshared fairly by the each source that passes the link.


Summarizing the above analysis, we can see that: The proposed &ow rate controller is very simple,which only consists of two proportional controllers, an integrator and pure time delays. Because of the dualclosed-loop control structure, the abilities against disturbance and model-mismatch can be improved greatlyby the separate design for disturbance. Moreover, the controller based on H∞ optimal control not only makesthe performance level be optimal but also implements Dahlin algorithm approximately, thus the designed &owrate controller has a better transient response and a better robustness. Besides, the di8erent bandwidth can beallotted to each source conveniently by the set of the bandwidth allotment weight coe?cients.

4. Simulation result

In ATM networks, the multiplex of several kinds of communication exists in links. The bandwidth used forthe ABR communication is changed frequently due to its lower priority. Now we take the control of the ABR&ow rate as an example to do the simulation research to verify the e8ectiveness of the proposed scheme.

We presume that all links in Fig. 1 have the same bandwidth, and there are several ABR communicationspassing the same switching node. Therefore, the node will become a bottleneck node. In the following simu-lation, the designed controller is used for controlling the bu8er queue level in the bottleneck node and makingit stabilized on a set-point quickly to avoid the congestion caused by the bu8er over&ow and utilize the linkbandwidth su?ciently due to the nonempty bu8er.

In order to evaluate the transient performance of the designed controller better, presuming that the ABRbandwidth does the most abrupt change, i.e. step-change, in the simulation. Besides, in ATM networks thepacket is called cell which has a 5xed length 53 bytes, so the cell is as an unit in the following simulation.

In Fig. 2, presumption:

1. The ABR bandwidth of the outgoing link of the bottleneck node is(a) d(t) = 30I(t − 200) cell=ms, where I(t) is an unit step signal, which is shown in Fig. 4.(b) d(t) is a periodical step signal shown in Fig. 12.

2. There are three ABR communications passing the bottleneck node.3. The estimated delays in each ABR communication are �̃1 = 5:0 ms, �̃2 = 10:0 ms and �̃3 = 12:0 ms, respec-

tively.4. The weight coe?cients for the ABR bandwidth allotment are �1 = 0:2; �2 = 0:5 and �3 = 0:3, respectively.5. The set-point of the bu8er queue level is qr = 500 cells.

0 100 200 300 400 5000

10

20

30

40

50

d(t

)(c

ell/

ms)

ABR bandwidth

time (ms)

Fig. 4. ABR bandwidth step-change.


0 100 200 300 400 5000

200

400

600

800

1000

q(t)

(cel

l)

New method Single PID controlSingle PID control

Actual delay = Estimated delayAmplitude margin 7.5 dB

time (ms)

Fig. 5. Bu8er queue levels under the control of the proposed scheme and single closed-loop PID, respectively, when model-match.

6. There are ABR cells to be sent on the source terminals all the time.7. The &ow rate controller has 7:5 dB amplitude margin.

When the ABR bandwidth is a step-change shown in Fig. 4, the transient responses of the bu8er queuelevels under the control of the proposed scheme and the single closed-loop PID, respectively, are shown inFig. 5. In the simulation according to the above presumption, the parameters in the proposed scheme can begot as follows: � = 13:2 ms, kP1 = 0:044=ms, kP2 = 0:044=ms; The transfer function of the single closed-loopPID controller is GPID(s) = KP + 1=TIs + TDs, where the parameters are KP = 0:044=ms, TI = 1600 ms andTD = 0:10 ms for the response shown by dashed curve, and KP = 0:044=ms, TI = 900 ms and TD = 0:10 ms forthe dotted curve.

From Fig. 5 we can see the following results. When there is no estimated error of the delay, in the 5rst nodisturbance 200 ms the set-point response is both speediness and non-overshoot, and in the subsequent timethe response to the step-changed disturbance can be stabilized on the set-point quickly under the control of theproposed scheme. The transient response is much better than the single closed-loop PID in respect of whetherthe rapidity or the amplitude. Then it can be seen that not only the required bu8er capacity is smaller, butalso the cell queuing delay and its jitter in the bu8er are all smaller. In this 5gure it is also shown that thebu8er is empty in a period under the single closed-loop PID control, which means the link bandwidth is notutilized su?ciently during the period due to the relatively slow response.

Under the control of the single closed-loop PID control, the disturbance response will become slower if theovershoot of the set-point response is reduced. The overshoot of the set-point response will be increased ifthe disturbance response is quickened. These have been shown su?ciently by the dashed and dotted responsecurve in Fig. 5.

Fig. 6 shows that under 7:5 dB amplitude margin the proposed scheme still has a very good transientperformance, namely a very good robustness, even if there is biggish di8erence between the actual delayand the estimated value. When the actual delays are 0.7 multiple of the estimated delays, i.e. �1 = 3:5 ms,�2 = 7:0 ms and �3 = 8:4 ms, the bu8er queue level is shown by the dashed curve; When the actual delaysare 1.3 multiple of the estimated delays, i.e. �1 = 6:5 ms, �2 = 13:0 ms and �3 = 15:6 ms, the queue level isshown by the dotted curve.

When there are estimated errors of the delays, presuming the actual delays are 1.3 and 0.7 multiple of theestimated delays, respectively, the bu8er queue levels under the proposed scheme control and the di8erentamplitude margins are shown in Figs. 7 and 8. Obviously, the more the amplitude margin, the better therobustness, but the slower the response. Contrarily, the less the amplitude margin, the worse the robustness,


0 100 200 300 400 5000

200

400

600

800

1000

q(t)

(cel

l)

Actual delay = Estimated delay Actual delay = 1.3 x Estimated delayActual delay = 0.7 x Estimated delay

New method Amplitude margin 7.5 dB

time (ms)

Fig. 6. Bu8er queue levels under the control of the proposed scheme when model-mismatch.

0 100 200 300 400 5000

200

400

600

800

1000

time (ms)

q(t)

(ce

ll)

Amplitude margin 7.5 dBAmplitude margin 9.0 dB

New methodActual delay = 1.3 x Estimated delay

Fig. 7. Bu8er queue levels under the proposed scheme control and the di8erent stable margins when model-mismatch.

0 100 200 300 400 5000

200

400

600

800

1000

q(t)

(cel

l)

Amplitude margin 7.5 dBAmplitude margin 9.0 dB

New methodActual delay = 0.7 x Estimated delay

time (ms)

Fig. 8. Bu8er queue levels under the proposed scheme control and the di8erent stable margins when model-mismatch.


0 100 200 300 400 5000

5

10

15

20

25

30

Flo

w R

ate

into

Buf

fer (

cell/

ms)

from the 1st sourcefrom the 2nd sourcefrom the 3rd source

New methodActual delay = Estimated delayAmplitude margin 7.5 dB

time (ms)

Fig. 9. ABR &ow rates entering the bu8er under the control of the proposed scheme when model-match.

0 100 200 300 400 5000

200

400

600

800

1000

time (ms)

q(t

)(c

ell)

New methodThere is no cell from the 3rd sourceafter the 300th ms.

Fig. 10. Bu8er queue level under the proposed scheme control when there is no cell from the third source after 300 ms.

but the faster the response. In the 5gures, the dashed curve is for 9:0 dB amplitude margin, i.e. � = 17:5 msand kP2 = 0:037=ms, and the solid curve is for 7:5 dB amplitude margin, i.e. �= 13:2 ms and kP2 = 0:044=ms.

When there are no estimated errors of the delays and under 7:5 dB amplitude margin, the &ow rates ofthe three ABR communications entering the bottleneck node bu8er under the control of the proposed scheme,namely the allotted status of the ABR bandwidth, are shown in Fig. 9. It can be seen that the ABR bandwidthof the outgoing link will be allotted fairly to all the ABR sources passing the link according to the bandwidthallotment weight coe?cients.

When there is no cell from the third ABR source after 300 ms, the bu8er queue level and the allotted statusof the ABR bandwidth under the control of the proposed scheme are shown in Figs. 10 and 11, respectively,which are under the condition of 7:5 dB amplitude margin and no estimated errors of the delays. Obviously,the control has a good robustness. After 300 ms, the bandwidth released by the third ABR communication isshared rapidly by other two ABR sources according to the weight coe?cients.

When the ABR bandwidth is a periodical step-change shown in Fig. 12, there are no estimated errors ofthe delays and the set-point is half of the bu8er capacity, the cell loss rate (CLR) and link utilization (LU)of the ABR communications under the control of the proposed scheme (7:5 dB amplitude margin) and thesingle closed-loop PID (KP = 0:044=ms, TI = 1200 ms, TD = 0:10 ms) respectively, are shown in Figs. 13 and


0 100 200 300 400 500

0

5

10

15

20

25

30

time (ms)

Flo

wR

ate

into

Buf

fer

(ce

ll/m

s)

from the 1st sourcefrom the 2nd sourcefrom the 3rd source

There is no cell from the 3rd sourceafter the 300th ms .

Fig. 11. ABR &ow rates entering the bu8er under the proposed scheme control when there is no cell from the third source after 300 ms.

0 400 800 1200 1600 2000 2400 2800 3200 3600 40000

10

20

30

40

50

d(t)

(cel

l/ms)

ABR bandwidth

time (ms)

Fig. 12. ABR bandwidth periodical step-change.

400 600 800 1000 1200 14000

5

10

15

20

25

30

Buffer Capacity (cell)

Ce

llLo

ssR

ate

(%)

New methodSingle PID control

Actual delay = Estimated delayAmplitude margin 7.5 dBSetpoint = 0.5 x Buffer capacity

Fig. 13. Cell loss rates under the control of the proposed scheme and single closed-loop PID, respectively, when model-match.


400 600 800 1000 1200 140050

60

70

80

90

100

Buffer Capacity (cell)

Lin

kU

tiliz

atio

n(%

)New methodSingle PID control

Actual delay = Estimated delayAmplitude margin 7.5 dBSetpoint = 0.5 x Buffer capacity

Fig. 14. Link utilizations under the control of the proposed scheme and single closed-loop PID, respectively, when model-match.

14. It is obvious that the proposed scheme has lower CLR and higher LU under the same bu8er capacity.De5nition: CLR = the lost ABR cells caused by bu8er over&ow/the ABR cells sent by sources; LU = theABR cells transferred by link/the ABR cells can be transferred by the ABR bandwidth of link.

It can be seen that the theoretical analysis is veri5ed by the simulation results.

5. Conclusion

The proposed dual closed-loop &ow rate controller can enhance the abilities against disturbance and model-mismatch. Moreover, the controller designed by H∞ optimal control theory not only makes the performancelevel optimal but also implements Dahlin algorithm approximately, which can overcome the adverse e8ect ofthe time delay better. Thus the source rates can follow the changes of the bandwidth more quickly, and thenthe &uctuation of the bu8er queue level will become smaller. Only smaller bu8er capacity is necessary forthe low packet loss rate and the high link utilization. Besides, the di8erent bandwidth can be allotted to eachsource conveniently by setting the bandwidth allotment weight coe?cients. The simpleness, e8ectiveness androbustness of the proposed scheme can meet the practical requirements of the network control well.

Acknowledgements

The author thanks the anonymous reviewers for their constructive comments, which helped to improve thequality of the paper.

References

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[8] M. Morari, E. Za5riou, Robust Process Control, Prentice-Hall, Englewood Cli8s, NJ, 1989.[9] F.-y. Ren, C. Lin, Y. Ren, PID controller design for congestion control in ATM networks, Chinese J. Comput. 25 (10) (2002)

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